Modeling variation in the refractive index of optical glasses

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5-1-1990

Modeling variation in the refractive index of optical

glasses

David Stephenson

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Recommended Citation

in the Refractive Index

of Optical Glasses

by

David Stephenson

B.S. Rochester Institute of Technology

(1982)

A thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science

at the Center for Imaging Science

in the College of Graphic Arts and Photography

of the Rochester Institute of Technology

May 1990

Signature of the Author

....,..

_

Accepted by

Name Illegible

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

M.S. DEGREE THESIS

The M.S. Degree Thesis of David Stephenson

has been examined and approved

by the thesis committee as satisfactory

for the thesis requirement of the

Master of Science degree

Dr.

Edward Granger, Thesis Advisor

Dr.

P. MourouliB

Professor Robert Novak

Date

ROCHESTER INSTITUTE OF TECHNOLOGY

COLLEGE OF GRAPHIC ARTS AND PHOTOGRAPHY

I, David Stephenson, prefer to be contacted each time a request for reproduction is

made. I may be reached at the following address:

David Stephenson

Melles Griot

Optics Division

55 Science Parkway

Rochester, NY 14623

Signature

in

the

Refractive Index

of

Optical Glasses

by

David Stephenson

Submitted to the

Center for

Imaging

Science in partial fulfillment ofthe requirements

forthe Master ofScience Degree atthe Rochester Institute of

Technology

Abstract

Experimental measurements that describe the dispersive behavior of production samples of optical glasses are fit with models of minimum_{complexity for}the purposeofin terpolation and extrapolation. Software to perform this procedureon a regularbasisispresented, and showntodis

tinguishbetweenmodels ofinappropriatecomplexity. Two degrees offreedom usually providea _{statistically} optimum fit to the data _contrary to the widespread practice offit

ting

ageneral, six term model to such measurements.

Using

specially developed analysis _tools, it is concluded that _{annealing does}not _{significantly} changethepartial dis

persion of the sample. Partial dispersion is established at the time the ingredients are combined in a melt and is in variantfrom one_annealingtoanother. This isan important result to consider when _planning the fabrication of optical systems whoseprescriptionchanges with changesinmateri al characteristics.

I wish to thank Mike

Mandina,

General Manager ofMelles Griot (Optics

Division),

in

Rochester

NY,

_{for sponsoring}this research; Andrea Hopkins and PublishEase for the

production ofthe

diagrams;

Schott Optical Technologies and Ohara

Corporation, Inc.,

for sample measurement _assistance; Gaertner Scientific Corporation for their

help

in

For my brother Chris

I.

Introduction

Description ₁

History

4

Research ₁₄

II.

Methods

Experimental Equipment ₁₆

Software ₁₈

III. Results

The Effect of

Annealing

on Dispersion 23

Melt Makes Subtle Differentiations ₂₃

Recommended Interpolation Model 24

IV. Discussion

Experimental Equipment 28

Software ₃₉

Output ₃₉

Input ₄₃

Algorithms 45

Analysis ofVariance ₄₈

Model Assessment ₅₈

Replications ₆₂

Chi-square ₆₅

Results ₆₇

V. Conclusions

Annealing

Effect on Dispersion 72

Interpolation Model ₇₂

Suggestions for Further Work 72

VI. Appendices

Appendix 1: Principal Dispersion Invariance with

Annealing

₇₄ Appendix 2:

Equivalency

ofSellmeier & Kettler-Drude Series 75

Appendix 3: Gaertner L123 Spectrometer ₇₆

Accuracy

76

Refurbishment ₇₆

Deficiencies ₇₇

Sources ₇₉

Adjustment & Operational Procedure ₈₀

Appendix 4: Index

Uncertainty

due toAngular

Uncertainty

₈₅

Appendix 6: Data Acquisition Software for HP-41C 92

Appendix 7: MELT User's Manual 100

Syntax Rules 100

Files and Locations 100

Command Syntax 102

Appendix 8: Matrix Algebra Approach 108

Appendix 9: Nested Model Assessment 1 14

Weighted Average in Sum of Squares

Partitioning

.... 117

Example MELT Analysis with Replicated Data 118

Melt Output for Figure 28 & Figure 29 126

Melt Program

Listing

130

Index 146

References 147

Appendix 10

Appendix 11

Appendix 12

Figure 1 Different Dispersions of SKI 6 and F2 Glass Types 2

Figure 2 Exaggerated Variations from Nominal Dispersion 4

Figure 3 Null

Hypothesis,

H0

15

Figure 4 Alternate

Hypothesis,

Hx

15

Figure 5 Gaertner L123 Spectrometer 17

Figure 6 MELT flowchart 19

Figure 7 Model Histogram at 95% Confidence Level 26

Figure 8 Pulfrich Refractometer 28

Figure 9 V-block Refractometer 30

Figure 10 Refraction through aPrism 32

Figure 1 1 Spectrometer Refractometer 32

Figure 12 Prism Sample Dimensions 34

Figure 13

Compensating

Verniers 37

Figure 14 Example ofPlotted Output from MELT {Sample

1)

40

Figure 15 Example ofPrinted Output from MELT {Sample

1)

42

Figure 16 Example Input file for Melt {Sample

1)

44

Figure 17 MELT.MEL

Setup

File for

MELT,

part 1 45

Figure 18 MELT.MEL

Setup

File for

MELT,

part 2 46

Figure 19 Same as Figure

14,

Except that _n0{X) is LAFN2 {Sample

1)

49 Figure 20 ANOVAfor Figure 14 and Figure 15 Data {Sample

1)

51

Figure 21 ANOVAfor a Model thatis too Complex {Sample

2)

53

Figure 22 Plot for a Model that is too Complex {Sample

2)

56

Figure 23 Plot for aModel that is not Complex Enough {Sample

2)

57

Figure 24 F+

FQ

Ratio 59

Figure 25 Melt

Searching

for the Optimum Model {Sample

2)

60

Figure 26 Plot for a Model that is Judged "Optimum"

by

MELT {Sample

2)

. . . 61 Figure 27 ANOVA for aModel whichis ofthe

Wrong

Form {Sample

3)

64 Figure 28 Plot

Showing

CharacteristicsofAnneal #313 {Sample

4)

68

Figure 29 Plot

Showing

Characteristics ofAnneal #314 {Sample

5)

69

Figure 30

Complexity Frequency

71

Figure 31 Model Histogram at99% Confidence Level 70

Figure 32 FilarEyepiece 76

Figure 33 Abbe-LamontEyepiece 78

Figure 34 Sample Adjustment 81

Figure 35 Curved Prism Faces 88

Figure 36 Partial

Listing

ofSCHOTT.TXT 101

Figure 37 D:\SdP\GLASDATA\NOMTNAL\LAF2.MEL 102

Description

As optical objectives become more complex with performance _routinely expect

ed at or near the diffraction

limit,

competition is

forcing

companies to reduce material

costs,

labor,

and lead time required to manufacture them. The tolerance budget is di

rectly affected. Few companies can afford to _{specify extraordinarily} tight tolerances

on optical components and subassemblies. While thispractice will _generally guarantee

ahigh percentage of good assemblies, it also increases material cost,

labor,

and fabri

cation time to an unacceptable level.

