 1582 MSI311.HLP, Generated by A3UTIL.EXE, 1994  (c) PCSCC Inc., 1993,01√#UúvÅ╤-qò»╥I¿{▒√az╝∩                                                                             ┌─────────────────────────┐                                             │╒════════════════════════╧╕                                            ││    Application File     ├─────────────────────────────────────       ││                         └──────────────────────────────────────      ││                                                                      ││            Minarad Spectrum's Interactive Manual of                  ││                                                                      ││             OPTICS, and IR RADIOMETRY Calculations                   ││                                                                      ││             By Nelson Engborg and Paul Murphy                        ││                                                                      ││            Technical Staff, Minarad Spectrum, Inc.                   ││                                                                      ││      Application File  Version 3.11    06-01-1994                                                                                                                                                                           (c) Copyright 1994 Minarad Spectrum, Inc.                               (c) Copyright 1981, 1994 PCSCC, Inc.                                                                                                                          All rights reserved.                                                                                                     Read and understand  WARRANTY INFORMATION . Type J twice to read.                                                                            Instructions on using help file are at end of file. Type K to read.                                                                                                                                                   ╒══════════╕                                                            │ CONTENTS └──────────────────────────────────────────────────────────                                                                          0.00                  Table of Contents                                                                                                               *    To quickly jump to Sections, position the Section line number      *    on the │JUMP│ line (line 12) and type J or type Lxxx (enter)       *    where xxx is line number. For example, to jump to line 139,        *    type L139 (ent).  Type S or R to skip back/forth to Sections.                                                                         SECTION                   TITLE                              PAGE│LINE                                                                           0.00 ... Table of Contents ................................... 1   31  1.00 ... Introduction ........................................ 2   81  2.00 ... An Introduction to Calculations Using ............... 4  159           AstroFormCalc 3.                                               3.00 ... Electromagnetic Radiation: The IR Source ............ 6  275           (Blackbody)                                                    4.00 ... Behavior of the Aspherical Refraction Surface ....... 9  370           (Cartesian Ovoid).                                             5.00 ... Behavior of the Spherical Refraction Surface ........ 9  395           (Paraxial Rays).                                               6.00 ... Thin Lenses, Lensmaker's Formula, Gaussian ......... 11  461           Formula and Newtonian Formula.                                 7.00 ... Thin lenses, Focal Planes and Lateral or ........... 13  553           Transverse Magnification.                                      8.00 ... Compound Thin Lenses, Effective Focal Lengths ...... 14  621           and Dioptric Power.                                            9.00 ... Thick Lenses, Principal Planes and the Paraxial .... 15  657           Approximation.                                                 10.0 ... Compound Thick Lenses, Principal Planes and ........ 15  683           Effective Focal Lengths.                                       11.0 ... Reflections from Planar, Aspherical and ............ 16  718           Spherical Mirrors.                                             12.0 ... Cassegrain and Gregorian Mirrors. .................. 18  789  13.0 ... Field of View (FOV) of IR Instrument. .............. 19  837  14.0 ... Figure of Merit of an IR System: Noise ............. 21  932           Equivalent Power Density (NEPD), Radiance (NEN)                         and Temperature (NET).                                         15.0 ... Circumference of an Ellipse: Calculation by ........ 25 1143           Series and Integration                                         16.0 ... About Minarad Spectrum. ............................ 27 1197  17.0 ... Glossary of Terms .................................. 28 1271  18.0 ... Product License and Warranty. ...................... 29 1307  19.0 ... User Support. ...................................... 30 1373  20.0 ... Order/Registration Form ............................ 31 1398  21.0 ... Optics and Infrared References. .................... 32 1464  22.0 ... Instructions for Manually Installing Package. ...... 33 1515  23.0 ... Help File: Summary of File Viewing Commands ........ 34 1556                                                                        ╒═══════════════╕                                                       │ INTRODUCTION  └──────────────────────────────────────────────────────                                                                         1.00          Introduction                                              ────────────────────────────────────────────────────────────────────────                                                                                 To change the display colors to high-intensity white on blue            type Q when the highlight bar is lighted on the EQU below.                                                                             'EQU                                                                    'COL HWHITE BLUE                                                        'HELP                                                                                                                                                    This document is an on-line application file which discusses            the theory and presents solved problems in Geometrical Optics           which includes lenses and mirrors and in infrared technology            (IR) which includes IR sources (blackbodies), fields of view,           figure of merits (NEPD, NEN, NET) and other topics..  Excellent         references, which can serve both as a supplement to an existing         course in Optics, IR and Radiometry or as stand-alone study or          self-help guides are listed at the end of this document in the          REFERENCE section.                                                                                                                              Geometrical optics deals with the behavior of "light" (called           Vis and closely related radiations like UV and IR in                    situations where the wave nature of the light is negligible.            Radiometry deals with the measurement of radiation such as UV,          Vis and IR "light", microwaves and (radio frequencies) RF.                                                                                      Personal computers (PC's) and their automation and calculation          capabilities were not available before 1981.  Proper use of the         PC can reduce or eliminate much of the tedium associated with           reducing theory into algebraic expressions and then into useful         day-to-day applications.                                                                                                                        The staff of Minarad Spectrum has written this application to           provide guidance to both students and professionals who work            with Optics and especially infrared (IR) Radiometry.  The               calculations discussed here can provide help in mastering the           execution of the necessary algebra and math upon which Optics           and UV Vis and IR Radiometry is heavily based.  The staff of            Minarad Spectrum will continue to update this file with new             discussions and calculations.  All registered users will be             kept updated about releases of new versions.  The user is               advised to register for the nominal fee (See below.).                                                                                           The application is driven by a versatile spreadsheet/calculator         call AstroFormulaCalculator 3 which exists as the file                  ACAL3.EXE.  Minarad Spectrum has obtained a distribution                license for ACAL3.EXE and its on-line helpfile ACAL3HLP.HLP.            from PCSCC Inc.                                                                                                                                 ACAL3.EXE is a compact program that will run on any PC from the         XT on up and on any DOS version from 2.1 up.  It contains a             highly structured on-line helpfile with countless interactive           examples on how to setup calculations.  This program is                 specifically tailored for analysis based on algebraic                   expressions, formulas or equations.  One of its main functions          is to eliminate the "programming-type overhead" in working with         known formulas and to allow meaningful learning to proceed              rapidly. It is also a virtually paperless program.  Its many            and versatile operations can be learned from its highly                 structured on-line helpfile with countless interactive examples         on how to setup calculations.                                                                                                                   AstroFormCalc 3 can also be used with any traditional outline           manual or text to develop equation templates that solve                 specific problems.  For technical support of AstroFormCalc 3,           the user is required to register for AstroFormCalc 3,                   separately.  See instructions in its on-line helpfile (type key         F1 from the equations display).                                                                                                                 This file is organized by topics with are summarized in the             Table of Contents.  The use of AstroFormCalc 3 is discussed in          the on-line help file which is accessed by typing (F1) from the         equations display.  This entire file can be searched using the          auto-search command "A".                                                                                                               ╒═══════════════════╕                                                   │ HOW TO CALCULATE  └───────────────────────────────────────────────────                                                                        2.00     An Introduction to Calculations Using AstroFormCalc 3.         ────────────────────────────────────────────────────────────────────────                                                                                 This Section provides an introduction to numbers, number                formats, variables, variable names, mathematical operations,            equations and related topics.  In general, in order to avoid            ambiguities, one must always express numbers, mathematical              expression and equations in an acceptable way when entering             them into a computer program.  These requirements lead to the           formulation of a set of rules which are summarized in the               on-line Help File for the ASTRO 3 program.  This help file can          be reached from the main equation display by typing the (F1)            key.                                                                                                                                            For example:  It is very common to have to evaluate an                  expression which might be written as:                                                                                                                                   456.789 x 0.345                                                                                                         As written, one would interpret this expression as:  Find the           product that results from multiplying the decimal 456.789 by            the decimal 0.345.  