Coherent thermal radiation of Fabry-Perot resonator structures
The coherent thermal radiation from semiconductor plane-parallel resonator structures is investigated both theoretically and experimentally. The coherent properties of thermal radiation from these objects are manifested by sharp spectral lines and wellpronounced lobe-like directional patterns. We...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2007
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| Цитувати: | Coherent thermal radiation of Fabry-Perot resonator structures / O.G. Kollyukh, V.P. Kyslyi, A.I. Liptuga, V. Morozhenko, V.I. Pipa, E.F. Venger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 94-102. — Бібліогр.: 17 назв. — англ. |
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irk-123456789-1183402017-05-30T03:05:07Z Coherent thermal radiation of Fabry-Perot resonator structures Kollyukh, O.G. Kyslyi, V.P. Liptuga, A.I. Morozhenko, V. Pipa, V.I. Venger, E.F. The coherent thermal radiation from semiconductor plane-parallel resonator structures is investigated both theoretically and experimentally. The coherent properties of thermal radiation from these objects are manifested by sharp spectral lines and wellpronounced lobe-like directional patterns. We investigated the dependences of the intensities of spectral lines and the angular distribution of thermal radiation on optical parameters of the structures. New optical effects are presented, namely, the modulation of the coherent thermal radiation spectrum by an external magnetic field and a uniaxial pressure. 2007 Article Coherent thermal radiation of Fabry-Perot resonator structures / O.G. Kollyukh, V.P. Kyslyi, A.I. Liptuga, V. Morozhenko, V.I. Pipa, E.F. Venger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 94-102. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS 42.25.Kb, 78.30.-j, 78.45.+h http://dspace.nbuv.gov.ua/handle/123456789/118340 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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| description |
The coherent thermal radiation from semiconductor plane-parallel resonator
structures is investigated both theoretically and experimentally. The coherent properties
of thermal radiation from these objects are manifested by sharp spectral lines and wellpronounced
lobe-like directional patterns. We investigated the dependences of the
intensities of spectral lines and the angular distribution of thermal radiation on optical
parameters of the structures. New optical effects are presented, namely, the modulation
of the coherent thermal radiation spectrum by an external magnetic field and a uniaxial
pressure. |
| format |
Article |
| author |
Kollyukh, O.G. Kyslyi, V.P. Liptuga, A.I. Morozhenko, V. Pipa, V.I. Venger, E.F. |
| spellingShingle |
Kollyukh, O.G. Kyslyi, V.P. Liptuga, A.I. Morozhenko, V. Pipa, V.I. Venger, E.F. Coherent thermal radiation of Fabry-Perot resonator structures Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Kollyukh, O.G. Kyslyi, V.P. Liptuga, A.I. Morozhenko, V. Pipa, V.I. Venger, E.F. |
| author_sort |
Kollyukh, O.G. |
| title |
Coherent thermal radiation of Fabry-Perot resonator structures |
| title_short |
Coherent thermal radiation of Fabry-Perot resonator structures |
| title_full |
Coherent thermal radiation of Fabry-Perot resonator structures |
| title_fullStr |
Coherent thermal radiation of Fabry-Perot resonator structures |
| title_full_unstemmed |
Coherent thermal radiation of Fabry-Perot resonator structures |
| title_sort |
coherent thermal radiation of fabry-perot resonator structures |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118340 |
| citation_txt |
Coherent thermal radiation of Fabry-Perot resonator structures / O.G. Kollyukh, V.P. Kyslyi, A.I. Liptuga, V. Morozhenko, V.I. Pipa, E.F. Venger // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 94-102. — Бібліогр.: 17 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T13:50:33Z |
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2025-07-08T13:50:33Z |
| _version_ |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 94-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
94
PACS 42.25.Kb, 78.30.-j, 78.45.+h
Coherent thermal radiation of Fabry-Perot resonator structures
O.G. Kollyukh, V.P. Kyslyi, A.I. Liptuga, V. Morozhenko, V.I. Pipa, and E.F. Venger
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine
41, prospect Nauky, 03028 Kyiv, Ukraine; e-mail: kollyukh@isp.kiev.ua
Abstract. The coherent thermal radiation from semiconductor plane-parallel resonator
structures is investigated both theoretically and experimentally. The coherent properties
of thermal radiation from these objects are manifested by sharp spectral lines and well-
pronounced lobe-like directional patterns. We investigated the dependences of the
intensities of spectral lines and the angular distribution of thermal radiation on optical
parameters of the structures. New optical effects are presented, namely, the modulation
of the coherent thermal radiation spectrum by an external magnetic field and a uniaxial
pressure.
