Dispersion curve fitting in the infrared
Item Type
text; Thesis-Reproduction (electronic)
Authors
Nissley, Joe Scott
Publisher
The University of Arizona.
Rights
Copyright © is held by the author. Digital access to this material
is made possible by the University Libraries, University of Arizona.
Further transmission, reproduction or presentation (such as
public display or performance) of protected items is prohibited
except with permission of the author.
Download date
26/04/2021 22:01:44
DISPERSION CURVE FITTING IN THE INFRARED
by
Joe Scott Nissley
A Thesis Submitted to the Faculty of the
COMMITTEE ON OPTICAL SCIENCES .(GRADUATE)
Tn Partial Fulfillment of the Requirements For the Degree of
MASTER OF SCIENCES
In the Graduate College
THE UNIVERSITY OF ARIZONA
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of re quirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judg ment the proposed use of the material is in the interests of scholar
ship. In all other instances, however, permission must be obtained from the author.
SIGNED:
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
W. L. WOLFE
ACKNOWLEDGMENTS
I would like to thank Professors W. L. Wolfe and 0. N.
Stavroudis for their guidance and support in the development of these
dispersion formulae. I sincerely appreciate the exemplary typing and
thoughtfulness of Norma Emptage. Also, I thank Drs. F. 0. Bartel1 and
E. L. Dereniak for their important constructive criticism in the
TABLE OF CONTENTS -Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . v ABSTRACT . . . . vi 1. INTRODUCTION . . . 1 2. TYPES OF FORMULAE. . . . . . .'. . . . . . . . . . . . . . . . 3 Cauchy1s Formula ... 3 Sellmeierrs Formula... 5 Herzberger1s Formula . . . 9
3. METHODS AND RESULTS. . . . . . . . . . . . . . . . . . . 11
Matrices for Finding Herzberger Formulae . . . 11
Matrices with Test Functions for Finding Sellmeier Formulae ... 12
Sellmeier Formulae by Other Means. . . . 17
4. COMPARISON AND DISCUSSION. . . . 20
APPENDIX A: GRAPHS OF RESIDUAL ERRORS'. . . . 24
APPENDIX B: COMPUTER PROGRAM LIST . . . 55
REFERENCES . . . 64
LIST OF TABLES
Table Page
1. Herzberger Formula Data. . . , . . ... 13
2. Sellmeier Formula Data . . . . . . . .... ... ... 18
3. Comparison of the Herzberger and Sellmeier Residual Errors . 21
4. Comparison of Sellmeier Resonances from Table 2 and from
the American Institute of Physics Handbook (1972). . . 23
ABSTRACT
A computer program for finding two-term Sellmeier dispersion
formulae in the infrared spectral region yields residual errors
comparable with the Herzberger formula while using fewer degrees of
CHAPTER 1
INTRODUCTION
Snell's Law states that when a ray of light enters a material
from a vacuum the ratio of the sines of the angles of incidence and re
fraction is a constant known as the index of refraction. Spectral
dispersion is the variation in the refractive index with the wavelength
of light. The refractive index is often described by tabulating its
values at many wavelengths for each material. A second way to describe
the index is to specify an equation known as the dispersion, formula.
This can be evaluated to find the index at any wavelength in a desig
nated spectral range. This thesis is concerned with finding dispersion
formulae to match tabular data for wavelengths greater than 0.7 microns
for common infrared-transmitting materials.
Data points from many wavelengths may be used to find a few
parameters for a dispersion formula describing the index. For this
formula to be useful it must accurately reproduce the data. One measure
of the accuracy of this reproduction is the root-mean-square residual
difference between the original experimental data and the dispersion
formula evaluated at the same wavelengths. This measure is a valuable
criterion for comparison between formulae for the same data.
The minimum number of parameters which are necessary to de
scribe the dispersion of a material is the number of degrees of freedom
2
by which materials differ from one another. Knowledge of these minimal
parameters can be helpful in determining which combinations of materials
may be fruitful in the design of achromatic lenses. Hence one tends to
favor formulae with the least number of parameters. Also, one tends to
favor formulae which are directly related to theories of the interaction
of light with matter.
Two forms of dispersion formulae are frequently used in the
infrared literature: the Herzberger form and the Sellmeier form. Also,
formulae of unusual forms are presented for many materials. Comparison
of materials by means of formulae of such different kinds is difficult.
One prefers to have dispersion formulae of the same form and units for
all materials.
One objective of this thesis is the presentation of two-term
Sellmeier formulae for a wide variety of infrared materials. Another
objective is to provide comparison between the Herzberger form.and the
two-term Sellmeier form. This will involve presenting the root-mean-
square residual errors obtained under like circumstances for both
CHAPTER 2
TYPES OF FORMULAE
The dispersions of most transparent materials in the visible
spectrum are similar. The index of refraction as a function of wave
length has a negative slope and positive curvature. Figure 1 shows a
typical dispersion curve. These similarities have led to the development
of dispersion formulae which describe the dispersions of many materials.
Attempts by Cauchy, Briot, Sellmeier and Helmholtz
to model dispersion preceded Einstein's theory of relativity. They
involve the concept of the ether. These formulae will be described in
chronological order as presented by Preston (1912).
Cauchy's Formula
Before Maxwell's theories were published, Augustin-Louis Cauchy
was engaged at describing light as a transverse deformation of the ether.
He followed Fresnel, with matter embedded in the ether and oscillating
with it. He never arrived at a model for light with which he was
satisfied. Nevertheless, in 1835 he produced a dispersion formula which
agrees with harmonic oscillator models in that the index is a function
of the square of the frequency. According to Preston (1912) this
4 i.6 6 0 0 0 1.65000 1.64000 1.63 00 0 1.62000 - -T*» 2.2 1.0 2.0
X
( M I C R O N S )This might not be the form which Cauchy originally proposed.
