OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-2 18-1    :  n  Section 18 n Dispersing Prisms    12 2  is the sum surfaces. the deviations at the two of is the sum sin sin sin cos sin        The ray deviation as a function of the input angle the of deviation as a function ray The The net ray ray deviation net The Dispersing Prism OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-4 18-3       2      2   1   n 1 12     1    .    2 MIN       cos sin 1 22 ° n n 1  12 80 n      1 12 2         n 122 sin sin sin cos sin   n           sin sin sin cos sin 211   211 ° 1        211 60 °  sin sin sin cos sin  sin sin cos sin sin cos sin 60   MIN   n1.5 ° 40 for which the ray deviation is minimized. the ray which for   n 1 MIN    ° ° °  1   n n 1 MIN 50 45 40        22 11          n 1 222 111 sinsin sin sin      1 (180 ) 180     ,,,,   sin sin ,, 22 12  11           12 12 12 nn       12 11 2 2  22 1              21 1 The corresponding deviation is the angle of minimum minimum deviation deviation is the angle of corresponding The There is some input angle There is some sin sin sin sin sin Snell’s Law: Snell’s Positive: Negative:  Deviation versus Input Angle Input versus Deviation Prism Deviation - Deviation Prism Derivation    OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-6 18-5 MIN  IN M   MIN     = 60°  For n n 2.0 -120° 1.31.41.5 -21.1° 1.6 -28.9° 1.7 -37.2° 1.8 -46.3° -56.4° -68.3°  n         MIN   gn convention, the deviation is negative for convention, gn 2 /  MIN   .   of the net deviation the net deviation of MIN   n   1  MIN   :  2sin sin /2  sin / 2  MIN   produce minimum minimum produce  ′      sin / 2    and MIN n     The ray path is symmetric the prism. is symmetric through path ray The Therefore, at Both equal. not are they but deviation, contradiction is a as there This cannot angles that input two different be minimum. the produce The minimum the is for deviation condition minimum The measurement of the index of refraction: At minimum symmetric: is not assume deviation, path At minimum At minimum deviation, the ray path through a dispersing prism is symmetric is ray The symmetric prism . a dispersing is path through the ray deviation, At minimum at each si equal amount By surface. bent an this prism orientation. angle The of minimum deviation is Minimum Deviation – Deviation Minimum Proof Minimum Deviation Minimum But both the forward and reverse But both the and reverse forward ray of magnitude the same paths produce be to assumed deviation, At minimum 50% deviation, At minimum occurs at each surface. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-8 18-7 0 Telescope min   . . α 2 Reticle   12 II .     . n 1 MIN     1 /2     2    sin / 2          12 1  or to measure the prism apex angle apex prism the measure or to     sin / 2        12 n n   n MIN    n  1     11   sin / 2     2sin sin /2 MIN sin sin Collimator on of prism deviation. prism on of  to determine the angle of minimum determine deviation. minimum to the angle of I /2 sin / 2  2   1        I  n   2 MIN 12  12  MIN from I from Slit sin / 2 sin / 2   1  min 2 1 Source         MIN 1    Accuracies of one part in the sixth decimal place can be obtained. diffraction effects. Limited the slit and by width - prism) I angle (no Measure the straight through - telescope The and the collimator are mounted to a rotation stage. - the telescope autocollimat Use as an - I Subtract - Insert the refracted observe the angle of the prism and beam - deviation and measure obtain minimum this angle I the prism Rotate to Minimum deviation provides an extremely extremely an Minimum deviation provides of index the measuring of accurate method material. a refraction of Refer back to derivati to back Refer Minimum Deviation – Deviation Minimum Derivation Index Measurement –Measurement Index Spectrometer Prism OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-9 18-10 S R n n    1  sin  C  orientation. The The orientation. rs at one surface. the rs at one In  RS S nn n Plano-Convex it should be oriented with the curved side be oriented with it should Dark C is often used for index measurement. is often used for index measurement. The duces the aberrations. Small duces the aberrations. Small the angles make bent the same at amount each lens surface. bent the same  in reflection to measure the index of ) –“convex-plano” the ometer. Some instruments also measure measure also instruments Some ometer. ray bending is split between the surfaces. This minimizes minimizes This surfaces. the split between is bending ray on, all of the ray occu on, the ray bending all of Convex-Plano Diffuse Illumination 1.5  n Reference Prism Transparent Sample Transparent This instrument is called an Abbe Refract Abbe This instrument is called an number Abbe of the material. the 0.0001. is measurement index of method this for accuracy typical The Critical angle techniques also be used can opaque samples. Note that in the minimum minimum Note that in the aberration condition, the like a prism at minimum the lens looks used of edge deviation –surfaces. ray at both equal bending is a slightly biconvex 1.5 = true minimum for n The The planolens. radius, the surface a long but has essentially is insignificant. difference In the plano-convex configurati In the plano-convex convex-plano the configuration, incidence the angles of at the surfaces and re situation as close to the assumptions paraxial situation the assumptions of optics possible. as close to as When focusing light with a plano-convex lens, a plano-convex light with focusing When towards the long conjugate (the the long object side towards The critical total angle for internal refraction sample is placed contact referencein with a higher index. prism of minimum minimum aberration the light is occurs when Infinity: Object at Index Measurement Index Measurement – Critical Angle Techniques Lens Orientation for Minimum Orientation for Minimum Spherical Lens Aberration 18-11 OPTI-502Optical Designand Instrumentation I

