Relationship of Internal Reflection Measurements to Bulk Optical Properties with a Transmission Comparison J. A. Horton and W . N . Hansen North American Aviation Science Center, Thousand Oaks, CaIV. 91360 Experimental data are presented to prove that reflection data taken using standard spectrophotometric procedures describe the bulk optical properties of homogeneous materials, and that reflection data obey Fresnel formulas even though coherent infinite plane waves assumed in them are not used. The procedures used included single and multiple internal reflection from various dye solutions as a function of wavelength, angle of incidence, polarization, and prism material. Transmission data for comparison were determined by using a variable path length cell to measure the absorption coefficient of the same dye solution used in the multiple reflection measurements. Error analyses are also given.
INTERNAL REFLECTION SPECTROSCOPY has developed since publications by Fahrenfort ( I ) and Harrick ( 2 ) appeared on the subject a few years ago, until it is now widely used in qualitative and quantitative investigations (3-7). If accurate spectral data are desired, reflectance must be measured accurately and related to the spectra of the sample. The reflection equations used (cf. Equations 1 and 2) assume an infinite plane wave (coherent) of known polarization, and two semi-infinite, perfectly homogeneous and isotropic phases with perfectly sharp plane boundary. In practice, the beam of the spectrometer used is rather narrow and has an angular and spectral spread. Samples are never perfectly homogeneous and boundaries always have roughness and exhibit continuous rather than discontinuous changes in properties. Thin films are also characteristic of the interface. In view of these deviations from ideality, with Equations 1 and 2 as the basis, it is important to know the accuracy we can expect of the reflection equations. Transmission spectra carefully taken by proved methods give true bulk properties of the sample. On the other hand, reflection spectra are determined by thin surface regions. In this research we have taken reflection and transmission spectra of the same sample for comparison. Since perturbation due to surface films, etc., has entirely different effects in the two cases, the degree of agreement indicates the degree of accuracy under the experimental conditions used. In a second test of reflection methods on a sample too opaque to be investigated by transmission, accuracy was checked by changing the angle of incidence and polarization and then noting to what extent the observed reflectance agreed with Equations 1 and 2. In a third test, the reflectance of a glass-water interface was measured at an angle less than critical and at the critical angle, and was compared with the reflectance calculated using (1) J. Fahrenfort, Spectrochim. Acta, 17,698 (1961). (2) N. J. Harrick, J. Phys. Chem., 64, 1110 (1960). (3) J. Fahrenfort and W. M. Visser, Spectrochim. Acra, 18, 1103 (1962). (4) A. C. Gilby, J. Burr, and B. Crawford, Jr., J. Phys. Chem., 70, 1525 (1966). (5) W. N. Hansen, Spectrochim. Acta, 21,815 (1965). 36,783 (1964). (6) W. N. Hansen and J. A. Horton, ANAL.CHEM., (7) P. A. Wilks, and T. Hirschfeld, “Applied Spectroscopy Reviews,” Marcel Dekker, New York, 1967, in press.
