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An amplitude Fourier spectrometer for infrared solid state spectroscopy
Volume 8, number 1
OPTICS COMMUNICATIONS
May 1973
AN AMPLITUDE F O U R I E R S P E C T R O M E T E R F O R I N F R A R E D SOLID STATE S P E C T R O S C O P Y J. GAST "f and L. GENZEL Max-Planck-lnstitut fiJr Fes'tkiSrperJbrschung, Stuttgart, German 3,
Received 30 March 1973
A Michelson interferometer for amplitude and phase Fourier spectroscopy in the infrared is described. It allows the determination of the optical constants n and k of relatively small crystal samples at high resolution and low temperatures. For demonstration, the spectra of InSb and lnAs in the middle and far infrared are reproduced.
1. Introduction With a conventional Fourier spectrometer, in which the sample is placed outside the interferometer itself, the Fourier transform of the measured interferogram yields the power spectrum of either reflectance or transmittance, respectively. In many cases, such a power spectrum does not give sufficient information about the material under study. Frequently, the spectra of the complex refractive index (n, k), or the dielectric function (e', e") are required. With the knowledge of only one power spectrum, or reflection for instance, the K r a m e r s - K r o n i g analysis presents, as is well known, the possibility of determining both optical constants. In order to make this analysis the power spectrum must be known for the entire spectral region. As a result, a Kramers Kronig analysis often yields uncertain results, especially when the reflection phase varies rapidly with frequency [ 1]. In contrast, amplitude and phase Fourier spectroscopy allows a direct and more accurate determination of the amplitude- and phase-spectrum of the sample even at frequencies for which the reflectivity or the transmission is very weak. In this method, the sample must be placed in one arm of the two-beam interferometer resulting in an interferogram function which is asymmetric with respect to the position of equal + Now at Bruker-Physik A.G., Karlsruhe-Forchheim, Germany. 26
geometrical beam lengths. Therefore, a sine and cosine transform of the interferogram are necessary, yielding the required amplitude- and phase-spectrum. Reports have been made about the theory of amplitude spectroscopy [2, 3], as well as about several asymmetric measurements with special interferonreters, like determination of refractive index of gases and liquids [3] for instance, or solid state spectroscopy with medium-sized samples at room temperature and moderate resolution [2, 4, 5]. In this paper we describe an asymmetric Fourier spectrometer which allows in particular solid state spectroscopy on small samples in the middle and far infrared at low temperatures and with high resolution.
2. The optical layout of the interferometer The asymmetric interferometer has been constructed to satisfy several demands: Small samples ( < 1 cm 2) should be measurable in reflection and at low temperatures. By changing the path difference, there may not appear any displacements of the end-foci of the radiation on sample and reference mirror. To avoid atmospheric absorption, the measurements should preferably be carried out in vacuum, and finally the resolution of the spectrometer should be comparable to conventional symmetric interferometers. The entire apparatus consists essentially of three
Volume 8, number 1
OPTICS COMMUNICATIONS
May 1973
i
Fig. 1. Optical layout of the Michelson interferometer. S1-$3, spherical mirros. P1-P4, plane mirrors. OP1, OP2, off-axis paraboloids. C1, C2, cube-corner mirrors. BS, beamsplitter. E, off-axis ellipsoid. W1, W2, windows. S, Hg-lamp. CH, chopper. R, reflection filters. GD, Golay detector. CR, cryostat. SC, sample chamber. D, drive. parts (fig. 1): the interferometer which is located in a large vacuum chamber. The source and detector parts are separately placed in two little vacuum chambers at the side of the main chamber for improvement of thermal stability and for more flexibility. The beam path of the interferometer is symmetrical with respect to the plane of the beamsplitter. Metallic mesh [6] and mylar foils [5] have been used as materials for beamsplitters. To avoid unwanted polarizarion of the radiation the angle of incidence at the beamsplitter has been minimized to 23 ° . This angle of incidence
