REFRACTIVE INDEX MEASUREMENTSOF MAGNESIUM OXIDE, SAPPHIRE, AND
AMTIR-1 AT CRYOGENIC TEMPERATURES
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Nofziger, Michael James
REFRACTIVE INDEX MEASUREMENTS OF MAGNESIUM OXIDE, SAPPHIRE, AND AMTIR-1 AT CRYOGENIC TEMPERATURES
The University of Arizona M.S. 1985
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REFRACTIVE INDEX MEASUREMENTS OF MgO,
SAPPHIRE, AND AMTIR-1 AT CRYOGENIC TEMPERATURES
by
Michael James Nofziger
A Thesis Submitted to the Faculty of the
COMMITTEE ON OPTICAL SCIENCES (GRADUATE)
In Partial Fulfillment of the Requirements For the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
19 8 5
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements 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 or her judgment the proposed use of the material is in the interests of
scholarship. In all other instances, however, permission must be obtained from the author.
2hLL̂ JL£L SIGNED:
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
HUt.,,.. L W. L. WOLFE' 'Date
Professor of Optical Sciences
DEDICATION
Dedicated, with love, to my family.
Their constant support gives me the courage to persevere.
iii
ACKNOWLEDGMENT
Many people have contributed to the successful completion of
this work, including the students who have worked on this project in the
past. Without their fine efforts this would have been a much more
difficult task.
A special thank you needs to be extended to those most closely
associated with this thesis. Professor W. L. Wolfe, advisor and friend,
has taught and guided me through as much personal growth as a knowledge
of optics. What more can one ask of a teacher? His confidence in me at
the start of this project was a Christmas gift never to be forgotten.
An equal thanks goes out to Dr. Jim Palmer, Jerry DeBell, and
Lang Brod. When mechanical and experimental problems arose, they always
had a solution. The real "nuts and bolts" success of the measurements
was achieved only as a result of their professional guidance.
Among the numerous students who offered suggestions, one in
particular gave more of his time and knowledge than could ever have been
expected. To my friend and consulting handyman, Chris Thompson, I say
thank you.
Countless hours have been spent on this manuscript by Anne
Damon, Kathy Seeley, and Bill Wolfe. They have gone above and beyond
their call of duty to help complete the job. Thank you.
Finally, but most importantly, is the gratitude I owe to my
fiancee, Judy. The completion of this thesis is a direct result of her
iv
V
constant love and support. May I be able to return the same in our
lives together.
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS viii
LIST OF TABLES ix
ABSTRACT xi
1. INTRODUCTION ' 1
2. EXPERIMENTAL PROCEDURE 3
Method of Perpendicular Incidence 3 Test Setup 5
System Alignment 8 He-Ne Coalignment 8 System Signal-to-Noise Ratio (SNR) 10
Test Procedures 10 Monochromator Calibration ........................ 11 Prism Apex Angle 11
Prism Alignment 14 Beam Profile 14 Window Correction 16 Data Measurement 17
3. RESULTS 18
Data Reduction Procedures 18 Magnesium Oxide Results 20
Comparison with Other Measurement 27 Sapphire Results 27
Comparison with Other Measurements 32 AMTIR-1 Results 32
Comparison with Other Measurements 44
4. ERROR ANALYSIS 60
Random Errors 60 Systematic Errors 61
Calculation of (Ad)p 64 Combined Errors 66
Temperature Coefficient 66 System Repeatability 7 2
Short-term Precision 72 Long-term Precision 74
vi
vii
TABLE OF CONTENTS—Continued
Page
APPENDIX A: PRISM SPECIFICATIONS 75
APPENDIX B: SYSTEM ALIGNMENT 77
APPENDIX C: He-Ne COALIGNMENT 80
APPENDIX D: SYSTEM SNR 82
APPENDIX E: PRISM ALIGNMENT ERROR ANALYSIS 86
REFERENCES 93
LIST OF ILLUSTRATIONS
Figure Page
1. Method of Perpendicular Incidence 4
2. Test Setup Layout Drawing 6
3. Prism Alignment Axes 9
4. Beam Profile 15
5. MgO (300K) n(x) 21
6. MgO (77K) n(x) 22
7. MgO n(T) 28
8. NBS vs UA MgO (300K) n(x) 30
9. Sapphire (300K) n0 (X) and ne (X) i . 33
10. Sapphire (77K), nQ (X), and ne (X) 34
11. Sapphire n0(T) * 39
12. Sapphire ne(T) 40
13. NBS vs UA Sapphire (300K) nQ (X) 42
14. AMTIR-1 (230K, 300K, 328K), n(X) 45
15. AMTIR-1 n(T) at 8 pm 52
16. AMTIR-1 n(T) at 10 urn 53
17. AMTIR vs UA AMTIR-1 (300K) n(x) 56
18. dn/dT Error Analysis 71
19. Prism Specifications 76
20. Prism/Dewar Alignment 79
viii
LIST OF TABLES
Table Page
1. System Components 7
2. Monochromator Calibration 12
3. Prism Apex Angle 13
4. Typical Notebook Entry 19
5. Sellmeier Fit for MgO at 300K 23
6. Herzberger Fit for MgO at 300K 24
7. Sellmeier Fit for MgO at 77K 25
8. Herzberger Fit for MgO at 77K - . 26
9. MgO : n(T) at 3.5 ym 29
10. NBS vs UA Data : MgO at 300K 31
11. Sellmeier Fit for Sapphire (nQ) at 300K 35
12. Herzberger Fit for Sapphire (nQ) at 300K 36
13. Sellmeier Fit for Sapphire (ne) at 300K 37
14. Herzberger Fit for Sapphire (ne) at 300K 38
15. Sapphire : n0(T) and ne(T) at 3.5 urn 41
16. NBS vs UA Data : Sapphire at 300K 43
17. Sellmeier Fit for AMTIR at 230K 46
18. Herzberger Fit for AMTIR at 230K 47
19. Sellmeier Fit for AMTIR at 300K 48
20. Herzberger Fit for AMTIR at 300K 49
21. Sellmeier Fit for AMTIR at 328K 50
ix
X
ige
51
54
55
58
59
62
63
65
67
68
69
73
85
LIST OF TABLES—Continued
Herzberger Fit for AMTIR at 328K . . .
AMTIR-1 : n(T) at 8 p
AMTIR-1 : n(T) at 10 um
AM vs UA Data : AMTIR-1 at 300K
AM vs UA Data : AMTIR-1 dn/dT values
Error Analysis
Description of Symbols
Variables for Error Analysis
Combined Errors: MgO at 300K
Combined Errors: Sapphire n0 at 300K
Combined Errors: AMTIR-1 at 300K . . .
System Repeatability
Variables Used in SNR Calculation
ABSTRACT
The measurement of the index of refraction of three infrared
transmitting materials, magnesium oxide, sapphire, and AMTIR-l, is
described in this thesis. The index of refraction of each material was
measured both as a function of wavelength and temperature.
Experimental data were taken over wavelength ranges of 2 to 5 jam and 7
to 12 ym, while the temperature varied over a range of 80 to 350K. A
desired accuracy in the index of five parts in the fourth decimal place
was generally achieved. Also described are the equipment, alignment and
measurement procedures, results, and a complete summary of the sources
and magnitudes of errors.
xi
CHAPTER 1
INTRODUCTION
Infrared optical systems typically utilize transmissive elements
in at least one or more of the sub-system assemblies. Cryogenic cooling
of these elements is frequently required to reduce self-emission and
increase the system signal-to-noise ratio. In the design of these
system elements one needs to know accurately both the index of
refraction and dn/dT, the change in index with temperature.
Unfortunately, for all but the most commonly used infrared materials,
the index of refraction and associated dn/dT are either not known
accurately enough or have never been measured. The research program
described in this thesis provides the infrared design engineer with a
source of obtaining index of refraction and dn/dT data for previously
unmeasured materials.
Many techniques have been devised to measure a material's index
of refraction. The method of perpendicular incidence, as described here,
provides for quick yet accurate measurements on samples over a
temperature range of 4K to at least 500K.
The index of refraction and associated dn/dT of three materials
have been measured at various infrared wavelengths and temperatures.
Data are presented for magnesium oxide (MgO) and sapphire over a
wavelength region of 2 to 5 p and a temperature range of 80K to 300K.
In addition, a Ge-As-Se type glass known as AMTIR-1 has been measured.
1
2
Data are presented over a wavelength region of 7 to 12 ym and a
temperature range of 170K to 350K.
Error analyses are presented for all three materials. In each case
the systematic and random errors are about 2x10"*, less than our
desired goal of SxlO-11. A two-term Sellmeier equation and a Herzberger
equation were fit to the data for each material. The results of this
curve fitting as well as comparison to published data are also,
presented.
CHAPTER 2
EXPERIMENTAL PROCEDURE
Many methods exist for measuring the index of refraction. Piatt
(1976) has compiled an excellent reference list on the subject. Most
methods are well-suited for room temperature measurements at visible or
infrared wavelengths. In particular, the minimum deviation technique
provides sixth decimal place accuracy over the visible portion of the
spectrum. However, for measuring the index and dn/dT of samples at
cryogenic temperatures, the required use of vacuum dewars prohibits easy
sample alignment. Any technique involving changes in sample position
during the measurements becomes difficult.
Method of Perpendicular Incidence
This method of measuring the index of refraction, introduced by
Piatt (1976) and more recently developed by Swedberg (1979), represents
a relatively easy yet accurate means of measuring samples at cryogenic
temperatures.
Swedberg provides detailed descriptions of the method and
equipment; because this work essentially parallels his, only a brief
summary of each will be given here.
The method of perpendicular incidence is illustrated in Figure 1.
The sample material with index n( A,T) is cut in the shape of a right-
triangular prism having apex angle a. A collimated beam of light is
3
REFERENCE BEAM
n ( A , T )
n's i n(a+d)
Fig. 1. Method of Perpendicular Incidence.
5
incident normal to the first surface of the prism; as a result, this
method has also been referred to as the autocollimation technique. The
incident beam is purposely made to be larger than the prism.
Approximately 30 percent of this beam is undeviated, providing a
reference beam with which to measure d, the prism's angle of deviation.
The remaining portion of the incident beam enters the prism through the
first surface. No refraction occurs here because the beam is at normal
incidence to this surface. Travelling parallel to the base of the prism,
the beam is refracted at the second surface. Applying Snell's Law at
this surface, the index of refraction of the prism material may be
expressed simply as
n'sin(a+d) n sin(a)
_ _ sin(a+d) ° " sin(a) (1)
where n' has been set equal to 1 for a prism sample in a vacuum. In
principle, the index of refraction is determined by measuring the prism's
apex angle a and the angle of deviation d.
Test Setup
The optical system used in these measurements has been designed
specifically for the method of perpendicular incidence. The system
provides a well collimated beam normal to the first surface of the
prism and a detection system to measure the prism's angle of deviation.
Figure 2 shows a layout of the optics, while Table 1 lists a brief
description of each of the components. More detailed information on
this same optical system may be found in Swedberg's thesis (1979).
PRISM DEWAR ASSEMBLY
[INFERENCE BEAM-
DETECTOR DEMAR ASSEMBLY-
M7<
ANGULAR REAOOl/r
ROTARY TAB I F
ORDKR-BI.OCKI NO FILTER.
SOURCE
CHOPPER
COLLIHATINC MIRROR
PRISM OP MATERIAL UNDER TEST
JARRELL-ASII MONOC11ROMATOR-
L.ASEP/ BEAM EXPANDER-
Test Setup Layout Drawing.