known indices. This last simple test is very sensitive. In a final test, we used a commercial multiple internal reflection sapphire plate unit, which had entirely different prism material, index of refraction, surface, crystal structure, and cell construction from previous cells. The purpose of this test was to determine precision of reflection measurements with definite handicaps. Several important points should be noted about the above tests. It is difficult to compare transmission and reflection data accurately. Transmission measurements are most accurate for lower values of the absorption coefficient, a, while internal reflection measurements are most accurate for higher values. Therefore, the concentration of the aqueous dye solution was chosen to give an intermediate value of the absorption coefficient. This coefficient permitted approximately equal but nonoptimum accuracy for the two methods. In this experiment the solution absorbed about 99% of the radiation at the absorption peak in a 25-micron transmission cell, while with a single reflection only about 6% was absorbed. This comparison partially illustrates the difficulties in comparing the two techniques. In the second test the dye concentration was chosen which would improve conditions for reflection measurements and would represent a considerable range of samples. The use of water in the third test created a situation which was most sensitive to depolarization and scattered light for the reflection test but was insensitive to depolarization and s cattered light for the critical angle measurements. The final point is that no great effort was made to use prisms of best available optical quality. THEORY
From Maxwell’s equations it can be shown (8)that
cos 8 Ril
=
~
f i 2 COS 8
-
ii2 - sin2 8
1’
+ 5fiz- sin* 8 i
=
RTM
(2)
where R L and R , I are reflectances for light polarized perpendicular and parallel to the plane of incidence. Here A = (n,/nl)(1 i K Z ) is the complex index of refraction, and 8 is the angle of incidence, where nl is the index of the transparent phase, n2 is the real part of the complex index of the absorbing phase, and K a is the absorption index (attenuation index) of the second phase. The above equations are exact and general, in that they describe reflection (of an infinite plane wave) from any homogeneous isotropic phase (with a plane boundary) regardless of its optical constants, at any angle of incidence, and with radiation incident from a transparent phase of any refractive index. Assuming that the absorption coefficient, a , can be measured by ordinary transmission methods ( K is small), n and K
+
(8) M. Born and E. Wolf, “Principles of Optics,” pp. 624 ff., Pergamon Press, Los Angeles, 1959. VOL. 39, NO. 10, AUGUST 1967
1097
can be determined, and therefore Rl and R I Ican be calculated. Since QI
=
4TnzKz/ho
where h o is the wavelength of light in vacuo, n2K2 can be determined from a. Since T = e-ab and A = loglo (l/T) where T is the transmittance, b is the path length, and A is the absorbance, we can write ho ( A In 10) H2Kz = b 4n ~
Since K~ is very small, dispersion theory tells us that the difference between nz and the index of the solvent at each is very small, and could be neglected without affecting results significantly. nawas actually determined for the experiment. Therefore, when n2 is determined either experimentally or theoretically, ho ( A In 10) Kz=-(5) 4nnz b With a knowledge of n2 and K2, R I and Rli can now be calculated. EXPERIMENTAL
Equipment used included a Cary 14R spectrophotometer with extra slits, a Limit variable ultraviolet transmission cell Model BC-14, one single and one multiple ATR cell constructed in the laboratory (6, 9), a Wilks Scientific multiple reflection unit Model 14C with sapphire plate, a Zeiss refractometer, and two Glan-Thompson polarizers of schlieren grade. A most important first step was to make certain that the spectrophotometer was reading correctly. The slidewire should be checked for linearity. The absorbance slidewires used in the experiment had been checked by the manufacturer and correction charts were supplied. (Most manufacturers will supply this service.) The spectrophotometer was then checked for linearity and proper operation to meet manufacturer’s specifications. Even higher precision can be obtained by following detailed instructions (IO, I I ) . The limit cell was calibrated using a helium light source by closing the cell t o a very low thickness (-3 microns) and observing the interference patterns. By observing the fringe movement (with a near-normal light source) it can be determined if the two windows are parallel and normal to the opening axis. Adjustments were made t o achieve this alignment. The path length change was determined accurately by counting fringes as the cell was opened. This change was repeated several times t o determine the reproducibility. The fringe count agreed with the cell calibration marks, and therefore these marks were used when the data were taken. The most accurate transmission data can be taken when a variablethickness cell is used (if b is known), because it eliminates the effects of surface layers on the windows, of reflection from the windows due t o refraction index differences between the windows and the sample, and of differences between cells. The thickness reproducibility was found t o be better than fringe, which at 25-micron cell thickness gives a maximum possible error of 1%. Slits were inserted into the chopper compartment of the Cary 14 t o collimate the beam t o a 30-minute spread and 1 X 5 mm maximum cross-sectional dimension. The polarizers (9) W. N. Hansen, ANAL.CHEM.,37,1142 (1965). (10) H. E. Bennett, Appl. Optics, 5, 1265 (1966). ( 1 1 ) D. C. Nelson and R. C. Hawes, Conference on Applied Spectroscopy and Analytical Chemistry, Pittsburgh, Pa., March 1964. 1098
ANALYTICAL CHEMISTRY
SAMPLE
Plate
(3)
-
Prism ‘Yighl
From Monochromator
Prism
/ ,/
L i g h t To Detectorla
Figure 1. Geometry of 11-reflection cell
were checked to see that they did not angularly deviate the beam as they were rotated. A multiple-reflection cell was using a constructed as proposed by Hansen and Horton BSC-2 (borosilicate crown nD = 1.5172) plate 2 mm thick and 2 45’45”-90” prisms, as shown in Figure 1. The prisms were joined by epoxy cement t o the plate on the geometrically calculated locations. The cell was aligned in the spectrophotometer using the “dove prism effect.”
