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Distortion of absorption profiles due to window reflection
InfraredPhysics,1973, Vol. 13, pp. 245-252. PergamonPress.Printedin Great Britain.
DISTORTION OF ABSORPTION PROFILES DUE TO WINDOW REFLECTION J. Laboratoire
QUAZZA
PH. MARTEAU*
des Interactions
Mol&daires
and H. Vu?
et des Hautes Pressions, France
(Received 22 March 1973) Abstract-The distortion of absorption profile due to multiple passes resulting from reflections at the windows of an optical cell is considered. A general method applicable to absorption in any spectral region is developed. In particular calculations are performed for collision induced far infrared absorption by rare gas mixtures. Such a calculation, in addition to the optical properties of the window material, needs the knowledge of the refractive index of the sample in the region of interest. It is shown that for rare gas mixtures, the translational contribution to the refractive index is negligible. The electronic contribution is obtained from the additivity of molar refractivities. The “true” absorption profile for Xe-Ar mixture is deduced from the observed profile obtained using a cell with diamond windows. This “true” profile in turn is used to obtain the expected profiles of the mixture with silicon and germanium windows. The suitability of this procedure will be demonstrated in a subsequent communication in which the results on integrated absorption for translational spectra of several rare gas mixtures will be presented. INTRODUCTION VALIDITY of Beer-Lambert’s law is generally assumed even when absorption spectroscopic measurements are carried out using an optical cell. In fact, for most of the materials used as cell windows, the reflective power is so small that the intensity of the reflected beam is very weak and consequently may be neglected in the calculation of the total transmitted intensity received at the detector. But in the far infrared region, highly reflecting windows and light pipes techniques are often used so that an important part of the beam reaches the detector after multiple passes through the cell. The purpose of this paper is to investigate the distortion of the absorption profiles by these phenomena. Three types of materials, useful in the far infrared region are considered: diamond, silicon and germanium. The calculations are simplified by introducing the assumption of normal incidence. However, all the results should probably remain valid for incidence angles of up to 20”. Furthermore the reflection coefficient determination requires the knowledge of the refractive index of the absorbing medium. So, in the first part of this paper we carry out a calculation of the refractive index of rare gas mixtures and, in the second part the distortion of the absorption profiles of these mixtures by multiple pass through optical cells is investigated.
CALCULATION
OF THE REFRACTIVE INDEX OF RARE IN THE FAR INFRARED REGION
GAS
MIXTURES
The observed absorption bands in binary mixtures of rare gases are mainly of two types : in the far ultraviolet region, intrinsic absorption corresponding to the electronic transitions of each of the atoms, and in the far infrared region of translational absorption band, induced by heteronuclear atomic collisions.“-5’ * Present address; L.I.M.H.P., C.S.P., Place du 8 Mai 1945, 93206 Saint Denis, France. To whom all correspondence should be addressed. t L.I.M.H.P., C.N.R.S., 92 Rellevue, France. 245 1.P. 1314-A
246
J. QUAZZA, PH. MARTEAUand H. Vu
The refractive index at a frequency v0 may thus be expressed as n(v,) -1 =
In
(VO)
-
1
I elec
+
In
(VO)
-
1 I tram1
(1)
where v0 can take any value from 0 to co. However, we shall confine ourselves to the region of O-600 cm-’ in which the measurements were made. Since v,, is far from the electronic transition frequencies, the electronic term in equation (1) can be estimated by an approximate method. The translational contribution will be obtained from absorption data by using a Kramers-Kriinig dispersion relation :c6)
(2) (here the integral is a Cauchy’s principal value). Calculation of the electronic contribution
Let us consider a mixture of two gases (designated 1 and 2); their respective densities are p1 and pZ expressed in amagat units; the molar refractivity of a gas is given by (7) Al = fJ
CL1
(3)
here JV is the Avogadro’s number and a1 the polarizability of the gas 1. According to the additivity of the contribution due to each of the gases, the molar refractivity of the mixture is A=---.---3 Pl
+
P2
(PI al +
PZ a2).
(4)
This quantity may also be expressed as n2-I -
V.
A=--_
PI +
~2
n2 +
(5)
2
here V, is the Avogadro’s volume, n the refractive index of the mixture, the total density of which is (pl + p2). Combining equations (4) and (5) we obtain: n2-1
-=y#p
n2 + 2
4aN 1 al +
~2
4.