The optical _{shop is} most efficient when

fabricating

to commercial manufactur

ing

tolerances. Commercial parts often have too much _variability to guarantee that

they

will lead to diffraction-limited assemblies, however. Methodsof_compensating

for the use ofthese parts must be implemented ifattractive

delivery

time and price are

to be attained. This compensation must take the form of adjustments to the optical

and mechanical designs to allow the use ofthese commercial-grade components to

create an objective thatforms a perfect image. This analysis and adjustment must

often be done on an _{assembly-by-assembly basis.}

Optical components depart fromtheirnominal characteristics in two ways.

Power errors are those which result in rotationally symmetric image _{quality degrada}

tions. Element surface curvatures, thicknesses, separations, and refractive indices all

cause _{varying degrees} ofsymmetrical degradations since thepowerbalance ofthe

objective is upset

by

their departure fromnominal. The power balance can often be

restored

by deliberately

_changing another source of power error to add an equal and

opposite amount. The offending parameteroften goes uncorrected, in the true sense.

Asymmetrical image quality degradations those that are not_rotationally

symmetric about the optical axis are the othersymptom of optical component

departures fromnominal. Surface cylindrical

irregularity,

element wedge or

decenter,

often, performance is restored

by

correcting

the _offender, though _compensating

strategies are

occasionally

adopted.

This researchhas sought tocharacterize one cause ofpower errors: the depar

ture ofthe bulk refractive index ofthe optical component from its nominal, expected

value. _{When departures from nominal are}

small, compensation forthe variation in

raw material refractive index is accomplished

by

_changing airspaces. For departures

of greater_magnitude, curve changes _{may be} required. This _adjustment, done priorto

fabrication,

is referred to as the melt recomputation and _may result in a production

nominal that is different from the design nominal. This is often repeatedjustprior to

assembly when all other characteristics ofthe components are known.

The melt recomputation is complicated

by

the dispersive nature ofoptical

glasses. Refractive index varies in a nonlinear fashion with wavelength. Figure 1

shows the dispersion function for two differentoptical glass types: SK-16 and F-2.

DISPERSION OF DIFFERENT OPTICAL GLASS TYPES

X W D 1.64

2

I- 1.62 O < DC u. LU a.

i.60

1.61

\

\

\ \

\

\300 ~A00 ~4S0 .BOO ToBO .600 .630 .700 .780 .600 .800 .900 .830 1.000 1.030 WAVELENGTH (MICRONS)

TRANSLATION

COMMON MELT-TO-MELT VARIATIONS IN DISPERSION CURVE CHANOE

\

^

^

X

^X\

UJ Q

Z

\N^O"

^ UJ

>

\

^v>~;^t_

~~""~ -U

<

O.

U.

^^^

UJ *--. *"""* '

.

a. ""--* "* *"

-^.

"~

*^h~*~--^_ **** "**" "-.

^"^-^.r^

WAVELENGTH

Figure 2 Exaggerated Variations from Nominal Dispersion

Figure 2 shows, in exaggerated scale, three common changes of_n(K) from

melt-to-melt and annealing-to-annealing forthe same glass type. In practice, of

course, allthree changes occur simultaneously, but one is often the dominant effect. A shift or translation ofthe _n(k) curve relative to nominal requires fewer compensat

ing

changes to the optical design than a change in thefunctional form or shape of n(k). Changes in the shape ofn(k) almost always require element curvature changes;

any techniqueused forthe meltrecomputation should alert the optical designer to

changes in the shape of_n(k) to allow aproperchoice of variables and suggest possible

performance problems.

The nominal dispersive behavior ofoptical glasses has been well catalogued

by

glass The values reported are the result of experimental

necessary forthe optical designer to determine what the actual dispersion_n{X) is for

everymaterial in the optical pathifmaximum performance is to be attained.

It is often not possible todetermine n at some

X

ofinterest

by

directex

perimentation dueto the _{unavailability} of suitable radiation sources. This implies that

the melt recomputation method must be able to_{reliably interpolate} over experimental

data,

smoothing experimental _errors, and

fitting

with a nonlinear dispersion model.

Minimizing

the amount of experimental data requiredforreliable interpolation is an

important goal since thisproblem is one encountered in a production _situation, not an

academic one.

History

Modeling

the dispersion of optical glass is a well studiedtopic. Most workers

have concentrated either upon _extracting maximum _{accuracy in index} over the widest

possible wavelength _{interval using} the fewestpossible _{coefficients,} or upon

developing

a power series modelfor dispersion on which atheoreticalmodel ofthe dispersive

behavior of optical systems would be based.7 Some have also investigated or pro posed suitable models for dispersion in regions ofthe spectrum where the manufactur

ers do not _supply

data,

_namely the short ultraviolet

(UV)

region8

and the

long

infrared

OR).9

In order to _study the dispersive nature ofoptical systems _using aberration

coefficients it is necessary to express_n{X) as a truncated power series. Standard

methods of series manipulation cannot be utilized, nor is derivative manipulation feas

ible,10

ifanotherform is used. Buchdahl proposeda suitable equation11

that was

based on an analysis of thetwo-term Hartmann model,

X-X

n =

n0

+

a(O+b(D2

+c(D3

+ -, Q) =

.

(1)

1 +_.jiA,

Robb and_{Mercado recently} analyzed the Buchdahl model, Eq.

(1),

andhave found

that, while it is not as accurate as the model adopted

by

the glass manufacturers, its accuracy is to allow the theoretical _modeling of thedispersive behavior of optical systems without_{producing misleading} results12.

The accuracy of alternative dispersion models _{is usually} evaluated

by

com paring the index predicted

by

themodel in question withthe index predicted

by

the manufacturers'

dispersion formula,13

2 a a \2

^2

A3

^4

^5

(2)

nl

= An+A,X + + + + . vA

01 _{X2 X4 X6 Xs}

This Laurent series, which can be derived from either the classical or quantum disper sion

models,14

is validover the wavelength interval from 0.365 |im to 1.014 |im with

an index accuracy of 0.000005. Over the restricted range of0.400 |imto 0.750 |im,

it is accurate to 0.000003. The six coefficients

A0,

A

...,

A5

are determined

by

a least squares fit to equation

(2)

ofexperimental

data,

averaged over_many samples and melts. It represents the

manufacturers' best estimate ofthe nominal dispersive charac

teristics ofthe glass type in question. A set ofthese six coefficients is supplied for each ofthe more than 800 optical glass types _currently availiable.

The importance of_{using least} squares methodsto determine thecoefficients

from experimental data is often understated in the literature. Most modelformulations

are polynomials. Exact

fitting

of apolynomial through datapoints thathave ex

perimental error often introduces unsatisfactory oscillatory structure into the

inter-polant.15

When

basing

the coefficients on more experimental data points than there are

coefficients, a set ofsimultaneous equations cannot be cast and solved; the least

squares procedure is the _only practical method. Withmore datapoints thanparame

ters, its use results inthe _{interpolant passing} near each ofthe experimental data points,

but not _{necessarily passing} through _them, such that the sum of squared errors between

the observed and expected values isminimized. _{Buchdahl originally determined} the

coefficients for Eq.