Preston (1912) also states that Briot modified this in 1864 to become
n ^ = a ^X2‘+a.Q+a.^X~^+a.2\ ' t + a . ^ k ~ ^ + a . ^ X ~ ^ +
Sellmeier's Formula
Sellmeier's formula of 1872 and Helmholtz's modification of
it are also described in Preston. Preston (1912) states that W.
Sellmeier proposed that the molecules were elastically linked to the
ether. He arrived at a formula relating the index to the frequency
responses of harmonic oscillators as
" - 2
9 m A.X
n -1 = I
^ ~ 2
CD
j=l X -X.
When X approaches Xj the j term in this formula approaches
infinity. The problem of the infinities was alleviated by Helmholtz.
Helmholtz noted that if the molecules reached a resonance then their
large motions would be damped by interactions with other molecules.
These damping interactions would draw energy from the light; they would
absorb it.
The Sellmeier formula is consistent with modern theory of the
interaction of light with matter. Sellmeier did not involve charged
particles in his model as does modern theory. He arrived at the same
formula because harmonic oscillator characteristics determine the
The classical modern theory will be described for an isotropic
material without absorption, following Jackson (1975). For'such a
material, the square of the index is the dielectric constant e in
2
n E = eE = £+417 P
Here E is the electric field and P is the polarization due to this
electric field. ■ •
Solving for the ratio P/E yields
2 .
p/E-Tr
:
C2)
The polarization P is approximately the summation of individual
dipole moments per unit volume as in the equation below : where’ is the
number per unit volume of the type of dipole.
p = I NiPi j J 3
The individual dipoles are due to the electric field action on
elastically bound charged particles. At visible wavelengths the primary
contributors are electrons.
Neglecting any differences between the macroscopic electric
field and the local field acting on an electron, the force driving the
electron is the charge times the electric field qE^e1^'*'. For a parti
cle of mass m bound by a force -kx with damping -bx and driven by a
harmonic field of Eoelmt, the equation of motion is
Solving for x yields „ iwt - 1 , 2 2 . -1 x = qEQe m -xcoyj where = k/m J y = b/m
The dipole moment due to this oscillation is the displacement x
times the electron’s charge q as in
q2E e1"* P = qx = ---2 ^ 2
---m(w.-w -itoy) J
At this point a factor f is introduced since the j ^ resonance
actually arises from a quantum-mechanical transition rather than a
classical electron oscillation: f is called the oscillator strength.
In the summation of dipole moments to obtain the polarization, f^
multiplies the contribution of the type of dipole. With the ratio
P/E expressed as the summation of the dipoles per unit volume divided
by the electric field, Eq. 2 becomes
T = Z "j C 2 q? j ■■
j J m(a).-m -iiiiy. J
When m is near Wj, the index of refraction has an imaginary
component. This corresponds to absorption. For spectral regions where
8
*2-i - 1 - A
■
c«
J w.-w
3 - : •
Setting w to 2-nc/X makes Eqs. 1 and 4 equivalent. Thus, the
correspondence between the Sellmeier formula and modern classical theory
has been shown.
For Eq. 3, it was assumed that the effective field acting on the
electron was the same as the macroscopic electric field. The Lorenz-
Lorentz relation used on the left-hand side of Eq. 5 takes into account
the difference between these two entities. Born and Wolf (1964) offer
a derivation of the Lorenz-Lorentz relation. If the range of values for
the index is small, however, the Sellmeier formula can produce similar
dispersion curves with either form, the Lorenz-Lorentz form or the form
2 ‘
n -1. This is possible by using different values for the parameters in 2
the formula. The form n -1 is often retained for computational ease.
' 4 ^ * M c ®
n +2 j Wj-w
With the Lorenz-Lorentz relation one might expect to find a
frequency of strongly resonant absorption by using Eq. 5 and experimental
data for the index of refraction. This is not a valid procedure in this
thesis. Each term of the two-term Sellmeier formula can represent
multiple absorption resonances in special regions far from the data.
The g o
.
1 s which are found to fit such data cannot be interpreted as wave-^ ’ . '
lengths at which absorption resonances actually Occur. Henceforth the
Herzberger1s Formula
M. Herzberger and C. Salzberg (1962) proposed the formula of
Eqs. 6 and 7 allowing easy computation of infrared dispersion. For
these equations X is in units of micrometers.
n = A + BL •+ CL2 + DA2 + EX4 (6)
L = 1/(X -0.028) (7)
Several expansions of harmonic oscillator terms are required to
understand this formula as an approximation of a Sellmeier formula. A
single Sellmeier term from Eq. 1 can be rewritten as
A.A2 A .X?
f e = f e + A l (8)
J 3
The difference between two resonances with the same numerator
occurring at slightly different resonance wavelengths (Aq and x p is
approximately
:(xr x2>
B B (9)
x2-x|
(x2-x2)2
Equations 8 and. 9 allow the dispersion due to a resonance at
10 and its. square times the difference in squares of the resonant wave
lengths as in
A ix2 .. . , A ixi ,
X2-x2 1 X2-X2 (X2-X2)2
This equation shows how the first three terms in the Herzberger formula
describe dispersion due to absorption resonances in the ultraviolet and
visible. For resonances at wavelengths longer than the data, a differ
ent expansion is
A A2
4-2 - -
A -A0aii
2/
xo
1“A /A0■ V x2/x>x4/4 ••• )
™
2 4
Thus -A and -A terms can approximately describe the contribu
tions of far infrared absorption resonances. This corresponds to the
CHAPTER 3
METHODS AND RESULTS
An advantage of the Herzberger form is that all parameters are
linear coefficients. Thus a Herzberger formula may be found by
straightforward solution by matrices. The Sellmeier form involves
parameters— the absorption resonance wavelengths— which are not as easily
found.
Matrix methods are used to find Herzberger formulae. Then a
technique is introduced which uses matrices to find Sellmeier formulae
with two terms. The resulting formulae are tabulated in this chapter
and compared in the next chapter. The residual errors are graphed in
Appendix A for the two-term Sellmeier formulae resulting from this
matrix technique.