L = lens, M = mirror.
7
Table 1. System Components
Element Description
Ml Flat, first-surface mirror
Thermal source Ceramic rod (1/8 in. diam x 1 in. long) Temperature around 1250°C
Cut-on infrared filter at 6.5 ym
Princeton Applied Research optical chopper
(Model No. 125A) modulation frequency at 170 Hz
6 in. diam, F/4 parabolic mirror
Flat, first-surface mirrors mounted on the monochromator
Filter
Chopper
M2
M3, M4
Monochromator
M5
M6
Prism dewar
assembly
Rotary table
M7
M8
Detector
Detection electronics
Temperature sensor and controller
Jarrell Ash, 0.25 m Ebert-type (Model 82-420) 5 ym blaze (1/d = 148 mm-1) grating was used for 2-5 ym measurements 10 ym blaze (1/d =» 50 mm-1) grating was used for the 7-12 ym measurements
3 in. diam, F/4 off-axis parabolic mirror
Flat, first-surface mirror (relocatable out-of-
beam path)
Double-chamber dewar with KRS-5 windows (dewar
designed for liquid helium)
Genevoise Rotary Table, Type 1-4
angular readout to 1 arc sec
4 in. diam, F/4 off-axis parabolic mirror
Flat, first-surface mirror
2-5 ym: 0.010 in. diam InSb photodiode 7-12 ym: HgCdTe photoconductor
Detector Preamp Princeton Applied Research lock-in amplifier
Si diode sensors: Lakeshore Cryotronics Model DT-500CV-DRC Controller: Lakeshore Cryotronics Model DRT-7C
8
System Alignment
Proper alignment of the optical system may be described by two
separate alignment procedures: (1) alignment of the optics used to
provide a collimated beam at the prism, and (2) alignment of the prism
so this collimated beam is normal to the prismas first surface. These
two steps form the essence of the method of perpendicular incidence.
Swedberg (1979) has provided a good discussion of how to achieve this
alignment. Appendix B of this thesis contains step-by-step procedures
used to align the system for these measurements.
He-Ne Coalignment
As outlined in Appendix B, the alignment of the incident beam
normal to the prism's first surface is achieved using an autocollimation
technique. Figure 3 shows the three degrees of freedom used in prism
alignment. Rotations of the prism about the alpha and beta axes change
the angle of reflectance of the incident beam off the first surface.
When the prism's first surface is adjusted normal to the incident beam,
the reflected portion will be imaged by mirror M5 directly back onto the
exit aperture. This may be observed by placing a microscope near the
exit aperture and looking at the reflected image.
In principle, this autocollimation technique appears simple; in
practice, it can become quite difficult. Problems arise because the
radiance at the exit aperture is low and transmission of the KRS-5
window is poor over the visible spectrum. The situation is much worse
for prism materials whose Fresnel surface reflectance is only 4 to 5
percent (materials with a low index of refraction over the visible
9
A1 pha Ax i s
D i hedra1
Edge Gamma Ax i s
Beta Ax i s
Fig. 3. Prism Alignment Axes.
10
spectrum). A solution, found by Swedberg (1979) and used in these
measurements, involves the replacement of the thermal source with a He-
Ne laser beam. This beam is focused to provide a real image at the
location of the (removed) thermal source. The reflected image is now
visible even for prism materials which reflect only a few percent of the
incident beam. Details of this He-Ne co-alignment are given in
Appendix C.
System Signal-to-Noise Ratio (SNR)
The instrument described in this thesis is adaptable for making
index of refraction measurements over a wide range of wavelengths and
temperatures (Piatt, et al.; Wolfe, DeBell, Palmer, 1980). Changing the
wavelength region typically implies using a different detector having the
proper spectral response. Once the detector is installed, the entire
system is aligned according to the procedures already given and a signal
is detected. At this point, a comparison of the measured and calculated
signal-to-noise ratios can be useful to determine if the alignment is
optimal and if the detector is functioning properly. The details for
calculating the overall system signal-to-noise ratio are given in
Appendix D.
Test Procedures
In addition to aligning the optical system, there are other
procedures which must be carried out before data are actually taken.
This section gives a description of each.
Monochromator Calibration
Initial calibration of the monochromator was carried out using
the green line of a mercury source. An easier method, used for this
work, replaced the mercury source with the co-aligned He-Ne laser.
Calibration of wavelength vs dial number on the monochromator was done
by rotating the grating through several orders, noting the dial number
each time. This was done for both the 5 urn and 10 pm blazed gratings
used in the measurements. A linear least-squares fit to the data was
carried out in each case, with a correlation coefficient of 0.999 being
typical. Table 2 presents data and values of fit for a calibration
representative of both gratings. Also calculated is a root-mean-squared
(rms) value for the estimated error in monochromator calibration. This
represents the average error between the actual wavelength (the
multiple orders of 0.6328 um) and the calculated wavelength (from the
linear fit) at each of the dial numbers noted during the calibration.
This value of will be used later in the error analysis.
Prism Apex Angle
It was shown in equation (1) that calculating the index of
refraction involves knowing the vertex angle a of the prism to be
measured. The vertex angles of each of the three prisms were measured
using a Wild No. 79 Precision Spectrometer. Several different readings
were taken for each prism. The average values and standard deviations
of the measurements are listed in Table 3. It is worth noting that no
error is introduced into the values of index if the prism has a pyramidal
angle. (This angle is present if the three planes containing the prism
12
Table 2. Monochromator Calibration
Order Actual Calculated Difference Number Wavelength Wavelength x 1000
1 0.6328 0.63257 -0.2230 2 1.2656 1.26553 -0.0690 3 1.8984 1.89824 -0.1504 4 2.5312 2.53906 -0.2313 5 3.1640 3.16392 -0.0774 6 3.7968 3.79687 +0.0764
7 4.4296 4.42959 -0.0044 8 5.0624 5.06231 -0.0853 9 5.6952 5.69503 -0.1662
10 6.3280 6.32798 -0.0123 11 6.6908 6.69607 -0.0932 12 7.5936 7.59342 -0.1740
13 8.2264 8.22637 -0.0202 14 8.8592 8.85909 -0.1011 15 9.4920 9.49181 -0.1819
16 10.1248 10.12477 -0.0281 17 10.7576- 10.75749 -0.1089 18 11.3904 11.39021 -0.1898 19 12.0232 12.02316 -0.0360 20 12.6560 12.65580 -0.1168
21 13.2888 13.28860 -0.1977
22 13.9216 13.92155 -0.0438 23 14.5544 14.55427 -0.1247
RMS T0TAL= 0.1300
13
Table 3. Prism Apex Angle
Material Number of Average Standard Readings Deviation
deg min sec deg min sec
Magnesium Oxide 10 31 00 13 .6 00 00 0.8
Sapphire 10 31 00 13 .1 00 00 1.4
AMTIR-1 3 15 54 18 .9 00 00 0.3
14
faces intersect in a point.) Proper alignment of the prism by rotation
about the gamma axis (Appendix B) orients the dihedral edge (Figure 3) in
a vertical direction, regardless of whether there is pyramidal angle
present or not.
Prism Alignment
Prism alignment was carried out using the autocollimation
technique already described. In the process of measuring dn/dT, however,
it was discovered that cooling and warming of the prism sample dewar
changed the alignment of the prism. To check this, the dewar was cooled
to 77K while the position of the retroreflected image (back onto the
exit aperture) was tracked. The change in prism alignment was
predominantly about the alpha axis (see Figure 3) and was estimated to
be 0.98 milliradians. Even though this in itself seems insignificant, the
error introduced in the absolute value of index is unacceptably high (see
the "Error Analysis" section). The solution was to use the
autocollimation technique and realign the prism after every change in
sample temperature during dn/dT measurements.
Beam Profile
The intensity profiles of both the reference and deviated beams
were carefully measured prior to taking a set of data or after
realigning the optics. A typical beam intensity profile of a properly
aligned system is shown in Figure 4. In determining absolute values of
the index, the overall accuracy is determined in part by the degree of
symmetry of each of the beam profiles. One of the goals of the
alignment process is to make the beam profiles as symmetric as possible.
15
<nU>
7
n
+
+
2
+
+
+
+ +
+
+
9
0
+
ROTARY TABLE POSITION (relat ive scale)
Fig. 4. Beam Profile.
16
Window Correction
Measurements made at cryogenic temperatures require the use of
windows on the evacuated prism sample dewar. The three KRS-5 windows
used for this each have wedged surfaces. Even though the windows were
mounted on the dewar with the wedges in the vertical direction, Swedberg
(1979) had measured a residual horizontal component. This introduces a
systematic error in the measurements. In addition, the deviated beam
does not enter the exit window at normal incidence. Unwanted refraction
occurs through this window and another systematic error is introduced.
Swedberg (1979) provides a more detailed description of these two
sources of error.
The procedure to eliminate this combination of errors in the
measurements is as follows. With none of the three windows in place,
the system was carefully aligned and the index of refraction was
measured as a function of wavelength at room temperature. All three
windows were installed, the dewar was evacuated, and again the system
was carefully aligned. The index of refraction was measured at the same
wavelengths and temperature. A difference was observed between the
sets of indices, constant to a few parts in the fourth decimal place.
An average of these differences was calculated and this number was then
used to correct any set of data for which windows were used on the
dewar.
17
Data Measurement
Data were taken under two different sets of conditions. In the
first, the temperature of the sample was held constant and the index
was measured as a function of wavelength, i.e., n(x). This is referred
to as a wavelength run. In the second, a wavelength was chosen and held
constant while the index was measured as a function of sample
temperature, i.e, n(T). This is referred to as a temperature run. The
change in index with a change in temperature, dn/dT, is calculated by
finding the slope of the n(T) curve at a given temperature. The curve
representing n(T) may or may not be linear; if not, then dn/dT will be a
function of temperature and not a constant.
A cross-check of dn/dT is essentially built into the measurement
process. By making a wavelength run at various temperatures, dn/dT (as
a function of wavelength) may be determined independent of the value
obtained from the temperature run.
The actual process of taking data involves measuring the prism's
angle of deviation, d, with respect to the reference beam (see Figure 1).
After having mapped out both beam profiles to insure symmetry, the
process is to locate the center of each beam. Due to the slit function
of the monochromator, the beam profiles are broadened and the center
may be difficult to locate directly. An easier method is to locate the
85 percent points of the beam profile and take the average to find the
center. The angle of deviation, d, is simply the difference between the
centers of the reference and deviated beams.
CHAPTER 3
RESULTS
The results of measurements of n( x ) and n(T) for magnesium
oxide, sapphire, and AMTIR-l are presented. Techniques used for data
reduction are first discussed, followed by the data and a comparison
with results from the literature.
Data Reduction Procedures
The actual process of taking data, although simple in approach,
turned out to be quite tedious in practice. After aligning the system as
outlined in the last chapter, wavelength and temperature runs were made
for each material.
A wavelength run was done by first setting the monochromator to
the desired wavelength. The rotary table was then used to rotate the
detection system to locate the 85 percent points of each beam profile.
Each angular position (in degrees, minutes, seconds) was recorded and the
temperatures directly above and below the prism were also noted. Table
4 shows a typical entry into the notebook. The four angles were
converted to decimal degrees and then averaged to find the beam centers.
The angles of the centers were subtracted to give the deviation angle,
d, and equation (1) was used to calculate the index. This reduction was
done quickly using a hand calculator. By doing it after every change in
wavelength, bad data points became apparent immediately.