(6)
0
Thus the refractive index of a mixture can now be extracted with the only assumption that the molar refractivity does not depend on the density. Strictly speaking molar refractivity is a function of density. However, some experimental data indicate that this dependence is so small that it may be safely neglected. The molar refractivity of oxygen in the gas phase at 1 atm pressure is 4.01 whereas it is 4.05 for the liquid phase. Moreover, as far as low frequency range is concerned it also seems to be a good approximation to use the static values of the polarizabilities@) (aHe = O-204; a Ne= 0.392; a.& = l-630; aKr = 2.465; axe = 4.010) in the numerical calculation involving equation 16. In fact, the refractive index of each of these atoms is close to their limiting d.c.-values even as far as out in the visible region. This can be noticed from the example of argon, the refractive index of which is equal to 1GJ0282’g’ (5893 A) and the value calculated with the aid of formula (6) is 1.000275.
Distortion
of absorption
profiles due to window reflection
247
Calculation of the translational contribution
Absorption profiles are well known for all the binary mixtures of rare gases (except for the He-Ne pair) so that the function A (v) can be introduced in the formula (2) for the purpose of the evaluation of the quantity In (v,J - 1 Jtrans,.However, it is very important to know the profiles with the best possible accuracy throughout the frequency range concerned. Because of the energy limitation of conventional far infrared sources this becomes a difficult task particularly at lower frequencies. (lo*l l) However, our main purpose here is to compare the relative magnitudes of the two contributions, viz. the electronic and the translational. For this purpose, equation (2) can be used to a good approximation if an analytic expression for A (v) is available. As it was suggestedC2’we write : A (v) = A, v2 exp (-flu).
(7)
Theoretical profile is compared in Fig. 1. with the experimental one for the Ne-Ar pair, obtained by Bosomworth and Gush; the best fit to experimental data is achieved with
Wave
FIG. 1. Translational
absorption
number,
cm“
profile: experimental values from Ref. (2) (broken line) and theoretical curve from formula (7) (full line).
following values for the parameters: A0 = 0511 x lOA cm and fi = 0.02115 cm. This profile was recorded with a long optical cell (3 m) closed by two polyethylene windows and we shall see in the second part of this paper that there is no deformation of the profile when this window material is used. This is the main reason of our choice of this profile without further correction. Inserting formula (7) in the relation (2), after performing the integration one obtains :(l’)
v. exp(- is,01 M-
I n(v,>- 1 ltransl=
bo> - &B~o>ll
or co
2$-I 1-
I nbo) - 1 ltrsnsl= -
8~~ev(-- bo>
c a=o
@o)2a+ 1 @I + l)! @I + 1) *
I
(9)
248
J.
QUAZZA,
PH. MARTEAUand H. Vu
The dispersion curve due to the translational absorption, according to the relation (9) is shown in Fig. 2 in which the theoretical absorption profile [relation (7)] is also shown. It may be observed that the maximum translational contribution to the refractive index (I.225 x lo-‘) is much smaller than that of the electronic contribution (1.7 x 10e4). 14’7
2 YI
E ,+
10.5
\
P
1’
t5
/I //
200
100
Wave number
,
300
---_
cm”
FIG. 2. Dispersion of the refractive index of a Ne-Ar mixture due to induced translational absorption. The broken line represents the corresponding theoretical profile from relation (7) with an arbitrary unit.
However, it may be remarked that the values have been calculated for a density of one amagat. Now the translational part is proportional to p2 in a large region of density values(i3) and the electronic contribution is linear with density. Nevertheless for a density of 400 Am the ratio of these two contributions is still only 7 X 10T4. For the purpose of calculating the effect of window reflections on the absorption profile, as described in the next section, it is thus, sufficient to consider only the electronic contribution to the refractive index of the medium. MODIFICATIONS
OF THE WINDOW
ABSORPTION REFLECTIONS
PROFILES
DUE
TO
Let us consider an optical cell closed with two identical windows of a material of refractive index n W.The refractive index of the enclosed absorbing medium, which in the present case is a rare gas mixture, is denoted by n. The index of the outside medium is 1.0. There are two reflection coefficients, one for the air-window interface RI and the other for the windowmedium interface R2 which, for normal incidence are R, =
(5;)’
R,
=
is)‘.