(a,
Any isosceles prism such as a 45 “-45 “-90” prism which has the normals of all three faces in a plane has the characteristic shown by Hansen and Horton (6) of transmitting the beam a t the same angle of incidence as the angle of incidence of the incoming beam. If the beam is parallel to the prism base [the sample area as shown by Hansen and Horton (6)],it will be transmitted parallel. An image inversion takes place, but since the spectrometer beam is symmetric, this inversion causes no problems. It is this characteristic of dove prisms that makes them so useful here. If the base is put against the multiple plate, it is therefore parallel with the plate. The multiple cell is shifted to allow light to pass through and over the dove prism (the prism is of such height that it intercepts only half of the beam) and then the cell is rotated until the light passing through the dove prism aligns with the light passing over the prism. When the light beam is parallel with the cell plate, there will be no deviation between the two beams. Any error of alignment is doubled by the dove prism, making the error very visible. The procedure will give as precise an alignment as the beam can be defined. The multiple cell was aligned to about 1 minute of arc. The liquid retainer cell was then joined by epoxy cement to the plate and prisms. The polarizers were aligned with the plane of incidence of the cell (the cell being between the polarizers) and a crossed polarizer check was performed. A crossed polarizer check is here defined as the absorbance change obtained when, after both polarizers were aligned with the plane of incidence, one polarizer is rotated 90” from the plane of incidence. A more complete coverage is given by Shurcliff (12). An absorbance of approximately 3.5 was obtained. With the polarizers still crossed, the cell was removed and the absorbance went to about 5.0, showing that the cell produces a depolarization of 0.0003 and negligible optical rotation. These steps are essential before accurate data can be taken. The need for the degree of polarized light varies with the sample being studied. If R l is approximately equal to Rll, it is obvious that a crossed polarizer check which gives an absorbance of 0 would still mean that the cell would give accurate data. But if R L 1 and R 0.001 and an attempt was made to measure R l l accurately, say t o 1 part per thousand, a crossed polarizer check of about 6.0 absorbance would be needed. The greater the difference between R l and R, of the sample, the greater the degree of polarization needed. An aqueous solution of Eosin B dye was used which contained 11.82 grams of Eosin B per liter. Eosin B was chosen because care must be taken to avoid a dye that adsorbs ex(12) W. A. Shurcliff, “Polarized Light, Production and Use,” Harvard University Press, Cambridge, 1962.
PRISM
__MOTION
.026 .024-
.022,020-
-I
',
+
I
21
=I
Position2
_____
'
0
I Light Beam
I
-----
1
L \
.olew
Position3
Light Beam
h;,
/
1
2 .Oi6-
a m cc
0 .Oi4-
cn
m a
Figure 2. Point of reflection scan across the sample and silver as unit as a whole is moved perpendicular to incoming light beam cessively onto the cell surface, such as methylene blue (unless one wants to study adsorption). A much more concentrated solution containing 195.8 grams of Eosin B per liter was also used with the single reflection cell. An aqueous Eosin Y solution, which contained 50.0 grams of Eosin Y per liter, was used with the sapphire multiple reflection cell. For the reflection measurements a 0 4 . 1 , 0.1-0.2 absorbance-expanded slidewire was used, which had a manufacturer's stated nonlinearity of less than 0.0005 A . Data were taken at 5 and 11 reflections. Both polarizations were used to measure A l and A [here A = loglo (l/R)]. The base lines for both polarizations were taken with the cell empty. The sample was poured into the 5 reflection side of the cell and the wavelength region of interest was then scanned slowly and stopped in 100-A increments to average machine noise and prevent pen lag. The machine slits varied between 0.01 and 0.05 mm, which represents a resolution of better than 2 A . The scan was repeated for 11 reflections and both polarizations. The cell was then