(1)

by

_solving simultaneous equations; Robb andMercado have shown that

by

_{using least}squares techniques it is possible to reduce the index error

by

a factoroffive overthe same interval.16

While the manufacturers'

dispersion formula

(2)

is almost an order of_mag nitude more accurate than the Buchdahl model

(1),

this does notdiminish the useful ness ofthe latter. Equation

(1)

is suitable for the academic _study of optical system dispersion

by

orders, just as the other_primary optical aberrations have been studied

(e.g.,

third-order spherical _{aberration, etc.);} Eq.

(2)

is not suitable for this type of study. But Eq.

(2),

and _any otherform that was derived from eitherthe classical or quantum theories of

dispersion,

has greater _{accuracy due} to its physical basis. It is intended to be used when

designing

and _{manufacturing} an objective. Both approaches

are accurate enough for theirintended applications.

Many

other functional forms that have been proposedfor dispersion modeling have been summarized in the literature. Some are empirical while others are based on

the physical and chemical phenomenathat give riseto thedispersive nature ofoptical glasses. A short list follows:

Cauchy17

Conrady

18

_V-*-

(3)

0 _{X2 X4}

Hartmann 1-term19

n =

___

(5)

1.2

Hartmann 2-term20

(*-**)

"=A0+T-4LTT

(X-X.f-2

(fi)

Hartmann 3-term21

n=A0+L-+ 2

(7)

(a

-a,)

(a-Aj)

Lorentz-Lorenz22

2-l

^e2

4,

A.

_i_ _....__ *

2+2 37tm _{v2-tf v2-v}

Helmholtz23

^(l-K2)

= 1+

1

+- ₊ i

+- ₊

____+

.-(8)

+

*-;?___

*-!;?___

(9)

12 i2 ₁₂ t,2

A

-A[ A A;

Sellmeier24

A,a2 Al2

n2=l+_J + +__ +

(10)

12 \2 _\2 \2

A

-Aj A

-A

Kettler-Drude25'26

"2=^o+-r-LT

+ - ₊

^4

+

binomial27

n2

= -+A_lX2+A0+AlX-z+A2X-4

+-+AjX-2j

+-

(12)

Herzberger

(visible)

28

n = AL2+BL+

C+DX2,

L=

L_,

a0

= _0.168um

(13)

A

-Aq

29

Herzberger

(infrared)

n = _AL2+BL+_C+DX2+

EX4,

L =

L,

a0

= _0.168um

(14)

A, _Aq

The empirical _equations, Eqs.

(3)

through

(7),

are _surprisingly accurate over the visible spectrum. All provide for one or more absorption bands

by

_{making index} approach at particular _{wavelengths, Aj,} where the glass molecules resonate.

They

differ in the location ofthese wavelengths and in how many absorption bands

are

provided.30'31

The

Cauchy

and

Conrady

forms provide _{for only} one absorption band located

at a wavelength of zero. It can be shown that the index of refraction at a wavelength of zero is

unity,32

however,

so one should not have high expectations for theperfor mance of such models. Absorption bands are known toexist on both sides ofthe visible region, onein the nearUV and anotherinthe farIR.33

The Hartmann

formulas,

Eqs.

(5)

through

(7),

provide for one or more such absorptionbands at nonzero wavelengths and, consequently, are ofimproved accuracy.

The Hartmann models _may, at

first,

seem _especially well suited for situations where littleexperimental information is available _regarding the actual dispersion of a

Algebraically,these singularities are caused

by

formulatingthemodel sothat thedenominator of one or moreterms goestozero atA.j, making indexapproachinfinity. In_actuality, of

course, index doesnotbecome infiniteatthese wavelengths; it is onlyanomaliesinthedisper sion curve thatoccur. Butsincelittleuse oftheoptical materialis made at or nearthese

absorption bands,the infinite indexartifact ofthealgebraicformulation doesnotdiminishthe

sample. Instead of

determining

the absorption wavelengths with theother coefficients

by

solving simultaneous equations or

by

leastsquares,

they

may simply be assigned values that are realistic _though, perhaps, not optimum. This reduces the numberof coefficients that must be determined from the scarce experimental data.

Forexample, ifforEq.

(5)

we assume _Aq = _{0.168 |im, which} is _{the mean}

absorption band central wavelength reported

by

Herzberger,34 itcan be rewritten

A0

=

(A.

_-0.168

|_n)

.

(I5)

Upon substituting one _{experimentally determined} wavelength and index pair, say Ad

and _nd, the coefficient

A0

is found. A rough estimate ofindex at some otherwave

length may then be performed.

As might be expected, such a simple-minded approach is oflow reliability.

Suppose,

for example, that two different glass types, each with the same _nd but

different reciprocal relative

dispersions, Vd,

are compared with this approach. The

interpolated indices will be equal.

Yet,

as shown in Figure

1,

their_{dispersions may be}

very different.

Dispersion _{is reportedly determined}

by

the chemical composition ofthe glass at

the time ofthe melt.35

During

annealing, stress is relieved,

homogeneity

is im

proved, andindex is adjustedto within a specified tolerance from nominal at one

wavelength

(usually

Ad). But the effect of_annealing on the reciprocal relative

dispersion,

the so calledAbbe number,

Inthecourse ofperformingthisresearch, it has beenfoundthatmuchbetterresults are obtainedifthequantity (Jl-0.168um)israisedto the0.012power ratherthan the 1.2power

when thewavelengthunits aremicrons. This istrueevenifthe0.168um valueis subsequent

ly

allowedto _{vary during}optimizationofEq. (5)toexperimental data. Equation₍₆₎benefits

Vd

- __L

,

(16)

n-nc

is minor. The quantity d-l changes with _annealing, but theprincipal

dispersion,

the

quantity nrnc, does not [see Appendix

1,

page 74]. While there are _any numberof combinations of_nF and _nc that could occur and

keep

the principal dispersion invariant

with _annealing, with_only a single experimental datapointthe opticaldesigner must assume _{that nF, nc,} and _nd have all changed

by

the same amount. The result is that

the actual _n{X) curve is parallel to the nominal _{n0{X) curve,} as shown

by

the translation

case in Figure

2*

An improvement in the _{interpolation accuracy} overthatprovided

by

empirical

equations occurs when chemical andphysical

theory

is used to suggest a dispersion

model. With theLorentz-Lorenz series, Eq.

(8),

index

is,

ingeneral, complex-val although _only thereal terms are shown here.36 The

imaginary

componentbe

comes significant inthe region of an absorptionband and _may, formost practical situ ations in the design andfabrication of optical objectives, be ignored. Note thatEq.

(8)

is a function of

frequency

instead of_wavelength, where x> = _{c/X and c} = 3 _x 108_m/s.

The Lorentz-Lorenz equation is the fundamental equation of classical dispersion

theory;37

e is the charge of an electron, m is the mass of an electron,

N0

is

Avoga-dro's number, andthe _quantityA- _is related to the strength ofthe absorption that

occurs at and aboutthe

frequency

D-. One term is carried foreach ofthe absorption

bands characteristic ofthe material. This equation_may be simplified when not used

Theexperimentalportion ofthisresearchwilltestthis conclusion, whichis basedonthesimple

analysis presentedin Appendix 1,

by

determiningwhether samples(whichare all fromthe samemeltbut differentannealings),all have n{X)curves which are parallel toone another

withinexperimental accuracy.