causes a measured polarization of nearly 10%. The partial beams, which are split by the beamsplitter, are reflected on the two cube-corners C1 and C2. For generating the path difference between the two partial beams, C2 is movable in the direction of the incident light. The room diagonal of the cube is parallel to the direction of incidence in order to take best advantage of the mirror planes. The path difference produced by the displacement of one of the cubes by a distance d is 4d because the radiation runs through the moving cube-corner twice. 27
Volume 8, n u m b e r 1
OPTICS COMMUNICATIONS
The drive unit which is placed outside the main vacuuln chamber allows a displacement of the cubecorner of + 7.5 cm, which corresponds to a maximum path difference of + 30 cm. The theoretically available resolution therefore is about 0.04 cm 1. For improving the images of the source on sample, end-nrirror, and detector, only mirrors with high accuracy of the surfaces were used in the inteferometer. (Maximum decline from the assigned plane: 500 A.) The shortest sampling distance which is determinable with the available Moire system is 4 Arm. Therefore, by choosing suitable sources, filters, and beanrsplitters spectra to at least 1000 cm 1 can be measured. The cube-corners provide for the parallel displacement of the radiation. Therefore, the partial beams are directed onto the two off-axis paraboloids which reflect both beams into the sample chamber through only one window which is inclined against the vertical. Although it has been shown [7] that an asymmetric background like a symmetric one does finally not falsify the results, it is advantageous to work with a symmetric background which is always guaranteed by this conception of crossing lhe beams at the position of the window. In the sample chamber the radiation is turned around by two small mirrors onto the horizontally lying end-mirrors. To obtain the sample interferogram, one of the end-mirrors is replaced by the sample. The reflecting surface of the sample has to be placed at exactly the same plane in which the surface of the mirror was lying before, in order not to disturb the phase information of the interferogram. The sample holder we used showed a displacement of the planes of less than 0.3 tim when replacing the sample by the mirror. The end foci are about 3 5 mm in dianreter depending on the diameter of the aperture. Therefore, samples with sizes down to at least 6 X 6 mm 2 can be measured. To avoid deranging reflections at the surfaces of the cryostat windows they are inclined against the vertical at an angle of about 10 °. The phase fronts of the two partial beams are bended by the windows in their outer parts more than in the central parts. To prevent the possible deterioration of the interference of the two beams at the detector some conditions must be considered. The angles of incidence of the axes of the partial beams at the surface of the window must be equal, the planes of incidence must be sym28
May 1973
metrical to the normal of the window, and the window itself has to be homogeneous and exactly plane parallel. For determination of the amplitude- and phasespectrum from the sampled interferogram two computing programs were used. One of them works with the method of the C o o l e y - T u k e y algorithnr t . The other program bases on the method of the direct sum t~
3. Measurements
As a performance test and as a demonstration of the range of applicability, a series of measurements were made for several crystals whose infrared spectra were fairly well known [7]. In this section the results of these measurements for InSb and lnAs at room temperature are presented. The spectral range was chosen from 40 to 300 cm -1 in order to cover both the lattice vibration spectra and the free carrier plasma edges. The samples used were fiat to within -+ 0.5/am over the 5 mm focus area and were 0.9 and 1.2 nma thick. The high absorption throughout the spectral range eliminated naultiple interference effects in the samples thus obviating the need for wedge-shaped sample geonretry. The measurements were made for various conrbinations of beamsplitters and filters, and yielded a reproducibility for the frequencies of sharp reflection edges or phase maxima of better than 0.3 cm 1 . Figs. 2a, b reproduce small central parts of two asymmetric interferograms of lnSb taken with different beamsplitters. Figs. 3 and 4 show in their upper parts the spectra of the power reflectivity [ r 12 and the reflection phase ~ of InSb and InAs which we found from the cosine- and sine-transforms of the interferograms. From these, the real and imaginary parts of the complex index of refraction (H, k) can be conrputed by the relations 1--jrl 2 tl
=
-
-
-
-
,
l+jrL2+2qr4cosO
-t We thank E.E. Bell for giving this program. "k+ In essential parts this program was drafted by C. Irslinger.