emptied and the base lines for both polarizations were repeated with water in the cell. This gives the machine drift and shows any adsorbed layer. For the transmission measurements a 0-1, 1-2 A slidewire was used (the nonlinearity is less than 0.002 A on the 0-1 scale and 0.005 A on the 1-2 scale) The machine drift was approximately 0.005 A per hour, which could be neglected. The polarizers were removed, but the extra slits were left in. After ensuring that the cell was normal to the beam, the cell was filled, taking care to avoid bubbles. The same solution was used in the transmission and the multiple-reflection cells. The base line was taken with the variable cell set to 3-micron thickness, and the wavelength region was then scanned using the above procedure. The run was repeated after opening the cell in 5-micron steps until 25 microns was reached. Then the cell was opened in 50-micron steps to a total opening at 125 microns. From these transmission data, a was determined as a function of Xo. All runs were at room temperature (23' C). A rerun of the starting point was made to check drift and cell reproducibility. Another reflection test was also made, using a single reflection cell of the type described by Hansen (9). This cell was checked between crossed polarizers and gave an absorbance reading of about 3.6. The three samples for this cell were the 195 grams per liter Eosin B solution, water, and air. The prism was a silver-plated BSC-2 (no = 1.5185) prism. The plating was such that one face was fully silvered and the other face was silvered except for a strip in the center, as shown in Figure 2. The reflectance of this silver was carefully measured against total reflection to allow us to vary
T E polarization I Reflection 0 measured reflection data 2 data calculated f r o m t r a n s m i s s i o n
.002/ = I
I
,45 .46 47
' '
measurements 1 I I I I 1 .48 49 .50 .51 .52 .53 .54 .55 Waveiength p
'
.56
,
Figure 3. Calculated reflection data and measured reflection data of 11.82 grams per liter Eosin B dye solution the angle of incidence past the critical angle and still have a base line. The sample was measured us. the silver and the silver was measured us. total reflection. The sum of the absorbances thus is the sample absorbance. The data were taken a t a single wavelength of 0.56 micron at incidence angles of from 22.78' to 75.30", and at both polarizations. By shifting the beam from the silver film to the sample and back, the difference can be accurately determined. The angle of incidence was then changed and the shift repeated. This procedure is very accurate and very fast. This cell was also used to measure the critical angle of the prism-air interface, the prism-water interface, and the reflection absorbance, A , of the prism-water interface at a 45 ' angle of incidence (all at 0.5893-micron wavelength). This is an extremely difficult test for a reflection cell because of the need for scattered-light control, a known angle of incidence, a well collimated beam, and near-perfect polarization control. The second prism face does not have to be silvered for this test. When determining the critical angle, allowances must be made for the beam spread of 30 minutes. (This gives less than 10-minute maximum deviation from the beam centerline at the sample.) A 0.07 absorbance increase from 0 was defined to indicate the critical angle. The water reflection measurement was done simply by setting the cell to 45', adding the water, and noting the absorbance change. Surface layers, small surface roughness, scattered light, and depolarization have little effect o n the critical angle determination, but can have a very large effect on the reflectance at any angle. The effect of surface roughness is shown by Bennett (13). A Wilks Scientific Model 14C sapphire multiple reflection unit was also tried. T o use this unit effectively, the beam height must be masked because of the small liquid holder height. The previous masking was sufficient. The cell when properly aligned gave a sample angle of incidence of 60" and 8 sample reflections. A crossed polarizer check gave ab(13) H. E. Bennett, Ind. Quality Control, 20, No. 8 (February 1964). VOL. 39,
NO. 10, AUGUST 1967
1099
..