Ditchbum describeswhythe indexmustbecomplexinordertobuilda_relationshipthatdefines

thepolarization vector ofthewaveinthedielectric_(glass)as afunctionofthe incidentelectric

to model dispersion nearthese absorption bands. The Sellmeier series, Eq.

(10),

re

sults.38'39

The Helmholtz series, Eq.

(9),

gives the real component ofindex as a function

of wavelength with or without absorption bands in its range.

It,

_{too, has} one term for

each ofthe absorption bands that are characteristic of the material. The A coefficients

have the same interpretation as when_using the Lorentz-Lorenz equation. Used in

ranges far _{from any} absorption

bands,

the extinction coefficient, K, and the terms

multiplied

by

the

damping

_{coefficients, gj,} ofthe atomic oscillators _{become negligibly}

small _{reducing this equation, also,} to the Sellmeier series, Eq. (10).40

The Sellmeier series is themost

frequently

used formulation fordispersion modeling. It is a valid approximationofboth the classical Lorentz-Lorenz and

Helmholtz equations when the range is not in the neighborhood of an absorption

band.41

Forthe visible spectral _range, this includes all materials that are _visually

transparent. In the classical _sense, the same number ofterms are carried and the same

interpretation is given to the

A;

coefficients. When quantum mechanics is used as the

theoretical

basis,

Eqs.

(8)

through

(10)

are still _valid, but the number of oscillators

(and thus the number of_terms) is found to be

infinite,

and the

Aj

coefficients must be

given adifferent interpretation.42

Fortunately,

acceptable _{accuracy in}the interpolated index can _{usually be}

obtained even when

truncating

the series after a reasonable number of_terms,

AA2 A2X2 AX2

n2=\+ + + .

(17)

t,2 }2 _^2 T,2 _^2 t,2

A _Aj A _A2 A _A3

The samephysical interpretation of

Xj

and

Ay

should not be extended to atruncated

Sellmeier series. While a 3-term

formulation,

Eq.

(17),

will often be more than

adequate as an interpolant

following

a least squares determination of

Av

A2, A3,

a,,

A^,

assume that the strengths ofthese absorption bands are proportional to the

Aj

coeffi

cients.43

All terms ofthe series are required forthis interpretation.

Also,

it is

important to remember that none ofthe absorption_{bands may} occur within or nearthe range over which the Sellmeier series is defined forit to be valid.

It follows that the least squares solution forthe truncated series, Eq.

(17),

will be better if the

a;

parameters are determinedrather than assigned values based on experimental absorption data. This means that lessprecision will _generally result ifan attempt is madeto reduce the amount of experimental data required

by

_assigning known values to

Xu

Xj,

and

A3

followed

by

the determination of

A,, A2,

and

A3

by

linear least squares. This is unfortunate because Eq.

(10),

andtruncated forms such as Eq.

(17),

are awkward to solve with least squares due to their nonlinearity. The Sellmeier forms can be linearized

by

_{clearing fractions} so that standardlinear least squares _{may be}

used.44

Butthis transformation causes unequal _weighting to be applied todata of various values of

X,

which is difficult to defend. Least squares solutions to transformed equations do not_necessarily constitute abest fit solution to the original untransformed case and should be avoided.

When the Kettler-Drude series, Eq.

(11),

is discussed in the literature it is always presented _{separately from}the Sellmeier series, Eq.

{10).45'46

Appendix 2 [page

75]

shows that the two series are equivalent. When Eq.

(11)

is expanded with the binomial theorem, Eq.

(12)

results.47

Comparing

Eqs.

(2)

and

(12)

reveals that

Eq.

(2)

is a special truncatedcase ofEq.

(12),

with the coefficients ofEq.

(2)

renum

bered to correspond tothe

manufacturers'

usage.

The glass manufacturershave chosen this portion ofthe infinite series, Eq.

(12),

to representindex overthe wavelength range from 0.365 Mm to 1.014 Mm

with adequate accuracy. Ifmore accuracy had been required in theultraviolet than Eq.

(2)

provides, more low-orderterms in

X

would beincluded from Eq.

(12) [i.e.,

Herzberger has proposed Eqs.

(13)

and

(14)

tomodel dispersion in the visible and infrared regions ofthe _spectrum, respectively. The forms ofthe equations are similar to truncated forms ofthe SellmeierorKettler-Drude equations,48

but the

Herzberger equations are linear in their coefficients.

Accuracy

suffers from the _necessary approximations that allow this simplifi cation.

Indeed,

Eq.

(13)

is noted

by

its author49

to be an approximation ofthe

nonlinear

form,

A. A,

n =

!_

₊

i_

,

(18)

A

-Aj X

-X^

which

is, itself,

_only an approximation ofthe Kettler-Drude series [note thatEq.

(11)

yields n2, not n]. Herzberger laternotes50

that a5-coefficient truncated Sellmeier

series

[e.g.,

Eq.

(18)

with a constant term _added] provides abetter fit over the entire wavelength range from 0.58 Mm to 11.9 Mm than a piece-wiseinterpolant

involving

three separate invocations ofEq. (14).

Historically,

computational ease appears to

have been a constraint that leadto this development. This is ofless importance

currently.

In summary, equations

(10),

(11),

and

(12),

each

being

based on the Lorentz-Lorenz andHelmholtz equations, exhibit the same asymptotic behaviorat

several Equation

(12)

doesn'tjust have one absorption band at a wavelength ofzero, as a casualinspection may suggest. Truncated versions of

Eq.

(12),

such as Eq.

(2),

approximate this behavior. _{It may be} seen, then, that the

manufacturers choice ofEq.

(2)

has a good basis in

theory

and it may be expanded in

range ineither direction as the need arises.

Further,

although the form ofthe Sellmeier series is attractive since the

Xj

and

A-parameters have physical significance, this cannot be claimed forthe _truncated,

often-used, two- orthree-term Sellmeier equations. No less information is conveyed

*

by

the truncated binomial expansion, Eq.

(2),

aboutthe properties ofthe dispersive

material. It is also linear in its coefficients. With this in mind, there seems tobe little

incentive to use _{anything but} the form used

by

the _{manufacturers,} Eq. (2).

Research

It is possible to increase the_accuracy of a statement that ismade about the ac

curacy of experimental data

by

the proper mathematical treatmentofthe data and

proper experiment design.51 The_primary purpose ofthis research was the

develop

ment of a procedure thatwould allow optical designers toestimate refractive indices

of optical glasses to a much greater_accuracy than_{previously attainable,}

thereby

enhancing the usefulness oftheir scarce experimental data.

The current state-of-the-art in

fitting

a curve to _{experimentally determined}

index of refractiondata involves the

fitting

of a general dispersion model. All coeffi

cients ofthe model are allowed as degrees offreedom.

The situationis similar to the following. Suppose that an experimentis

performed which tested some dependent variable_y that was known to _vary with some

independent variablex in a linearfashion. It would be incorrect forthe analyst tofit

the _{resulting data}to the general _polynomial,

y = A0+Alx+A2x2+AyK3+

-,

(19)

of which the linear model is a special case, with the expectation that the coefficients

ofthe high-orderterms would be zero ifthe data

truly

described alinearrelationship.

It is _exactly this type of approach that is taken when

fitting

a general dispersionmodel

to experimental refractive index

data,

however.