Because the letter symbols are most often           associated with variable names, the use of "x" as a                     mathematical operator for multiplication and the                        equally-acceptable use of "x" as a variable name in an equation         such as "y=x" would result in confusion for the                         expression-interpreter of the program.  As a result, the                multiplication operator is commonly the "*" symbol and the              "by-the-rules" way to enter the above expression would be               456.789 * 0.345 (enter) In the language of "computerese",               "(enter)" means to type the enter key of the keyboard and               usually results in some sort of computer action.                                                                                                The following examples illustrate some common ways to write             mathematical expressions and the "by-the-rules" way they are            entered into the ASTRO 3, and many other, computer programs.            These examples exist as equations templates that can be                 "called-up" by scrolling the screen until the highlighted bar           appears over the appropriate 'EQU line.                                                                                                         After the templates are loaded, follow the instructions and             always type (F2) from the main equation display to return to            this application helpfile immediately after the 'EQU reference.                                                                        Topic # 1: ENTERING NUMBERS and EXPRESSIONS                                                                                                     Discussion: The following examples show common representations for       expressions and the proper way to enter these into this program. These  expressions can be entered at the ENTER variable below.                                                                                            commonly used     ASTRO 3 way     Result          Reason                                                                                         24.1 x 56.7       24.1*56.7      1366.47        * means multipl.                                                                                     5                                                                       -            5/180          2.77E-2 or    / means division             180                          2.77*10^-2                                                                                                        6 ∙ 3 ∙ 4      6*3*4          72           * not ∙ for multipl.                                                                                 2(-7)5         2*-7*5         -70            implied multipl.                                                        not allowed.                                                                                       25÷5           25/5           5            / is division operat.                                                                                                                                                         2      5                                                             14 -  + 8 -  (14+2/7)+(8+5/9)   22.84           express compound           7      9                                     fractions as sums                                                                                                                                                                                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAWRIT      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 2: ENTERING EXPONENTS and SQUARE ROOTS                                                                                                  Discussion: The following examples show common representations for       expressions and the proper way to enter these into this program. These  expressions can be entered at the ENTER variable below.                                                                                            commonly used     ASTRO 3 way     Result          Reason                    ___                                                                    √144         (space) sqr(144)  12         use sqr(..) function                                                    to take square root.           2.3                                                                   10             10^2.3           199.5       ^ means raise to power               4                                                              5.67 x 10      5.67e4            56700      e means base 10                 4     3                                                             3x10 -2x10     3e4-2e3 or        28000      calculation in                             3*10^4-2*10^3                scientific notation.            2 -1                                                               (2x10 )        (2e2)^-1          0.005    no implied exponentiation                                                                              ___                                                                    √ 6                                                                     ------    (space) sqr(6)/(4*sqr(3))   .3535  must match parentheses     4√3                                                                                                                                                                                                                                                                                                                                                                'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAEXPO      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The remainder of the on-line manual studies many scientific             concepts which when grouped together are commonly called                "Optics, IR and Radiometry".                                                                                                           ╒════════════╕                                                          │ RADIATION  └─────────────────────────────────────────────────────────                                                                         3.00       Electromagnetic Radiation: The IR Source (Blackbody)         ────────────────────────────────────────────────────────────────────────                                                                                 Electromagnetic radiation propagates through space in                   wavefronts of fluctuating magnetic and electric fields.  The            frequency, or wavelength, of this radiation identifies it as            radio waves, microwaves, infrared, visible UV and so on.  The           first equation template identifies the kind of radiation from           its wavelength or frequency.  Geometrical optics and Radiometry         is most applicable to radiation in and around the visible (Vis)         which includes ultra-violet (UV) and infrared (IR) radiations.                                                                         Topic # 3: WAVELENGTH, RADIATION TYPE, FREQUENCY                                                                                                Problem # 3.1: Enter wavelength in microns.                              (a) What radiation has a wavelength of 400 nm (400E-3 microns)?         (b) What radiation has a frequency of 400 THz (400*10^14 Hz)?                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRARAIN      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          An IR source (blackbody) is the "perfect" absorber of all               incident electromagnetic radiation.  It is also the "perfect"           radiator because it is in equilibrium at a fixed temperature            and so must emit all its incident energy.  However, the                 wavelength distribution of its emitted radiation follows                Planck's Law, Setfan's Law and Wien's Law.  Emittance is mainly         in the infrared region and is felt as heat, hence the term              "blackbody".  Above 600°C objects begin to glow "red" as the            emitted energy moves into the visible region.  Understanding            these Laws is fundamental to understanding radiation in                 general.  However, the distribution of illuminance from a               visible source rarely shows the IR-source distribution.  For            example, a tungsten filament heated to 3000 K will show a               "graybody" Planckian spectral distribution, but only in the             transmission region of its quartz enclosure.                                                                                           Topic # 4: RADIATION, POWER EMITTED vs ABSORBED, STEFAN'S LAW, BLACKBODY                                                                        Problem # 4.1: A household-type 'radiator' has emissivity of 0.6 and     area of 2.0 m^2                                                         (a) At what rate is radiation absorbed by the radiator when the walls   of the room are at 27°C?                                                (b) At what rate is radiation emitted from the radiator when its        temperature is 50°C?                                                    (c) What is the net radiation rate from radiator?                                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA1946      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The relative radiance or power in any wavelength interval can           be calculated by integrating Planck's equation.  The total              power is found from Stefan's law which is the closed integral           of Planck's equation over all wavelengths.  The following               calculation shows how one estimates the percentage of                   blackbody radiation in any wavelength relative to the maximum           theoretically possible.                                                                                                                Topic # 5: PLANCK'S LAW, EMISSIVE POWER, RADIATION IN WAVELENGTH         INTERVAL                                                                                                                                       Discussion: Calculate the emissive power in the far IR (1000 to 5.       6microns), middle IR (5.6 to 1.5 micron) and near IR (1.5 to 0.72       microns) for objects at 1000 and 100 °C.                                                                                                       'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAPLLW      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          Finally, one can estimate the radiation cone of an IR source            which is its "field of view".  Its FOV is based on a                    point-source located at the end of its cavity.  The rays are            limited, or stopped, by its exit aperture.  One half of its FOV         is its divergence angle.  IR will irradiate anything within the         cone.  If a plane surface is placed perpendicular to the                optical axis at a given working distance from the source, it            will cast a spot on that plane.  Note, atmospheric effects are          not included.                                                                                                                          Topic # 6: SPOT DIAMETER, SIZE, FOCUS AT INFINITY, CONE ANGLE, WORKING   DISTANCE, IR POINT SOURCE                                                                                                                      Discussion: What is the spot size ( = spot diameter) at a working        distance of 2 inches from a cylindrical IR source with an aperture      diameter of 1 inch and a cone angle of 12°?                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRASFOV      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ ASPHERICAL REFRACTION  └──────────────────────────────────────────────                                                                        4.00       Behavior of the Aspherical Refraction Surface (Cartesian              Ovoid).                                                        ────────────────────────────────────────────────────────────────────────                                                                                 The following problem studies the behavior of a spherical wave          diverging from a point source and impinging on an aspherical            refracting surface known as a Cartesian Ovoid.  These surfaces          include the ellipsoid, hyperboloid and paraboloid.                                                                                     