As the extinction coefficient due to the translational absorption of the rare gas mixtures is always less than 10W4,n can be equated to its real part. If one considers all the reflections on the two faces of one window the reflectance of a window is given by
R=
RI
i-
R,
1-
-
2R1 R,
R, R,
(11)
Distortion of absorption profiles due to windowreflection
249
and the transmittance T is equal to 1 - R since the absorption of the window material in the region of interest may be neglected. The intensities ofthe transmitted and reflected beams are calculated using standard methods. (r4) Z, is the total intensity of the beam which arrives at the first window of the cell, the intensity of the nth wave, emerging, after 2n - 1 passes through the cell is: i,, = T* I, RZne2 exp [-(2n - 1) AZ].
(12)
Where A is the absorption coefficient, I is the geometrical length of the cell. The total intensity of the emerging beam is consequently:
c co
I=
n=l
(1 - R)* exp(- AZ) Z In = 1 - R2 exp(- 2AZ) ” *
(13)
In terms of RI and R, [cf. equation (1 I)] the total intensity may also be expressed as
I=
(1 - RJ*(l - R,)’ exp(- AI) (1 - RI R2)* - (RI + R2 - 2R1 R2)* exp(-
I 2AI) se
(14)
The experimental measurements concern only Z and I,; Z,, is the total intensity the detector receives when the cell is empty. In this case, A (v) is equal to zero and n is equal to 1 and R,=R,so 1 -
z, = ~
RI
1 + 3R,
1s
(15)
Z is the measured intensity when the cell is filled with the medium interest. The apparent transmittance, Z/Z,,is given by: Z
-= 10
(1 - R2)* (1 - R,) (1 + 3Rd exp(- Al) (1 - R, R2)’ - (R, + R, - 2R1 R2)* exp(- 2Al)’
(16)
If the windows are not too reflecting (n ry 5 1.5) this expression may be simplified to the well known Beer-Lambert’s law: i
# exp(-Al).
(17)
But, if diamond (n w = 2.417) silicon (nw = 3.41) or germanium (nw = 4) are used as window materials one must use the equation (16) to define the true absorption coefficient A.(r5*r6) When a value of nw = 2.417 (measured in the non-dispersive visible region) for diamond is used one obtains a value of 29 per cent for the reflectivity using relation (11) whereas our measurements indicate a value of 32.5 per cent corresponding to a refractive index of 2.54. Note that measurements were not done strictly at normal incidence. Also it is entirely possible that due to small but finite contribution due to multiphonon processes the refractive index of diamond in the far infrared region may in fact be slightly higher than its value in the visible region. Unfortunately no far infrared data of this quantity is available. For the purpose of the present calculation we have used the value nw = 2.417. As will be noted later, the higher the value of nw the greater is the distortion of absorption profile.
J.
250
PH. MARTEAUand H. Vu
QUAZZA,
Analysis for the case of a non-absorbing medium In the limited spectral region (O-600 cm-‘), no absorption has ever been observed for a pure rare gas, even under high pressure. In the present case, the experimental quantity Z/Z0 is not a function of frequency, but it still depends on the density of the gas since the R2 coefficient in relation (16) is sensitive to it. Experimental measurements of this quantity have been carried out for pure Ar at various densities using an optical cell with diamond windows. The results are compared in Fig. 3 (circles) to the theoretical calculations derived
0.10
---
$005 -__ b
0
i
‘r 1
600
2 O(J
Density,
FIG. 3. Variation
Amagot
mt
I
of transmitted energy vs density for pure Ar. Full line is theoretical relation (16). Circles are experimental measurements.
curve from
from relation (6), (10) and (16) (full line). It may be noted that both experimental data and calculated curve show the remarkable fact that a transmission of more than 100 per cent is obtained and which steadily increases with the density of the rare gas. This of course results from the fact that the reflectivity of diamond window decreases in the presence of the rare gas as compared to its value in vacuum. Analysis for the case of an absorbing medium When the cell is filled with an absorbing medium, the analysis becomes more complex because the apparent optical density does not follow Beer-Lambert’s law. The optical path length is not equal to the geometrical length of the cell. It is not possible to define an apparent path length because firstly this is different for each of the emerging waves, and secondly, their relative contribution to the total intensity of the emerging beam depends on the value of the absorption coefficient. The determination of the exact value of the absorption coefficient is possible if one introduces in relation (16) the experimental value of the ratio Z/Z0 for each frequency. With nearly frequency independent values of n, and n, one can draw calibration curves connecting the measured values of Z/Z,, with the true values of optical density. These curves are shown in Fig. 4 in which a value of n = l-06 and IZw corresponding to those of diamond, silicon and germanium have been used. The broken line of Fig. 4 is the limiting case when Beer-Lambert’s law is rigorously obeyed because the windows are non-reflecting. It may be remarked that the departures from the Beer-Lambert’s law become more serious higher the refractive index of the window; n = I.06 used in Fig. 4 corresponds to the refractive index of a mixture of Xe and Ar with respective partial densities of 50 and 110 amagats. The experimental collision induced absorption profile, obtained with diamond
Distortion
Al
of absorption
profiles due to window reflection
251
I
log, r,/l
FIG. 4. Calibration curves of the absorption as a function of the experimental quantity In IO/Z for one mixture (n = 1.06) and three types of windows (1) diamond n W = 2.42; (2) silicon nR = 3.41; (3) germanium nW = 4.