It was

proposed52

that the nominal dispersion curve, 0(a), forthe glass type

under testshould be fit to experimental

data,

notthe general dispersion model. The

complicated, nonlinear shape of_n0(X) is both provided and _maintained, which is intu

itively

appealing since _{annealing is only} supposed to translate the _n0(X) curve in the

direction ofthe ordinate. It isnot supposed to change its shape. Ifthe general

dispersion model werefit to the data

instead,

it would be difficult to constrain the

resulting curve shape tobe the same as n0{X).

Such a procedure has been committed to software and will be used to test the

following

nullhypothesisH0:

Annealing

causes a simple translation _of the nominal

curve, n0(X), in the_{direction of}theordinate, but doesnot

significantly alter the shape _ofthe curve thatis charact

eristic _ofthe material.

Figure 3 Null

Hypothesis,

H0

Specifically,

if it can be shown that samples have significantly different curve shapes,

even though

they

are known to be from the same melt(but different annealings), then

H0

must be rejected andthe alternate hypothesis

Hx

accepted:

Annealing

maysignificantly change the shape _ofthedis persion _{curve, n(X),} so that a simple translation _of the

nominal_{curve, n0(X),} inthe _{direction of}theordinatedoes

not _adequately explain the observed variation.

Figure 4 Alternate

Hypothesis,

Hx

The procedure allows more complicated departures of_n{X) from _n0(X) than a

simple _{translation, but} it is important to know when a simple translation is inadequate.

When n(X) is allowed tochange shape it _{may be necessary} to altertheradius of

curvature on one or more optical elements in the objective to compensate. Simple

translationscan _{usually be} accommodatedwith simple airspace changes. The optical

designer should be aware ofwhat departures n(X) has takenfrom _n0{X) so that a

properchoice ofvariables _{may be} made.

When

fabricating

diffraction-limited

_objectives, optical glass is

typically

purchased to amuch more liberal tolerancethan what is required for fabrication. For

example, when _procuring the material it _{may be} required that _d be within 0.001000

ofthe _catalog nominal, yet the actualindex must often be known to 0.000010 for

successful fabrication ofthe objective. This large

disparity

betweenthe _accuracy

needed

by

the glass _vendor, and thatofthe end user often makes _{it necessary for} the

end user to have index of refraction measurement capability. Such was thecase for

our Company. In _sponsoring this research, Melles Griot has refurbished a spectro

meter that is capable of_measuring refractive indexto the fifth decimal place.

Experimental

Equipment

A spectrometer is a precision instrument thatisused tomeasure prism angles

anddeviation angles oflight which passes through a prism. Ofprime importance in

any spectrometer is the circular reference scale which is divided into

degrees,

arc

minutes, or arc seconds. For_{this research,} an old Gaertner LI 23 spectrometerwas

acquired. The steel reference scale is 7 inches in diameterand is engraved_{every 10}

arc minutes. With the aid ofits microscopes and filareyepieces, it is able to measure

prism angle to 5 arc [see Appendix

3,

page

76],

minimum deviation angle

to 5 arc _{seconds, and, therefore,} refractive index to 0.000025 [for n =

1.65;

_see

Appendix

4,

page 85].

The spectrometer is shownin Figure 5. The sample isprepared in the shape of

a prism and placed onthe platform above the rotation axis ofthe instrument.

Spectral lamps areplaced at the slit aperture ofthe collimator, located to the right.

The radiation is collimated and passed through theprism sample. After_emerging

from the sample, it is still collimated, though deviated

by

refraction. The observation

telescope is _swung about its axis until the radiationenters its aperture whereupon the

Figure 5 Gaertner LI 23 Spectrometer

slit is now visible with the aid ofthe eyepiece. As the observation telescope is

pivoted, the two microscopes which permit _viewing ofthe circular scale move with it.

By

subtracting the scale _reading obtained when the sample is in place, from the

reading obtained when the sample is _withdrawn, the angular deviation ofthe radiation

due to the prism is determined. Ifthe prism angle is measured_carefully, and the

angle ofincidence thatthe radiation makes with the prism face is known

(directly

or

indirectly),

then the refractive index ofthe sample at the wavelength ofthe spectral

lamp

may be determined [see page 33].

Various elemental spectral lamps are used as radiation sources. Each emits

characteristic line spectra at wavelengths known tohigh precision.53'54

Overthe

wavelength interval of

interest,

a set of spectral lines is chosen and the test sequence described above is performed. A table of refractive indices atthese test wavelengths

results. These data are used as the basis offurther _analysis, in which experimental

errors are smoothed, and interpolation ofindex at wavelengths otherthan the test

Software

Two categories of software were written. The first assists with dataacquisition

when _operatingthe _{spectrometer;} aHewlett-Packard 41C programmable, hand-held

calculator isused. Actual analysis and

curve-fitting

is performed

by

a program written

in Fortran for theMS-DOS or VAX/VMS environment.

ATON Angle to

Index;

This HP-41C program is the front-end for the other

two programs. It providesthe user_{interface for reducing} the datafrom

the spectrometerto prism _angles, minimum deviation angles, and

refractive index. It also shows the difference between microscope

readings ofthe dividedcircle so thatpotential _reading errors can be

spotted.

Sad Spectrometer Angle and

Difference;

Takes the two microscope readings

ofthe divided circles, in

degrees,

minutes, and _seconds, and determines

the average _reading and the difference ofthis average with the last

average.

ADN Prism Angle

A,

Minimum Deviation Angle

D,

andRefractive Index

N;

Written in the interchangeable solutionformat where you _{supply any}

two ofthe parameters and the third is computed. Aton sets _{up A} and

D as known parameters and N is solved for. As a stand-alone _program,

ADNis useful for ranning whatif? problems where prism angleA is

perturbed _slightly andthe effect on index N is noted.

When using the spectrometer, the datais reduced

immediately

from microscope

readings ofthe divided circle to prism angle and minimum deviation angles so that

ATON's diagnostics can be usedto spot _anypotential _misreading ofthedivided circle

or filareyepieces. The design ofthe filar eyepieces makes _reading errors of 1 arc

minute likely. Errors of 10 arc minutes are less

likely,

though possible. The diagnos

tics provided

by

theseprograms minimizethe chancethat erroneous data will go

undetected atthe time oftest. The Fortran-based analysis software Melt would detect

the erroneous

data,

but,

ifpostponed until analysis, correcteddata may be more

difficultto obtain. It is best to do some data checking as the experiment is

being

The task ofthe Fortranprogram

Melt is more complex. Figure 6 _shows, in the form of a flow _{chart, the} general

procedure.

First,

the experimental data are

read from files and/orthe keyboard. Commands that instruct Meltwhat to

do withthe data are input. Itthen car

ries them out without further input. Foreach datapoint, the experi

mental _wavelength, observed refractive

index,

experimental _uncertainty that

should be assigned to the

index,

and the

nominal

(expected)

index is specified.

Ofthe

four,

_only the firsttwo are re

quired.

Uncertainty,

ifspecified, is used to weightdata in proportionto its preci

sion

during

curve-fitting.

Any

specified

nominal index is used in preference to the calculated value that Melt supplies.

See page 106 forthe input data syntax.