Keywords: thermal radiation, coherent radiation, plane-parallel structure, IR emitter.
Manuscript received 03.12.07; accepted for publication 19.12.07; published online 26.02.08.
1. Introduction
Thermal radiation (TR) emitted from heated solids is
commonly viewed as incoherent radiation, i.e. that with
the broad spectrum and a quasiisotropic angular
distribution of the intensity. However, it has been
demonstrated in a number of papers [1–11] that TR of
some special sources possesses well-pronounced
coherent properties. These are sharp spectral peaks
(temporal coherence) and narrow angular lobes (spatial
coherence). Coherent TR is generated via the following
mechanisms: (i) excitation of surface waves (surface
plasmons or surface phonon polaritons) that are coupled
to propagating electromagnetic waves due to a surface
grating [1–3]; (ii) wave interference effects in plane-
parallel structures [4–9]; and (iii) control of emissivity
due to the existence of a photon band structure (photonic
crystals [10–11]). The researches of coherent TR
attracted a substantial interest due to its potential
applications for the creation of novel controllable
radiation sources intended for middle and far IR.
Here, we investigate the coherent TR of
semiconductor plane-parallel resonator structures.
Contrary to (i) and (iii) mechanisms (that may be
responsible for the generation of coherent TR in some
narrow spectral regions determined by the material
properties), the interference effects in the resonator
structures make it possible to develop coherent TR
sources for wide spectral ranges. In addition, the
structures studied by us have one more advantage: their
manufacturing technology is easy.
We study the dependences of the spectral lines
intensity and the angular distribution of TR on optical
parameters of the structures. The effects of the
modulation of a coherent TR spectrum by an external
magnetic field [12, 13] and a uniaxial pressure are
presented. These new optical effects are caused by
induced optical anisotropy and are realized only for TR
with well-pronounced coherent properties.
2. Some theoretical considerations
Let us consider a semiconductor plane-parallel layer
( )lz ≤≤0 placed on a substrate ( )lz > ; the surface z = 0
is contacting with vacuum. We label these media
(vacuum, layer, and substrate) with the subscript j = 1, 2,
and 3, respectively. The system is at a temperature T.
The semiconductor has refractive index n and absorption
coefficient α. Let dJ = J (λ, 1ϑ ) cos 1ϑ dλ dΩ dS be the
intensity of radiation in the wavelength range dλ from
the area dS of the surface z = 0 into an element dΩ of the
solid angle (oriented at an angle 1ϑ to the normal).
According to Kirchhoff’s principle, J (λ, 1ϑ ) is given by
the expression
( )( )1exp
2
),(
5
2
1
−λλ
ϑλ=
kThc
hc
AJ , (1)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 94-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
95
where A is the radiator emissivity (that is equal to its
absorptivity). The absorptivity is defined as a fraction of
the radiation energy flux (incident on the surface z = 0)
that is absorbed by the system. The quantity J coincides
with the blackbody radiation intensity if A = 1. The
polarization properties of TR are described with the
partial emissivities As and Ap (and A = (As + Ap) / 2).
Here, the subscripts s and p label the quantities related to
the s- and p-polarizations, respectively.
The coefficients As and Ap for the structure
considered were calculated in [6]. For a thick
( 14 >>λπnl ) layer with weak absorption
( λπ<<α n4 ), one can present the partial emissivities of
the structure with a transparent substrate as
( )( )( )
νννν
νν
ν
η+δη+
η−η+−
=
21
2
2312
21
cos21
111
RRrr
RR
A . (2)
Here, ps,=ν , 2
121 νν = rR and 2
232 νν = rR are the
coefficients of reflection from the front (radiating) and
back surfaces of the structure, respectively; ν12r and
ν23r are the real parts of the Fresnel reflection
amplitudes, ( )2cosexp ϑα−=η l is the transmission
factor; 2cos4 ϑ⎟
⎠
⎞⎜
⎝
⎛
λ
π=δ nl ; and the angle 2ϑ is related to
the external angle 1ϑ by the relation 21 sinsin ϑ=ϑ n .