Matrices for Finding Herzberger Formulae
Solving simultaneous linear equations, m data points are re
quired to determine m coefficients. The data-ppint matching equations
are given by
m
f(^i) = I c a (A )
1 i=l 3 J 1
2
n-1, n -1 or the form of the Lorentz-Lorenz relation may be
used for f(A.). Since the results of this section will be compared 1 .
with those of the next section, n -1 was used in both sections. The
term in the dispersion equation evaluated at X. is a. (XI). Its
1 3 1
coefficient is' c.. Using matrix notation, this is written J
F= A C
-1
Multiplying both sides on the left by A produces
A F = C
From this equation we see that the coefficients may be found “1
from the data and the inverse matrix A by
c. = I a (j,i) f(A )
3 i=l 1
Table 1 presents Herzberger coefficients for data from the
American Institute of Physics Handbook (1972). Table 2 (p. 18) will
present two-term Sellmeier formulae found from the same data. Both
formulae match the data at both ends of the spectral range and at
points--three for the Herzberger and two for the Sellmeier— spaced
in between.
Matrices with Test Functions for Finding Sellmeier Formulae
A two-term Sellmeier formula has two resonance wavelength
parameters as well as two coefficients. Only the coefficients can be
found by a simulatneous linear equation (matrix) approach. A tech
13
Table 1. Herzberger Formula Data.
Material A B C D E RMS ‘ MAX RANGE
BqF eCdS oCalcite 2.159246 5.735698 2.693489 808 -8757 2164 -9 10689 • -79 -178319 -28750823 -1303763 -8412 602971828 -4885359 42 308 14 119 648 43 0.7-10 0.8- 1 0.7- 2 eCalci ts CcF CsSr 2.182359 2.833544 2.733916 1034 621 3117 -67 -12 -31 -183689 -319701 -20122 -870811 -29862 -158 8 23 48 22 63 132 0.7- 2 0.7-10 1.0-39 Csl oQucrtz eQuartz 3.937232 2.345634 2.335276 4932 2472 1902 26 -501 10 -12943 -853357 -1241491 -55 -2347574 80522 9 4533 12 28 15899 34 1.0-41 0.8- 7 0.8- 2 Silica Ge xtal Irtranl 2.10332O 16.600455 1.898244 947 309265 298 -34 156505 115 -886587 -17185 -419023 -1266348 9453 -81577 21 500 95 59 1465 190 0.7- 4 2.1-13 1.0- 9 Irtrar>2 Irtran3 Irtran4 5.G93S51 2.033523 5.940323 14890 685 19634 356 -66 3235 . -237401 -321335 -153065 -23591 -26291 4223 52 101 293 154 212 601 1.0-13 1.0-11 1.0-29 IrtranS IrtranS LiF 2.968338 7.194490 1.925481 1701 64264 502 104 18821 -14 -1089362 -60424 -641093 -153938 1024 -88393 341 537 2049 850 1442 5499 1.0- 9 1.0-16 0.8-18 foo K S r KC1 2.955523 2.351309 2.174164 2415 2340 1934 -224 -16 -212 -1093476 -31153 -47843 -85149 -535 -3987 207 6 8191 479 17 24661 1 . 0 - 5 9.7-25 0.8-29 KI oSajjphire Si 2.643317 3.681777 11.653393 4232 1716 98352 -70 -39 7635 ' •-23418 -1634727 -495 -380 -592103 2792 201 69 249 611 189 701 0.7-29 9.7- 6 1.4-11 AsCI HaCI HiF 4.033241 2.327353 1.743339 7913 5305 79 134 -5733 182 -85127 -85983 -200112 -1971 -5370 -16070 2 2583 28147 5 9463 114344 1.7-21 1.4-27 9.7-24 TIBr T13r~Cl TIBr-I 5.141032 4.823557 5.663359 2631196 17762 29333 -1395487 1437 1021 68391 -88201 -45641 -5794 -77 -182 0 625 18 0 1641 46 0.8-24 1.4-23 2.8-38 oRutile GcAs InSb 6.086175 10.383023 15.524333 6563 13057935 2902704 6512 -7555713 67904294 -4585038 -1946493 -281474 4823431 165037 9101 14151 26332 2010 36393 78095 3832 0.7- 5 0.3-19 7.9-21 SrTi03 oTe ele 5.211333 23.195727 33.765187 15752 483193 2613123 721 21609502 5165963 -2755090 -334110 -5488 -389045 84359 -19331 625 1107 951 2031 2037 1968 0.7- 5 4.0-14 4.0-14 AsSeGlass AsSZglass PbF 6.142520 5.815865 2.994934 44213 29427 ' 3979 6326 2007 12 -16188 -125454 -316330 1373 -22107 -78 63 58 3 140 157 5 1.0-12 1.2-11 0.8-11
Note: The following data can be used to find values for the refractive index approximately matching data from the American Institute of Physics Handbook (1972). Data for PbF is from the Handbook of Optics (1978). Refractive index may be found from
n = A + BL + CL2 + DA2 + EA4 2
where L is 1/(A -0.028) with A in microns. RMS represents the root mean square residual error times 10&. MAX represents the worst residual error times IQ^. RANGE represents the spectral range in microns, e designates the extraordinary index, and o the ordinary.
including two extra terms in the two-term Sellmeier formula as test
functions and iteratively adjusting the wavelengths until the coeffi
cients of these extra terms found by matrix solution are negligibly
small„ With these final values of the resonance wavelengths, the
formula without the two extra terms matches the data on the four points
used in solving the matrix.