Afterall input ofdata and

corn-Read setup file Readsamplespecificfile

Read keyboard

Resolveomittednominalvalues

Adjustreported n

Compute weighting factors Initializesimplexverticies

Trialsolution

fornextmodelIn list

Fit bestorspecifiedmodel

Perform ANOVA Writereport

Write plot(s) Update_historyfiles

Figure 6 Meltflowchart

mands, Melt fitsthe specified model to the experimental data and analyzes the results. Ifthe model relies on anominal dispersion function _{n0{X) then the} manufac

turers'

catalogues willbe used toresolve _any omissions thatthe user_{may have} made

in _providing nominal values. This isnormally how thenominal indices are _provided;

only when the experiment is performed at wavelengths where the

manufacturers'

dispersion formulais

invalid,

or when the material undertest does not have coeffi

Over 30 differentmodels and 4 different merit functions are provided. Some

models are composite functions ofthe manufacturers'

nominal dispersion

formula,

Eq.

(2),

and some donot involve a nominal dispersion model. The latter are provided

forcompleteness and convenience since Melt is also useful _{for creating} adispersion

model when one is not available in the literature. This is not the subject ofthis _thesis, however. Thisresearch relies on a

known,

nominal dispersion model. Over 20 such models are provided.

The "goodness-of-fit" ofthe model to the experimental data is judged mathe

matically with a meritfunction. Itmeasuresthe agreement betweenthe data and the model for a particular choice of parameters. Four different merit functions are pro vided, each formulated so that small values represent close agreement between thedata

and the model. Guidelines for using one in preference to another are givenon page 46.

Weighted sum of squares

->m_

(20)

Weighted sum of absolute

Jmax

nrn(xr

W-Aj

(21)

Weighted maximum deviation

/

= max

Wj

nj-nfXj;

Chi-square

An

/-_

;=i

;-(a.;

bvb2,...,bkJ

An

(23)

The term n;-is _{the observed}

response, and _(a;;

ft,, 62,

...,

6J

isthe model's response.

The

w>7

factor is the weight assigned to thejth data point, and is equal to the reciprocal

relative An (thepoint with the smallest An in the data set has a weight ofunity, and

all others have a smaller weight). Datapoints

having

the smallest stated uncertainty,

An,*

have the greatest potential for

impacting

the value ofthe merit function. The

model _n{Xj,

bv

b2,

_...,

b^J

is afunction of wavelength _a, and parametric in the coeffi

cients

bx, b2,

...,

b^

which MELT seeks todetermine. This model _{may be linear} or

highly

nonlinear in these coefficients.

Minimizing

the value ofthe merit function

implies optimizing the model to the data.

The minimization method used

by

Melt is the downhillsimplex

ofNelder & Mead.55'56 It does not require knowledge ofderivatives _{only func}

tionevaluations are required. It adapts to the function

/

being

minimized

by

reflec

tion, extension, contraction, or shrinkage ofthe simplex in responseto characteristics

ofthe surface.

A simplex is a geometric figure that has one more vertex than the spacein

which it is defined has

dimensions.57

On a plane, whichis a _{two-dimensional space,}

a simplex would be a triangle; in three-dimensional space it would be atetrahedron.

The function

/

being

minimized describes a surface. It is evaluated at the vertices of

the simplex. The vertices are rankedfrom best to worst, andthe worst is replaced

with abetterestimate

by

_{moving away from} thehigh function value thatit represents.

In this way, the simplex moves towards the function's minimum

by

_{moving away}

*

Ideally,

Arij

isthestandarddeviationoftheestimateofn},a,,

**

Nottobeconfused withthesimplex methodoflinearprogramming whereboththefunction

beingoptimized,andthe constraints,happen tobelinearfunctionsoftheindependentvariables.

from high values. Most othermmimization procedures attempt tomove in the direc

tion ofthe minimum

by

_{moving in a straight line towards} it,58

with the fastest algo

rithms ofthis type _utilizing partial derivatives to point the way. But with this speed

comes the _possibility ofdivergence. The simplex algorithm cannot diverge. Though

notthe most _{efficient, requiring} on theorder of;max2

storage and _many more function

evaluations than _most, its can _easily outweigh its speeddisadvantage for

jobs werethe computational burden is small.

After the merit function

/

is nunimized, the difference betweenthe optimized

model _n{Xj,

bx, b2,

...,

b^J)

and the experimental values

n;

is evaluated and summa

rized in tabular, _graphical, and statisticalform. Ifthe fit ofthe model to the experi

mental data is judged

by

MELTand the analyst to be adequate, the indices that MELT

computes for the design wavelengths _may then be used forthe melt recomputation.

By

considering the model that was _required, the optical designercan often make a

betterchoice of variables: if it is known thatthe partial dispersion ofthe material has

deviated from its expected value,the _{designer may} allowcurve changes. Not

knowing

this about _{the material, only} airspace changes might have been allowed. This knowl

edge can save time andlead to a superiorfabrication solution.

It cannotdiverge. It is insensitiveto initialvalues thesolutiondoesnot needto besurround ed orbracketedas withthegolden sectionsearch,orBrent'smethod. Derivativesdo not need

tobe knownor computednumerically,as withMarquardt's,Powell's, or othermethodsbased ontheNewton-Raphsonalgorithm. Almostno special assumptions are madeaboutthenonlin earfunction

being

minimized. Itis not_necessary tolinearizeit

by

variable_{transformations,} which canleadto undesirable weighting. Thoughnotneededforthisapplication,thefunction

being

minimized_maybe formulatedsothatthe simplex staysoutof certain solution areas [Olsson, p.45 (ref. 58)]. Thisconstrainedoptimization_maybe accomplished

by

_adding a pen

To examine the behavior of a large number ofsamples, it was _necessary to

write flexible _software, sophisticated enough so that_{it is easily} usedin a production

environment. The results of_many trials with different interpolation models are

summarized in this

Section;

greater detail is presentedin Section IV.

The Effect

of

Annealing

on

Dispersion

No reasonhas been found to rejectthenull hypothesis

H0

in favor ofthe

alternate hypothesis

Hx

[see page 15]. Multiple annealings of material from the same

melt have always been observed tohave parallel dispersion functions _n(X) that are

simply shiftedin the direction ofthe ordinate n from one another.

This does not constitute proof of

H0

but,

_rather, lack ofdisproof. As is often

the case,

H0

is difficult to prove but easy to disprove with _only a single counter-exam

ple. Such an example has notbeen encountered. Rejection of

H0

is not required on

the basis ofthe data collected and reviewed.

Melt Makes Subtle

Differentiations

Although

H0

has not been

disproven,

this is not to _say that all samples have

been observed tohave their _n(X) curve parallel to the nominal n0{X) curve.

Indeed,

somehave been different enough from their expectedcurve shape that_they should be

classified as separate and distinctglass types!

Spotting

curve shape changes is

difficult withoutMelt. Ifthe designer failsto realize that such a material is planned

for use in

fabrication,

an inappropriate choice of variables _may be made

during

the

melt recomputation.