sorbances of 0.3. Since sapphire is birefringent, and the optical axis was unknown, natural light was used by turning the polarizer t o halfway between 1 and /I polarization. The data were taken a t 0.52-micron wavelength with monochromator slits a t 0.01 mm. RESULTS
The results of the transmission us. reflection test are shown in Figure 3. To compare single reflection data calculated from transmission data with reflection data, the reflection data, obtained with the 11-reflection cell, were divided by 11 t o give data for a single reflection. The agreement is within 1 over most of the band. There are slightly larger differences in the lower readings, mainly because of the difficulty of measuring such low reflection absorbances. The parallel polarization showed an equivalent behavior, and therefore no figure is given. As 11 reflections were used, the effect of a surface layer is multiplied by 11. Since the effects of surface layers are very sensitive to polarization (14), the fact that both polarizations agree means that these effects can be accounted for or neglected. The results of the single reflection experiment using the more concentrated dye solution are shown in Figure 4. From the two indicated data points, the optical constants were calculated using an IBM 360 computer and the mathematical solution derived by Fahrenfort and Visser (3), giving n = 1.4093 and K = 0.1847. The optical constants were then used to calculate reflectance curves for both polarizations as a function of angle. Although it is not clear from Figure 4, on the T M curve on the large angle side of Brewster's angle, four points are off the curve, These errors are mostly because of the effect of imperfect polarization, showing the importance of proper polarization. Data were taken later using two polarizers (one before and one after the cell), causing these four points to move onto the calculated line, thus proving the problem to be polarization. Using the latter data for these points, the previous data for the TE curve and the rest of the TM curve, the mean per cent deviation (the per cent that a point deviated from the calculated curve is the per cent deviation; the sum of the absolute values of the per cent deviations of all the points is divided by the number of points, and this factor is the mean per cent deviation) was 0 . 3 2 z for the TE curve and 0.31 for the TM curve. These curves are very sensitive to n and K . An over-all change in these curves equal to the mean deviation will correspond to about 0.0010 change in the index of refraction, and about 0.0004 change in K . The results of the critical angle or prism-air interface measurements indicated that the index of the prism was 1.5182, while the Zeiss refractometer indicated 1.5185. (The Zeiss refractometer had been checked with a 1.5170 standard to prove accuracy.) The prism-water critical angle indicated a n index of 1.3336 for water, which is t o be compared with a reading of 1.3329 on the refractometer. This comparison indicates either a very small error in angle (less than 2 minutes) or a small error in the calculated 0.070 absorbance correction for a 30-minute beam spread. It could also indicate a beam that is not homogeneous in intensity as a function of angle from the center of the beam. Using the I polarization, the reflection at 45" from water gave a n absorbance of 1.640, which indicated the index of water to be 1.3338. This difference is an apparent error of approximately 0.005 A , assuming the refractometer reading is correct. The reading should have been 1.635 absorbance
\y
z
z
(14) W. N. Hansen, North American Aviation Science Center, unpublished data. 1100
ANALYTICAL CHEMISTRY
I c-
- Calculated dola
I-
--
0 h'easured data a 3010 points used for c o l c d l o t i o n
-
5-
t 0
t,
I
I
I
I
I
L L l & L u L 3 ~ 4 ~ d I L I_U uL I
50
IO Angle
01
60
I J-170
?c dence
Figure 4. Measured and calculated reflection data, from a concentrated 195.8 grams per liter Eosin B solution, as a function of angle of incidence
units for the reflection calculated from the refractometrically determined index of water. The readings on the sapphire cell are not presented here to show the precision of internal reflection measurements, because the optical axis of the sapphire plates used was unknown. The data are presented to show that even with such a handicap, plus some angular alignment difficulties, the data are still close to the theoretical calculations. The measured absorbance was 0.624 A , and the calculated one 0.633 A , assuming eight reflections. The optical constants of the Eosin Y dye solution at 0.52-micron wavelength as measured by Hansen ( 9 ) were 1.353 for n and 0.02137 for K . The n of sapphire was assumed to be 1.7720. CONCLUSIONS
The comparison of results of absolute transmission and reflection measurements indicates that the theory and procedure of internal reflection spectroscopy are accurate to 1 or better. This comparison assumes optical elements, beams, etc., which differ from the ideal no more than typical for the visible spectral region. A prism which depolarizes the beam and has thick chemically formed layers-typical of KRS-5, for example-will cause greater inaccuracies. Coupled with the other reflection measurements on more optimum samples, the data show that, in the low to high CY region, optical constants determined by internal reflection are as accurate as those determined by transmission in the zero to low CY region. The accuracies of both methods are about equal at a = 1000 cm-l. The results also demonstrate an internal consistency within a few parts per thousand absorbance, as did the absolute measurements of the glass-water interface. Together, these results indicate that even with the use uf ordinary optical components and a small, partially coherent beam, the basic reflection equations are closely obeyed.
z
ACKNOWLEDGMENT The authors thank Tornas Hirschfeld and Arnold Prostak for valuable discussions and encouragement.
RECEIVED for review March 13,1967. Accepted May 8,1967.