Volume 8, number 1
OPTICS COMMUNICATIONS
I
May 1973
r
I
I
I
I
,-_ 1.0
I
InSb (
@
~ 08
¢
¢3
=o o.~
0'20~ O0
50
100
I
I
.
10.0
~-k
150
250
200
300
I
A
I
lnSb I
(a)
=: 20 ~" 1.0 o
05 0.2
50
100
150
(b)
Fig. 2. Asymmetric interferograms of InSb. (a) Interferogram for the 100-240 cm --1 spectral range. 1000/inch metal mesh beamsplitter. (b) Interferogram for the 15-110 cm- 1 and 140 280 cm - i spectral range. 25 um mylar beamsplitter.
200
Fig. 3. Upper half: reflectivity Ir 12 and reflection phase 0 of InSb. Lower half: the corresponding real and imaginary part of the complex refractive index (n, k). -~_
InAs
1.0
e
0.8
ir,2
- 2 Ir t sin~
k =
1 +lrl 2 +2trlcosq~
The lower parts of figs. 3 and 4 give the spectra of n and k showing in the high-frequency region the fundamental lattice absorption and below the plasma edge due to free carriers. The position of the minimum of the plasma edge determines [9] either the effective mass, rn*, or the free carrier concentration, N, depending upon which of these parameters is already known. The appropriate relation is
300
250
(crn q)
wavenumbers
11-i2 0'2 F
000
m.ol 5.0
50
100
150
I
I
r
200
250
300
t
InAs
/k
,k
J
i
~n a¢ 213 c 1.0 05
2
=
4nNe2
C°min r n * ( e - 1) "
0.2
50
100 wavenumbers
The inclusion of the finite damping 1/r of the free carriers introduces a negligible correction to COmin. Since the plasma edge for both samples lies well below
150
200
250
300
(cm -I)
Fig. 4. Upper half: reflectivity Ir 12 and reflection phase 4> of InAs. Lower half: the corresponding real and imaginary part of the complex refractive index (n, k). 29
Volume 8, number 1
OPTICS COMMUNICATIONS
the f u n d a m e n t a l lattice vibration, one can use the static dielectric constant, % , for e. With the values rn* = 0.0115 me, e 0 = 17.68, Umin = 79 cm -1 for InSb we obtained a free carrier c o n c e n t r a t i o n o f N = 1.8 × 1016cm 3 in almost perfect agreement with the value given by the manufacturer of the crystal. For InAs, the plasma edge is less p r o n o u n c e d than in InSb, yielding some u n c e r t a i n t y for Pmin" Using the values m* = 0.024 me, e 0 = 15.15 and Vmin = 93 cm l, we obtained a free carrier concentration o f N = 3.3 X 1016cm 3 which has to be c o m p a r e d with 3.4 X 1016cm 3 given by the manufacturer. These m e a s u r e m e n t s confirm the published spectra of l r 12, qS, n and k for InSb [9, 10] while corresponding measurements for InAs have, to date, not been found.
30
May 1973
References [1] R. Geick, Z. Physik 166 (1962) 122. [2] E.E. Bell, Infrared Phys., 6 (1966) 57. [3] J.E. Chamberlain, J.E. Gibbs and H.A. Gebbie, Infrared Phys., 9 (1969) 185. [4] J.E. Chamberlain, J.E. Gibbs and It.A. Gebbie, Nature 198 (1963) 874. [5] E.E. Russell and E.E. Bell, Infrared Phys., 6 (1966) 75. [6] P. Vogel and L. Genzel, Infrared Phys., 4 (1964) 257. [7] J. Gast, Diplomarbeit, Freiburg (1972). [8] C. hslinger, Diplomarbeit, Freiburg (1968). [9] H. Yoshinaga and R.A. Oetjen, Phys. Rev. 101 (1956) 526. [ 101 E.E. Bell, Handbuch der Physik, Vol. 25, Bd. 2a, ed. S. Flfigge (Springer, Berlin, 1967) p. 20.