Topic # 7: ASPHERICAL REFRACTING SURFACES, CARTESIAN OVOID                                                                                      Problem # 7.1: Write an expression for the Cartesian ovoid with image    and object distances of 15 and 12 cm, respectively. Ovoid made of       glass Ng=1.5 and surrounded by air Na=1.                                (a) Calculate the ray length from ovoid surface to its image if that    point is 14 cm from the source object.                                  (b) What does a negative distance mean?                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA402       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ SPHERICAL REFRACTION   └──────────────────────────────────────────────                                                                        5.00       Behavior of the Spherical Refraction Surface (Paraxial                Rays).                                                         ────────────────────────────────────────────────────────────────────────                                                                                 The basis of most geometrical optics is the spherical                   refracting surface which is of higher symmetry than its                 aspherical cousins, discussed above.  This higher symmetry              allows many basic formulas to be derived.  The following                problems deal with spherical surfaces.                                                                                                 Topic # 8: SPHERICAL REFRACTING SURFACES, PARAXIAL RAYS                                                                                         Problem # 8.1: The input end of a wide glass (Ng=1.6) fiber is ground    to a convex hemisphere of 1.5 cm radius.                                (a) If a point source is located 4 cm to the left of the hemisphere's   vertex, where will its image be?                                                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA405       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 9: SPHERICAL REFRACTING SURFACES, PARAXIAL RAYS                                                                                         Problem # 9.1: The input end of a wide glass (Ng=1.6) fiber with convex  hemisphere of 1.5 cm radius is dipped in water (Nw=1.32).               (a) If a point source is located 4 cm to the left of the vertex, where  will its image be?                                                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA406       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 10: SPHERICAL REFRACTING SURFACES, OBJECT FOCAL LENGTH                                                                                  Problem # 10.1: An interface separates air on the left and glass (Ng=1.  44) on the right.                                                       (a) What is the object focal length if its radius of curvature is 6     cm?                                                                                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA408       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 11: SPHERICAL REFRACTING SURFACES, IMAGE FOCAL LENGTH                                                                                   Problem # 11.1: An interface separates air on the left and glass (Ng=1.  44) on the right.                                                       (a) What is the image focal length if its radius of curvature is 6 cm?                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA408A      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 12: SPHERICAL REFRACTING SURFACES, OBJECT FOCAL LENGTH                                                                                  Discussion: What is the radius of curvature of a glass fiber (Ng=1.48)   required to focus light rays 95 cm in front of its spherical interface  in a non-aqueous media of isopropanol (Nm=1.38)?                                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA409       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════╕                                                         │ THIN LENSES └─────────────────────────────────────────────────────────                                                                        6.00       Thin Lenses, Lensmaker's Formula, Gaussian Formula and                Newtonian Formula.                                             ────────────────────────────────────────────────────────────────────────                                                                                 The majority of calculations in geometrical optics make two             basic assumptions:  1) the impinged-upon surface is a sphere            (spherical approximation) and 2) the impinging light strikes            that surface at small angles with respect to its optical axis           (paraxial approximation). In general practice, however, one             usually invokes these formulas for any lens calculation.                The following problems deal with "thin" lenses. Such a lens             is infinitely thin and all distances are measured from their            interior principal plane which passes through its center.                                                                                       An illustration of a "thin" lens is shown below.                        Its center is marked by "||".  All distances are measured               from it.                                                                                                                                                    ss                                                                     s  s                                                                    s  s                     M[T] + means erect image     ╦ Y[O]           s    s                                                 ║······o·········s·||·s········o··········o  S source point             S     F[O]       s    s       F[I]        ║  P image point                                s  s               Y[I] ╩  S[O] Source distance       |·X[O]·|···f····· s  s·····f···|····X[I]··|  S[I] Image distance                           ss                                                   |--------S[O]------|-----------S[I]-------|                                                                                                   Sign Convention:  X[O] + means its left of F[O]        focal_length < 0                   X[I] + means its right of F[I]           means                   Y[I], Y[O] + means above optical axis        diverging lens.                                                                                  The following problems illustrate a number of calculations              involving thin lenses.                                                                                                                 Topic # 13: THIN SPHERICAL LENS EQUATION, LENSMAKER'S FORMULA                                                                                   Discussion: The source and image points are located 5 and 7 cm,          respectively, from a spherical concave lens with source radius of -3    cm and image radius of 4 cm. What should the index of refraction of     the glass be?                                                                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA410       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 14: THIN SPHERICAL LENS EQUATION, LENSMAKER'S FORMULA                                                                                   Problem # 14.1: A source is 20 cm from a glass (ng=1.6) convex-plano     lens with a radius of curvature of 4 cm. Where is the image located     (a) when its flat surface faces source and                              (b) when its flat surface is away from source?                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA411       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 15: THIN SPHERICAL LENS EQUATION, FOCAL LENGTH vs N[G], N[L]                                                                            Discussion: Calculate the focal length of a bi-concave thin lens made    of glass (Ng=1.6) and immersed in isopropanol (Na=1.35). The source     radius is -9 and the image radius is 18 cm.                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA413       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 16: LENS EQUATION, FOCAL LENGTH vs INDEX REFRACTION, RADIUS CURV                                                                        Problem # 16.1: A bi-concave (R1=-8 cm, R2=12 cm) thin lens is made of   hepa-fluor (Ng=1.37) and is immersed in solvent alpha-k-1 (Nm=1.77).    (a) What is its focal length?                                           (b) is it converging or diverging?                                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA414       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 17: MEASURING INDEX REFRACTION in MEDIUM from FOCAL LENGTHS                                                                             Problem # 17.1: A bi-convex lens of index 1.6 has a focal length of 60   cm in air and 240 cm when immersed in a transparent fluid.              (a) What is the index of refraction of the transparent fluid?                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA416       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════╕                                                   │ THIN LENS IMAGING └───────────────────────────────────────────────────                                                                        7.00       Thin lenses, Focal Planes and Lateral or Transverse                   Magnification.                                                 ────────────────────────────────────────────────────────────────────────                                                                                 Predicting or estimating the location of an object, also called         a source or target, and the location of its image are very              important.  The following problems deal with this topic.                                                                               Topic # 18: THIN LENS IMAGERY, CONVEX, TRANSVERSE MAGNIFICATION                                                                                 Problem # 18.1: A thin bi-convex lens of index 1.6 has equal radii of    40 cm in air.                                                           (a) Compute the focal length.                                           (b) If the image is to be projected full-size on a white wall, what     are the image and object distances?                                                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA419       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 19: THIN LENS IMAGERY, CHARACTERISTICS for REAL OBJECTS,         GAUSSIAN                                                                                                                                       Problem # 19.1: A 10 cm tall statue of Curly Howard is sitting 100 cm    from a positive (converging) lens of 30 cm focal length.                (a) Describe Curly's image.                                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA420       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 20: THIN LENS IMAGERY, CHARACTERISTICS for REAL OBJECTS,         NEWTONIAN                                                                                                                                      Problem # 20.1: A 9 cm tall statue of Mo Howard is sitting 100 cm from   a positive (converging) lens of 30 cm focal length.                     (a) Describe Mo's image.                                                                                                                       'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA421       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 21: THIN LENS IMAGERY, MAGNIFICATION, OBJECT, IMAGE DISTANCES                                                                           Problem # 21.1: An object positioned 38 inches to the right of a         positive lens is imaged 8 inches to its left.                           (a) Where will the object appear if its image is moved to 2 inches      from the lens?                                                          (b) Describe it.                                                                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA423       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 22: THIN LENS IMAGERY, FIND F from R1, R2, BI-CONVEX, GAUSSIAN                                                                          Problem # 22.1: A thin bi-convex glass (N[g]=1.6) with radii of 20 and   70 cm casts a half-sized image of a plant pot on a white wall.          (a) What is its focal length?                                           (b) What are the lens to plant and lens to wall distances?                                                                                     