For a layer on the absorbing substrate (when the
whole structure is non-transparent), one obtains
( )( )
νννν
νν
ν
η+δη+
η−−
=
21
2
2312
2
2
1
cos21
11
RRrr
RR
A . (3)
Equations (2) and (3) describe the quickly
oscillating dependence of emissivity on the radiation
wavelength λ at a fixed observation angle 1ϑ . For a
given λ, the intensity is an oscillating function of the
angle 1ϑ . The positions of extrema are determined by
the interference conditions
m
n
l
4
cos 2
λ
=ϑ , (4)
where m >> 1 are integers: even numbers for peaks and
odd ones for minima (if r12ν r23ν < 0), or odd numbers for
peaks and even ones for minima (if r12ν r23ν > 0).
Conditions (4) coincide with those determining the
positions of peaks and minima of the intensity of light
passing through a Fabry-Perot interferometer.
Equations (2) and (3) are valid also for a layer with
mirrors (dielectric or metal coatings) deposited onto its
large faces (these mirrors change the reflection
coefficients). In such a structure, due to additional phase
jumps at the layer faces, the positions of oscillation
peaks differ from those given by Eq. (4). However, the
oscillation amplitudes are determined by the same
equations. From here on, we refer to the interfaces with
reflection coefficients R1 and R2 as mirrors 1 and 2,
respectively.
Let us define the amplitude of the emissivity peaks
for each polarization as ∆A = Amax − Amin (the subscript ν
is omitted), where Amax and Amin are the emissivity values
in the peak and the adjacent minimum, respectively. A
TR line corresponds to every emissivity peak. The
difference between the intensities in a peak and in an
adjacent minimum will be referred to as the line
amplitude ∆J. Let us determine the radiation line width
(∆λ) at a half amplitude:
22 minmax
JJJ ∆
=−⎟
⎠
⎞
⎜
⎝
⎛ λ∆
+λ . (5)
Taking into account that ∆λ << λmax (for thick
layers), one obtains from Eq. (5):
( )
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
η+
η
π
λ
=λ∆
21
2
21
21
2
max
1
2
arccos
2 RR
RR
ln
. (6)
One can see from Eq. (6) that an increase of the
parameter ( ) 21
21RRη leads to the narrowing of the
radiation line width.
We also studied how the TR spectrum depends on
some external actions such as a magnetic field and a
uniaxial deformation. Let us consider TR emitted along
a normal to the layer in a constant magnetic field B =
(0, 0, B) (the Faraday configuration). Inside the layer,
the normal waves are circularly polarized waves with
complex refractive indices 21)( xyxx in εε=± m . The
labels “+” and “−“ are referred to waves with the right
and left polarizations, respectively; the components of
the dielectric tensor ),( Bωεαβ obey the conditions
yxxyyyxx ε−=εε=ε , .
In order to explain the magnetic field effect more
clearly, we do not use Kirchhoff’s law but obtain the
spectral intensity of TR from a direct calculation. We
assume that the layer radiates due to thermal fluctuations
of the electric current density [14]. For the TR intensity,
we use Eq. (1), where now
A = (A+ + A− ) / 2. (7)
Here, A± is the plate emissivity attributed to the
mode “±”. Note that A(B) = A(−B) and the emissivity
coincides with the layer absorptivity. Hence, the
intensity obtained obeys Kirchhoff’s law.
The spectral intensity of radiation is determined by
the sum of the partial intensities attributed to the “+” and
“−“ modes, and the interference conditions (4) are valid
for each mode. Since −+ ≠ nn , the positions of the
extrema in the “±” spectra generally do not coincide, and
this may lead to a “beat effect” resulting in a low-
frequency modulation of the total spectrum.