18
Table 4. Typical Notebook Entry.
Material: Magnesium Oxide
REF. o i i , — o i //
Dev. QBA* o / 'f — * t ft
T ToP
T &PiS£
d n
2. o
1
o 3? 5"S" ' o Ljtj s~i
1
l<? "35" 2/ J to 01 2i •z°it ZW 29. to So r /• 7® ??o
3.c 1
1 1
1 5~. ( J
1
1
1
1
1
20
A curve-fitting program published by Swedberg (1979) was used to
fit the n(X) data to both a two-term Sellmeier equation and a Herzberger
equation. The measured indices, the indices calculated from the
equations, and the differences are presented.
A temperature run was done in basically the same way as a
wavelength run, except the wavelength was held constant and the
temperature of the sample varied. Data were taken and reduced in the
manner just described. Sample temperature was changed using the
Lakeshore Cryotronic heater controller. Sufficient time was allowed
after each temperature change for the prism sample to reach the new
equilibrium temperature. The temperature gradient across the prism was
never greater than 2K.
For the three materials measured, the index of refraction was
plotted as a function of temperature. For AMTIR-l it appeared as if the
data closely followed a straight line. A linear least-squares fit to
the data was done to determine the dn/dT coefficient. The deviations
from a straight line for the other two materials are. discussed later in
the Temperature Coefficient section.
The following data are listed without error bars. An error
analysis, presented in the next chapter, provides a detailed calculation
and listing of the separate errors.
Magnesium Oxide Results
The refractive index as a function of wavelength, n( x ) , was
measured for magnesium oxide. A wavelength run was done from 2 to 5 pm
for sample temperatures of 300K (room temperature) and 77K (liquid
3 4 MAUELENGTH <nicroneters)
Fig. 5. MgO (300K) n(X).
22
I N D E X
0
F
R
E F R a c T I 0
H
1.72_
1.71
1.70
1.69
1.68
1.67
1.66
1.65
1.64
1.63
* *
-t
*
* *
*
*
I I I I >
3 4 WAVELENGTH (nicroneters)
Fig. 6. MgO (77K) n(A)
Table 5. Sellmeier Fit for MgO at 300K
UAUELENGTH MEASURED CALCULATED DIFFERENCE <UM> INDEX INDEX
2.0 1.70880 1.70889 -0.00009 2.1 1.70734 1.70739 -0.00005 2.2 1.70587 1.78584 0.00003 2.3 1.70431 1.70426 0.00005 2.4 1.70271 1.70262 0.00009 2.5 1.70101 1.70094 0.00007 2.6 1.69926 1.69920 0.00006 2.7 1.69750 1.69741 0.00009 2.8 1.69560 1.69556 0.00004 2.9 1.69372 1.69364 0.00008 3.0 1.69168 1.69167 0.00001 3.1 1.68967 1.68963 0.00004 3.2 1.68758 1.68752 0.00006 3.3 1.68537 1.68534 0.00003 3.4 1.68311 1.68309 0.00002 3.5 1.68081 1.68078 0.00003 3.6 1.67840 1.67839 0.00001 3.7 1.67593 1.67592 0.00001 3.8 1.67336 1.67338 -0.00002 3.9 1.67077 1.67076 0.00001 4.0 1.66808 1.66807 0.00001 4.1 1.66528 1.66529 -0.00001 4.2 1.66268 1.66243 0.00025 4.3 1.65932 1.65949 -0.00017 4.4 1.65648 1 •65647 0.00001 4.5 1.65337 1.o533b 0.00001 4.6 1.65020 1.65016 0.00004 4.7 1.64691 1.64687 0.00004 4.8 1.64358 1.64349 0.00009 4.9 1.64012 1.64002 0.00010 5.8 1.63663 1.63646 0.00017
ROOT MEAN SQUARE DIFFERENCE3 1.424156603E-5
COEFFICIENTS OF THE FORMULA
Al=l.95438519646 A2=4.66452641279 U1=0.0168154985595 U2=»444.923611359
Table 6. Herzberger Fit for MgO at 300K
HAUELENGTH MEASURED CALCULATED DIFFERENI (un> INDEX INDEX
2.0 1.70880 1.70880 0.00000 2.1 1.70734 1.70736 -0.00002 2.2 1.70587 1.70586 0.00001 2.3 1.70431 1.70430 0.000Q1 2.4 1.70271 1.70269 0.00002 2.5 1.70101 1.70101 0.00000 2.6 1.69926 1.69928 -0.00002 2.7 1.69750 1.69748 0.00002 2.8 1.69560 1.69563 -0.00003 2.9 1.69372 1.69371 0.00001 3.0 1.69168 1.69172 -0.00004 3.1 1.68967 1.68967 0.00000 3.2 1.68758 1.68756 0.00002 3.3 1.68537 1.68537 0.00000 3.4 1.68311 1.68312 -0.00001 3.5 1.68081 1.68080 0.00001 3.6 1.67840 1.67840 0.00000 3.7 1.67593 1.67593 0.00000 3.8 1.67336 1.67339 -0.00003 3.? 1.67077 1.67077 0.00000 4.0 1.66808 1.66807 0.00001 4.1 1.66528 1.66529 -0.00001 4.2 1.66268 1.66244 0.00024 4.3 1.65932 1.65950 -0.00018 4.4 1.65648 1.65649 -0.00001 4.5 1.65337 1.65339 -0.00002 4.6 1.65020 1.65020 0.00000 4.7 1.64691 1.64693 -0.00002 4.8 1.64358 1.64358 0.00000 4.9 1.64012 1.64013 -0.00001 5.0 1.63663 1.63660 0.00003
ROOT MEAN SQUARE DIFFERENCE" 1.021974922E-
COEFFICIEHTS OF THE FORMULA
A«1.71842051314 D=-0.0030329773527 B*0.0149027044262 E=-1.051138757E-5 C=-0.0169470306715
Table 7. Sellmeier Fit for MgO at 77K.
MAUELENGTH MEASURED CALCULATED DIFFERENl <UM> INDEX INDEX
2.0 1.707t4 1.70739 -0.00025 2.1 1.70576 1.70585 -0.00009 2.2 1.70435 1.70427 0.00008 2.3 1.70280 1.70266 0.00014 2.4 1.70116 1.70101 0.00015 2.5 1.69942 1.69931 0.80011 2.6 1.69764 1.69756 0.CU008 2.7 1.69591 1.69575 0.00016 2.8 1.69391 1.69389 0.00002 2.9 1.69208 1.69197 0.00011 3.0 1.69008 1.68998 0.00010 3.1 1.68800 1.68793 0.00007 3.2 1.68588 1.68582 0.00006 3.3 1.68370 1.68364 0.00006 3.4 1.68142 1.68138 0.00004 3.5 1.67909 1.67906 0.00003 3.6 1.67665 1.67666 -0.00001 3.7 1.67417 1.67419 -0.00002 3.8 1.67162 1.67165 -0.00003 3.9 1.66897 1.66902 -0.00005 4.0 1.66627 1.66632 -0.00005 4. 1 1.66347 1.66354 -0.00007 4.2 1.66087 1.66067 0.00020 4.3 1.65750 1.65772 -0.00022 4.4 1.65467 1.65468 -0.00001 4.5 1.65157 1.65156 0.00001 4.6 1.64835 1.64835 0.00000 4.7 1.64510 1.64504 0.00006 4.8 1.64178 1.64165 0.00013 4.9 1.63838 1.63816 0.00022 5.0 1.63486 1.63457 0.00029
ROOT MEAN SQUARE DIFFERENCE* 2.166125959E-5
COEFFICIENTS OF THE FORMULA
Al«l.94740252201 U1=0.0199465938176
A2=4.15008386479 U2=400.091773703
Table 8. Herzberger Fit for MgO at 77K
HAUELENGTH MEASURED CALCULATED DIFFERENCE <un) INDEX INDEX
2.0 1.70714 1.70717 -0.00003 2.1 1.70576 1.70577 -0.00001 2.2 1.70435 1.70430 0.00005 2.3 1.70280 1.70275 0.00005 2.4 1.70116 1.70113 0.00003 2.5 1.69942 1.69945 -0.00003 2.6 1.69764 1.69770 -0.00006 2.7 1.69591 1.69589 0.00002 2.8 1.69391 1.69402 -0.00011 2.9 1.69208 1.69208 0.00000 3.0 1.69008 1.69008 0.00000 3.1 1.68800 1.68801 -0.00001 3.2 1.68588 1.68587 0.00001 3.3 1.68370 1.68367 0.00003 3.4 1.68142 1.68140 0.00002 3.5 1.67909 1.67906 0.00003 3.6 1.67665 1.67665 0.00000 3.7 1.67417 1.67417 0.00000 3.8 1.67162 1.67161 0.00001 3.9 1.66897 1.66898 -0.00001 4.0 1.66627 1.6662? 0.00000 4.1 1.66347 1.66349 -0.00002 4.2 1.66087 1.66063 0.00024 4.3 1.65750 1.65769 -0.00019 4.4 1.65467 1.65468 -0.00001 4.5 1.65157 1.65158 -0.00001 4.6 1.64835 1.64840 -0.00005 4.7 1.64510 1.64513 -0.00003 4.8 1.64178 1.64178 0.00000 4.9 1.63838 1.63835 0.00003 5.0 1.63486 1.63483 0.00003
ROOT MEAN SQUARE DIFFERENCE* 1.147062802E-5
COEFFICIENTS OF THE FORMULA
A»l.7166703965 D=-0.00306729781252 0=0.0193441942796 E®-9.416368853E-6 C=-0.0307656552071
27
nitrogen). Measurements were made every 0.1 um in wavelength. The data
are presented in Figures 5 and 6 and Tables 5,6, 7 and 8.
The refractive index as a function of temperature, n(T), was
measured for magnesium oxide. A temperature run was done at a constant
wavelength of 3.5 um for sample temperatures of 80K to 290K.
Measurements were made approximately every 25K. A plot of the
refractive index vs temperature is shown in Figure 7. Table 9 lists the
values of index and the corresponding sample temperatures.
Comparison with Other Measurements
The index of refraction of magnesium oxide has previously been
measured by NBS (Stephens and Malitson, 1952; Strong and Brice, 1935).
Data were taken for samples only at or near room temperature, over
visible and near-infrared wavelengths. Figure 8 and Table 10 show a
comparison of the 300K data reported by Stephens and Malitson (1952) and
the 300K data given in this thesis. To provide a more readable graph,
only some of the data from this thesis (Table 5) have been plotted. An
excellent agreement between the two sets of data is clearly seen.
Sapphire Results
The anisotropic crystalline structure of sapphire causes optical
birefringence. Depending on the orientation of the crystal with respect
to the incident electromagnetic field, one of two indices of refraction
may be measured. The sample of sapphire that was measured had its
optical axis contained in the plane of the first surface of the prism.
As a result, both the ordinary and extraordinary indices of refraction
could be measured without reorientation of the prism sample.
28
<n<T>-"n<80> >£1E4
18
16
14
12
18
8
6
•Jv -h
80 100 120 140 160 180 200 220 240 260 280 360 TEMPERATURE <K>
Fig. 7. MgO n(T).
Table 9. MgO : n(T) at 3.5 um
29
Temperature Measured (K) Index
80 1.68144
85 1.68146
90 1.68145
98 1.68146
106 1.68149
110 1.68157
121 1.68167
151 1.68182
177 1.68200
203 1.68219
226 1.68245
251 1.68277
276 1.68304
292 1.68323
Note: The above data have not been corrected for the effects of the dewar windows. However, the correction is constant with temperature and does not affect dn/dT, the slope of the curve.