OPTICS COMMUNICATIONS
May 1973
AN AMPLITUDE F O U R I E R S P E C T R O M E T E R F O R I N F R A R E D SOLID STATE S P E C T R O S C O P Y J. GAST "f and L. GENZEL Max-Planck-lnstitut fiJr Fes'tkiSrperJbrschung, Stuttgart, German 3,
Received 30 March 1973
A Michelson interferometer for amplitude and phase Fourier spectroscopy in the infrared is described. It allows the determination of the optical constants n and k of relatively small crystal samples at high resolution and low temperatures. For demonstration, the spectra of InSb and lnAs in the middle and far infrared are reproduced.
1. Introduction With a conventional Fourier spectrometer, in which the sample is placed outside the interferometer itself, the Fourier transform of the measured interferogram yields the power spectrum of either reflectance or transmittance, respectively. In many cases, such a power spectrum does not give sufficient information about the material under study. Frequently, the spectra of the complex refractive index (n, k), or the dielectric function (e', e") are required. With the knowledge of only one power spectrum, or reflection for instance, the K r a m e r s - K r o n i g analysis presents, as is well known, the possibility of determining both optical constants. In order to make this analysis the power spectrum must be known for the entire spectral region. As a result, a Kramers Kronig analysis often yields uncertain results, especially when the reflection phase varies rapidly with frequency [ 1]. In contrast, amplitude and phase Fourier spectroscopy allows a direct and more accurate determination of the amplitude- and phase-spectrum of the sample even at frequencies for which the reflectivity or the transmission is very weak. In this method, the sample must be placed in one arm of the two-beam interferometer resulting in an interferogram function which is asymmetric with respect to the position of equal + Now at Bruker-Physik A.G., Karlsruhe-Forchheim, Germany. 26
geometrical beam lengths. Therefore, a sine and cosine transform of the interferogram are necessary, yielding the required amplitude- and phase-spectrum. Reports have been made about the theory of amplitude spectroscopy [2, 3], as well as about several asymmetric measurements with special interferonreters, like determination of refractive index of gases and liquids [3] for instance, or solid state spectroscopy with medium-sized samples at room temperature and moderate resolution [2, 4, 5]. In this paper we describe an asymmetric Fourier spectrometer which allows in particular solid state spectroscopy on small samples in the middle and far infrared at low temperatures and with high resolution.
2. The optical layout of the interferometer The asymmetric interferometer has been constructed to satisfy several demands: Small samples ( < 1 cm 2) should be measurable in reflection and at low temperatures. By changing the path difference, there may not appear any displacements of the end-foci of the radiation on sample and reference mirror. To avoid atmospheric absorption, the measurements should preferably be carried out in vacuum, and finally the resolution of the spectrometer should be comparable to conventional symmetric interferometers. The entire apparatus consists essentially of three
Volume 8, number 1
OPTICS COMMUNICATIONS
May 1973
i
Fig. 1. Optical layout of the Michelson interferometer. S1-$3, spherical mirros. P1-P4, plane mirrors. OP1, OP2, off-axis paraboloids. C1, C2, cube-corner mirrors. BS, beamsplitter. E, off-axis ellipsoid. W1, W2, windows. S, Hg-lamp. CH, chopper. R, reflection filters. GD, Golay detector. CR, cryostat. SC, sample chamber. D, drive. parts (fig. 1): the interferometer which is located in a large vacuum chamber. The source and detector parts are separately placed in two little vacuum chambers at the side of the main chamber for improvement of thermal stability and for more flexibility. The beam path of the interferometer is symmetrical with respect to the plane of the beamsplitter. Metallic mesh [6] and mylar foils [5] have been used as materials for beamsplitters. To avoid unwanted polarizarion of the radiation the angle of incidence at the beamsplitter has been minimized to 23 ° . This angle of incidence
causes a measured polarization of nearly 10%. The partial beams, which are split by the beamsplitter, are reflected on the two cube-corners C1 and C2. For generating the path difference between the two partial beams, C2 is movable in the direction of the incident light. The room diagonal of the cube is parallel to the direction of incidence in order to take best advantage of the mirror planes. The path difference produced by the displacement of one of the cubes by a distance d is 4d because the radiation runs through the moving cube-corner twice. 27