windows is shown in Fig. 5. In this figure we also show the exact profde after corrections for the window reflectivity. Expected profiles for silicon and germanium windows are calculated using the exact profile. These are also shown in Fig. 5. Thepeakposition ofthe band does not of course change by the choice of the window, although the width and shape change significantly. One notes also that the measurements of the absorption coefficient on the wings is particularly difficult for reasons discussed in the case of non-absorbing media.
Wave FIG.
5.
Exact absorption
number,
cm“
profile (1) and distorted ones for three types of windows: (2) diamond (nw = 2.42); (3) silicon (nW = 3.41); (4) germanium (nW = 4).
Furthermore, the distortion of the recorded profile increases when the index of absorbing medium is close to unity. It is hoped that for such cases the corrections indicated in this paper will be carried out before. CONCLUSION
In this paper we have shown that multiple passes through an optical cell due to finite reflectivity of the windows may distort spectral profile. In particular, when the refractive index of the window material is high, as is often the case with many materials used in far
252
J. QUAZZA, PH. MARTEAU and H. Vu
infrared spectroscopy, and when the refractive index of the medium is close to unity, a situation not uncommon with low density gases, this distortion can indeed be very significant. Correction for this effect should be made before drawing conclusions from such recorded spectra. Our present discussion has been confined to media of low absorption coefficients. It is expected that in cases where the extinction coefficient is large enough to make the refractive index a complex quantity, additional corrections might be necessary. Although in this paper we have discussed the problem of far infrared absorption by mixed rare gases, the derived relations are however, quite general and may be useful in other spectral regions as well. For example, introduction of pure Ar (500 amagats) in an optical cell with sapphire windows increases its transmission in the near infrared region (l-5 pm) by about 6 per cent. This phenomenon can also be understood by the use of equation (16). Acknowledgements-The authors’ sincere thanks are due to Dr. S. S. Mitra and Dr. H. Damany for their many helpful discussions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
KISS, Z. J. and H. L. WELSH, Phys. Rev. Lefts 2, 166 (1959). BOSOMWORTH,D. R. and H. P. GUSH, Can. J. Phys. 43, 751 (1965). MARTEAU, PH., R. GRANIER, H. Vu and B. VODAR, C.R. Acad. Sci., Paris 265B, 685 (1967). MARTEAU, PH., H. Vu and B. VODAR, J. Quunt spect. Radiat. Trans. 283, 10 (1970). BUONTEMPO,U., S. CONSULOand G. JACCUSI,Phys. Lefts 31A, 128 (1970). NUDELMAN,S. and S. S. MITRA, Optical Properties of Solids, p. 144. Plenum Press, New York (1969). BORN, M. and E. WOLF, Principles of Optics, p. 88. Pergamon Press, Oxford (1970). DALGARNO, A. and A. E. KINGSTOM,Proc. Roy. Sot. A259,425 (1960). COOK, G. A., Argon, Helium and the Rare Gases, p. 239. Interscience, New York (1961). MARTEAU, PH., H. Vu and B. VODAR, C.R. Acad. Sci., Paris 266B, 1068 (1968). V&N KRANENDONK,J., Can. J. Phys. 46, 1173 (1968). ERDELYI, A., Znt. Trans. Z, p. 135. McGraw-Hill, New York (1953). QUAZZA, J., These 3e cycle, Paris, April (1972). MILLER, K. D. and W. G. ROTHSCHILD,Fur Infrared Spectroscopy, p. 415. Wiley, New York (1971). ARONSON,J. R. and H. G. MCLINDEN, Phys. Rev. 135A, 785 (1964). DECAMPS,E. and A. HADNI, C.R. Acad. Sci., Paris 250, 1827 (1960).