30
I N
D E X
0
F
R E F R A C T I 0
H
1.72
1.71
1.70
1.69
1.68
1.67
1.66
1.65
1.64
1.63
1.62 1 1 1 i i I I I • j * * * 1 * • » * • » • • * * * * » • •
1.5 2 3 4 5
WAUELEHGTH <nicroneters>
• * • * * • * • * • * • •
Fig. 8. NBS vs UA MgO (300K) n(X). * = data, + = NBS data.
Table 10. NBS vs UA Data : MgO at 300K
31
X(jim) n(UA) n(NBS) An(UA-NBS)
2.0 1.70880 1.70852 +0.00028
2.5 1.70101 1.70079 +0.00022
3.0 1.69168 1.69161 +0.00007
3.5 1.68081 1.68076 +0.00005
4.0 1.66808 1.66809 -0.00001
4.5 1.65337 1.65347 -0.00010
5.0 1.63663 1.63673 -0.00010
n(UA) n(NBS)
= measured experimental data
= values from NBS curve-fit equation
32
A wavelength run was done from 2 to 5 pm to measure the
ordinary index, nQ(X), and the extraordinary index, ne(x). Data were
taken for sample temperatures of 300K and 77K, at every 0.1 um in
wavelength. The data are presented in Figures 9 and 10 and Tables 11
through 14.
The refractive indices as functions of temperature, nQ(T) and
ne(T), were measured as well. A temperature run was done at a constant
wavelength of 3.5 ym for sample temperatures of 80K to 294K.
Measurements were made at ten different temperatures over this range.
A plot of nQ vs temperature is shown in Figure 11, and ne vs temperature
in Figure 12. Table 15 lists the values of n0 and ne measured at the
corresponding sample temperatures.
Comparison with Other Measurements
The refractive indices of sapphire have been measured at visible
wavelengths and room temperature by Jeppesen (1958). Data for the
ordinary index were extended to the near-infrared spectral region by
Malitson, Murphy, and Rodney (1958) and Malitson (1962) for samples at
room temperature. A comparison of the nQ data at 300K reported by
Maltison (1962) and the nQ data at 300K given in this thesis is presented
in Figure 13 and Table 16. As with magnesium oxide, an excellent
agreement is observed between the two sets of room-temperature data.
AMTIR-1 Results
AMTIR is an acronym for Amorphous Material Transmitting Infrared
Radiation. Manufactured by Amorphous Materials, Inc., AMTIR-1 is a Ge-
As-Se type of infrared glass designed to transmit over both atmospheric
33
1-75
1 .70
1 .65
*****
**** *****
**** **:* **!* * ! * . o - R A Y
* * * e-RAY *1*
** *** ±
*** ***
** ** ** *
1 • 60' * * * ' * * * * * * • • * * * • • * * * • * * * * * * * * •
3 4 MAUELENGTH (nicroMeters>
Fig. 9. Sapphire (330K) nQ (A) and ng (A).
4
34
1.75
I N D E X
0
F
R E F R A C T I 0
N
* t • * *
t
1.70
1.65
t ** t £ .
* * * * * * *!**
* * ? * o - R A Y * ± * *
e-RAY * * * t
Kt
' * * *
**
* l ± * t
** *
1.60'* * i 1 ,
2 3 4 5
WAUELEHGTH (nicrometers)
Fig. 10. Sapphire (77K), nQ (A) and ng (A).
Table 11. Sellmeier Fit for Sapphire (nQ) at 300K
HAUELENGTH MEASURED CALCULATED DIFFERENCE <un> INDEX INDEX
2.9 1.73725 1.73791 -0.00066 2.1 1.73536 1.73551 -0.00015 2.2 1.73323 1.73309 0.00014 2.3 1.73082 1.73062 0.00020 2.4 1.72850 1.72810 0.00040 2.5 1.72593 1.72552 0.00041 2.6 1.72332 1.72287 0.00045 2.7 1.72061 1.72014 0.00047 2.8 1.71773 1.71732 0.00041 2.9 1.71487 1.71441 0.00046 3.0 1.71046 1.71140 -0.00094 3.1 1.70864 1.70828 0.00036 3.2 1.70539 1.70506 0.00033 3.3 1.70203 1.70172 0.00031 3.4 1.69852 1.69827 0.00025 3.5 1.69489 1.69469 0.00020 3.6 1.69101 1.69098 0.00003 3.7 1.68708 1.68714 -0.00006 3.8 1.68309 1.68316 -0.00007 3.9 1.67891 1.67904 -0.00013 4.0 1.67465 1.67477 -0.00012 4.1 1.67016 1.67034 -0.00018 4.2 1.66606 1.66576 0.00030 4.3 1.66066 1.66101 -0.00035 4.4 1.65605 1.65608 -0.00003 4.5 1.65103 1.65099 0.00005 4.6 1.64582 1.64569 0.00013 4.7 1.64038 1.64021 0.00017 4.8 1.63496 1.63453 0.00043 4.9 1.62928 1.62864 0.00064 5.9 1.62343 1.62253 0.00090
ROOT MEAN SQUARE DIFFERENCE3 7.001953312E-5
COEFFICIENTS OF THE FORMULA
Al-2.05997409902 A2=2.63385602966 U1=0.0402780100594 U2=177.885878829
Table 12. Herzberger Fit for Sapphire (nQ) at 300K
NAUELENGTH MEASURED CALCULATED DIFFERENCE <un> INDEX INDEX
2.0 1.73725 1.73732 -0.00007 2.1 1.73536 1.73532 0.00004 2.2 1.73323 1.73316 0.00007 2.3 1.73082 1.73086 -0.00004 2.4 1.72850 1.72844 0.00006 2.5 1.72593 1.72590 0.00003 2.6 1.72332 1.72326 0.00006 2.7 1.72061 1.72052 0.00009 2.8 1.71773 1.71767 0.00006 2.9 1.71487 1.71472 0.00015 3.0 1.71046 1.71166 -0.00120 3.1 1.70864 1.70850 0.00014 3.2 1.70539 1.78522 0.00017 3.3 1.70203 1.70184 0.00019 3.4 1.69852 1.69833 0.00019 3.5 1.69489 1.69471 0.00018 3.6 1.69101 1.69097 0.00004 3.7 1.68708 1.68709 -0.00001 3.8 1.68309 1.68309 0.00000 3.9 1.67891 1.67895 -0.00004 4.9 1.67465 1.67467 -0.00002 4.1 1.67016 1.67025 -0.00009 4.2 1.66606 1.66568 0.00038 4.3 1.66066 1.66096 -0.00030 4.4 1.65605 1.65609 -0.00004 4.5 1.65103 1.65105 -0.00002 4. 6 1.64582 1.64586 -0.00004 4.7 1.64038 1.64049 -0.00011 4.8 1.63496 1.63495 0.00001 4.9 1.62928 1.62924 0.00004 5.0 1.62343 1.62334 0.00009
ROOT MEAN SQUARE DIFFERENCE® 4.469451364E-5
COEFFICIENTS OF THE FORMULA
A« i.7465386602 D*-0.90411312848259 0=0.056796400994 E=-3.595372842E-5 C»-0.102339136788
Table 13. Sellmeier Fit for Sapphire (ne) at 300K
HAUELENGTH MEASURED CALCULATED DIFFERENCE <UM> INDEX INDEX
2.0 1.72954 1.72972 -0.00018 2.1 1.72752 1.72758 -0.00006 2.2 1.72545 1.72536 0.00009 2.3 1.72320 1.72306 0.00014 2.4 1.72074 1.72066 0.00008 2.5 1.71823 1.71817 0.00006 2.6 1.71564 1.71559 0.00005 2.7 1.71297 1.71290 0.00007 2.8 1.71015 1.71011 0.00904 2.9 1.70730 1.70722 0.00008 3.0 1.70427 1.70422 0.00005 3.1 1.70116 1.70110 0.00006 3.2 1.69795 1.69788 0.00007 3.3 1.69462 1.69454 0.00008 3.4 1.69116 1.69109 0.00007 3.5 1.68754 1.68751 0.00003 3.6 1.o8376 1.68381 -0.00005 3.7 1.67993 1.67999 -0.00006 3.8 1.67601 1.67605 -0.00004 3.9 1.67190 1.67197 -0.00007 4.0 1.66774 1.66776 -0.00002 4.1 1.66335 1.66342 -0.00007 4.2 1.65936 1.65894 0.00042 4.3 1.65409 1.65431 -0.00022 4.4 1.64959 1.64955 0.00004 4.5 1.64470 1.64464 0.00006 4.6 1.63964 1.63958 0.00006 4.7 1.63437 1.63436 0.00001 4.8 1.62910 1.62899 0.00011 4.9 1.62360 1.62346 0.00014 5.0 1.61793 1.61776 0.00017
ROOT MEAN SQUARE DIFFERENCE3 2.118300887E-5
COEFFICIENTS OF THE FORMULA
Al*2.05133862591 'A2=6.0387957247 U1«0.0I21325438922 U2=371.914900274
Table 14. Herzberger Fit for Sapphire (ne) at 300K
HAUELENGTH MEASURED CALCULATED DIFFERENCE <uti> INDEX INDEX
2.0 1.72954 1.72956 -0.00002 2.1 1.72752 1.72754 -0.00002 2.2 1.72545 1.72540 0.00005 2.3 1.72320 1.72313 0.00007 2.4 1.72074 1.72076 -0.00002 2.5 1.71823 1.71827 -0.00004 2.6 1.71564 1.71568 -0.00004 2.7 1.71297 1.71299 -0.00002 2.8 1.71015 1.71019 -0.00004 2.9 1.70730 1.70728 0.00002 3.0 1.70427 1.70427 0.00000 3.1 1.70116 1.70115 0.00001 3.2 1.69795 1.69791 0.00004 3.3 1.69462 1.69456 0.00006 3.4 1.69116 1.69110 0.00006 3.5 1.68754 1.68752 0.00002 3.6 1•68376 1.68382 -0.00006 3.7 1.67993 1.67999 -0.00006 3.8 1.67601 1.67604 -0.00003 3.9 1.67190 1.67197 -0.00007 4.0 1.66774 1.66776 -0.00002 4.1 1.66335 1.66342 -0.00007 4.2 1.65936 1.65894 0.00042 4.3 1.65409 1.65433 -0.00024 4.4 1.64959 1.64958 0.00001 4.5 1.64470 1.64468 0.00002 4.6 1.63964 1.63963 0.00001 4.7 1.63437 1.63444 -0.00007 4.8 1.62910 1.62909 0.00001 4.9 1.62360 1.62359 0.00001 5.0 1.61793 1.61792 0.00001
ROOT HEAN SQUARE DIFFERENCE" 1.699808284E-5
COEFFICIENTS OF THE FORMULA
A*l.74413385049 D=-0.00446991590245 B=0.0264199282485 E=-2.471157894E-5 C=-0.0466218640364
39
<no<T>-no<80)>*1E4
16
14
12
18
8
6
0
A
-»—i—i—h -4 h
80 100 120 140 160 180 200 220 240 260 280 300 TEMPERATURE <K>
Pig. 11. Sapphire nQ(T).
40
I N
D E X
0
F
R E F R A C T I 0
H
<ne <T>-ne <80))*1E4
18
16
14
12
18
8
4
2
0 *•**•' *
80 100 120 140 160 180 200 220 240 260 280 300 TEMPERATURE <K>
Fig. 12. Sapphire ne(T).