The number of parameters in the final formula is four. The
matrix is a four by four array. With each iteration the ratio of the
coefficients of One Sellmeier term and of its corresponding test func
tion. provides information allowing one to adjust the resonance wavelength
in that term. With two terms, the Sellmeier formula can be written as
2 2
, A1 X A„A
• n -1 = A — ^ +
A 2 - X 2 - * =
To find them,, two test functions are introduced. • A and A^
are the resonance wavelength parameters. With these and the original
two terms a matrix solution finds four coefficients to match four data
points simulatneously, as shown in Eq. 11. The first and third terms
are the two original Sellmeier terms. The resonance wavelength
parameters A^ and Ay are iteratively adjusted until a matrix solution
15 For a resonance at shorter wavelength than the data, an
appropriate adjustment of the third and fourth terms of Eq. 11 is
Xy - xUy - C(4)/C(3) C12)
Equation 12 arises from inspection of Eq. 9 for the difference
between two resonances at different wavelengths. The fourth term in
Eq. 11 is intended, to have the form of this difference. By changing
notation, Eq. 9 can be rewritten
When the matrix finds the coefficient of this term C(4), the coefficient
that the coefficient of the third term is already nearly correct. The
ratio of the fourth coefficient to the third coefficient is taken to
parameter which will fit the four data points without the fourth term.
These resonance wavelength parameters are not to be taken as
absorption peaks though they are related to absorption. Many absorp
tion peaks may be approximated as one for our purposes. A single
absorption peak may not produce an absorption resonance parameter equal This involves the assumption
2
16 to its wavelength. The parameters are merely numbers to fit the four
data points. The relation to absorption is sufficiently strong that
initial guesses for the resonance wavelength parameters are chosen near
spectral regions where most materials absorb strongly. The ultraviolet
resonance wavelength parameter Xy is initially set at 0.4 microns. The
infrared resonance wavelength parameter XT is initially set at sixty
V .
mi crons... .
A different technique is used to adjust the infrared absorption
parameter Xj. The above technique was found to be unstable when applied
to infrared resonances. The problem was that if the resonance param
eter got too close to the data, the coefficients became overly large.
For this reason, the second term was chosen to simulate a resonance at
a fixed wavelength of infinity. If the second coefficient is positive
the resonance is adjusted toward longer wavelength. For a resonance at
longer wavelength than the data, the first two terms of Eq. 11 can
be adjusted by using Eq. 13. The factor X^ was found to result in an
optimum rate of convergence.
h ->Jxj-xJ*cc2)/cci) (13)
A second technique is used for many semiconductors. These often
require that both resonances be at shorter wavelengths than the data.
To simplify the solution, the resonance at X^ is fixed at zero wave
length where it contributes only a constant. The second term in Eq. 11
17 For most materials application of one of the preceding methods
yields a two-term Sellmeier formula. Tests have been found which select
the appropriate method. A computer program employing these techniques
produced the results presented in Table 2.
The root mean square residual errors and maximum residual
errors in Tables 1 and 2 were not found for all available data points of
each material. They were evaluated for about twenty data points almost
evenly spaced throughout the spectral range including the points matched
exactly by the matrix in arriving at the formula.
2 For a few materials, the program never converges. A plot of n
versus the square of the frequency is helpful in understanding the
reason. In this plot each Sellmeier term contributes a hyperbola. The
dispersion of these materials does not fit a hyperbola.
Apparently there is some absorption in the spectral region of
the data. Another possibility is that the data are inaccurate. For
IRTRAN 6, however, data from different sources show the same irregu
larity.
Sellmeier Formulae by Other Means
The major problem in finding Sellmeier formulae is finding the
resonance wavelengths. Three methods are commonly employed for finding
them. The first method finds the resonance wavelengths from separate
data, such as ellipsdmetry data taken at or near the resonance wave
18
Table 2. Sellmeier Formula
Data-Material X COEF X COEF RMS N MAX RANGE
CaF CsBr Csl 8.0774 8.1321 8.1566 1.038651 1.783944 2.037885 34.5663 120.7626 157.0205 3.827828 2.943255 3.183485 3 5 3 5 36 5 7 6 108 0.71- 9.7 1.80-39.2 0.70-41.0 BaF eCdS oCalcite 8.0349 8.3701 0.1033 1.158151 1.686334 1.695461 46.4466 0.8800 7.0328 3.338603 2.574836 0.633589 33 5 453 3 17 5 101 884 35 0.71-10.3 9.70- 1.4 0.71- 2.2 Ge xtal Irtranl Irtran 2 1.1940 8.0683 0.1919 1.616543 8.898359 4.093115 0.6000 25.6909 34.5302 13.336552 2.662148 2.843330 3832 4 182 4 164 4 6906 381 395 2.06-13.0 1.00- 9.0 1.00-13.0 ©Quartz eQuartz Si 1ica 0.0339 8.0395 0.0394 1.356791 1.334136 1.104070 10.4250 17.3049 9.8869 1.212025 3.592269 0.895317 2231 3 18 5 2 6 • 6931 41 3 0.77- 7.0 0.77- 2.1 0.71- 3.7 Irtran 3 IrtranS LiF 8.0763 8.8934 0.0727 1.633555 1.959486 0.925538 37.6455 28.4521 29.2642 4.538882 8.840152 5.521413 124 4 521 4 2024 3 276 1140 5185 1.60-11.8 1.00- 9.0 0.80- 9.8 HsO KBr . KC1 8.1032 0.1317 0.1153 1.953165 1.361257 1.174777 43.1471 77.6011 45.8645 20.582303 1.878158 1.196923 194 5 50 5 7547 3 457 110 19823 1.01- 5.4 0.71-25.1 0.77-23.8 KI oSaophlre Si 0.1573 8.0950 0.1631 1.648433 2.G32131 35.987577 85.5566 18.6031 0.0090 1.725954 5.332772 -25.316033 222 4 41 5 454 4 535 ' 129 1091 0.71-29.0 0.71- 5.6 1.36-11.0 AgCI HaCl HaF 0.1660 8.1175 0.0793 3.035856 1.329342 0.743801 65.1620 51.6474 44.2288 3.563091 2.448176 4.040437 152 5 2220 4 . 34143 3 273 7334 127134 0.70-20.5 0.70-27.3 0.71-24.0 SrTi03 oTe ThBr-Cl . 0.2368 1.2257 0.2298 4.204369 14.586814 3.829663 22.8335 0.0000 104.9237 14.822377 7. 196746 9.228518 855 4 2043 3 665 4 2322 5929 1579 0.71- 5.3 4.00-14.0 0.79-23.0 ThBr-I AsS3glass PbF 8.2591 8.2531 8.1432 4.660420 4.807748 1.994480 136.3244 .29.4939 168.1232 8.247948 0.429528 88.986516 394 5 721 5 55 5 655 1415 112 0.70-37.5 0.70-11.4 0.78-10.5
Note: The following data can be used to find values for the refractive index approximately matchine data from the American Institute of Physics Handbook (1972). Index values for Pbf are from the Hand book of Optics (1978). Refractive index may be found from
, V - COEF X2 n = , 1 + 1 2--- o—
/ j+l( *- LAMBDA2)
Where is the wavelength where index is found. LAMBDA is the resonance from table above. COEF is coefficient from column following LAMBDA. RMS is root-mean-square residual error times
106. N is the number of digits after decimal in handbook data. MAX is the worst residual error times 10^. RANGE is spectral range in microns, e designates extraordinary index, and o the ordinary.