'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA425       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒══════════════════════╕                                                │ COMPOUND THIN LENSES └────────────────────────────────────────────────                                                                        8.00       Compound Thin Lenses, Effective Focal Lengths and                     Dioptric Power.                                                ────────────────────────────────────────────────────────────────────────                                                                                 Lens combinations are very useful.  The following problems deal         with combinations of "thin" lenses.                                                                                                                                                                            Topic # 23: COMPOUND THIN LENSES, MAGNIFICATION, FRONT BACK FOCAL        LENGTHS                                                                                                                                        Problem # 23.1: A compound lens consists of two bi-convex lenses of      focal lengths 15 and 30 cm, respectively, separated by 60 cm.           (a) Describe the image generated by a statue 10 cm tall placed 12 cm    in front of the first lens.                                             (b) What separation distance is required for unity magnification.                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA427       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 24: COMPOUND THIN LENSES, MAGNIFICATION, MICROSCOPE                                                                                     Problem # 24.1: A low cost microscope is fabricated out of a bottom      lens of 2 cm focal length, a 10 cm cardboard tube and a top lens with   a 8 cm focal length.                                                    (a) Locate the image of an ant 3 cm from the bottom lens.               (b) Describe the ant.                                                                                                                          'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA431       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒══════════════╕                                                        │ THICK LENSES └────────────────────────────────────────────────────────                                                                        9.00 Thick Lenses, Principal Planes and the Paraxial Approximation.     ────────────────────────────────────────────────────────────────────────                                                                                 The basic thin lens formalism and equations can be applied to           "thick" lenses if effective focal lengths and principal planes          (from which distances are measured) are properly defined.  The          single principal plane of the thin lens is split into two               principal planes H1 and H2 separated by the lens thickness D.           The next problem deal with this topic.                                                                                                 Topic # 25: THICK LENS, GAUSSIAN FORMULA, PARAXIAL APPROXIMATION                                                                                Problem # 25.1: A double convex lens has radii of 6 and 24 cm, a         thickness of 3 cm and index of 1.6.                                     (a) Locate both the principal and focal points and compute the image    distance for an object 15 cm in front of V1.                            (b) determine the magnification of the image and                        (c) determine the f.f.l. and b.f.l.                                                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA442       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════════╕                                               │ COMPOUND THICK LENSES └───────────────────────────────────────────────                                                                        10.0       Compound Thick Lenses, Principal Planes and                           Effective Focal Lengths.                                       ────────────────────────────────────────────────────────────────────────                                                                                 As with "thin" lenses, "thick" lenses can also be used in               combinations. It should be noted that if the thickness of the           individual lenses, as measured between their principal planes,          approaches 0.0, then the solutions are the same as for the              "thin" lens combinations discussed above.  The following                problems deals with this topic.                                                                                                        Topic # 26: LENS COMBINATIONS, THICK LENSES, GAUSSIAN FORMULA                                                                                   Discussion: Two identical bi-convex lenses are separated by 30 mm. Each  lens has a radius of 50 and 30 mm, a thickness of 20 mm and an index    of 1.6. Calculate individual and system focal lengths.                                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA443       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 27: LENS COMBINATIONS, THICK LENSES, GAUSSIAN FORMULA                                                                                   Discussion: A bi-concave lens (radius=-50 mm, index=1.6) is positioned   150 mm up a plastic tube in front of a plano-convex lens (radius 50 mm, index 1.6). Calculate the image location of a 9 mm carpenter ant 90 mm  in front of the device.                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA446       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════╕                                                         │ MIRRORS     └────────────────────────────────────────────────────────                                                                         11.0       Reflections from Planar, Aspherical and Spherical                     Mirrors.                                                       ────────────────────────────────────────────────────────────────────────                                                                                 Lenses are refracting surfaces and mirrors are reflecting               surfaces.  There exists a correspondence between the equations          that predict the behavior of mirrors and those of lenses if one         takes into account sign conventions, etc.  A spherical convex           mirror is shown below.  Mirror surface is represented by 's'.           The convex mirror has a "virtual" focal point and image.  The           virtual image lies behind or "in" the mirror between the mirror         and the focal point.  The concave mirror has both the image and         object in the "real" domain on the same side of the mirror.  In         general, image/object distance are considered positive if they          are "real" and negative if they are virtual.  Thus, the convex          mirror has one positive and one negative distance while the             concave mirror has two positive distances.                                                                                                            °                                                                           °      s                                                                   · o·                   S = source object                         ·  °    s °                   V = vertex of sphere               ·  °          s   ° P     C          P = image (between F and C)S ·°· · · · · · · · ·o· · ·°o · · o          R = 2 * F                  |                   V|s     |  ·°                                       |                    | s   ·|°  R      S[O] = distance S to vertex      |                    |  s°  |         S[I] = distance vertex to image   |-------S[O]---------|-S[I]-|    C = center of sphere R = radius curvat.                                                                        Sign Convention: S[O] + means its left of vertex V,                                      S[I] + means its left of vertex V,                                      R    + means center C is right of V                                          Convex mirror has negative focal length                                 Concave mirror has positive focal length                                                                                           The following problems deal with mirrors.                                                                                              Topic # 28: SPHERICAL REFRACTING MIRRORS, PARAXIAL RAYS                                                                                         Problem # 28.1: A convex mirror (17 cm radius) casts the image of a      bird on a screen 20 cm distant.                                         (a) Where is the bird located and                                       (b) what will it look like?                                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA460       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 29: SPHERICAL REFRACTING MIRRORS, IMAGE vs MAGNIFICATION                                                                                Problem # 29.1: A spherical mirror casts an inverted half-sized image    of an object that is located 110 in from its vertex.                    (a) Where will the image be located?                                                                                                           'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA461       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 30: SPHERICAL REFRACTING MIRRORS, IMAGE vs MAGNIFICATION                                                                                Problem # 30.1: An eraser 5 cm tall sits 150 cm in front of a concave    mirror of focal length 400 cm.                                          (a) Describe the image.                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRA495       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════╕                                                          │ CASSEGRAIN └──────────────────────────────────────────────────────────                                                                        12.0       Cassegrain and Gregorian Mirrors.                            ────────────────────────────────────────────────────────────────────────                                                                                 Mirror optics offer broadband spectral characteristics and are          useful as collectors and as transfers in instruments designed           for broadband operation such as the Minarad SA-200 and SA-1200          spectroradiometrs.  The Cassegrain and Gregorian are very               similar.  They both have concave primary mirrors.  The                  Cassegrain has a diverging or convex secondary while the                Gregorian has a converging or concave secondary.  The                   Cassegrain style with its diverging secondary mirror allows a           longer optical path from the detector to the secondary mirror.          Thus, it is more accommodating to the radiometer's IR hardware          such as Minarad's inline blackbody reference cavity, field stop         and filter/CVF.  Minarad uses the Cassegrain in its SA-200 and          SA-1200 spectroradiometers                                                                                                                      The following problems deal with Cassegrain and Gregorian               mirror systems.                                                                                                                        Topic # 31: CASSEGRAIN OPTICS, SPHERICAL THIN MIRRORS, MAGNIFICATION                                                                            Discussion: A Cassegrain optic has a large concave mirror of FL=12 in.   and smaller convex of FL=4 in., respectively, initially separated by    10 in. Focus the system on a 3 inch tall target 42 inches in front of   the primary and describe the image located 2 inches behind the primary  mirror's hole.                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRACASS      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 32: GREGORIAN OPTICS, SPHERICAL THIN MIRRORS, MAGNIFICATION                                                                             Problem # 32.1: A Gregorian optic has a large concave mirror of FL=12    in. and smaller convex of FL=6 in., respectively, initially separated   by 8 in.                                                                (a) Focus the system on a 3 inch tall target 42 inches in front of the  primary and describe the image located 3 inches behind the primary      mirror's hole.                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAGREG      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════╕                                                       │ FIELD OF VIEW └───────────────────────────────────────────────────────                                                                        13.0       Field of View (FOV) of IR Instrument.                        ────────────────────────────────────────────────────────────────────────                                                                                                                                                         What an IR instrument "sees" is usually characterized by its            field of view, FOV.  The FOV is twice its acceptance angle.  If         an IR detector is located at the focal point of an optical              system containing a single converging (convex lens), the                illustration, below, can be used to identify terminology:                                                                               FOV/2 is 1/2 of FIELD OF VIEW                                                │ ∙   <----- FOV/2                    or ACCEPTANCE ANGLE               │......°.....∙                                                        S │            |    °     ∙          |                                  P │    OPTIC-->|              °      |<-- DETECTOR                      O │            |          ∙   °      |                               T │...... .... ∙ ° Detector at focal point of                                │ ∙    °                                 optical system.                |---WORK_DIS-|--------S[DET]-------|                                                 |---------FL----------|                                                                                                               Detector:  The IR detector's diameter, if circular, or the                         length of its sides, if square or rectangular.               S[DET]:  Distance from the center of the lens to the detector.          OPTIC:   Diameter of the lens  FL:  The focal length of lens            WORK_DIS:  the distance from the center of the lens to a plane,         perpendicular to the optical axis and containing an image of            the detector called its "spot".                                         SPOT: The diameter of the image of the detector at WORK_DIS.            FOV: Field of view in degrees, Acceptance angle is FOV/2                                                                                        The shape of the spot is assumed to be the same as that of the          detector.  A circular detector has a circular spot (and the             same FOV in all directions perpendicular to the optical axis).          A rectangular one has a rectangular spot (and two FOV's), a             square detector has a square spot (but only one FOV).                                                                                           If the detector is a perfect point, then the systems FOV is 0°.         For any sized detector, a collimating location exists, measured         from the lens, at S[DET] = FL + F# * DIAM_DET. For example, a 1         mm detector in an F/2 system will have a 0° FOV if the detector         is located 2 mm from the focal point. A negative FOV implies            that the detector's image is focussed in front of the lens.                                                                                     The following equations can be used to calculate FOV's working          distances, spot sizes and F# for any simple system with a               detector at an arbitrary location behind the lens.                                                                                     Topic # 33: FOV OPTICAL SYSTEM, DETECTOR VS DISTANCE FROM LENS                                                                                  Problem # 33.1:                                                          (a) Calculate the FOV of a radiometer with a 0.3 inch lens, a 0.6 inch  FL and a 4 mm detector 8 mm behind its focal point.                     (b) What is the spot diameter at 400 inches in front of the lens?                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAFDET      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The next set of equations can be used to make system                    calculations when the detector is located at the optical                focal point, a very common location in IR systems.                                                                                     Topic # 34: FOV OPTICAL SYSTEM, DETECTOR AT FOCAL POINT                                                                                         Problem # 34.1:                                                          (a) Calculate the FOV of a radiometer with a 0.3 inch F/2 lens and a 4  mm detector at its focal point.                                         (b) What is the spot diameter at 4 inches in front of the lens?                                                                                'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAFFOC      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The FOV (or acceptance or divergence angle) and lens diameter           can be measured experimentally by measuring the spot diameter           vs. the working distance and using linear regression. The               following example illustrates this procedure.                                                                                          Topic # 35: SPOT DIAMETER vs WORKING DISTANCE, FOV, DIAMETER OPTIC,      LINEAR REGRESSION                                                                                                                              Discussion: A quality department wants to measure its laser's            divergence angle (FOV/2) and its exit aperture by measuring the spot    diameter vs working distance and using linear regression. If data are   (11 mm, 10 ft), (23,20), (49, 50) and (95.1, 100), what are the FOV     and aperture diameter?                                                                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRASVSW      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════╕                                                     │ FIGURE OF MERIT └─────────────────────────────────────────────────────                                                                        14.0       Figure of Merit of an IR System: Noise Equivalent Power               Density (NEPD), Radiance (NEN) and Temperature (NET).          ────────────────────────────────────────────────────────────────────────                                                                                 The performance of an IR instrument depends on many factors.            Some of the more important are:  field of view (FOV), usually           from 1 mrad to 20°, spectral bandwidth typically from 1.3 to 2          % of its nominal wavelength if a CVF is used, detector type,            such as InSb, MCT or a sandwich combination of these, and               electrical bandwidth (from 1 to 20 KHz or wider, depending on           the acquisition rate).  Performance is usually evaluated in two         different ways depending upon whether the target fills the              detector field of view or whether the target is essentially a           point source within the detector field.                                                                                                         At least three useful figures of merit are available for                evaluating an IR instrument's ability to sense target radiation         above its electrical noise.  1) The noise equivalent power              density (NEPD) which characterizes its ability to detect a              target (essentially a point-source) that does not fill the              detector field of view.  2) The noise equivalent radiance (NEN)         which indicates the instrument's ability to detect a target             that fills the instrument's FOV.  3) The noise equivalent               temperature (NET) which is the smallest different in target             temperature that can be detected.                                                                                                                                                                                       Definitions of calculated and supplied parameters follow:                                                                                       PEAK_WAVE - The wavelength of maximum spectral radiance as              predicted by the Wien Displacement Law.                                                                                                         FOV[MRAD] - The field of view (FOV) in milliradians.  Usually           the FOV is express in degrees if FOV>0.5° and in mrad if                FOV<0.5°.                                                                                                                                       W_R[STER] - The solid angle subtended by the FOV.  The FOV is           the apex angle of a plane bisecting the FOV cone.  For a FOV            that can be approximated as a right-circular cone, the solid            angle is equal to 2*pi*(1-cos(FOV/2)) or pi*tan(FOV/2)^2.                                                                                       NEP[W] - The noise equivalent power in Watts at the detector.           NEP is calculated from the detector D* (dee-star) at the                specified wavelength, the detector area DT_AREA in mm and the           detector bandwidth DT_BW in Hz.                                                                                                                            NEP= sqr(DT_AREA*DT_BW)/D_STAR.                                                                                                      NEP[W] is used to supply "the noise" in most figure of merit            calculations.                                                                                                                                   NEPD[W] - The noise equivalent power density at the entrance of         the collecting optic in Watts÷cm².  The NEPD (also sometimes            called the noise equivalent irradiance or NEI) is the target            energy density required to equal the detector noise scaled to           its energy density at the entrance of the collecting optic.             With NEPD, the instrument achieves a signal-to-noise ratio of           1.  The smaller the NEPD, the better the performance of the IR          instrument.  NEPD is derived from a point source target which           does not fill the instruments field of view and is always               measured above the background irradiance.                                                                                                       NEN[W] - The noise equivalent radiance at the entrance of the           collecting optic in Watts÷cm²÷ster.  The NEN is the target              radiance required to equal the detector noise scaled to its             equivalent radiance at the entrance of the collecting optic.            With NEN, the instrument achieves a signal-to-noise ratio of 1.         The smaller the NEN, the better the performance of the IR               instrument.  NEN is always derived from a target that fills the         instrument FOV.                                                                                                                                 N_EFF - the effective radiance at the entrance to the                   collecting optics for a blackbody target at temperature TEMP[C]         and in the specified spectral bandwidth SP_BW[HZ]                                                                                               WET - The power at the detector equivalent to N-EFF.                                                                                            S_N - the system signal-to-noise ratio.  Calculated for a               blackbody target filling its FOV.  It is calculated from the            ratio of the target power at the detector WET to the noise              equivalent power NEP.                                                                                                                           NET[C] - The blackbody temperature difference (always at a              specific temperature TEMP[C]) equal to the detector/electronic          system noise.  NET is directly related to the system's NEN and          the filter region/spectral bandwidth of the filter used in the          system.  NET = NEN / (dNEN/dT at TEMP[C]).                                                                                                                                                                              