Table 15. Sapphire : nQ(T) and ng(T) at 3.5 ym
41
Temperature Measured Measured (K) Index ne Index n0
80.2 1.68909 1.69657
84.8 1.68909 1.69656
89.5 1.68909 1.69655
94.5 1.68911 1.69656
102.5 1.68915 1.69659
122.5 1.68925 1.69668
129.5 1.68929 1.69679
149.7 1.68938 1.69683
201.3 1.68973 1.69718
293.5 1.69076 1.69806
Note: The above data have not been corrected for the effects of the dewar windows. However, the correction is constant with temperature and
does not affect the dn/dT, the slope of the curve.
42
1.74
1.73
1.72
1.71
1.70
1.69
1.68
1.67
1.66
1.65
1.64
1.63
1.62
+*
*
.*
*
*
* ' 1 I I 1 1 ' ' 1 1 ' J f ' 1 ' • ' ̂ » ' ' • '
MAUELENGTH <nicroneters>
Fig. 13. NBS vs UA Sapphire (300K) nQ (X). * = UA nQ data,
+ = NBS n data. o
Table 16. NBS vs UA Data : Sapphire at 300K
A (um) n0 (UA) n0 (NBS) An (UA-NBS)
2.0 1.73725 1.73773 -0.00048
2.5 1.72593 1.72609 -0.00016
3.0 1.71056 1.71224 -0.00178
3.5 1.69489 1.69534 -0.00045
4.0 1.67465 1.67524 -0.00059
4.5 1.65103 1.65159 -0.00056
5.0 1.62343 1.62397 -0.00054
n (UA) = measured experimental data
n (NBS) = values from NBS curve-fit equation
44
windows (3 to 5 vim and 8 to 12 yin) as well as the YAG laser wavelength
of 1.064 ym.
A sample in the form of a prism was obtained for measurements.
A wavelength run was carried out from 7 to 12 ym for sample
temperatures of 230K (-43°C), 300K (room temperature), and 328K (+55°C).
Measurements were made every 0.1 pm in wavelength. The data are
presented in Figure 14 and Tables 17 through 22.
As with the other samples, the index of refraction vs
temperature, n(T), was measured. Two temperature runs were done at
constant wavelengths of 8 and 10 ym for sample temperatures ranging
from 170K to 350K. Fourteen data points were taken over this
temperature range. A plot of n(T) at 8 ym is given in Figure 15 and n(T)
at 10 ym is shown in Figure 16.
Lists of the indices and corresponding temperatures at the two
wavelengths appear in Tables 23 and 24. For each set of n(T) data, a
least-squares linear fit was done to calculate dn/dT. The results and
coefficients of fit are also given. In each case, dn/dT was found to be
76 x lO-^C-1.
Comparison with Other Measurements
The only previously available source of refractive index data
for AMTIR-1 is that published by Amorphous Materials, Inc. (1983). They
have used the minimum deviation technique to measure the room
temperature refractive index from 1 to 14 ym. Values for dn/dT were
also measured over a temperature range of 25°C to 65°C at 1.15, 3.39,
and 10.6 ym in wavelength.
45
INOEX
2.51000r
1 *•*<?*< m
e o Ul >
•5
« u
u. 41
(J
2.48000
***U
****** x?v *****. ^*o*t
. 328 K
300 K
s 230 K
a 9 10 11 12
WAVELENGTH (u)
Fig. 14. AMTIR-1 (230K, 300K, 328K) , n(X).
Table 17. Sellmeier Fit for AMTIR at 230K
HAUELENGTH MEASURED CALCULATED DIFFERENCE (an) INDEX INDEX
7.9 2.49921 2.49^46 -0.00025 7.1 2.49902 2.49916 -0.00014 7.2 2.49860 2.49886 -0.00026 7.3 2.49840 2.49857 -0.00017 7.4 2.49836 2.49829 0.00007 7.5 2.49778 2.49801 -0.00023 7.6 2.49785 2.49773 0.00012 7.7 2.49765 2.49746 0.00019 7.8 2.49737 2.49719 0.00018 7.9 2.49715 2.49692 0.00023 8.0 2.49687 2.49666 0.00021 8.1 2.49664 2.49639 0.00025 8.2 2.49636 2.49613 0.00023 8.3 2.49606 2.49586 0.00020 8.4 2.49579 2.49560 0.00019 8.5 2.49553 2.49534 0.00019 8.6 2.49525 2.49507 0.00018 8.7 2.49496 2.49481 0.00015 8.8 2.49466 2.49454 0.00012 8.9 2.49442 2.49428 0.00014 9.0 2.49402 2.49401 0.00001 9.1 2.49381 2.49374 0.00007 9.2 2.49352 2.49347 0.00005 9.3 2.49321 2.49319 0.00002 9.4 2.49294 2.49291 0.00003 9.5 2.49259 2.49263 -0,00004 9.6 2.49231 2.49234 -0.00003 9.7 2.49202 2.49205 -0.00003 9.8 2.49171 2.49175 -0.00004 9.9 2.49134 2.49145 -0.00011 10.0 2.49097 2.49114 -0.00017 10.1 2.49067 2.49083 -0.00016 10.2 2.49037 2.49051 -0.00014 10.3 2.49002 2.49018 -0.00016 10.4 2.48972 2.48985 -0.00013 10.5 2.48937 2.48951 -0.00014 10.6 2.48906 2.48915 -0.00009 10.7 2.48866 2.48879 -0.00013 10.8 2.48825 2.48842 -0.00017 10.9 2.48784 2.48804 -0.00020 11.0 2.48746 2.48764 -0.00018 11.1 2.48720 2.48724 -0.00004 11.2 2.48682 2.48682 0.00000 11.3 2.48638 2.48638 0.00000 11.4 2.48596 2.48593 0.00003 11.5 2.48554 2.48547 0.00007 11.6 2.48529 2.48499 0.00030 11.7 2.48475 2.48448 0.00027 11.8 2.48441 2.48396 0.00045 11.9 2.48395 2.48342 0.00053 12.0 2.48343 2.48285 0.00058
ROOT HEAH SQUARE DIFFERENCE3 2.7 85060643E-5
COEFFICIENTS OF THE FORMULA
Al=5.22571012898 U1=0.330659293792
A2=0.0613323313319 U2=264.67358834
Table 18. Herzberger Fit for AMUR at 230K
UAUELENGTH MEASURED CALCULATED DIFFERENCE <un> INDEX INDEX
7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8 . 8 8.1 8.2 8.3 8.4 8.5 8 . 6 8.7 8 . 8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
10.0 10 .1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 1 1 . 1 1 1 . 2 11.3 11.4 11.5 1 1 . 6 11.7 1 1 . 8 11.9 1 2 . 0
2.49921 2.49902 2.49860 2.49840 2.49836 2.49778 2.49785 2.49765 2.49737 2.49715 2.49687 2.49664 2.49636 2.49606 2.49579 2.49553 2.49525 2.49496 2.49466 2.49442 2.49402 2.49381 2.49352 2.49321 2.49294 2.49259 2.49231 2.49202 2.49171 2.49134 2.49097 2.49067 2.49037 2.49002 2.48972 2.48937 2.48906 2.48866 2.48825 2.48784 2.48746 2.48720 2.48682 2.48638 2.48596 2.48554 2.48529 2.48475 2.48441 2.48395 2.48343
2.49913 2.49893 2.49872 2.49850 2.49828 2.49805 2.49782 2.49758 2.49734 2.49709 2.49684 2.49658 2.49632 2.49666 2.49579 2.49552 2.49524 2.49497 2.49468 2.49440 2.49411 2.49382 2.49352 2.49323 2.49292 2.49262 2.49231 2.49200 2.49168 2.49136 2.49103 2.49070 2.49037 2.49003 2.48969 2.48934 2.48899 2.48863 2.48827 2.48790 2.48753 2.48715 2.48677 2.48638 2.48599 2.48559 2.48519 2.48477 2.48436 2.48393 2.48350
0.00008 0.00009
-0.00012 -0.00010 0.00008 -0.00027 0.00003 0.00007 0.00003 0.00006 0.00003 0.00006 0.00004 0.00000 0.00000 0.00001 0.00001
-0.00001 -0.00002 0.00002
-0.00009 -0.00001
0.00000 -0.00002 0.00002 -0.00003
0.00000 0.00002 0.00003 -0.00002 -0.00006 -0.00003 0.00000 -0.00001 0.00003 0.00003 0.00007 0.00003 -0.00002 -0.00006 -0.00007 0.00005 0.00005 0.00000
-0.00003 -0.00005 0.00010 -0.00002 0.00005 0.00002 -0.00007
ROOT MEAN SQUARE DIFFERENCE3 8.781856593E-6
COEFFICIENTS OF THE FORMULA
A=2.49439374217 B=0.647967078296 C=-12.5684088088
D=-4.768709281E-5 E=-3.817914938E-7
48
Table 19. Sellmeier Fit for AMUR at 300K
WAUELENGTH MEASURED CALCULATED DIFFERENCE <un> INDEX INDEX
7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 9.9 8 . 1 8.2 8.3 8.4 8.5 8 . 6 8.7 8 . 8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 19.0 19.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 1 1 . 1 1 1 . 2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 1 2 . 0
2.50506 2.50470 2.50429 2.50412 2.50405 2.50351 2.50357 2.50334 2.50302 2.50277 2.50259 2.50232 2.50201 2.50176 2.50149 2.50118 2.50090 2.50061 2.50029 2.50006 2.49976 2.49949 2.49915 2.49877 2.49848 2.49819 2.49784 2.49754 2.49724 2.49690 2.49666 2.49621 2.49587 2.49555 2.49518 2.49490 2.49453 2.49412 2.49376 2.49338 2.49303 2.49262 2.49226 2.49184 2.49148 2.49104 2.49068 2.49026 2.48979 2.48941 2.48907
2.50551 2.50512 2.50475 2.50440 2.50405 2.50371 2.50339 2.50307 2.59277 2.50247 2.50218 2.50189 2.50161 2.50134 2.501O7 2.50081 2.50055 2.50029 2.50004 2.49979 2.49954 2.49929 2.49905 2.49880 2.49855 2.49831 2.49305 2.49780 2.49754 2.49728 2.49702 2.49674 2.49646 2.49617 2.49586 2.49555 2.49521 2.49486 2.49449 2.49410 2.49367 2.49321 2.49270 2.49215 2.49153 2.49084 2.49006 2.48916 2.48812 2.48687 2.48537
-0.00045 -0.00042 -0.00046 -0.00028
0.00000 -0.00020
0.00018 0.00027 0.00025 0.00030 0.00041 0.00043 0.00040 0.00042 0.00042 0.00037 0.00035 0.00032 0.00025 0.00927 0.00022 0.00020 0.00010 -0.00003 -0.00007 -0.00012 -0.00021 -0.00026 -0.00030 -0.00038 -0.00036 -0.00053 -0.00059 -0.00062 -0.00068 -0.00065 -0.00068 -0.00074 -0.00073 -0.00072 -0.00064 -0.00059 -0.00044 -0.00031 -0.00005