19 The second finds the resonance wavelength by minimizing the .
resultant root mean square residual difference from the experimental
data one is fitting.
The third method dodges the problem by including resonances
at many wavelengths in every pertinent spectral range. These wave
lengths cease to be variable parameters. The many coefficients contain
the information about the material. •
None of these methods were applied here. All three appear to
have been used in finding dispersion formulae for the American
CHAPTER 4
COMPARISON AND DISCUSSION
In the preceding chapter, Herzberger formulae and two-term
Sellmeier formulae were developed for about thirty materials. The
two-term Sellmeier has two coefficients and two resonance wavelength
parameters, while the Herzberger has five coefficients.
The most important criterion for comparison of these forms is
the difference between the resulting values for the index and the
experimental data. This is the residual error which is inherent in the
disagreement between the form of the equation and the experimental data.
A given technique may or may not produce the optimum values for
the parameters. The matrix methods presented in the last chapter match
the data exactly at a few points without regard to the remaining points.
If two forms are to be compared fairly by means of the residual errors
in formulae found by these matrix methods, then the number of parameters
must be similar. The five parameter Herzberger form has a slight
advantage over the four parameter Sellmeier form in this respect
since the Herzberger matrix solution involves one more data point.
For this comparison. Table 3 lists both the root mean square residual
error and the worst residual error between any data point and the
formula for both formulae for all materials in Table 2. No advantages
are evident on either side from comparing the residual errors. It is
not necessarily a definitive comparison.
21
Table 3. Comparison of the Herzberger and Sellmeier Residual Errors.
Root Mean-Square Wofst ,
Residual Errors * 10 ' Residual Errors • 10 Material Herzberger Sellmeier Herzberger Sellmeier
BaF 42 33 119 101 eCdS 308 458 640 884 oCdS 14 17 40 35 CaF 23 3 68 7 CsBr 48 3 132 6 Csl ; • 9 36 28 108 oQuartz 4533 2231 15899 6981 eQuartz 12 18 34 41 Silica 21 2 50 3 Ge xtal 500 3832 1465 6906 Irtran 1 96 182 190 381 Irtran 2 52 164 154 395 Irtran 3 101 124 212 276 Trtran 5 341 521 850 .1140 LiF 2049 2024 5499 5185 MgO 307 194 479 457 KBr 6 50 17 110 KC1 8191 7547 24661 19023 Ki ' 301 222 611 585 oSapphire 60 41 189 129 Silicon 249 454 701 " 1091 AgCl 2 152 5 273 NaCl 2583 2220 9468 8334 NaF 28147 34143 114344 ; 127134 SrTiCL 625 855 2031 2322 oTe 1107 2048 2007 5920 TlBr-Cl 625 665 1641 1579 TIBr-I 18 394 46 655 AsS„ glass 58 721 157 1415 PbF 3 55 5 112
22 Residual errors may arise from experimental inaccuracy,
incorrect form of the equations, or from the limitations of the matrix
methods. Examination of Table 4 reveals that resonance wavelength
parameters generated by matrices with test functions agree with those
generated by different techniques. The agreement, however, is not high
ly accurate. Also, values of the root mean square residual error were
found to depend on the choice of data points to be matched. These facts
suggest that limitations of the matrix methods working with imperfect
data are the major factor in determining the root mean square residual
error of the resulting formulae. Perhaps the best use of these formulae
is as starting points for iterative root mean square minimization programs
which utilize all of the data.
If the matrix methods are accepted as the standard of comparison
the two forms produce similar root mean square errors. There are
materials for which the Herzberger formula can be found when no corre
sponding two-term Sellmeier formula has been found. When the fifth
term in the Herzberger formula is positive, Eq. 10 no longer relates
*
the fourth and fifth terms to an infrared Sellmeier resonance. The
majority of the materials for which the program for finding Sellmeier
formulae does not converge have positive Herzberger terms. The
Sellmeier formula may require a third resonance, possibly within the
23
Table 4. Comparison of Sellmeier resonances from Table 2 and from the American Institute of Physics Handbook (1972). : .
MATERIAL
ULTRAVIOLET .