The effect of a low-frequency modulation of the
TR spectrum by an external magnetic field is analogous,
in its nature, to that observed in plane-parallel plates
with anisotropic (due to a uniaxial strain) permittivity. In
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 94-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
96
the plate ( lz ≤≤0 ) of a uniaxial crystal with the optical
axis oriented along the x-axis, the light waves
propagating along the z-axis are the two waves with
different indices of refraction: ||n and ⊥n for the waves
with the electric vectors E = (E, 0, 0) and E = (0, E, 0),
respectively. In the linear (in the strain) approximation,
Pnn γ+=⊥ , Pnn σγ−=|| . (8)
Here, γ is a phenomenological coefficient, P
pressure, and σ the Poisson’s ratio.
The polarizations of these waves are mutually
perpendicular, and the interference between them does
not occur. Hence, the TR spectrum is the sum of the
partial spectra attributed to each type of waves, and a
low-frequency modulation of the total spectrum can take
place.
3. Experimental results
In our experimental studies of TR, we used the plane-
parallel plates of n-GaAs (the concentration of
uncompensated donors Nd – Na = 1×1018 cm−3), n-Si
(Nd – Na = 6×1017 cm−3 and 6×1016 cm−3), n-InAs (Nd –
Na = 1.4×1018 cm−3) single crystals and the
Pb0.8Sn0.2Te/BaF2 films of р-type (Na − Nd =
2×1018 сm−3). The plane-parallel plates were fabricated
using a mechanical grinding with a subsequent polishing
of their broad faces. The deviation from plane-
parallelism was estimated from the transmission spectra.
The plates with the biggest difference of intensities in
the interference maxima and minima of the transmission
spectra were chosen for the measurements. The plates,
for which the interference effect was negligible (e.g.,
wedge-shaped plates), were used in our experiments for
comparison. In what follows, we refer to these samples
as non-plane-parallel plates.
The plates had lateral dimensions of about
10×15 mm and thickness ≈ 100 µm. The thickness of the
films obtained by laser evaporation [15] was 1.1 µm.
The measurements were performed in the spectral region
of free charge carrier absorption using a Fourier-
spectrometer with a spectral resolution of 1 cm−1. The
aperture of the optical equipment inlet did not exceed
2.5○. The intensity of radiation was normalized to the
blackbody radiation measured under the same
experimental conditions.
3.1. TR angular distributions
The angular distributions of the TR from the plane-
parallel Si plate (Nd – Na = 6×1017 cm−3) are shown in
Figs. 1a and b. The solid curves show the calculated
ratios between the emissivities )( 1ϑνA and )( 1ϑνA of
the plane-parallel and non-plane-parallel plates,
respectively, with different directions of polarizations
(λ = 10.57 µm). One can see that, because of the
multiple-beam interference, the radiation directional
pattern has a number of pronounced lobes. The angular
dependences )()( 11 ϑϑ νν AA for the s- and p-polarized
emission (Fig. 1b and the left-hand side of the pattern in
Fig. 1a) differ markedly from each other. In the 55°−75°
range, the extrema for the p-polarized radiation virtually
disappear; this is related to a drop (almost to zero) of the
reflectance Rp. The angular dependences for the s-
polarized and unpolarized emissions are similar in many
respects, see Fig. 1a. One can see that the calculated and
experimental data are in good agreement (in our
calculations, we took the dependences α(λ) and n2(λ)
obtained from our measurements into account).
Fig. 1. Angular distributions of TR from n-Si plane-parallel
plates for λ = 10.57 µm and l = 92 µm. The solid lines
correspond to the results of calculation, while the dots
correspond to the experimental results. (a) and (b) − the ratios
between the emissivities of the plane-parallel and identical
non-plane-parallel plates; (c) − the emissivities of plane-
parallel plates. (a): the left-hand side corresponds to s-
polarization; the right-hand side, to unpolarized radiation. (b)
corresponds to p-polarization. (c): to s-polarization. The
parameters used in the calculation: (a, b) − η = 0.93, n = 3.4;
(c) − plates with l = 179 µm and n = 3.4 (right-hand side),
n = 6 (left-hand side of the pattern), η = 0.87.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 94-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
97
A more pronounced TR pattern can be obtained in
the angular dependences if R1,2 of the plane-parallel plate
are increased. This effect can be seen from Fig. 1c that
shows the calculated angular dependences for the
emissivities As( 1ϑ ) of a plate with the refractive index
n = 6 (the left-hand side of the pattern) and that with n =
3.4 (the right-hand side of the pattern). For both plates,
l = 179 µm and η = 0.87 (for 1ϑ = 0).