0.00020 0.00062 0.00110 0.00167 0.00254 0.0037O
ROOT MEAN SQUARE DIFFERENCE3 1.117799703E-4
COEFFICIEHTS OF THE FORMULA
Al=5.21978984951 U1=0.574085835591
A2=0.00983376568743 U2=166.25768891
Table 20. Herzberger Fit for AMTIR at 300K
MAUELENGTH MEASURED CALCULATED DIFFERENCE <un) INDEX INDEX
7.0 2.50506 2.50490 0.00016 7.1 2.50470 2.50468 0.00002 7.2 2.50429 2.50445 -0.00016 7.3 2.50412 2.50422 -0.00010 7.4 2.50405 2.50399 0.00006 7.5 2.50351 2.50375 -0.00024 7.6 2.50357 2.50351 0.00006 7.7 2.50334 2.50326 0.00008 7.8 2.50302 2.50302 0.00000 7.9 2.50277 2.50277 0.00000 8.0 2.50259 2.50251 0.00008 8.1 2.50232 2.50225 0.00007 8.2 2.50201 2.50199 0.00002 8.3 2.50176 2.50172 0.00004 8.4 2.50149 2.50145 0.00004 8.5 2.50118 2.50118 0.00000 8.6 2.50090 2.50090 0.00000 8.7 2.50061 2.50062 -0.00001 8.8 2.50029 2.50033 -0.00004 8.9 2.50006 2.50004 0.00002 9.0 2.49976 2.49975 0.00001 9.1 2.49949 2.49945 0.00004 9.2 2.49915 2.49915 0.00000 9.3 2.49877 2.49884 -0.00007 9.4 2.49848 2.49853 -0.00005 9.5 2.49819 2.49822 -0.00803 9.6 2.49784 2.49790 -0.00006 9.7 2.49754 2.49757 -0.00003 9.8 2.49724 2.49725 -0.00001 9.9 2.49690 2.49692 -0.00002
18.0 2.49666 2.49658 0.00008 18.1 2.49621 2.49624 -0.00003 10.2 2.49587 2.49590 -0.00003 10.3 2.49555 2.49555 0.00000 10.4 2.49518 2.49520 -0.00002 10.5 2.49490 2.49484 0.00006 10.6 2.49453 2.49448 0.00005 10.7 2.49412 2.49412 0.00000 10.8 2.49376 2.49375 0.00001 10.9 2.49338 2.49338 0.00000 11.0 2.49303 2.49300 0.00003 11.1 2.49262 2.49262 0.00000 11.2 2.49226 2.49224 0.00002 11.3 2.49184 2.49185 -0.00001 11.4 2.49148 2.49146 0.00002 11.5 2.49104 2.49106 -0.00002" 11.6 2.49068 2.49066 0.00002 11.7 2.49026 2.49026 0.00000 11.8 2.48979 2.48985 -0.00006 11.9 2.48941 2.48944 -0.00003 12.0 2.48907 2.48903 0.00004
ROOT MEAN SQUARE DIFFERENCE3 8.437407477E-6
COEFFICIENTS OF THE FORMULA
A=2.51671869904 D=-l.84483594E-4 B=-0.218392136146 E=9.514793892E-9 C=3.98731608444
Table 21. Sellmeier Fit for AMTIR at 328K
MAUELENGTH MEASURED CALCULATED DIFFERENCE <un>
7.9 7.1 7 .2 7 .3 7 .4 7 .5 7 .6 7 .7 7.8 7.9 8.9 8.1 8 . 2 8.3 8.4 8.5 8 .6 8.7 8 . 8 8.9 9.8 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 19.9 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 11.9 1 1 . 1 11.2 11.3 11.4 11.5 11.6 11.7 1 1 . 8 11.9 12.0
ROOT MEAN SQUARE DIFFERENCE3 2.197092122E-4
COEFFICIENTS OF THE FORMULA
INDEX INDEX
2. 59753 2. 51093 -0. 00250 2. 59739 2. 59936 -0. 00197 2. 59689 2. 50872 -0. 00192 2. 50649 2. 50810 -0. 00161 2. 59644 2. 50751 -0. 00107 2. 59598 2. 50695 -0. 00097 2. 59598 2. 50641 -0. 00043 2. 59578 50589 -0. 00011 2. 50546 2! 50539 0. 00007 2. 50516 2. 50496 0. 00026 2. 50501 2. 50444 0. 00057 2. 50468 2. 50400 0. 00068 2. 50438 2. 50357 8. 00081 2. 50405 2. 50315 0. 00090 2. 50382 2. 50276 0. 00106 2. 50354 2. 50237 0. 00117 2. 58326 2. 50200 0. 00126 2. 50295 2. 50165 0. 00130 2. 50261 2. 50130 0. 00131 2. 50243 2. 50097 0. 00146 2. 50203 2. 50065 0. 00138 2. 50183 2. 50034 0. 00149 2. 50151 2. 50005 0. 00146 2. 50116 2. 49976 0. 00140 2. 50087 2. 49^48 0. 00139 2. 50058 2. 49921 0. 00137 2. 50025 2. 49895 0. 00130 2. 49991 2. 49870 0. 00121 2. 49964 2. 49946 9. 00118 2. 49926 2. 49823 9. 00103 2. 49888 2. 49801 9. 00087 2. 49859 2. 49780 0. 00079 2. 49827 2. 49760 0. 00067 2. 49795 2. 49741 0. 00054 2. 49756 2. 49724 0. 00032 2. 49726 2. 49709 0. 00017 2. 49686 2. 49697 -0. 00011 2. 49646 2. 49691 -0. 00045 2. 49612 2. 49700 -0. 00088 2. 49574 2. 49778 -0. 00204 2. 49541 2. 48883 0. 00658 2. 49499 2. 49453 0. 00046 2. 49466 2. 49490 -0. 00024 2. 49426 2. 49493 -0. 00067 2. 49387 2. 49488 -0. 00101 2. 49346 2. 49479 -9. 00133 2. 49306 0 . 49468 -0. 00162 2. 49271 2. 49457 -0.00186 2. 49236 2. 49445 -0. 00209 2. 49200 2. 49433 -0. 00233 2. 49154 2. 49421 -0. 00267
A1=5.18249059371 Ul=l.088694994
A2=-I.268093627E-4 U2al20.564074729
Table 22. Herzberger Fit for AMTIR at 328K
HAUELENGTH MEASURED CALCULATED DIFFERENCE (un) INDEX INDEX
7.0 2.50753 2.50748 0.00005 7.1 2.50739 2.50721 0.00018 7.2 2.50680 2.50695 -0.00015 7.3 2.50649 2.50669 -0.00020 7.4 2.50644 2.50643 0.00001 7.5 2.50598 2.50617 -0.00019 7.6 2.50598 2.50592 0.00006 7.7 2.50578 2.50566 0.00012 7.8 2.50546 2.50540 0.00006 7.9 2.50516 2.50515 0.00001 8.0 2.50501 2.50489 0.00012 8.1 2.50468 2.50462 0.00006 8.2 2.50438 2.50436 0.00002 8.3 2.50405 2.50409 -0.00004 8.4 2.50382 2.50382 0.00000 8.5 2.50354 2.50354 0.00000 8.6 2.50326 2.50326 0.00000 8.7 2.50295 . 2.50298 -0.00003 8.8 2.50261 2.50269 -0.00008 8.9 2.50243 2.50240 0.00003 9.0 2.50203 2.50211 -0.00008 9.1 2.50183 2.50181 0.00002 9.2 2.50151 2.50150 0.00001 9.3 2.50116 2.50120 -0.00004 9.4 2.50087 2.50089 -0.00002 9.5 2.50058 2.50057 0.00001 9.6 2.50025 2.50025 0.00000 9.7 2.49991 2.49993 -0.00002 9.8 2.49964 2.49960 0.00004 9.9 2.49926 2.49927 -0.00001 10.0 2.49888 2.49893 -0.00005 10.1 2.49859 2.49859 0.00000 10.2 2.49827 2.49825 0.00002 10.3 2.49795 2.49790 0.00005 10.4 2.49756 2.49755 0.00001 10.5 2.49726 2.49720 0.00006 10.6 2.49686 2.49684 0.00002 10.7 2.49646 2.49648 -0.00002 10.8 2.49612 2.49612 0.00000 10.9 2.49574 2.49575 -0.00001 11.0 2.49541 2.49538 0.00003 11.1 2.49499 2.49501 -0.O0002 11.2 2.49466 2.49464 0.00002 11.3 2.49426 2.49426 0.00000 11.4 2.49387 2.49388 -0.00001 11.5 2.49346 2.49350 -0.00004 11.6 2.49306 2.49311 -0.00005 11.7 2.49271 2.49273 -0.00002 11.8 2.49236 2.49234 0.00002 11.9 2.49200 2.49195 0.00005 12.0 2.49154 2.49155 -0.00081
ROOT MEAN SQUARE DIFFERENCE3 9.059281275E-6
COEFFICIENTS OF THE FORMULA
A=2.53322308243 D=-2.963718118E-4 B=-l.01062327641 E=3.392131754E-7 C=20.621222699
52
<n<T)-n<179))*1E4
148
129
199
99
69
49
29
0 i .i
169 139 299 228 249 269 299 399 329 349 369 TEMPERATURE <K>
Fig. 15. AMTIR-l n (T) at 8 ym.
53
I H D E X
0
F
R E F R A
C T I 0 H
<n<T)-n<179)>S1E4
140 r
129
190
39
£0
40
29
9 '
160 139 299 229 240 2S9 289 399 329 349 369 TEMPERATURE <K>
Fig. 16. AMTIR-1 n(T) at 10 ym.
Table 23. AMTIR-1 : n(T) at 8 um
Temperature (K)
Measured Index
169.5 2.50104
179.5 2.50184
189.5 2.50228
199.5 2.50292
209.0 2.50391
219.5 2.50444
239.5 2.50611
259.0 2.50766
278.5 2.50925
299.0 2.51092
310.0 2.51160
323.5 2.51256
330.0 2.51317
Least-squares linear data slope = dn/dT - 76.4x10 °C' correlation coefficient = (
Table 24. AMTIR-1 : n(T) at 10 vm
Temperature Measured (K) Index
169.5 2.49532
178.5 2.49603
188.5 2.49653
198.5 2.49713
209.0 2.49816
219.5 2.49892
239.5 2.50018
259.0 2.50189
278.5 2.50350
299.0 2.50518
312.0 2.50583
324.0 2.50694
350.0 2.50902
Least squares linear data fit: slope = dn/dT = 76.3x10 °C-1
correlation coefficient = 0.9994
56
2.52 r
I H D E X
0
F
R E F R A C T I 0
N
2.50
2.48
* + *
$
+ *
* t
* + *
2.46 i i i i »
8 10
i i » 11 12
WAUELENGTH <ni c rone tars >
Fig. 17. AMTIR vs UA AMTIR-1 (300K) n(A). * = UA data, + = AMTIR data.
A graphical comparison of their data with the data presented in
this thesis is given in Figure 17. There is an apparent offset between
the two sets of data. Table 25 lists the index values and these
differences for six wavelengths. Our data are found to be lower than
the published data by an average value of -0.00084. While this value is
substantially higher than the calculated systematic error of 0.00050 for
these measurements (see Error Analysis chapter), it is well within the
batch-to-batch variation in index of 0.0030 reported by Amorphous
Materials, Inc. A material analysis was not available for the specific
sample measured. However, the offset in the compared data would seem
to indicate a slight variation in glass composition or structure as
compared to the sample measured by the manufacturer.
A comparison of the dn/dT data is given in Table 26. Good agreement
is observed, even though the data from the manufacturer covers the
temperature range of 25°C to 65°C only.