AIPH TABLE 2 AIPH
INFRARED TABLE 2 BaF 0.0578,0.1097 0.0849 . 46.3864 46.4466 oCdS 0.4063 0.3701 — — — — CaP 0.0526,0.1004 0.0774 34.65 34.5608 CsBr 0.158 0.1321 119.96 120.763 Csl 0.02,0.15,0.18 0.1566 161. 157.02 Silica 0.0684,0.1162 0.0894 9.896 9.8869 MgO 0.1195 0.1032 _ — 43.1471 KBr 0.180 0.1317 — — 77.6011 KC1 0.1153 57.38 46.8645 oSapphire 0.1107,0.0615 0.0900 17.926 18.003 AgCl 0.2141 0.1660 — — 65.162 KRS 5 — 0.2591 164.59 136.324 ' Ti02 0.2834 OB «=o «*>. ZnS 0.27055 ■ -- -AsSg glass 0.2581 27.386 20.4939
APPENDIX A
GRAPHS OF RESIDUAL ERRORS
The computer program in Appendix B. produced the material for
Table 2. With the "go to" statement of line 9530 moved to line 10155
this program also produced this appendix. The heading for each graph
is the corresponding line of Table 2. This gives the pertinent facts
about the dispersion formula. Each page has two graphs with the abscissa
representing wavelength plotted on the same scale. Zero wavelength is
at the left edge of the graph, with tic-marks on the horizontal axis
one micron apart. The first graph has consistently negative slope and
represents the dispersion n(X) of the experimental data. The bottom of
the page corresponds to the smallest value for the index. Vertical tic-
marks represent a difference of 0.01 in this graph of the refractive
index. The scale of the second graph is magnified ten times so that
each tic-mark represents 0.001. This second graph is the residual error
between the formula from Table 2 and the experimental data. The
residual error axis is in the middle of the page and is suppressed.
That is, a straight horizontal line down the middle of the page would
designate a good fit. The end points of this graph and two intermediate
points were forced to zero by the matrix matching of four data points.
MATERIAL
BaF
LAMBDA
COEF
LAMBDA
0.0849 1.150151
46.4466
INDEX
1.396360
TO
1.471770
AND 10X "
RESIDUAL
COEF
RMS N MAX
RANGE
3.838608
33 5
101 0.71-10.3
LAMBDA,0 TO 10.346 MICRONS
Is)
MATERIAL
LAMBDA
COEF
LAMBDA
eCdS
0.3701 1.686334
0.0000
INDEX
2.321000 -
TO
2.432080 .
AND 10X
RESIDUAL
COEF
RMS N MAX
RANGE
2.574086 458 3
884 0.70- 1.4
LAMBDA,0 TO 1.4 MICRONS
MATERIAL
LAMBDA
COEF
LAMBDA
oCalcite
0.1083 1.695461
7.0328
INDEX
1.620990
TO
1.652070
AND 10X
RESIDUAL
COEF
RMS N MAX
0.683589
17 5
35 (
RANGE
.71- 2.2
*LAMBDA,0 TO 2.172 MICRONS
hJ ^1" £ ! R , M -
8R0??4
U%¥<*t
L§3!?g88
l?l|7028 m
3 3 "6X? 0 . ? n . 7
INDEX
1.3Q7560
TO
1.431670
AND 13X
RESIDUAL
LAMBDA,0 TO 9.724 MICRONS
tx) 05“
L
1.783944
l?I$3255 T , 5 "AX6
INDEX
1.561190
TO
1.677930
AND 10X
RESIDUAL
LAMBDA,0 TO 39.2 MICRONS
MATERIAL
Csl
LAMBDA
COEF
LAMBDA
8.1566 2.037085 157.0205
COEF
RMS H
3.183485
36 5
MAX
RANGE
108 0.70-41.0
INDEX
1.674570
TO
1.773230
AND 10X
RESIDUAL
LAMBDA,0 TO 41 MICRONS
Crl oMATERIAL
oQuartz
LAMBDA
COEF
LAMBDA
0.0889 1.356791
10.4250
INDEX
1.167000 :
TO
1.539030 :
AND 10X -
RESIDUAL :
JL ▲COEF
RMS N MAX
RANGE
M£TE RIAL
LAMBDA
COEF
LAMBDA
eOuartz
0.0895 1.384136
17.3049
INDEX
1.528230
TO
1.547940
AND 10X
RESIDUAL
COEF
RMS N MAX
RANGE
3.592209
18 5
41 0.77- 2.
MATERIAL
LAMBDA
COEF
LAMBDA
Silica
0.8894 1.184070
9.8869
INDEX
1.399389 -
TO
1.455145
AND 18X
R E SIDUAL
COEF
RMS N MAX
RANGE
0.895317
2 6
3 0.71- 3.7
LAMBDA,0 TO 3.7067 MICRONS
tzi O-J
• r a w i s ?
INDEX
4.602100 -
TO
4.101600
AND 10X "
RESIDUAL
COEF
RMS N
13.386552 3832 4
MAX
RANGE
6906 2.06-13.0
MATERIAL
LAMBDA
COEF
LAMBDA
Irtranl
0.0683 0.898359 25.0900
INDEX
1.226900
TO
1.377800
AND 10X
RESIDUAL
COEF
RMS H MAX
RANGE
2.662148
182 4
381 1.00- 9.0
LAMBDA,0 TO 9 MICRONS
CN
MATERIAL
Irtran2
LAMBDA
COEF
LAMBDA
0.1919 4.093115 34.5802
INDEX
2.150800
TO
2.290700
AND 10X
RESIDUAL
COEF
RMS N MAX
RANGE
2.843830
164 4
395 1.00-13.0
LAMBDA,0 TO 13 MICRONS
l/j
MATERIAL
Irtran3
LAMBDA
COEF
LAMBDA
0.8760 1.038955 37.6455
INDEX
1.269400
TO
1.428908 .
AND 10X
RESIDUAL
-COEF
RMS N MAX
RANGE
4.580882
124 4
276 1.00-11.0
LAMBDA,0 TO 11 MICRONS
Vi ^1
MATERIAL
IrtranS
LAMBDA
COEF
LAMBDA
0.0984 1.959486 28.4521
INDEX
1.406000
TO
1.722700
AND 10X
RESIDUAL
COEF
RMS H MAX
RANGE
8.840152 521 4
1140 1.00- 9.0
LAMBDA,0 TO 9 MICRONS
Ul CO
MATERIAL
LiF
e.9°5638 L2 § ^ 42
INDEX
1.109000
TO
1.388960
AND 10X
RESIDUAL
COEF
RMS N MAX
RANGE
5.521413 2824 3 5185 0.80- 9.
MATERIAL
LAMBDA
COEF
LAMBDA
MgO
0.1032 1.958165 43.1471
INDEX
1.624040 *
TO
1.722590 .