DISTORTION OF ABSORPTION PROFILES DUE TO WINDOW REFLECTION J. Laboratoire
QUAZZA
PH. MARTEAU*
des Interactions
Mol&daires
and H. Vu?
et des Hautes Pressions, France
(Received 22 March 1973) Abstract-The distortion of absorption profile due to multiple passes resulting from reflections at the windows of an optical cell is considered. A general method applicable to absorption in any spectral region is developed. In particular calculations are performed for collision induced far infrared absorption by rare gas mixtures. Such a calculation, in addition to the optical properties of the window material, needs the knowledge of the refractive index of the sample in the region of interest. It is shown that for rare gas mixtures, the translational contribution to the refractive index is negligible. The electronic contribution is obtained from the additivity of molar refractivities. The “true” absorption profile for Xe-Ar mixture is deduced from the observed profile obtained using a cell with diamond windows. This “true” profile in turn is used to obtain the expected profiles of the mixture with silicon and germanium windows. The suitability of this procedure will be demonstrated in a subsequent communication in which the results on integrated absorption for translational spectra of several rare gas mixtures will be presented. INTRODUCTION VALIDITY of Beer-Lambert’s law is generally assumed even when absorption spectroscopic measurements are carried out using an optical cell. In fact, for most of the materials used as cell windows, the reflective power is so small that the intensity of the reflected beam is very weak and consequently may be neglected in the calculation of the total transmitted intensity received at the detector. But in the far infrared region, highly reflecting windows and light pipes techniques are often used so that an important part of the beam reaches the detector after multiple passes through the cell. The purpose of this paper is to investigate the distortion of the absorption profiles by these phenomena. Three types of materials, useful in the far infrared region are considered: diamond, silicon and germanium. The calculations are simplified by introducing the assumption of normal incidence. However, all the results should probably remain valid for incidence angles of up to 20”. Furthermore the reflection coefficient determination requires the knowledge of the refractive index of the absorbing medium. So, in the first part of this paper we carry out a calculation of the refractive index of rare gas mixtures and, in the second part the distortion of the absorption profiles of these mixtures by multiple pass through optical cells is investigated.
CALCULATION
OF THE REFRACTIVE INDEX OF RARE IN THE FAR INFRARED REGION
GAS
MIXTURES
The observed absorption bands in binary mixtures of rare gases are mainly of two types : in the far ultraviolet region, intrinsic absorption corresponding to the electronic transitions of each of the atoms, and in the far infrared region of translational absorption band, induced by heteronuclear atomic collisions.“-5’ * Present address; L.I.M.H.P., C.S.P., Place du 8 Mai 1945, 93206 Saint Denis, France. To whom all correspondence should be addressed. t L.I.M.H.P., C.N.R.S., 92 Rellevue, France. 245 1.P. 1314-A
246
J. QUAZZA, PH. MARTEAUand H. Vu
The refractive index at a frequency v0 may thus be expressed as n(v,) -1 =
In
(VO)
-
1
I elec
+
In
(VO)
-
1 I tram1
(1)
where v0 can take any value from 0 to co. However, we shall confine ourselves to the region of O-600 cm-’ in which the measurements were made. Since v,, is far from the electronic transition frequencies, the electronic term in equation (1) can be estimated by an approximate method. The translational contribution will be obtained from absorption data by using a Kramers-Kriinig dispersion relation :c6)
(2) (here the integral is a Cauchy’s principal value). Calculation of the electronic contribution
Let us consider a mixture of two gases (designated 1 and 2); their respective densities are p1 and pZ expressed in amagat units; the molar refractivity of a gas is given by (7) Al = fJ
CL1
(3)
here JV is the Avogadro’s number and a1 the polarizability of the gas 1. According to the additivity of the contribution due to each of the gases, the molar refractivity of the mixture is A=---.---3 Pl
+
P2
(PI al +
PZ a2).
(4)
This quantity may also be expressed as n2-I -
V.
A=--_
PI +
~2
n2 +
(5)
2
here V, is the Avogadro’s volume, n the refractive index of the mixture, the total density of which is (pl + p2). Combining equations (4) and (5) we obtain: n2-1
-=y#p
n2 + 2
4aN 1 al +
~2
4.