3.2. TR spectral dependences
Oscillating curve 1 in Fig. 2 shows the spectral
dependence (measured at the angle 1ϑ = 60° to the
normal) of the TR intensity for a plane-parallel n-GaAs
plate with an Al layer on its back face. (In this case, s-
polarization is considered.) One can see that TR from a
plane-parallel plate is drastically modified as compared
to that from a blackbody (curve 2). The radiation
intensity in peaks practically approaches that of a
blackbody.
Figure 3 shows the experimental spectral depen-
dences for the emissivities of the plane-parallel GaAs
plate (curve 1) and the non-plane-parallel plate (steadily
increasing curve 2). Curve 3 shows the measured
variance of the optical thickness al for the plane-parallel
GaAs plate. The oscillations are pronounced better in the
wavelength area where absorption is weak. For
λ > 14 µm, when absorption becomes strong (αl > 2.5,
see Fig. 3, curve 3), the multibeam interference is
suppressed, and the oscillations practically vanish in the
noise. One can see that the maxima and minima of the
emission from the plane-parallel plate are arranged
virtually symmetrically relative to the curve of TR from
the non-plane-parallel plate.
Fig. 2. Spectral dependence of s-polarized TR: 1 − from a
plane-parallel n-GaAs plate with an Al layer on its back face.
The viewing angle 1ϑ = 60°, R1S ≈ 0.51, T = 365±2 K. 2 − from
a blackbody under the same experimental conditions. The inset
shows the same spectra in the λ = 7.95−8.15 µm area. Marks –
experimental J(λ) values, the curve – the result of the fitting to
the experimental values.
The curves in Fig. 4 present the emissivity of a thin
Pb0.8Sn0.2Te film; TR is emitted along the normal to the
film surface. In the calculation of A(λ), we took the
dispersion of α and n in the film into account that was
determined from our independent measurements (for
Pb0.8Sn0.2Te films, n ≈ 6). Note that the thick substrate
(L = 1.5 mm) with high transparency and small
refraction coefficient (n3 ≈ 1.4) contributed practically
nothing to TR. One can see from Figs. 2 and 4 that a
good agreement is obtained between the measurements
and the theory for both thick (λ /n << l, plates) and thin
(λ /n ≈ l, films) layers.
3.3. The amplitudes of TR lines
To investigate the dependence of the amplitudes of
emissivity peaks on the coefficients of absorption and
reflection, we chose n-GaAs (the concentration of
uncompensated donors Nd – Na = 1.1×1018 cm−3). Its
absorption coefficient beyond the fundamental
Fig. 3. Experimental spectral dependences of emissivity A(λ)
(curves 1, 2) and optical thickness α l (curve 3) for the n-
GaAs plate; 1 is a plane-parallel plate, and 2 is the identical
non-plane-parallel one. The viewing angle 1ϑ = 0°.
Fig. 4. Spectral emissivity of a Pb0.8Sn0.2Te film (l = 1.1 µm).
1 – experiment, 2 – theory.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 94-102.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
98
absorption edge is characterized by strong dispersion:
α(λ)∝ λ3 [16]. This fact ensures the variation of the
transmission factor η(λ) from η ≈ 0.98 down to η ≈ 0.01
in the spectral range 3−17 µm used by us. One can
follow the variation of the radiation line amplitude in a
wide transmission range by presenting the radiation
spectrum as a function of η(λ).
The curves in Fig. 5 present the theoretical and
experimental dependences of the amplitudes of
emissivity peaks, ∆A, on the transmission factor η at
different mirrors on the faces. Figure 5а demonstrates
∆A(η) for a symmetric structure with non-absorbing
mirrors (R1 = R2). The asymmetric cases – the plates with
absorbing (e.g., metal) mirror 2 and non-absorbing
mirror 2 – are illustrated by Figs. 5b and с, respectively.