Table 25. AM vs UA Data : AMTIR-1 at 300K
(um) n(UA) n(AM) n(UA-AM)
7.0 2.50506 2.5057 -0.00064
8.0 2.50259 2.5034 -0.00081
9.0 2.49976 2.5005 -0.00074
10.0 2.49666 2.4976 -0.00094
11.0 2.49303 2.4936 -0.00057
12.0 2.48907 2.4904 -0.00133
Average difference = -0.00080 Standard deviation =» 0.00027
NOTE: AM stands for Amorphous Materials, Inc.
Table 26. AM vs UA Data : AMTIR-1 dn/dT values
Source Wavelength (jim) dn/dT x 10S/°C
AM 1.15 101.0
AM 3.39 77.0
UA 8.0 76.4
UA 10.0 76.3
AM 10.6 72.0
Note: AM stands for Amorphous Materials, Inc. The data reported by Amorphous Materials cover the temperature range from 25 to 65°C only.
CHAPTER 4
ERROR ANALYSIS
Any claim to having measured absolute values of refractive index
requires a thorough knowledge of all the experimental errors involved.
In fact, the ability to precisely understand and calculate the errors is
just as important as the experimental skills needed to make the
measurements. Swedberg (1979) has described in great detail the errors
associated with the method of perpendicular incidence. The equations he
derived will be used in this chapter to calculate the overall errors
associated with these measurements.
Various types of errors inherently limit the absolute accuracy
of any physical measurement. Included in this analysis are errors of
both a random and systematic nature. In addition, overall system
repeatability and errors due to prism misalignment and beam asymmetries
are discussed in some detail.
Random Errors
The random errors associated with this measurement process are
those which lead to a random deviation of the index from some true,
absolute value. If large enough, these errors can mask the true
functional relationship between index and wavelength or temperature.
This is most critical in determining whether or not dn/dT (the slope of
the n(T) curve) is linear or nonlinear with respect to temperature.
60
For these measurements, three random errors have been
identified. They arise from inaccuracies in: (1) measuring the deviation
angle of the prism, (2) setting the monochromator to an exact
wavelength, and (3) maintaining a constant prism temperature. The
equations describing these random errors and an explanation of the
symbols used are given in Tables 27 and 28, respectively.
Systematic Errors
The systematic errors associated with this measurement process
are those which lead to a constant deviation, or offset, of the index
from the absolute value. Although these errors leave the shape of the
n(x) and n(T) curves intact (adding only a constant term to the curves)
they are greater in magnitude and harder to minimize than the random
errors. Most of the experimental effort involved with calibration and
system alignment is aimed at reducing the systematic errors.
For these measurements, six systematic errors have been
identified. They result from inaccuracies in: (1) aligning the prism, (2)
measuring the prism apex angle, (3) refraction due to window wedge and a
vacuum-air interface at the dewar windows, (4) the finite spectral
bandwidth of the monochromator, (5) the calibration of the monochromator
dial vs wavelength, and (6) the measurement of the prism temperature.
The equations describing these systematic errors and an explanation of
the symbols used are given in Tables 27 and 28, respectively.
One additional systematic error could involve the rotary table
used to rotate the detector. If this rotation included a systematic
error in the drive mechanism, an error in measuring the prism deviation
Table 27. Error Analysis.
62
Source Equation
Deviation Angle
Wavelength
Temperature
Prism Alignment
Prism Apex Angle
Slit Width
Wavelength
Temperature
Random
An(dfc) fcdb - -f< Ud)b 3n
An(X^) = gj UX)d
An(Tp) = (AT)p
Systematic
4n(V " [iisW "n'],/! <4d)
[afcr -f - • An(a) cos(a)
sin(a) Aa
3n An(Xb) =» — (AX)b
An(Xc) =• — (AX)C
An(Tc) = § (AT)c
63
Table 28. Description of Symbols
Symbol Description
a Prism apex angle
Aa Error in measuring the prism apex angle
n Index of refraction (min. and max.)
3n/3T Slope of the n(T) curve (min. and max.)
3n/3X Slope of the n(X) curve (min. and max.)
(Ad)^ Total combined error in measuring the reference and deviated beam centers
(Ad)p Error in measuring the prism deviation angle due to prism alignment errors
( AT)C Calibration error of the temperature sensor
(AT)p Average fluctuation of the prism temperature
(AX)|j Monochromator bandwidth
(AX)C Monochromator calibration error
(AX)j Wavelength uncertainty due to the resolution
of the monochromator dial
64
angle and therefore an error in index would result. It is believed that
this error source is negligible, and the accuracy of rotation was not
verified experimentally. However, it is a possible source of error that
should be measured in future work.
Calculation of (Ad)p
The effect of prism misalignment about the alpha, beta, or gamma
axes (Figure 3) is to introduce a change, (Ad)p, in the deviation angle
measured between the reference and deviated beams. This in turn leads
to an error, An(dp), in the index. Proper calculation of (Ad)p requires
that rays be traced through the prism (in a three-dimensional sense) for
arbitrary rotations of the prism about its three axes. This was done
using the ACCOS ray-trace program.
The actual calculation of (Ad)p was carried out as follows. A
reference deviation angle was first calculated assuming the prism had no
tilt errors. Under this ideal condition the deviation angle was found by
applying Snell's Law at the second surface of the prism.
A real ray was then traced through the prism after the
appropriate amounts of rotation were introduced. The rotation, or tilt,
errors about each of the three prism axes were estimated experimentally
and are listed in Table 29. These errors were introduced by tilting the
prism using the TILT command in ACCOS. The combined effect of this
three-dimensional rotation is to change the prism deviation angle (i.e.,
the projection of the real ray on the horizontal detector plane of
rotation). These deviation angles appear in the ACCOS output as shown in
Appendix E.
Table 29. Variables for Error Analysis
Variable MgO Sapphire AMTIR-1
a 31.00378 31.00364 15.90526 deg
Aa 0.0042 0.0068 0.0012 mr
d (X min) 29.10806 29.79194 28.56333 deg
d (X max) 26.45681 25.73958 28.22000 deg
Ad (ref) 5.9 10.8 2.9 sec
Ad (dev) 5.9 10.8 2.9 sec
Aa 0.075 0.075 0.075 mr
A3 0.075 0.075 0.075 mr
AY 2.0 2.0 2.0 deg
n (min) 1.63663 1.61793 2.48907
n (max) 1.70880 1.72954 2.50506
(Ad)b 0.0572 0.1047 0.0291 mr
|(Ad)p| (X min) 0.149 0.149 0.157 mr
|(Ad)p| (X max) 0.147 0.146 0.157 mr
(AX)jj (bandwidth) 0.0261 0.0261 0.066 um
(AX)C (calibration error)
0.0001 0.0001 0.0006 Vim
(AX)j (min. scale division)
0.0008 0.0008 0.0023 ym
(AT)(j (sensor cal.) 2 2 2 K
(AT)p (stability) 2 2 2 K
3n/3T NA NA 0.000076 K-1
3n/3X (X min) -0.0146 -0.0202 -0.0036 um-1
3n/3X (X max) -0.0349 -0.0567 -0.0034 urn-1
66
Finally, (Ad)p was found by taking the difference in deviation
angle between the reference ray and the real ACCOS ray. These values
are listed in Table 29 for each prism.
Combined Errors
Using the assumption of Icenogle (1975) and Swedberg (1979), the
experimental errors are considered to be statistically independent. This
allows the total random and systematic errors to be calculated as the
root-sum-square (RSS) of the individual errors. The variables and
calculated errors for each of the prisms are shown in Tables 29-32.
Note that the systematic error due to the window wedge and vacuum-air
interface has not been calculated. This error was accounted for in the
data by the window correction procedure described in Chapter 2.
All of the total errors except two fall close to the desired
accuracy of SxlO-1'. These two are maximum systematic errors for
magnesium oxide and sapphire, which are both large due to the spectral
bandwidth error. This error itself is high due to a large slope dn/dx
for each material.
Temperature Coefficient
The procedure for making a temperature run was discussed in
Chapter 3. An assumption was made, based on information available, that
the changes in temperature did not cause mechanical movement inside the
dewar. This in turn could lead to changes in prism alignment.
Unfortunately, it wasn't until after the first two prisms were measured
that this assumption was discovered to be wrong. In fact, measurements
67
Table 30. Combined Errors: MgO at 300K
Random Errors:
Max. Min.
Deviation angle An(db) 0.000060 0.000053
Wavelength An(xd) 0.000028 0.000012
Temperature An(Tp) 0.000018 0.000018
RSS Total 0.000063 0.000057
Systematic Errors:
Prism alignment An(d p) 0.000192 0.000179
Prism apex angle An(a) -0.000008 -0.000007
Spectral bandwidth An( Xfc) -0.000911 -0.000381
Monochromator
calibration An(Xc) -0.000002 -0.000001
Temperature An(Tc) 0.000018 0.000018
RSS Total 0.000931 0.000421
Table 31. Combined Errors: Sapphire n0 at 300K
Random Errors:
Deviation angle
Wavelength
Temperature
Systematic Errors:
Prism alignment
Prism apex angle
Spectral bandwidth
Monochromator calibration
Temperature
Max. Min.
An(db) 0.000112 0.000092
An(Ad) 0.000045 0.000016
An(Tp) 0.000016 0.000016
RSS Total 0.000122 0.000095
An(d p) 0.000256 0.000121
• An(a) -0.001480 -0.000011
An(Ab) -0.001479 -0.000527
An(Xb) -0.000002 -0.000001
An(Tc) 0.000016 0.000016
RSS Total 0.001502 0.000541
Table 32. Combined Errors : AMTIR-1 at 300K
69
Random Errors:
Max. Min.
Deviation angle An(db) 0.000078 0.000077
Wavelength An(\d) 0.000008 0.000008
Temperature An(Tp) 0.000152 0.000152
RSS Total 0.000171 0.000171
Systematic Errors:
Prism alignment An(d p) 0.000558 0.000439
Prism apex angle An(a) 0.000008 0.000007
Spectral bandwidth An(xc) 0.000238 0.000224
Monochromator
calibration An(Xc) -0.000001 -0.000001
Temperature An(Tc) 0.000152 0.000152
RSS Total 0.000625 0.000516
dn/dT Error:
Random errors plus prism alignment An
RSS Total 0.000584 0.000471
showed considerable prism rotation, mainly around the alpha axis. This
appears to explain the deviations of n(T) from a straight line for
magnesium oxide and sapphire. The n(T) curve for AMTIR-1 was taken by
realigning the prism after each change in temperature. The higher
degree of linearity in the data points appears to reflect this
correction.
The error analysis for n(T) data follows that given by Swedberg.
The maximum RSS random error An is considered as an error bar at the
end points of the n(T) curve. This sets a limit (error) to the minimum
and maximum values of the slope, given by:
(dn/dT)max - dn/dT + T*i\
(dn/dT)min = dn/dT - (2)
The actual error in slope is therefore . A graphical explanation 2 ~
of this equation is shown in Figure 18.
One important change from Swedberg's definition of the RSS
random error An is to be noted. By having to realign the prism after
every temperature change, the error due to prism alignment now must be
considered as a random error. The expression for An becomes:
(An)2 = [An(db) ]2 + [An(Xd)]2 + [An(Tp)]2 + [An(dp)]2 (3)
For the AMTIR-1 prism, An(max.) = 0.000584. The error in dn/dT is
calculated to be =, +7.3 x K-1. 2 ~ M
71
I t l D E X
n2
n l
j j r ( m a x . )
5T (min->
1 i n e a r f i t t o t h e d a t a
T 1
H e i g h t o f e r r o r b a r s = 2 A n
T 2
T E M P E R A T U R E
Fig. 18. dn/dT Error Analysis.