MELT has been instrumental _{in spotting} substitutions of

"equivalent" glass

types

by

well-meaning vendors. For example, Schott LaFN-2 and LaF-2 are consider

ed equivalent since their

nd

and

Vd

values are the same; therate of change ofindex

with _wavelength,

dn/dX,

_{is LaFN-2 is called for in}

a

fast,

_complicated, diffrac

tion-limited objective that is

intended

for usage at alaser wavelength and anearby,

broad wavelength interval. The objective is _particularly intolerant of changes in the

derivative ofindex with respect to _wavelength, yet _very tolerant of shifts ofthe

dispersion function n{X) in thedirection ofthe ordinate n. The vendor's data sheet

indicated thatthe material was

LaFN-2,

though analysis performed

by

_{Melt clearly}

indicates that it is LaF-2. Ifthis hadnot beenrecognized, many hours may have been

wasted

trying

to perform themeltrecomputation _using

ineffective

variables.

Recommended Interpolation

Model

The refractive index of optical glass is a_very nonlinearfunction of wavelength.

Expressed as a function of

frequency,

_{x>, it is} more linear. It should not be _surprising,

then, that the simplest interpolation models are those which involvereciprocal

wavelength terms since

frequency

and wavelengthhave the reciprocal_relationship

a =

, where c = 3.0xl08m/s .

(24)

t)

Best results are obtained when a compositefunction is formed ofthe nominal

dispersion model for the material under_{test, n0{X),} and one or more perturbation

terms. As recommended on page 13 ofthe Introduction _section, the manufacturers'

dispersion model, Eq.

(2),

isused as this nominal model. The perturbation terms

represent the departure ofthe actual material from the nominal material.

Comparing

therelative partialdispersionsofLaFN-2andLaF-2 is also a usefulmethodof tellingthetwo apart. Itisnot_necessaryto examinethe_derivative,dn/dX. The relativepartial dispersion

Px

ymay be definedas

P =_____

x'y

%-"c

Both LaFN-2andLaF-2 havethe same principaldispersionp-nc, andthesame_n6 value,but

The least complex perturbation wouldbe

n{X) =

n0{X)+

B1

(25)

which would _simply translate the nominal model _n0in the direction ofthe ordinate n.

More complicateddepartures from nominal are _possible; the

following

models are

recommended and should be applied in order of

increasing

complexity:

n(X) =

n0(X)+Bl

+

^

(26)

n(X) =

nQ(X)+Bl

+

^l

₂

(27)

A'

n(X) =

n0(X)+Bl

+

?i.

(28)

A*3

n(A) =

n0(A)+_,+_2A2

+

f|

(29)

X

The output from Melt can be usedto decide when the model _complexity shouldbe

changed.

Generally,

the simplerthe model the better.

Allowing

too _many degrees of

freedom may leadto unwarranted changes in curve shape and should be avoided, as

50

40

30

20

10

26 1 1

21 2 2

19 2 2

11 MODEL 1 Terms 2 Variables

Figure 7 Model Histogram at95% Confidence Level

Melt shouldfirst berun _{using Eq.}

(26)*

as the interpolationmodel. The problemis then rerun_{using Eq.}

(27)**

as themodel; thenEq.

(28)***

is used (and

so on). To decide which is

best,

thevalue ofthe "F TEST FOR SIGNIFICANCE OF THEREGRESSION TERMS EXCLUDING THE MEAN"is examined.

Beginning

with the least complexmodel, thisFtest will increase to a peak value asthe optimum interpolation model isreached, and thendecline as the modelbecomestoo complex. IftheF testis always

increasing

(neverpeaks), thenmore _{complexity may be}

TheMeltcommandisMODEL2.

The MeltcommandisMODEL4.

necessary. If it is always

decreasing,

thenthe least complex model _{may be} too

complex.

The computed value oftheF test must always be greater than or equal to the

critical value oftheF distribution. If itis not, then theterm(s) are insignificant and

Eq. (25)"

is indicated a single term _representingthe mean.

Melt may be run sothat it examines a list ofmodels and selects the bestone

based on this It will

display

the merit function and F test values foreach

one, and then perform the full analysis _using the one chosen. This eliminates the need

formultiple runs and is therecommended mode of operation.

The histogram presented in Figure 7 summarizes how often a model has been

found to be optimum in a production situation. Models

4, 21,

19,

and 2 all have two

degrees of

freedom;

model 26 has only one.

Clearly,

models ofminimal _complexity

are favored.

The Meltcommandis MODEL 26.

The Meltcommandto examinethesemodels automatically is MODEL 2 4 24; an

arbitrary number of models may be listedinthisfashion. IftheFtestisalwayslessthan

the criticalvalue, thenthe analysiswill beperformedas if MODEL 26had beenspecified.

In the _precedingtwo _sections, an overview ofthemethods

by

which optical

materials are characterized has beenoffered, and the results of experiments _using these

methods have been summarized. This section expands onthese summaries.

Experimental Equipment

Refractometers maybe

broadly

grouped into two categories: those which

measure

by

criticalangle _methods, and spectrometer methods whichdo not. Ofthese

two, the spectrometer is capable ofthe _{highest accuracy} and Critical angle

methods _{generally involve less} sample preparation and_{may be} accomplished with

samples of minimal dimensions. Instruments whose design allows such methods are

indispensable tools forthe nondestructive

testing

of work inprogress, for theneed to

quickly determinethe index of refraction of small samples

frequently

arises.

The Pulfrich refractometer operates

by

critical angle measurement. It only

requires that a single surface of a small sample be polished flat. The specimen

having

unknown index function n is placed in contact with a reference block

having

known

index function Nsuch that n <N. Imperfect specimen polish can be accommodated

by introducing

a liquidof_{slightly higher}

index than N between the two. The arrange

ment is shown inFigure 8.

A diffuse source illuminates the

specimen such that somerays strike the

boundary

ofthe specimen andthe blockat

grazing incidence. Such rays arerefracted

atthe critical angle into the

block;

their

angle ofrefraction intothe medium ofindex

N may be no larger. The telescope is used

Figure 8 Pulfrich Refractometer diffuse

monochromatic source

n

W

\

V/<t>

Telescope

*

Accuracyhastodo withthecloseness to the"true"valueofthe_{quantitymeasured;}precision

to observe the source

by

looking

through both media. A demarcation line is visible,

marking the critical angle boundary. On one side ofthis

line,

the field is

bright;

on

the other side, it is dark.

No rays _{may haveangles of emergence greater than <)). The telescope is} rota

ted to determine the angle at whichthe demarcation line occurs. This angle ((> is a

function of

X,

_and, when several wavelengths are used,59

n = _\/^2-sin2<t)

(30)

gives the dispersion ofthe unknown material.

Use ofEquation

(30)

requires that the angle, a, between the two faces of the

reference block be precisely 90. The error introduced ifit is not varies with n and

N,

and can be Thegeneral equation forn, of which Eq.

(30)

is a special

case, allows for circumstances where 90 is not an accurate value for

a,60

n = _{sinay/V2-sin2(|)}

-cosa sin(j) .

(31)

A spectrometer_may be operated in a manner which duplicates thefunction of a Pulfrich refractometer. A previously measured prism

having

known dispersion func tionNis used as the reference

block,

chosen such that n <Nas before. The spec trometer isused to measure a and <(> _very accurately. Equation

(31)

yields theindex

ofthe specimen, n. This may seem tobe a poorutilization ofthe spectrometer's potential, butproduction situations occurfor which it is impossible to fabricate a sampleprism fornormal analysis

by

spectroscopic means.