AND 10X
RESIDUAL
COEF
RMS H MAX
20.582308
194 5
457
RANGE
.01- 5.4
MATERIAL
KBr
LAMBDA
0.1317 1.361267 77.6011
COEF
LAMBDA
COEF
1.878158
RMS M
50 5
MAX
RANGE
118 0.71-25.1
INDEX
1.463240
TO
1.552447
AND 10X
RESIDUAL
LAMBDA,0 TO 25.14 MICRONS
MATERIAL
LAMBDA
COEF
LAMBDA
KOI
0.1153 1.174777 46.8645
INDEX
1.226808
TO
1.433778
AND 18X
RESIDUAL
COEF
RMS H MAX
RANGE
1.196928 7547 3 19023 0.77-28.8
LAMBDA,8 TO 28.8 MICRONS
-c*
MATERIAL
KI
LAMBDA
0.1578 1.648433 85.5566
COEF
LAMBDA
COEF
1.725054 222 4
RMS N MAX
585 0.71-29.0
RANGE
INDEX
1.557100
TO
1.653700
AND 10X
RESIDUAL
LAMBDA,0 TO 29 MICRONS
4^
MATERIAL
LAMBDA
COEF
LAMBDA
oSepphire 0.8909 2.082131
18.8831
INDEX
1.586389
70
1.763833
AND 18X
RESIDUAL
COEF
RMS H MAX
RANGE
5.332772
41 5
129 0.71- 5.6
LAMBDA,0 TO 5.577 MICRONS
MATERIAL
Si
LAMBDA
COEF
LAMBDA
6•168135.987677
8.0608
INDEX
3.417639
T Q
3.497589
AND 16X
RESIDUAL
COEF
RMS M MAX
RANGE
-25.316833 454 4
1691 1.36-11.6
LAMBDA,0 TO 11.04 MICRONS
-h.
MATERIAL
AgCl
LAMBDA
COEF
LAMBDA
0.1660 3.906856 65.1620
INDEX
1.981490
TO
2.845989
AND 18X
RESIDUAL
COEF
RMS H MAX
RANGE
3.563091
152 5
273 0.70-20.5
LAMBDA,0 TO 20.5 MICRONS
-u c\
MATERIAL
LAMBDA
COEF
LAMBDA
COEF
RMS N MAX
RANGE
HaCI
6.1175 1.329342 51.6474
2.443176 2220 4 7334 0.78-27.
INDEX
1.175888
TO
1.538818
AND 1QX
RESIDUAL
LAMBDA,8 TO 27.3 MICRONS
M ATERIAL
KaF
6J3793 0.743881
4?04843734143 3127134 0.71-24.0
INDEX
8.240888
TO
1.323720
AND 18X
RESIDUAL
LAMBDA,0 TO 24 MICRONS
COMATERIAL
Srii03
S
s
4.20486S Li :“ §3S .4?1I2377 %
2 ^
INDEX
2 c 183400
TO
2.353483
AND 18X
R E SIDUAL
LAMBDA5 0 TO 5.3334 MICRONS
-U VOMATERIAL
LAMBDA
COEF
LAMBDA
COEF
RMS N MAX
RANGE
0,3
1.225714.586814
8.8808
7.196746 2048 3 5929 4.08-14.9
INDEX
4.785800
TO
4.929800
AND 10X
RESIDUAL
LAMBDA,0 TO 14 MICRONS
MATERIAL
TIBr-Cl
LAMBDA
COEF
LAMBDA
0.2298 3.820600 104.9237
INDEX
2.086900
TO
2.298200
AND 10X
RESIDUAL
COEF
RMS H
9.228510 665 4
MAX
RANGE
1579 0.70-23.0
MATERIAL
T1Br-I
LAMBDA
0.2591 4.660423 136.3244
COEF
LAMBDA
COEF
8.247948
RMS N MAX
394 5
655 0.78-37.5
RANGE
INDEX
2.232810
TO
2.523850
AND 16X
RESIDUAL
LAMBDA,0 TO 37.5 MICRONS
Ln hOMATERIAL
LAMBDA
COEF
LAMBDA
AsS3glass 6.2581 4.867748 28.4939
8.429528 721 5
COEF
RMS N MAX
1415 0.78-11,
RANGE
INDEX
2.378180
TO
2.551986
AND 18X
RESIDUAL
LAMBDA,6 TO 11.4 MICRONS
MATERIAL
PbF
LAMBDA
COEF
LAMBDA
COEF
RMS N MAX
RANGE
0.1432 1.994488 163.1292 88.988516
55 5
112 0.70-18.5
INDEX
1.626738
TO
1.755828
AND 18X
RESIDUAL
LAMBDA,0 TO 18.5 MICRONS
APPENDIX B
CQMPOTER PROGRAM;LIST
. The computer program listed in this appendix produced the
materials for Table 2 and Appendix A. This program executes the process
described in the section "Matrices with Test Functions for Finding
Two-Term Sellmeier Formulae." Lines one through 1600 are data input,
and output headings. Lines 1610.through .1730 select the original
estimates of the resonance wavelength parameters. Lines 1900 through
2195 choose logarithmically spaced data points to be matched exactly by
the dispersion formula. Lines 2200 through 6550 setup and solve the
matrix to find the coefficients C(l) through C(4). Lines 7000 through
7500 calculate the point by point residual errors and the root mean
square residual error. Lines 9000 through 9210 adjust the resonance
wavelength parameters and test for convergence. Lines 9500 through
11080 are various outputs.