(6)
0
Thus the refractive index of a mixture can now be extracted with the only assumption that the molar refractivity does not depend on the density. Strictly speaking molar refractivity is a function of density. However, some experimental data indicate that this dependence is so small that it may be safely neglected. The molar refractivity of oxygen in the gas phase at 1 atm pressure is 4.01 whereas it is 4.05 for the liquid phase. Moreover, as far as low frequency range is concerned it also seems to be a good approximation to use the static values of the polarizabilities@) (aHe = O-204; a Ne= 0.392; a.& = l-630; aKr = 2.465; axe = 4.010) in the numerical calculation involving equation 16. In fact, the refractive index of each of these atoms is close to their limiting d.c.-values even as far as out in the visible region. This can be noticed from the example of argon, the refractive index of which is equal to 1GJ0282’g’ (5893 A) and the value calculated with the aid of formula (6) is 1.000275.
Distortion
of absorption
profiles due to window reflection
247
Calculation of the translational contribution
Absorption profiles are well known for all the binary mixtures of rare gases (except for the He-Ne pair) so that the function A (v) can be introduced in the formula (2) for the purpose of the evaluation of the quantity In (v,J - 1 Jtrans,.However, it is very important to know the profiles with the best possible accuracy throughout the frequency range concerned. Because of the energy limitation of conventional far infrared sources this becomes a difficult task particularly at lower frequencies. (lo*l l) However, our main purpose here is to compare the relative magnitudes of the two contributions, viz. the electronic and the translational. For this purpose, equation (2) can be used to a good approximation if an analytic expression for A (v) is available. As it was suggestedC2’we write : A (v) = A, v2 exp (-flu).
(7)
Theoretical profile is compared in Fig. 1. with the experimental one for the Ne-Ar pair, obtained by Bosomworth and Gush; the best fit to experimental data is achieved with
Wave
FIG. 1. Translational
absorption
number,
cm“
profile: experimental values from Ref. (2) (broken line) and theoretical curve from formula (7) (full line).
following values for the parameters: A0 = 0511 x lOA cm and fi = 0.02115 cm. This profile was recorded with a long optical cell (3 m) closed by two polyethylene windows and we shall see in the second part of this paper that there is no deformation of the profile when this window material is used. This is the main reason of our choice of this profile without further correction. Inserting formula (7) in the relation (2), after performing the integration one obtains :(l’)
v. exp(- is,01 M-
I n(v,>- 1 ltransl=
bo> - &B~o>ll
or co
2$-I 1-
I nbo) - 1 ltrsnsl= -
8~~ev(-- bo>
c a=o
@o)2a+ 1 @I + l)! @I + 1) *
I
(9)
248
J.
QUAZZA,
PH. MARTEAUand H. Vu
The dispersion curve due to the translational absorption, according to the relation (9) is shown in Fig. 2 in which the theoretical absorption profile [relation (7)] is also shown. It may be observed that the maximum translational contribution to the refractive index (I.225 x lo-‘) is much smaller than that of the electronic contribution (1.7 x 10e4). 14’7
2 YI
E ,+
10.5
\
P
1’
t5
/I //
200
100
Wave number
,
300
---_
cm”
FIG. 2. Dispersion of the refractive index of a Ne-Ar mixture due to induced translational absorption. The broken line represents the corresponding theoretical profile from relation (7) with an arbitrary unit.
However, it may be remarked that the values have been calculated for a density of one amagat. Now the translational part is proportional to p2 in a large region of density values(i3) and the electronic contribution is linear with density. Nevertheless for a density of 400 Am the ratio of these two contributions is still only 7 X 10T4. For the purpose of calculating the effect of window reflections on the absorption profile, as described in the next section, it is thus, sufficient to consider only the electronic contribution to the refractive index of the medium. MODIFICATIONS
OF THE WINDOW
ABSORPTION REFLECTIONS
PROFILES
DUE
TO
Let us consider an optical cell closed with two identical windows of a material of refractive index n W.The refractive index of the enclosed absorbing medium, which in the present case is a rare gas mixture, is denoted by n. The index of the outside medium is 1.0. There are two reflection coefficients, one for the air-window interface RI and the other for the windowmedium interface R2 which, for normal incidence are R, =
(5;)’
R,
=
is)‘.