*

Asan example: IfNis 1.74and n is 1.58,a fivearc second errorinthe angleawill cause a 0.000020errorinn; ifnis 1.70,a fivearc second error causes a 0.000090errorinn.

The principle of operation of an Abbe refractometer is the same as thatof a

Pulfrich. The demarcationline locates the rays _entering the referenceblock at _grazing

incidence anddefine an angle of_emergence, <|). As

before,

dispersion N and angle a

ofthe reference blockmust be known. The design ofthe instrument makes it well

suited for the measurement ofliquids. It is _rarely used for high accuracy measurement

of solids.

Critical angle refractometers depend on correct _positioning ofthe eyepiece

reticle on the demarcation line _{separating light} anddark areas inthe field of view.

This lack of_symmetry contributes to a reduction in the _accuracy that an observer can

attain. With a _{spectrometer,} the eyepiece reticle is superimposed on an image ofthe

slit; the _{viewing symmetry} allows for more accurate positioning.

Sample preparation

difficulty

is a serious drawback ofthe spectrometer. On a

relatively large specimen, two faces must be polished flat to high accuracy. Critical

angle _measuring refractometers relax the sample preparation

burden,

but _{they rely}

uponthe knowledge ofthe dispersion function Nto an _accuracy atleast five times

greaterthan that desiredofn. Formeasurements into the 6th decimal place, the refer

ence block musthave known dispersion characteristics to an _accuracy measured inthe

1th decimal place. This is a tallorder indeed!

The Hilger-Chance refractometer

(sometimes called a

"V-block"

refractome

ter)

was developed to avoid some ofthe

drawbacks ofboth the spectrometerand cri

tical angle _{measuring devices.} This refrac

tometer still relies on the knowledgeofthe

dispersion function Nof areference

block,

but a slit is observed

by

the telescoperather

than a demarcation line between light and

dark. The reference blockis made from

Figure 9 V-block Refractometer slit

,

1

specimen

\\A

Telescope

two prisms that were cutfrom the same material. Figure 9 indicates how

they

are

united to form a V-shaped chamber.

Ifa specimen

having

the same refractive index as the block were introduced

into _{the chamber, the ray} path would beundeviated.

Introducing

a sample

having

a

different index deviates the path of the rays. _The amount ofthis angular deviation is

used to compute theunknown index n,61

ViV2

+

cosY

V-n = _yAr+

cosy

y/v*2-cos2y

, or

= _7V+___-___

2 SN

( \

1 1

+ 12

j

Ay'

(32)

(33)

Though it is possible, in principle, to use a single reference

block,

this is a poor

practice. It is best to _{have many V-block} chambers available and, for

testing

a sample

having

refractive index _n, to choose the reference suchthatN= _{This reduces the}

method's _sensitivity to fabrication errors ofthe sample, particularly in achieving a90

angle between the two faces. _{Index matching fluid may be} used to accomodate an

imperfect right angle or unpolished surfaces.

As n >

N,

_y >

90

and

Ay

0. The series expansion for Eq.

(32),

shown as

Eq.

(33),

may be truncatedafterthe third term ifthereference is chosen so that

I n-N I < 0.000500 since

Ay

= 0.001 _radian (further_{terms are} 1 _x 10"6).62

A spectrometerdoes not require areference block that has known dispersion

properties; its measurements are _absolute, ratherthan relative. Thoughthe test is the

most intuitive of all the_{refractometry} methods, it is also the mostlaborious to per

form. Figure 10 andFigure 11 show the test

Sincethe deviceis not_measuringcritical angle,it is not_necessarythat thesampleindex nbe lessthan thereferenceblock index N.

Aphotograph ofthe spectrometerusedforthisresearchis shownin Figure 5 on page 16. The

Theprism sample is fabricated from

the material

having

_{unknown refractive in}

dex n. Faces AB andAC are ground and

polished

flat;

face BC is riot used and _may

be of _arbitrary surface quality. The two

polished faces are separated

by

theprism

angle, a. _{A ray} incident atface AB at angle

Ij

traverses the prism and emerges from face

AC at an angle l2'. The angleD between

the initial and final directionsofthe _{ray is}

the angle _ofdeviation. _{It may be}

Figure 10 Refraction through a Prism

shown63'64

thatD is minimized when

/,

=

I2

'. This is the so called minimum devia

tion condition where rays pass through the prism symmetrically. Minimum deviation

may be set to high precision with a _{spectrometer,} which is one ofthe principalreasons

why it allows n to be determined tosuch high precision.

K>\

//

A /.

Eyepiece

Atminimum

deviation,

//

-I2

= _{a/2, which}

immediately

suggests that a sam

ple ofhalfthe size could be used

by bisecting

_{the full} prism angle _a, and

by

autocol-limating

offthe new face so that the rays retrace their path.65

With such an

auto-collimating spectrometer, the angle ofincidence

/;

is measured for each

X.

The dis

persion function n wouldthen be

described,

using66

sin

/,

" =

a

'

(34)

sm

2

where a/2 is the measured prism angle. But Tilton67

builds a concrete case for

preferring minimum deviation _spectrometry overthe autocollimation approachbased

on _maximizing the tolerance of angular measurements. Minimum deviation spectrom

etry does not require themeasurement of

If,

the _geometry ofthe condition allows

Dmin

to be measured

instead,

indirectly

_providing the angle ofincidence. When at

minimum

deviation,

the general analyticalformula

describing

thepassage of a _ray

through a prismreduces to the spectrometerformula,6*

n =

sin __

a sm

2

(35)

which is identical to Eq.

(34)

whenit is recognized that

/,

=

/'

-

DmiD

+a

.

(36)

l i i

The errors which occur in thepractice of precise prism _{refractometry may be}

classified into two categories:

First,

having

todo with the prism sample and its

relationship to the _{spectrometer; second,}

having

to do with the spectrometer _only,

and its use as a goniometer. These two

sources will be considered in rum.

Total internal reflection will occurif

the prism angle is made too large. This

constraint _{may be}expressed as

a < 2arcsin_

n

(37)

n a BC

1.4

71

23.2mm 1.5 67 22.1 1.6 64 21.2 1.7 61 20.3 1.8 58 19.4 1.9 56 18.8

\

20mm

'4over <(>20

Figure 12 Prism Sample Dimensions

As this limit is approached, the

intensity

ofthe slit image will _{be greatly reduced,} and

a serious increase in aberration will occur.

Less obvious are theconstraints on angle measurement tolerance. The value

chosen for a has a _{strong influence} on whetherthe full precision and _accuracy ofthe

spectrometer are attained.

By

_adhering to the careful sample design described

below,

the full potential ofthe _{device may be} realized.

Determination ofindex to _{0.000010 accuracy} requires that a and

__, be

measured on the orderofarc seconds. The tolerance can be relaxed onthese angular

measurements, while _{minimizing the uncertainty} in computed

index,

by

optimum sam

ple design.

Tilton69

has performed the definitive error analysis of minimum devia

tion refractometry. Examination ofthe partial derivative ofthe spectrometer

formula,

Eq.

(35),

with respect toprism angle _a, and then with respect to deviation angleD (in

. D sm

dn

2

doc ~ ._2a

2sirr 2

, and

(38)

cos

dn

n

a+D

(39)

dD

~^ a+D ~ . a