72
System Repeatability
The combined effects of the random errors act to limit the
overall precision of the measurements. For this discussion, precision
may be thought of as the degree to which the same index value is
determined in a set of repeated measurements and conditions. The words
system repeatability and precision will therefore be used to mean the
same thing. Both a short-term and long-term precision have been
measured.
Short-term Precision
The short-term precision is that level of system repeatability
attainable during the time required for one wavelength run. This
precision was determined by first aligning the system and then making
repeated index measurements, alternating between two different
wavelengths. In this manner, the only uncertainties introduced by the
measurement process were those previously identified as the random
errors.
Five measurements at both 7 and 11 ym were made using the
AMTIR-1 prism. The standard deviation was calculated for each set of
index values and used as a measure of the precision. Results are shown
in Table 33. For each wavelength, the standard deviation is substantially
less than the RSS random error of 0.00017 calculated for AMTIR-1.
However, both standard deviations agree well with the deviation-angle
random error of 0.00007 calculated for AMTIR-1. The predicted
temperature random error of 0.00015 was not a factor in the measured
precision, indicating good temperature stability during the data-taking
Table 33. System Repeatability
Time of Measurement n(7 iim) n(12 um)
2:30 PM 2.50503
2:40 PM — 2.48902
2:45 PM 2.50506 —
2:49 PM — 2.48901
2:52 PM 2.50491 —
2:56 PM — 2.48894
3:00 PM 2.50507 —
3:04 PM — 2.48905
3:07 PM 2.50511 —
3:11 PM 2.48902
(7 um) = 0.00007
(12 um) - 0.00003
process. The short-term precision is therefore limited mainly by the
accuracy in measuring the deviation angle.
Long-term Precision
The long-term precision is that level of system repeatability
attainable from one day to the next. This precision was observed by
comparing index values taken on successive days, using the AMTIR-1 prism.
For the measurements made, the index at 11 and 12 pm changed by +0.00016
and +0.00022, respectively. Given that all factors affecting the
systematic errors remained constant, these changes should be accountable
by one of the random errors. In fact, both observed differences agree
well with the temperature random error of 0.00015 for AMTIR-1. This
temperature error was calculated assuming a change in prism temperature
of 2°C, a value well within the average daily temperature fluctuations
in the laboratory. The long-term precision is therefore limited by room
temperature stability, at least for prism materials where the
temperature error dominates the random errors. By measuring the
temperature fluctuation in a repeatable manner and knowing dn/dT, this
error may be corrected.
APPENDIX A
PRISM SPECIFICATIONS
75
.125 *.005
SURFACE I 5URFACE2
^0° ±20 ARC 5EC
see NOTE(2)
•SURFACE 3
.500 ±.005
0̂°
ARC SEC
UNIVERSITY OF ARIZONA OPTICAL SCIENCES CENTER
INFRARED LABORATORY
.000 ±.00E>
NOTES:
(1) Dimensions in inches (2) Apex angles ( in degrees)
MgO Sapphi re AMTIR-I
31.00 31 .00 15-90
MATERIAL: Ms°i Sapphire, AMTIR-1
SURFACE FLATNESS 1.2: ^ALLOWABLE CURVATURE
JjAFIGURE TOL. 3 -
(X=632.8 nmj
SURFACES MARKED "P" POLISHED ALL OTHERS f.ROUND
SURFACE QUALITY 1,2; 80/S0
3: 120/70 CHAMFER .020x45°, ALL EDGES
Fig. 19. Prism Specifications. c*
APPENDIX B
SYSTEM ALIGNMENT
Described in this appendix is a step-by-step procedure used to
align the optical system. Refer to Figure 2 in Chapter 2 to identify the
system components.
(1) Measure the height from the top of the table to the center
of the entrance slit.
(2) Adjust the center of the thermal source to this height.
(3) Install a pair of slits in the monochromator, each one
masked down to a 50 ym x 50 um hole centered in each of the slits.
(4) Adjust mirror M2 to focus the source onto the center of the
entrance slit.
(5) Rotate the grating to zero-order, and if necessary, align the
grating so the beam exits the monochromator at the center of the exit
slit.
(6) Adjust mirrors M4 and M5 to position the beam at the center
of the dewar entrance window.
(7) Install mirror M6 in the beam and adjust it until the beam
falls right beside the hole in the exit slit.
(8) Set up the 20x microscope to view the image. Translate
mirror M5 until best focus is achieved. If excessive astigmatism is
present, mirror M5 may not be in a proper off-axis position. When best
focus is obtained, the beam entering the dewar is collimated.
77
Refer to Figure 20 for the next section of prism alignment.
(9) Install the prism in the dewar, position the dewar on the
rotary table, and remove the three windows. Remove mirror M6.
(10) Use the adjusting screw (1) and the rotary table to position
the prism so the reflected beam falls back directly onto the hole in the
exit slit. If the reflected beam is too weak to see visually, use the
He-Ne beam as described in Appendix C. At this point the beam has been
autocollimated off the front surface of the prism.
(11) Rotate the grating to an infrared wavelength appropriate
for the prism being measured. Position the rotary table so the
reference beam falls on the detector.
(12) Adjust mirrors M7 and M8 to maximize the signal of the
reference beam.
(13) Position the rotary table so the deviated beam falls on the
detector. Use the adjusting screw (2) to maximize the signal of the
deviated beam. This step positions the prism dihedral edge vertical to
the plane of detector rotation, eliminating any effects due to prism
pyramid.
(14) Recheck step 10 to insure that the beam reflected off the
prism is still directly centered in the exit slit.
79
ADJUSTING SCREW (2)
DEV. BEAM
FIXED BALL PIVOT ADJUST I MR SCREW (1)
TOP VIEW
Fig. 20. Prism/Dewar Alignment.
APPENDIX C
He-Ne COALIGNMENT
This appendix describes a procedure used to coalign a He-Ne
laser beam to the thermal beam used for measurements. For prisms
having a low index of refraction over the visible spectrum, the beam
reflected off the first surface may be very difficult to see. The use
of a laser beam can make the autocollimatlon procedure much easier.
(1) Complete steps 1-8 in Appendix B.
(2) Remove the thermal source.
(3) Set up the laser, beam expander, and lens LI as shown in
Figure 2.
(4) Position lens LI so the focused laser spot falls in the
physical location normally occupied by the thermal source.
(5) Adjust mirror Ml so that mirror M2 focuses the laser beam at
the center of the entrance slit. Do not adjust mirror M2 in the
process. Lens LI should be chosen so the focused laser spot overfills
the opening in the entrance slit.
(6) Through an iterative process of tipping, tilting, and
translating mirror Ml, adjust the laser beam so it exits the
monochromator. Again, the opening in the exit slit should be overfilled.
(7) The prism alignment may now be carried out according to
steps 9 and 10 in Appendix R.
80
81
(8) Replace the thermal source and complete steps 11-13 in
Appendix B.
APPENDIX D
SYSTEM SNR
An expression for the overall signal-to-noise ratio (SNR) of the
system is derived in this section. By actually measuring the SNR and
comparing it to this calculated value, an indication of system
performance is obtained.
Certain assumptions are made for the calculation:
(1) The source is a blackbody source.
(2) The FHi of the beam input to the monochromator matches the
F/// of the monochromator.
(3) The spectral bandwidth of the monochromator is very small.
(4) The quantum efficiency of the detector is independent of
wavelength.
(5) The focused beam just fills the detector area. A description
of the variables used below may be found in Table D-l. The SNR at the
detector may be written as:
* 4»d • Dj
SNR - V (D-l) (Ad • B)*/*
The power on the detector, is calculated using the radiative transfer
equation:
T(system) • L,| ^ • A<j • A0 • cosQ^ • cosej
4>d - 1 s-i (D-2) •t n • 0
Assume that 0^ = e0 = 0 :
82
<|>d = |r(system) • l|>x • Aj • A0J/f02
83
(D-3)
The radiance of the source is calculated using the Planck equation:
s ® Le \ • A X = —75 e»A TT Xs
exp M-- r - l
A X (D-4)
where
2dW AX = —— • cos6
rm (D-5)
and
9 = arc sin [t] (D-6)
The overall system transmission may be written as:
t(system) =» r(filter) • t(chopper) • p(M2) • p(M3) • x(monochromator)
• p(M4) • p(M5) • t(window) • t(window) • p(M7)
• p(M8) • t(window) (D-7)
Combining (D-l), (D-3), and (D-4) gives:
[D* • x(system) • C, • A0 • (A^)1/2 • e • AX] SNR =
exp XT
- 1
(D-8)
f„2 ' (B)l/2 • Xs
As an example, the expected SNR for the HgCdTe detector system
may be calculated using equation (D-8). Consider a wavelength of 7.0 ym
and the following values:
T(system) = (0.9)(0.45)(0.9)(0.9)(0.5)(0.9)(0.9)(0.8)(0.8)(0.9)(0.9)(0.8)
= 0.06
C, = 3.74 x 10"12 W cm1
A# = 5.1 cm2
= 0.002 cm2
X = 7.0 ym
AX = 0.078 um
C2 = 14387.86 \im K
T - 1500 K
e = 0.8
F0 = 40.64 cm
B = 1 Hz
The calculated SNR is 6000. No experimental values were measured.
85
Table 34. Variables Used in SNR Calculation
Variable Description
active area of the detector
Aq area of the beam at mirror M7
B bandwidth of the electronics
Cj first radiation constant = 2irhc2
C2 second radiation constant = hc/k
d grating spacing between lines
D*^ figure-of-merit for particular detector in use
fm focal length of the monochromator
fQ focal length of mirror M7
Lse>x spectral radiance of the source
m order number of the grating
e emissivity of the source
w monochromator slit width
p reflection coefficient
t transmission coefficient
X wavelength selected by the monochromator
AX spectral bandwidth of the monochromator
(frj power (in watts) on the detector
Qd angle of the detector surface normal relative to the optic axis
90 angle of the mirror M7 surface normal relative to the optic axis
APPENDIX E
PRISM ALIGNMENT ERROR ANALYSIS
86
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87
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REFERENCES
Amorphous Materials, Inc., 1983, AMTIR-1 Product Information, Garland,
Texas.
Icenogle, Harmon Weston, 1975, "The Refractive Index of Silicon, Germanium, and Lithium lodate as a Function of Temperature and Wavelength," MS Thesis, University of Arizona.
Jeppesen, Myron A., 1958, "Some Optical, Thermo-Optical, and Piezo-Optical Properties of Synthetic Sapphire," J. Opt. Soc. Am. 48,
629.
Malitson, Irving M., 1962, "Refraction and Dispersion of Synthetic
Sapphire," J. Opt. Soc. Am. 52, 1377.
Malitson, Irving M., Frederick V. Murphy, Jr., and William S. Rodney, 1958, "Refractive Index of Synthetic Sapphire," J. Opt. Soc. Am. 48, 72..
Piatt, Ben C., H. W. Icenogle, J. E. Harvey, R. Korniski, and William L. Wolfe, 1975, "Technique for Measuring the Refractive Index and its Change with Temperature in the Infrared," J. Opt. Soc. Am.
65, 1264.
Piatt, Benjamin Curtis, 1976, "Instruments for Measuring Properties of
Infrared Transmitting Optical Materials," Ph.D. Dissertation,
University of Arizona.
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