LIS 1,2830
1 INIT
2 GO TO 10
3 PRINT " MATERIAL
LAMBDA
COEF
LAMBDA
COEF
"5
4 PRINT “RMS
N
MAX
R A N G E u
18 READ F
12 DATA 6 3 , 6 4 , 6 5 , 6 6 , 6 7 , 6 3 , 6 9 , 7 0 , 7 1 , 7 2 , 7 3 , 7 4 , 7 5 , 7 6 , 7 ? , 7 8 , 7 9 , 8 8
13 DATA 8 1 , 8 2 , 8 3 , 8 4 , 8 5 , 8 6 , 8 7 , 8 8 , 8 9 , 9 0 , 9 1 , 9 2 , 2 0 9
15 DATA 5 , 1 0 , 1 2 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 8 , 2 1 , 2 2 , 2 3 , 2 5 , 2 7 , 2 8 , 2 9 , 3 8 , 3 1 , 3 2
16 DATA 3 3 , 3 4 , 3 5 , 3 6 , 3 7 , 3 3 , 4 1 , 4 2 , 4 9 , 6 0 , 2 8 8
17 DATA 13,24,26,39,4 3 , 4 3 , 4 4 , 4 5 , 4 6 , 4 7 , 4 8 , 2 0 0
28 IF F<99 THEN 1080
30 END
48 REM
500 REM
501 PAGE
582 PRINT “USING L=WAVELENGTH AT WHICH N IS DESIRED (MICRONS)"
583 PRINT "LAMBDA=RESONANCE FROM TABLE BELOW, 12 REPRESENTS E X P O N E N T “
534 PRINT °COEF=COEFFICIEHT FROM COLUMN FOLLOWING LAMBDA"
565 PRINT
518 PRINT 0NT2-i=SUM 0F<C0EF*Lt2/(Lt2-LAi1BDAt2)) FOR BOTH RESONANCES"
515 PRINT
529 PRINT »RMS=ROOT MEAN SQUARE RESIDUAL ERR0R$1GT6, N=# OF DIGITS
IN
521 PRINT "DATA"
522 PRINT
" M A X = W O R S T R E S I D U A L E R R O R * 1 0 t 6 , R A N G E = S P E C T R A L R A N G E , M I C R O N S
539 PRINT
559 GO TO 3
10:3 DELETE
A,C,E,H,L,N,P
1029 14=0.7
1669 FIND F
1078 INPUT 033 : Q , D , 1$
LIS 1871,3869
1188 IF Q=1 THEN 1589
1185 DIM L(Q),N(Q),E(Q)
1118 J=1
1128 FOR 1=1 TO Q
1138 INPUT 833:L(J>
1143 INPUT 9 3 3 2 N<J>
1159 IF L(J)=>W THEN 1189
1168 Q=Q-1
1173 GO TO 1198
1189 J=J+1
1198 NEXT I
1195 S = L < 1>
1196 G1=L(Q>
1289 GO TO 1618
1589 INPUT 0 3 3 : S , Q 1 ,SI
1518 Q=(Q1-S)/S1+1
1515 DIM LCQ),NCQ),ECQ)
1528 J=1
1538 FOR 1=1 TO Q
1548 L(J)=S+S1*(I-1)
1559 INPUT 033:H(J)
1569 IF L(J)>W THEN 1598
1573 D=D-1
1508 GO TO 1688
1530 J=J+1
1689 NEXT I
1618 REM CHOOSE R E S O NANCE WAVEL E N G T H S
1611 G6=0
1612 G4=0
1613 M=4
LIS 1614,3889
1615 65=8
1516 Y=68
1617 T = 0 . 4
1629 REM CHECK FOR IR
1659 G = ( H ( I H T ( Q / 2 ) ) T 2 - ( N ( G - l > - i e f - D > f 2 ) / ( L ( I N T ( Q / 2 > ) T - 2 - L ( Q - l ) f - 2 )
1652 C = 6 - ( ( N ( G - l ) - 1 0 t - D ) f 2 - N ( Q ) 1 2 ) / ( L ( Q - l ) f - 2 - L ( Q ) t - 2 )
1668 IF G<8 THEN 1788
1665 Y=8
1666 64=1
1667 11=3
16S9 60 TO 1888
1789 REM CHECK FOR VIS
1710 G = ( N < l ) T 2 - H ( 2 ) T 2)/(L(l )t-2-L(2)f-2)
1711 G=G-(H(2) t 2 - H ( 3 ) 1 2 ) / ( L ( 2 ) t - 2 - L ( 3 ) t - 2 )
1712 G =G/CN(2)t2-NC3)t2)
1715 6=G$(L( l ) 1 - 2 - L ( 5 ) f - 2 ) / ( L ( l ) T - 2 - L ( 3 ) t - 2 )
1725 IF 6/0.5 THEN 1739
1738 66=1
1883 G0SU8 1989
1818 GO TO 2203
1939 REM
1910 DIM A ( M , M*2),P(M),H(M>
2888 REM CHOOSE LAMBDAS
2183 H(1)=L(1)
2185 F = E X P ( L 0 G ( Q 1 / S ) / ( M ~ D )
2165 6 = S / F f 3.24
2110 P(l)=H(l)f2-l
2115 FOR 1=2 TO M-l
2116 IF I O M - 1 THEN 2129
2117 G=GSFt0.5
Ln COLIS 2 1 1 8 , 7 0 8 0
2128
G
~
h $
G
2125
0=0
2139
FOR J=2 TO Q
2140
IF L ( J X = H ( I - 1 ) THEN
2168
2145
IF 0=1 THEN 2168
2159
HCI>=L(J)
2154
P(I)=H(J)t2-l
2155
IF H C I X G THEN 2160
2156
0=1
2160
NEXT J
2178
NEXT I
2188
H(M)=L(Q)
2199
P(M)=H(Q)t2-l
2195 RETURN
22G0
REM SET UP MATRIX
2218
FOR 1=1 TO M
2249
X = H ( I ) t 2
2268 G=Yt2
2279
A(I,1)=X/(X-T$T)
2289
A ( I , 2 ) = X / ( X - T $ T ) t 2
2299
A(I,3)=X/(X-G)
2295
A(I , 4 > = - X
2389
NEXT I
2318
GOSUB 6099
2315
G0SU3 7609
2329
GO TO 9088
6683
REM MATRIX INVERSION
OF A
6169
FOR 1 = 1 TO fl
6119
FOR J=M+1 TO
m2
6129
A< I , J ) = 9
VI