The following parameters are known for the instrument's                 intended design.                                                                                                                                CL_A[CM²] - The net collecting area in cm² of the entrance              optic reduced for any obscuration such as from the secondary            mirror of the Cassegrain collector.                                                                                                             DT_A[MM²] - The area of the detector in square mm.                                                                                              DT_BW[HZ] - The electrical bandwidth of the system in Hz.                                                                                       D-STAR - The detector detectivity in cm sqr(Hz)/Watt.  Usually          supplied with the detector specifications.                                                                                                      FOV[°] - The instruments FOV in degrees.  Generally, based on           the limiting aperture which may be the detector or a field              stop.                                                                                                                                           OPTEFF - The optical efficiency of the instrument.  The product         of the individual efficiencies of collecting optics, filters,           transfer optics and demodulating circuits.                                                                                                      SP_BW[HZ] - The spectral bandwidth in Hz.  (Usually the width           of the 50% transmission or power points relative to the center          of the filter.)                                                                                                                                 TEMP[C] - The target temperature in °C.  The target is always           assumed to be a blackbody.                                                                                                                      WVLNG[u] - The spectral wavelength in microns.  The center of           the filter band.                                                                                                                                An equation template that calculates NEPD, NEN, S_N and NET for         a blackbody target that fills the instrument FOV follows.                                                                                                                                                      'EQU               ≡ ** If highlighted, type Q to get equations      ** 'EGE IRRASENS.EQU  ≡ ** discussed above.  Scroll down to highlight.  **                                                                                                                                                          For an instrument with an InSb detector at 77K and CVF spectral         selector with 2% nominal resolution, use this template:                                                                                Topic # 36: INFINITY FOCUS, FULL FOV, FIGURE OF MERIT, INSB DET(77K),    2% CVF                                                                                                                                         Discussion: Calculate NEPD, NEN and NET for an InSb, CVF radiometer      with the following: Detector area 1 sq mm, det bandwidth 1 Hz, optical  efficiency of 0.06, collecting area of 120 sq cm, FOV of 2.5 mr,        target temperature of 90°C at wavelengths of 1, 3 and 5 microns.                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAINSB      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          For an instrument with an MCT (HCT, Mercury Cadmium                     Telluride, mercad) detector at 77K and CVF spectral                     selector with 2% nominal resolution, use this template:                                                                                Topic # 37: INFINITY FOCUS, FULL FOV, FIGURE OF MERIT, MCT DET(77K), 2%  CVF                                                                                                                                            Discussion: Calculate NEPD, NEN and NET for an MCT, CVF radiometer with  the following: Detector area 1 sq mm, det bandwidth 1 Hz, optical       efficiency of 0.06, collecting area of 120 sq cm, FOV of 2.5 mr,        target temperature of 90°C at wavelengths of 6, 12 and 14 microns.                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAMCT       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          A plot of estimated signal-to-noise vs wavelength for a Minarad         Spectrum SA-1200 narrow FOV (2.5 mr), CVF radiometer with a 6           inch Cassegrain collector and MCT detector is shown below.                                                                                                                                                                                                                              1658-                                                        k              |         Signal to noise vs. wavelength              k                 |         MSI SA-1200 Radiometer                   k     k              |         6 inch Cassegrain Collector          k           k            |         MCT (77K) detector                k              k        1243-         2.5 mr FOV                      k                             |                                       k                    k          |                                     k                      k          |                                   k                                   |                                  k                           k    829.-                                k                             k        |                              k                                 k      |                             k                                         |                            k                                   k      |                          k                                        414.-                        k                                              |                       k                                               |                     k                                                 |                   k                                                   |                k                                                 +0.00-k k   k   k  k                                                         ||--------------|--------------|--------------|--------------|-----      1             4.0            7.0            10.0           13.0                                   wavelength (microns)                                                                                                                                                                             To plot this spectral response use the following template.              Type comma first to read the comment for instructions.                                                                                 Topic # 38: INFINITY FOCUS, FULL FOV, FIGURE OF MERIT, MCT DET(77K), 2%  CVF, PLOT SIGNAL to NOISE VS WAVELENGTH                                                                                                        Discussion: Plot the signal-to-noise for an MCT, CVF radiometer with     the following: Detector area 1 sq mm, det bandwidth 1 Hz, optical       efficiency of 0.06, collecting area of 120 sq cm, FOV of 2.5 mr,        target temperature of 90°C from 1 to 14.9 microns.                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAPSN       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                                                                                         ╒════════════════════════╕                                              │ SERIES and INTEGRATION └──────────────────────────────────────────────                                                                        15.0       Circumference of an Ellipse: Calculation by Series and                Integration                                                    ────────────────────────────────────────────────────────────────────────                                                                                 Calculations involving integration and summation of formulas            are relevant to all math-based theories.  The following                 examples illustrate these methods.                                                                                                              The circumference of an ellipse with axes lengths of "a" and            "b" as measured from its origin cannot be expressed in a simple         "closed" form by a formula.  The approximate value is                   calculated as:                                                                                                                                         l ≈ 2 * pi * sqr( (a^2 + b^2)/2 )                                                                                                        Thus if a = 50 and b = 10, the circumference, l ≈ 226.5.  The           circumference can be express "exactly" by use of the elliptic           integral of the second kind E(k,Φ):                                                                                                                    l = 4 * a * E(k,Φ)   k= sqr( abs(a^2-b^2) )/a                                                                                            There are at least two methods to obtain a higher degree of             accuracy:  the first is to use a series expansion for E(k,Φ)            and the second is to numerically integrate E(k,Φ).  The                 following two examples show how AstroFormCalc 3 can be used to          sum up a series or to accurately integrate an equation.  The            methods are very general and are applicable to many problems in         Optics as well as virtually all other disciplines.  It should           be noted that the "approximation" formula, given above, is only         accurate to about 8%.                                                                                                                  Topic # 39: SERIES, ARC LENGTH (CIRCUMFERENCE) OF ELLIPSE                                                                                       Discussion: Estimate the circumference of an ellipse with axes lengths   of 50 and 10 cm as measured from its origin. Use the series             approximation.                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAELSE      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 40: INTEGRATION, ARC LENGTH (CIRCUMFERENCE) OF ELLIPSE                                                                                  Discussion: Estimate the circumference of an ellipse with axes lengths   of 50 and 10 cm as measured from its origin. Use the elliptic integral  method.                                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE IRRAELIN      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════════╕                                               │ MINARAD SPECTRUM INFO └───────────────────────────────────────────────                                                                        16.0         About Minarad Spectrum.                                    ────────────────────────────────────────────────────────────────────────                                                                                 As a matter of background, Minarad Spectrum designs and                 manufactures a complete line of electro-optical products                including:                                                                                                                                      Radiometers, spectroradiometers, IR spectrometers and IR                thermometers,                                                                                                                                        Radiometers measure absolute radiance or irradiance                     relative to their internal reference blackbody over an                  entire wavelength interval.  Spectroradiometers perform                 the same functions but at many discrete wavelengths across              a spectral interval.  IR spectrometers usually measure the              transmittance and absorbance of the target relative to a                reference over many discrete wavelengths.  IR thermometers              actually measure radiance but display the equivalent                    blackbody temperature.                                                                                                                     