As the extinction coefficient due to the translational absorption of the rare gas mixtures is always less than 10W4,n can be equated to its real part. If one considers all the reflections on the two faces of one window the reflectance of a window is given by
R=
RI
i-
R,
1-
-
2R1 R,
R, R,
(11)
Distortion of absorption profiles due to windowreflection
249
and the transmittance T is equal to 1 - R since the absorption of the window material in the region of interest may be neglected. The intensities ofthe transmitted and reflected beams are calculated using standard methods. (r4) Z, is the total intensity of the beam which arrives at the first window of the cell, the intensity of the nth wave, emerging, after 2n - 1 passes through the cell is: i,, = T* I, RZne2 exp [-(2n - 1) AZ].
(12)
Where A is the absorption coefficient, I is the geometrical length of the cell. The total intensity of the emerging beam is consequently:
c co
I=
n=l
(1 - R)* exp(- AZ) Z In = 1 - R2 exp(- 2AZ) ” *
(13)
In terms of RI and R, [cf. equation (1 I)] the total intensity may also be expressed as
I=
(1 - RJ*(l - R,)’ exp(- AI) (1 - RI R2)* - (RI + R2 - 2R1 R2)* exp(-
I 2AI) se
(14)
The experimental measurements concern only Z and I,; Z,, is the total intensity the detector receives when the cell is empty. In this case, A (v) is equal to zero and n is equal to 1 and R,=R,so 1 -
z, = ~
RI
1 + 3R,
1s
(15)
Z is the measured intensity when the cell is filled with the medium interest. The apparent transmittance, Z/Z,,is given by: Z
-= 10
(1 - R2)* (1 - R,) (1 + 3Rd exp(- Al) (1 - R, R2)’ - (R, + R, - 2R1 R2)* exp(- 2Al)’
(16)
If the windows are not too reflecting (n ry 5 1.5) this expression may be simplified to the well known Beer-Lambert’s law: i
# exp(-Al).
(17)
But, if diamond (n w = 2.417) silicon (nw = 3.41) or germanium (nw = 4) are used as window materials one must use the equation (16) to define the true absorption coefficient A.(r5*r6) When a value of nw = 2.417 (measured in the non-dispersive visible region) for diamond is used one obtains a value of 29 per cent for the reflectivity using relation (11) whereas our measurements indicate a value of 32.5 per cent corresponding to a refractive index of 2.54. Note that measurements were not done strictly at normal incidence. Also it is entirely possible that due to small but finite contribution due to multiphonon processes the refractive index of diamond in the far infrared region may in fact be slightly higher than its value in the visible region. Unfortunately no far infrared data of this quantity is available. For the purpose of the present calculation we have used the value nw = 2.417. As will be noted later, the higher the value of nw the greater is the distortion of absorption profile.
J.
250
PH. MARTEAUand H. Vu
QUAZZA,
Analysis for the case of a non-absorbing medium In the limited spectral region (O-600 cm-‘), no absorption has ever been observed for a pure rare gas, even under high pressure. In the present case, the experimental quantity Z/Z0 is not a function of frequency, but it still depends on the density of the gas since the R2 coefficient in relation (16) is sensitive to it. Experimental measurements of this quantity have been carried out for pure Ar at various densities using an optical cell with diamond windows. The results are compared in Fig. 3 (circles) to the theoretical calculations derived
0.10
---
$005 -__ b
0
i
‘r 1
600
2 O(J
Density,
FIG. 3. Variation
Amagot
mt
I
of transmitted energy vs density for pure Ar. Full line is theoretical relation (16). Circles are experimental measurements.
curve from
from relation (6), (10) and (16) (full line). It may be noted that both experimental data and calculated curve show the remarkable fact that a transmission of more than 100 per cent is obtained and which steadily increases with the density of the rare gas. This of course results from the fact that the reflectivity of diamond window decreases in the presence of the rare gas as compared to its value in vacuum. Analysis for the case of an absorbing medium When the cell is filled with an absorbing medium, the analysis becomes more complex because the apparent optical density does not follow Beer-Lambert’s law. The optical path length is not equal to the geometrical length of the cell. It is not possible to define an apparent path length because firstly this is different for each of the emerging waves, and secondly, their relative contribution to the total intensity of the emerging beam depends on the value of the absorption coefficient. The determination of the exact value of the absorption coefficient is possible if one introduces in relation (16) the experimental value of the ratio Z/Z0 for each frequency. With nearly frequency independent values of n, and n, one can draw calibration curves connecting the measured values of Z/Z,, with the true values of optical density. These curves are shown in Fig. 4 in which a value of n = l-06 and IZw corresponding to those of diamond, silicon and germanium have been used. The broken line of Fig. 4 is the limiting case when Beer-Lambert’s law is rigorously obeyed because the windows are non-reflecting. It may be remarked that the departures from the Beer-Lambert’s law become more serious higher the refractive index of the window; n = I.06 used in Fig. 4 corresponds to the refractive index of a mixture of Xe and Ar with respective partial densities of 50 and 110 amagats. The experimental collision induced absorption profile, obtained with diamond
Distortion
Al
of absorption
profiles due to window reflection
251
I
log, r,/l
FIG. 4. Calibration curves of the absorption as a function of the experimental quantity In IO/Z for one mixture (n = 1.06) and three types of windows (1) diamond n W = 2.42; (2) silicon nR = 3.41; (3) germanium nW = 4.