Volume 8, n u m b e r 1
OPTICS COMMUNICATIONS
The drive unit which is placed outside the main vacuuln chamber allows a displacement of the cubecorner of + 7.5 cm, which corresponds to a maximum path difference of + 30 cm. The theoretically available resolution therefore is about 0.04 cm 1. For improving the images of the source on sample, end-nrirror, and detector, only mirrors with high accuracy of the surfaces were used in the inteferometer. (Maximum decline from the assigned plane: 500 A.) The shortest sampling distance which is determinable with the available Moire system is 4 Arm. Therefore, by choosing suitable sources, filters, and beanrsplitters spectra to at least 1000 cm 1 can be measured. The cube-corners provide for the parallel displacement of the radiation. Therefore, the partial beams are directed onto the two off-axis paraboloids which reflect both beams into the sample chamber through only one window which is inclined against the vertical. Although it has been shown [7] that an asymmetric background like a symmetric one does finally not falsify the results, it is advantageous to work with a symmetric background which is always guaranteed by this conception of crossing lhe beams at the position of the window. In the sample chamber the radiation is turned around by two small mirrors onto the horizontally lying end-mirrors. To obtain the sample interferogram, one of the end-mirrors is replaced by the sample. The reflecting surface of the sample has to be placed at exactly the same plane in which the surface of the mirror was lying before, in order not to disturb the phase information of the interferogram. The sample holder we used showed a displacement of the planes of less than 0.3 tim when replacing the sample by the mirror. The end foci are about 3 5 mm in dianreter depending on the diameter of the aperture. Therefore, samples with sizes down to at least 6 X 6 mm 2 can be measured. To avoid deranging reflections at the surfaces of the cryostat windows they are inclined against the vertical at an angle of about 10 °. The phase fronts of the two partial beams are bended by the windows in their outer parts more than in the central parts. To prevent the possible deterioration of the interference of the two beams at the detector some conditions must be considered. The angles of incidence of the axes of the partial beams at the surface of the window must be equal, the planes of incidence must be sym28
May 1973
metrical to the normal of the window, and the window itself has to be homogeneous and exactly plane parallel. For determination of the amplitude- and phasespectrum from the sampled interferogram two computing programs were used. One of them works with the method of the C o o l e y - T u k e y algorithnr t . The other program bases on the method of the direct sum t~
3. Measurements
As a performance test and as a demonstration of the range of applicability, a series of measurements were made for several crystals whose infrared spectra were fairly well known [7]. In this section the results of these measurements for InSb and lnAs at room temperature are presented. The spectral range was chosen from 40 to 300 cm -1 in order to cover both the lattice vibration spectra and the free carrier plasma edges. The samples used were fiat to within -+ 0.5/am over the 5 mm focus area and were 0.9 and 1.2 nma thick. The high absorption throughout the spectral range eliminated naultiple interference effects in the samples thus obviating the need for wedge-shaped sample geonretry. The measurements were made for various conrbinations of beamsplitters and filters, and yielded a reproducibility for the frequencies of sharp reflection edges or phase maxima of better than 0.3 cm 1 . Figs. 2a, b reproduce small central parts of two asymmetric interferograms of lnSb taken with different beamsplitters. Figs. 3 and 4 show in their upper parts the spectra of the power reflectivity [ r 12 and the reflection phase ~ of InSb and InAs which we found from the cosine- and sine-transforms of the interferograms. From these, the real and imaginary parts of the complex index of refraction (H, k) can be conrputed by the relations 1--jrl 2 tl
=
-
-
-
-
,
l+jrL2+2qr4cosO
-t We thank E.E. Bell for giving this program. "k+ In essential parts this program was drafted by C. Irslinger.