One can see that the ∆A(η) curves are
nonmonotonic and have a peak, whose position (ηmax)
and height (∆Amax) depend on the reflection value. The
∆Amax value grows and shifts towards the higher
transparency region as the coefficient of reflection from
the radiating face increases.
This maximum is achieved at different values of
the transmission factor, depending on the reflection from
faces. One can see that ηmax increases with R1 and R2. If
good resonator conditions in a plate are ensured, then the
radiation redistribution over wavelengths is so strong
that the radiation line amplitude approaches the
blackbody radiation intensity.
However, one can see from Fig. 5b that, in the
case of a sample with an absorbing mirror at its back
side, ∆A(η) does not tend to zero as η→1. A nonzero
value of ∆A at η = 1 is due to the additional
illumination from the non-transparent substrate
(absorbing mirror) on the back surface. The substrate
emissivity is small: in our case, 1 − R2 = 0.05 for
curve 3 in Fig. 5b and 0.19 for curves 1 and 2 in
Fig. 5b. However, its contribution to TR is
considerable. Under sufficiently strict resonator
conditions, just this radiation may become predominant
in the total radiation from the structure. One can see
from curve 3 in Fig. 5b that the amplitude of radiation
lines for an absolutely transparent (η = 1) structure
with an absorbing mirror (R2 = 0.95) and a non-
absorbing mirror at the radiating face (R1 = 0.94)
practically approaches the intensity of blackbody
radiation. It should be noted that curve 3 in Fig. 5b is
also nonmonotonic. Its peak at η ≈ 1 is pronounced
very weakly and is not seen on the plot.
If non-absorbing mirrors are applied, then there is
no additional illumination effect, see Fig. 5с. Such
mirror coatings can be realized using an interference
multilayer structure made of IR-transparent materials.
The ∆A(η) curves for non-symmetric structures are
qualitatively similar to those in the symmetric case
(Fig. 5а). However, at high R2 value in a non-symmetric
structure, the amplitude of spectral lines is considerably
higher and may reach that for a blackbody.
Fig. 5. Dependences of the emissivity peak amplitudes ∆A on
the transmission factor η for plane-parallel plates with different
coefficients of reflection from the front (radiating) (R1) and
back (R2) faces. Curves – the results of calculation. Marks –
the results of analysis of the experimental spectra of s-
polarized TR from n-GaAs plates (Nd – Na = 1.1×1018 cm−3, l =
100 µm) at different viewing angles 1ϑ . T = 365±2 K. (а) –
mirrors 1 and 2 are non-absorbing, R1 = R2. 1 – R1 = 0.27; 2 –
R1 = 0.43; 3 − R1 = 0.64; 4 − R1 = 0.95. Marks – a free n-GaAs
plate, R1S=R2S: – 1ϑ = 0°, R1S≈0.27; − 1ϑ =50°, R1S ≈ 0.43;
− 1ϑ = 70°, R1S≈0.64. (b) – mirror 1 is non-absorbing, mirror
2 is absorbing. 1 – R1 = 0.27, R2 = 0.81; 2 – R1 = 0.52, R2 =
0.81; 3 – R1 = 0.94, R2 = 0.95. Marks – an n-GaAs plate with
an Al layer on the back face, R2=0.81. – 1ϑ = 0°, R1S ≈ 0.27;
– 1ϑ = 60°, R1S≈0.52. (c) – mirrors 1 and 2 are non-absorbing,
R2 = 1. 1 – R1 = 0.27; 2 – R1 = 0.65; 3 – R1 = 0.85.
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The ∆A(η) dependences (obtained from the
analysis of the experimental TR spectra for plane-
parallel GaAs plates) are shown in Figs. 5а and b by
marks. On the abscissa, the mean η value in the
λmax−λmin spectral range is plotted for each mark. Since
the η variation in those ranges did not exceed 6 %, one
can conclude that the error of such a representation is
insignificant. When calculating the J(η) dependence
from the experimental spectrum J(λ), we used the
dispersion dependence η(λ) that was determined by us
experimentally for the given material from its
transmission and reflection spectra.