windows is shown in Fig. 5. In this figure we also show the exact profde after corrections for the window reflectivity. Expected profiles for silicon and germanium windows are calculated using the exact profile. These are also shown in Fig. 5. Thepeakposition ofthe band does not of course change by the choice of the window, although the width and shape change significantly. One notes also that the measurements of the absorption coefficient on the wings is particularly difficult for reasons discussed in the case of non-absorbing media.
Wave FIG.
5.
Exact absorption
number,
cm“
profile (1) and distorted ones for three types of windows: (2) diamond (nw = 2.42); (3) silicon (nW = 3.41); (4) germanium (nW = 4).
Furthermore, the distortion of the recorded profile increases when the index of absorbing medium is close to unity. It is hoped that for such cases the corrections indicated in this paper will be carried out before. CONCLUSION
In this paper we have shown that multiple passes through an optical cell due to finite reflectivity of the windows may distort spectral profile. In particular, when the refractive index of the window material is high, as is often the case with many materials used in far
252
J. QUAZZA, PH. MARTEAU and H. Vu
infrared spectroscopy, and when the refractive index of the medium is close to unity, a situation not uncommon with low density gases, this distortion can indeed be very significant. Correction for this effect should be made before drawing conclusions from such recorded spectra. Our present discussion has been confined to media of low absorption coefficients. It is expected that in cases where the extinction coefficient is large enough to make the refractive index a complex quantity, additional corrections might be necessary. Although in this paper we have discussed the problem of far infrared absorption by mixed rare gases, the derived relations are however, quite general and may be useful in other spectral regions as well. For example, introduction of pure Ar (500 amagats) in an optical cell with sapphire windows increases its transmission in the near infrared region (l-5 pm) by about 6 per cent. This phenomenon can also be understood by the use of equation (16). Acknowledgements-The authors’ sincere thanks are due to Dr. S. S. Mitra and Dr. H. Damany for their many helpful discussions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
KISS, Z. J. and H. L. WELSH, Phys. Rev. Lefts 2, 166 (1959). BOSOMWORTH,D. R. and H. P. GUSH, Can. J. Phys. 43, 751 (1965). MARTEAU, PH., R. GRANIER, H. Vu and B. VODAR, C.R. Acad. Sci., Paris 265B, 685 (1967). MARTEAU, PH., H. Vu and B. VODAR, J. Quunt spect. Radiat. Trans. 283, 10 (1970). BUONTEMPO,U., S. CONSULOand G. JACCUSI,Phys. Lefts 31A, 128 (1970). NUDELMAN,S. and S. S. MITRA, Optical Properties of Solids, p. 144. Plenum Press, New York (1969). BORN, M. and E. WOLF, Principles of Optics, p. 88. Pergamon Press, Oxford (1970). DALGARNO, A. and A. E. KINGSTOM,Proc. Roy. Sot. A259,425 (1960). COOK, G. A., Argon, Helium and the Rare Gases, p. 239. Interscience, New York (1961). MARTEAU, PH., H. Vu and B. VODAR, C.R. Acad. Sci., Paris 266B, 1068 (1968). V&N KRANENDONK,J., Can. J. Phys. 46, 1173 (1968). ERDELYI, A., Znt. Trans. Z, p. 135. McGraw-Hill, New York (1953). QUAZZA, J., These 3e cycle, Paris, April (1972). MILLER, K. D. and W. G. ROTHSCHILD,Fur Infrared Spectroscopy, p. 415. Wiley, New York (1971). ARONSON,J. R. and H. G. MCLINDEN, Phys. Rev. 135A, 785 (1964). DECAMPS,E. and A. HADNI, C.R. Acad. Sci., Paris 250, 1827 (1960).