IR sources (blackbodies), heat-only and thermoelectric, conical         cavity and extended area, absolute and differential modes,                                                                                           The IR source or blackbody emits a spectral radiance                    distribution which closely approximates that predicted by               Planck's function.  It is the primary reference for most                IR work.                                                                                                                                   IR test sets, collimators, target wheels, target projectors and         targets/masks,                                                                                                                                       IR test sets usually consist of an IR source with an                    aperture wheel or variable slit, and optionally a filter                wheel, optical modulator (chopper) and possibly a shutter.              A collimator (off-axis parabolic mirror style) is used to               project parallel IR energy with a FOV which depends on the              widths of target masks located in its focal plane. A                    target projector consists of an IR test set, target wheel,              collimator and a series of apertures/targets/masks located              in its focal plane.                                                                                                                        IR microscopes and telescopes with remote sensing applications,                                                                                      IR microscopes are used for measuring small spots (10                   microns) at relatively short distances (0.5 to 3 inches,                typical).  IR telescopes are used to study larger spots (1              or 2 feet diameters) are very long distances ( 500 feet or              more).  Radiance, temperature or IR spectroscopy                        measurements result.                                                                                                                       Field-portable UV, Vis and NIR spectrometers,                                                                                                        Compact spectrometers and spectrophotometers                            (tungsten-halogen, xenon and deuterium sources included)                using Si CCD photodiode arrays for detectors and fiber                  optics.  Cover from the near UV (190 nm, blue enhanced Si               detector) to the near IR (1100 nm), this includes the                   visible (Vis) from 400 to 800 nm.                                                                                                          Custom instrument interface software using LabVIEW for Windows,                                                                                 Upgrades of older Barnes Engineering Corp (BEC) radiometers             such as the Mark II 12-550 spectroradiometer.                                                                                                   Many of the calculations presented in this application manual           are used in the design and operation of these instruments.                                                                                      If you need any guidance related to these instruments, please           contact the Minarad Staff at (203) 452-8310 or fax a note to            (203) 452-8311.                                                                                                                        ╒═══════════╕                                                           │ GLOSSARY  └───────────────────────────────────────────────────────────                                                                        17.0     Glossary of Terms                                              ────────────────────────────────────────────────────────────────────────                                                                               ║ Note, Use the Jump command to locate a glossary reference.    ║       ║ Scroll the screen until the reference is in the middle at line║       ║ 12.  The Jump box will light up.  Type J to locate.  Repeat   ║       ║ typing A, as needed, to continue the search.                  ║                                                                                                                                                Cartesian ovoid.  See IRRA402.                                                                                                                Compound thick lenses.  See IRRA443.                                                                                                          Compound thin lenses.  See IRRA427.                                                                                                           Integration.  See IRRAELIN.                                                                                                                    R Source (blackbody).  See IRRA1946.                                                                                                         Lens Imagery.  See IRRA419.                                                                                                                   Mirrors.  See IRRA495.                                                                                                                        Series.  See IRRAELSE.                                                                                                                        Spherical refracting surfaces.  See IRRA405.                                                                                                  Radiation.  See IRRASLWV.                                                                                                                     Thick Lenses.  See IRRA442.                                                                                                                   Thin Lenses.  See IRRA410.                                                                                                                    ╒═════════╕                                                             │ LICENSE └─────────────────────────────────────────────────────────────                                                                        18.0         Product License and Warranty.                              ────────────────────────────────────────────────────────────────────────                                                                                       Software License Agreement  (Registered Users)                                                                                            If you are a registered user, Minarad Spectrum, Inc.  (MSI)             grants you the right to use this Software for a single-user             computer or a workstation not attached to a network.  You may           not rent or lease this Software without the expressed written           permission of MSI.  You may not decompile, disassemble, reverse         engineer, copy, create a derivative work, or otherwise use this         Software except as stated in this agreement.  The license               authorizes you to create and use an extra copy of the Software          on a home or laptop computer, as long as the extra copy is              never loaded at the same time this Software is loaded on the            primary computer on which you use this Software.                                                                                                If any provision of this agreement is found to be unlawful,             void, or unenforceable, then that provision shall be severed            from this Agreement and will not affect the validity and                enforceability of any of the remaining provisions.  This                agreement shall be governed by the laws of the State of                 Delaware.                                                                                                                                       The Software and accompanying materials are provided with               Restricted Rights.  Use, duplication, or disclosure by the              Government is subject to restrictions as set forth in                   subparagraph (c) (1) (ii) of The Rights in Technical Data and           Computer Software clause at DFARS 252.227-7013, or                      subparagraphs (c) (1) and (2) of the Commercial Computer                Software--Restricted Rights at 48 CFR 52.227-19, as applicable.         Contractor/manufacturer is Minarad Spectrum, Inc., at the               address listed below.                                                                                                                           Non registered users must comply with the Shareware policy              described elsewhere in this software Pacakage.                                                                                                            Type T to return to the Table of Contents.                                                                                                                 Warranty Information                                                                                                       Minarad Spectrum, Inc.  (MSI) hereby disclaims all warranties           relating to this Software, whether express or implied,                  including without limitation any implied warranties of                  merchantability or fitness for a particular purpose.  MSI will          not be liable for any special, incidental, consequential,               indirect or similar damages due to loss of data, loss of                profits, lost savings or any other reason, even if MSI or an            agent of MSI has been advised of the possibility of such                damages.  In no event shall MSI's liability for any damages             ever exceed the price paid for the license to use Software,             regardless of the form of the claim.  The person using the              Software bears all risk as to the quality and performance of            the Software.  This Software is licensed AS IS.                                                                                                 Any action or proceeding brought by either party against the            other arising out of or related to this agreement shall be              brought in a STATE or FEDERAL COURT of competent jurisdiction           located in Fairfield County, Connecticut.  The parties hereby           consent to in personam jurisdiction of said courts.                                                                                                       Type T to return to the Table of Contents.                                                                                   ╒══════════════╕                                                        │ USER SUPPORT └────────────────────────────────────────────────────────                                                                        19.0                    User Support.                                   ────────────────────────────────────────────────────────────────────────                                                                                 This program has undergone over 5 years of "beta testing"               around the world.  The algorithms have been in use since the            1960's, are proven and believed highly reliable.  However,              comprehensive programs of this nature may develop what are              called "bugs".  User support is provided to all registered              users for a minimum of 3 months without charge.  Registered             users can receive such help via phone/mail.  Registration also          enrolls the user for notification of "bug" fixes, upgrades, new         versions, new products and many more benefits.  The phone and           fax numbers for Minarad Spectrum are:                                                                                                                    Voice: 203 452-8310        Fax: 203 452-8311                                                                                           To jump to the registration section, position the jump line in          the center of the display and type J.  ∙Registration∙.                Support will be provided to non-registered users and users              beyond the free support period on a time and materials basis.                                                                                                                                                  ╒══════════════╕                                                        │ REGISTRATION └───────────────────────────────────────────────────────                                                                         20.0                 Order/Registration Form                            ────────────────────────────────────────────────────────────────────────                                                                                 If a printer is available, position the order form (part 1) on          the screen and type the "(print screen)" key to make a copy.            Then, advance the text and position the order form (part 2) on          the screen and type "(print screen)" again to finish.                                                                                        Minarad Spectrum Products Order/Registration Form  (part 1)                                                                                   Name:________________________________  Title:________________                                                                                   Company:_____________________________________________________                                                                                   Address:____________________________ City:___________________                                                                                   State:___________   Zip:_____________  Country:______________                                                                                   Date:_____/_____/_____           (Fill out form completely)                                                                                     ---------------------------------------------------------------         Product                     #     Price/ea     S&H       Total          ---------------------------------------------------------------                                                                                OPTICS, IR and RADIOMETRY   ___    _$29.95_    _$5.00_   _______        Calculations By Nelson                                                  Engborg and Paul Murphy                                                                                                                                                                                                                                                                      ------------------------------------------------------------------------------------------------------------------------------------------------           Minarad Spectrum Order/Registration Form  (part 2)                                                                                       * Canadian Orders, Add $2.00 per copy                   _______                                                                                 * International Orders, Add $6.00 per copy              _______                                                                                 * CT Residents add appropriate sales tax                _______                                                                                                                  TOTAL ENCLOSED      __________                                                                                 Disk Size: [ ] 5-1/4"   [ ] 3-1/2"   [ ] Check  [ ] Visa  [ ] MC                                                                                Card# __________________________________     Exp. Date ________                                                                                 Signature ______________________________                                                                                                        How did you hear about our Products?    [ ] BBS   [ ] Friend            [ ] Other _____________________________________________________                                                                                               Mail Completed form with Payment to:                                                                                                           Minarad Spectrum, Inc.   Dept. MSI311                                            126 Monroe Turnpike                                              Trumbull,  CT   06611   06404  USA                                                                                               Support ONLY,  203 452-8310     Fax ONLY, 203 452-8311                                                                                               [July 1994]  Prices subject to change without notice.                                                                                                 Type T to return to the Table of Contents.                                                                                       ╒════════════╕                                                          │ REFERENCES └──────────────────────────────────────────────────────────                                                                        21.0           Optics and Infrared References.                          ────────────────────────────────────────────────────────────────────────                                                                         M.  Born and E.  Wolf, "Principles of Optics", Pergamon, Oxford, 1980.                                                                          Dr.  M.  Foigel, Chief Editor, "The Optics Problem Solver", Research    and Education Association, 61 Ethel Road West, Piscataway, New Jersey,  08864, Revised Printing 1990                                                                                                                    E.  Hecht, "Optics", Addison-Wesley, Reading, Massachusetts, 1988.                                                                              Eugene Hecht, Ph.D., "Optics", Schaum Outline of Theory and Problems,   McGraw-Hill Book Company, New York, 1975                                                                                                        M.  V.  Klein and T.  E.  Furtak, "Optics", John Wiley & Sons, New      York, 1986.                                                                                                                                     R.  S.  Longhurst, "Geometrical and Physical Optics", John Wiley &      Sons, New York, 1964.                                                                                                                           F.  W.  Sears, " Optics", Addison-Wesley, Cambridge, Massachusetts,     1948.                                                                                                                                                          ------------- IR References -------------                                                                                        Nelson Engborg, Al Kozlowski and Paul Murphy, "IR polarization:         Applications to sources and radiometers", Technical Note #11, Minarad   Spectrum, Inc., May 1994.                                                                                                                       Nelson Engborg and Paul Murphy, "Discussion of Field of View (FOV),     spot diameter and working distance for an infrared system", Technical   Note #21, Minarad Spectrum, Inc., May 1994.                                                                                                     Nelson Engborg and Paul Murphy, "Non-contact temperature measurement    with the infrared microscope", Technical Note #31, Minarad Spectrum,    Inc., May 1994.                                                                                                                                 Al Kozlowski, Al Waldie, Nelson Engborg and Paul Murphy, "8 inch Target Projector:  Collimator, target wheel and IR source", Technical Note     #41, Minarad Spectrum, Inc., May 1994.                                                                                                          Gerald C.  Holst, "Testing and Evaluation of Infrared Imaging Systems", JCD Publishing Co., 2923 Cove Trail, Maitland FL 32751, 1993.                                                                                   W.  L.  Wolfe and G.  J.  Zissis, "The Infrared Handbook", Infrared     Information and Analysis Center (IRIA), Environmental Research Institue of Michigan, Revised Printing, 1985.                                                                                                           ╒═════════════════════╕                                                 │ MANUALLY INSTALLING └─────────────────────────────────────────────────                                                                        22.0           Instructions for Manually Installing Package.            ────────────────────────────────────────────────────────────────────────                                                                             The following are minimal instructions for manually installing          Minarad Spectrum's Optics and IR Radiometry Package.                                                                                                Minimum distribution files:  README.1ST, MSI311.ICO, ACAL3.EXE,         ACAL3HLP.HLP and MSI311.HLP with its *.EQU files.                                                                                               These files will probably be bundled in a compressed file such          as MSI311.ZIP, MSI311.SDN, or something similar.  Please                unload.                                                                                                                                         Create a DOS directory named MSI and change to it:  cd\ (enter)         md\msi (enter) cd \msi (enter)                                                                                                                  Copy ACAL3.EXE, ACAL3HLP.HLP, MSI.ICO and MSI311.HLP and all            *.EQU file into the directory MSI                                                                                                               Add these statements to AUTOEXEC.BAT in the base directory:             SET ASTRO3=C:\MSI. Append to existing PATH statement ;C:\MSI;                                                                            Running Under Windows as a Non-Windows Application                                                                                                     Follow the installation instructions and install on the hard            drive C.  Start WINDOWS (tm, MicroSoft).  Click on FILE then            click on PROPERTIES.  Enter the following information for each          of the fields.  Description:  Optics and Radiometry, Command:           ACAL3.EXE MSI311.HLP, Working Directory:  \MSI.  Click on               CHANGE ICON.  Enter Filename:  C:\mis\mis311.ico.  Click on OK          to exit and click on OK to exit FILE.  The MSI icon should              appear in the accessory window.  Click on it to run MSI311.                                                                              To start:  cd\msi (enter)    acal3 msi311.hlp (enter)                                                                                                  Access on-line help (file: ACAL3HLP.HLP) from equations                 display with (F1) key.                                                                                                                 ╒═══════════╕                                                           │ HELP FILE └───────────────────────────────────────────────────────────                                                                        23.0          Help File: Summary of File Viewing Commands               ────────────────────────────────────────────────────────────────────────                                                                            A - Auto-search of entire text for user-supplied string. Search is          in uppercase and is case insensitive. If string found,                  type A to continue or to enter new string, type (space) A.          B,N - Move back or ahead 18 lines.  Also use (backspace) for "B".       E, (esc) - Exit from help file to equations display.                    F,T,K - Jump to the first (beginning), T.O.C or last (end) line.        G - Skip to Glossary of terms or summary of functions, if present.      J,(ent) - Search line on │JUMP│, Reference is after . Jump to                    line number or perform autosearch ahead if text reference.    L,P - Jump to line/page. Type Lxxx(ent) or Pyy(ent) to jump to                line #xxx or page #yy. With + or -, ahead or back 56 lines.       S,R - Skip ahead (S) or back (R) in help file to major headings         W - Skip to Summary of useful calculations or warnings, if present      1 to 9 - Typing 1 to 9 advances the text by 1 to 9 lines.               (ctrl Q) - Exit from help file.  Same as command "E".                   Edit keys - Use edit keys: (Home) (up arrow) (PgUp) (left arrow)                    (right arrow) (End) (down arrow) (PgDn).                                                                                             Entry into the application file is at/near the last position.           Type: T to return to Table of Contents or E to exit the file.      