Lens Shape for Minimum Spherical Aberration © Copyright 2019 John E. Greivenkamp

There is no bending that completely eliminates spherical aberration. It can only be minimized. Different object/image conjugates require different bendings to minimize spherical aberration.

The optimum shape varies with index. At high index, as is often found in the IR, more bending occurs at the surface of the lens. For the same focal length as before, the front surface becomes flatter and the best shape is a meniscus.

Object at Infinity:

n  3

Meniscus

Equal bending occurs at both surfaces of the lens and edge of the lens looks like a prism used at minimum deviation.

18-12 OPTI-502Optical Designand Instrumentation I

Dispersion of a Prism © Copyright 2019 John E. Greivenkamp



MIN  n C d F

The details of the prism dispersion depend on the geometry used and the index dispersion

curve. However, assuming the prism is used at or near MIN, the average prism dispersion over a wavelength band (F to C) can be estimated.

d Dispersion of a prism  2sin1 n sin  /2 d MIN   d d dnd dn d 2sin /2 MIN MIN   0 ddnddnd  dn cosa  MIN / 2

d d  dn  0  0  Glass Dispersion 0 d d d

Blue light is deviated more than red light. As  increases,  increases, but  is negative. The magnitude of the deviation decreases. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-14 18-13      C   MIN   MIN d-C F-C d      MIN     F-d F       MIN  n          sin / 2      22 2 22 2 tion of wavelength, the glass dispersion can the glass dispersion wavelength, of tion sin / 2 n n sin / 2 sin / 2    n sn2o /2 1sin/2cos sn2sn/2 1sin/21sin n m m m     Index (BK7)Index (F2) Index    rent wavelength pairs: wavelength rent MIN    n 21 21     nn  1  m m -.0729/ m m -.1186/ m m -.1002/  22 finite differences: finite    n 2sin /2 mmm 1.52238 1.51680 1.51432 1.63208 1.62004 1.61503 2sin /2            2sin sin /2  1sin/2  BK7 F2 cos / 2     dn n  d MIN MIN dn dn   MIN d d    n/ d-C -.0361/ F-d -.0550/ F-C -.0474/ be approximated be approximated by Since the refractive index is a complex Since the refractive is a complex funcindex FdC .4861 Glass dispersion diffe for .5876 .6563  Dispersion of a Prism a Prism – of Dispersion Derivation Glass Dispersion – Example OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 18-16 18-15 3 Detector 2   1  ter dispersion by the m m m    Lens  Focusing Dispersing Prism C C to measure to measure the spectral content a of light detector plane. The slit images each slit images for The detector plane. is then collimated. Af is then collimated.  due to the lens diameter due to the lens diameter and prism size. over the width of the slit image. Each F F ource or the transmission/absorption ource or the transmission/absorption spectrum nn  0  m -.1002/ m 0.177/ m 10.2°/    MIN       = 60°) a MIN MIN 4.18°/ -38.7° rad-.675 -1.535-.0474/ -48.2° rad -.841 -1.705 0.073/ BK7 F2 .00418°/nm .0102°/nm  2sin /2      Sample dd Collimator cos / 2 Measurement   MIN d light F-C glass dispersion dn  d   IN IN M n      Slit dn M  d dddn n d d dn d dn dn    d Source Example: 60° ( prism The source is focused on a slit, the light and on The source is focused lens images prism, the slit the a second onto wavelength are a spectrum. displaced producing the spectrum Each wavelength in is spread diffraction by blurred wavelength is further The dispersing power of a prism be used a prism can of power dispersing The beam. This includes the spectrum of the s of a material of a material placed in the beam. Prism Spectrometer Prism Prism Dispersion – Dispersion Prism Example OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp 18-17 Mirror on Prism Face Detector Slit Source nd large diameter beam. Large prisms are prisms Large beam. diameter large nd Mirror ons use a double pass through the prism the prism for increased through pass ons use a double     R Detector Slit Source The Resolving Power of the spectrometer the spectrometer is defined as of Power Resolving The R can be increased a by using narrow slit a required for high-resolution spectroscopy. high-resolution for required Littrow Prism configurati dispersion: Resolving Power Resolving Power and Littrow Prisms