Volume 8, number 1
OPTICS COMMUNICATIONS
I
May 1973
r
I
I
I
I
,-_ 1.0
I
InSb (
@
~ 08
¢
¢3
=o o.~
0'20~ O0
50
100
I
I
.
10.0
~-k
150
250
200
300
I
A
I
lnSb I
(a)
=: 20 ~" 1.0 o
05 0.2
50
100
150
(b)
Fig. 2. Asymmetric interferograms of InSb. (a) Interferogram for the 100-240 cm --1 spectral range. 1000/inch metal mesh beamsplitter. (b) Interferogram for the 15-110 cm- 1 and 140 280 cm - i spectral range. 25 um mylar beamsplitter.
200
Fig. 3. Upper half: reflectivity Ir 12 and reflection phase 0 of InSb. Lower half: the corresponding real and imaginary part of the complex refractive index (n, k). -~_
InAs
1.0
e
0.8
ir,2
- 2 Ir t sin~
k =
1 +lrl 2 +2trlcosq~
The lower parts of figs. 3 and 4 give the spectra of n and k showing in the high-frequency region the fundamental lattice absorption and below the plasma edge due to free carriers. The position of the minimum of the plasma edge determines [9] either the effective mass, rn*, or the free carrier concentration, N, depending upon which of these parameters is already known. The appropriate relation is
300
250
(crn q)
wavenumbers
11-i2 0'2 F
000
m.ol 5.0
50
100
150
I
I
r
200
250
300
t
InAs
/k
,k
J
i
~n a¢ 213 c 1.0 05
2
=
4nNe2
C°min r n * ( e - 1) "
0.2
50
100 wavenumbers
The inclusion of the finite damping 1/r of the free carriers introduces a negligible correction to COmin. Since the plasma edge for both samples lies well below
150
200
250
300
(cm -I)
Fig. 4. Upper half: reflectivity Ir 12 and reflection phase 4> of InAs. Lower half: the corresponding real and imaginary part of the complex refractive index (n, k). 29
Volume 8, number 1
OPTICS COMMUNICATIONS
the f u n d a m e n t a l lattice vibration, one can use the static dielectric constant, % , for e. With the values rn* = 0.0115 me, e 0 = 17.68, Umin = 79 cm -1 for InSb we obtained a free carrier c o n c e n t r a t i o n o f N = 1.8 × 1016cm 3 in almost perfect agreement with the value given by the manufacturer of the crystal. For InAs, the plasma edge is less p r o n o u n c e d than in InSb, yielding some u n c e r t a i n t y for Pmin" Using the values m* = 0.024 me, e 0 = 15.15 and Vmin = 93 cm l, we obtained a free carrier concentration o f N = 3.3 X 1016cm 3 which has to be c o m p a r e d with 3.4 X 1016cm 3 given by the manufacturer. These m e a s u r e m e n t s confirm the published spectra of l r 12, qS, n and k for InSb [9, 10] while corresponding measurements for InAs have, to date, not been found.
30
May 1973
References [1] R. Geick, Z. Physik 166 (1962) 122. [2] E.E. Bell, Infrared Phys., 6 (1966) 57. [3] J.E. Chamberlain, J.E. Gibbs and H.A. Gebbie, Infrared Phys., 9 (1969) 185. [4] J.E. Chamberlain, J.E. Gibbs and It.A. Gebbie, Nature 198 (1963) 874. [5] E.E. Russell and E.E. Bell, Infrared Phys., 6 (1966) 75. [6] P. Vogel and L. Genzel, Infrared Phys., 4 (1964) 257. [7] J. Gast, Diplomarbeit, Freiburg (1972). [8] C. hslinger, Diplomarbeit, Freiburg (1968). [9] H. Yoshinaga and R.A. Oetjen, Phys. Rev. 101 (1956) 526. [ 101 E.E. Bell, Handbuch der Physik, Vol. 25, Bd. 2a, ed. S. Flfigge (Springer, Berlin, 1967) p. 20.