Figure 5а demonstrates the symmetric case (the
reflection coefficients at both faces are the same). The
GaAs plate surfaces were clean, and the R1 and R2 values
were varied by changing the angle of outgoing radiation
viewing. In Fig. 5b, we show the ∆A(η) dependences for
an n-GaAs plane-parallel plate with an aluminum mirror
deposited onto the back face. In this case, the coefficient
of reflection from the back face was R2 = 0.81 (this value
was determined by us experimentally).
The curve in Fig. 6 presents the dependence of the
relative radiation line width ( 2
maxλλ∆ ) on the
parameter η(R1 R2)1/2. The marks show the 2
maxλλ∆
values determined from the experimental TR spectra for
the n-GaAs plates (with free faces and with an aluminum
mirror on the back face).
We determined ∆λ for radiation lines in a narrow
spectral range (λ = 8−9 µm), where n varies insigni-
ficantly (in our calculations, we used the n value
averaged over the above spectral range: n = 3.18). When
analyzing the line parameters, we applied the fitting to
the experimental results. There is a good agreement
between the experimental results and those of theoretical
calculations.
Fig. 6. Dependence of the relative radiation line width
∆λ/∆λ2
max on the parameter η(R1R2)1/2; n2 = 3.18; l = 100 µm.
The curve is the result of calculation. Marks are the results of
analysis of experimental data in the λ = 8−9 µm range: –
1ϑ = 0°, R1 = R2 ≈ 0.27; – 1ϑ = 50°, R1S = R2S ≈ 0.43; –
1ϑ = 70°, R1S = R2S ≈ 0.64; – 1ϑ = 0°, R1 ≈ 0.27, R2 = 0.81;
– 1ϑ = 60°, R1S ≈ 0.52, R2 = 0.81.
3.4. Influence of a magnetic field on the TR spectrum
For the measurements of spectra, the plane-parallel
plates made of n-InAs were used (this material is
isotropic in the absence of a magnetic field). The high
concentration of free electrons (Ne = 1.4×1018 cm−3)
made it possible to carry out the measurements in a
classical magnetic field: the Landau splitting energy ħωc
is small as compared with kT (ωc is the cyclotron
frequency). The 10×13 mm samples had the thickness
l = 100 µm; the deviation from plane-parallelism was no
more than a few seconds of arc.
The plate was placed between the magnet poles so
that the magnetic field was directed normally to the
broad faces of the sample, and TR emitted along the
magnetic field was measured.
In Fig. 7, we present the experimental TR spectra
taken without magnetic field (Fig. 7a) and in the
magnetic field B = 1.3 T (Fig. 7e). In the magnetic field,
a bottleneck appears at nearly λ ≈ 9.2 µm where the
interference extrema practically disappear. As the
wavelength grows, the interference becomes apparent
again; the spectrum extrema, however, are in antiphase,
as compared with those at B = 0.
The polarization properties of TR were
demonstrated by its transformation into the linear
polarized radiation. For this purpose, we used a Ge total
internal reflection prism with base angles of 42°56′.
Since the polarization planes of right- and left-hand
polarized radiation are mutually perpendicular, we
separated the required radiation mode by applying a
polarizer-analyzer.
Figures 7b and 7c show the spectra of the right- and
left-hand circular polarized modes of TR, respectively.
Since n+ ≠ n−, the positions of extrema of the spectrum in
Figs. 7b and 7c generally do not coincide. Such a
difference in the partial spectra leads to a “beat effect”
resulting in a low-frequency modulation of the total
spectrum. One can see that the oscillation phase of the
right-hand circularly polarized mode takes the lead over
that of the left-hand circularly polarized mode, and they
are in antiphase at λ ≈ 9.2 µm. The oscillations in the
total spectrum (recorded without a polarizer-analyzer,
Fig. 7e) are not observed practically in this wavelength
range. In Fig. 7d, we give the arithmetic sum of the
spectra of the right-hand and left-hand circularly
polarized modes; it agrees well with the total TR
spectrum shown in Fig. 7e. This result proves that the
mode “+” (“−”) had a right-hand (left-hand) circular
polarization in our experiments. Some insignificant
distinction between the spectrum amplitudes in Figs. 7e
and d is due to the losses introduced by a polarizer-
analyzer. The theoretical spectra agree well with the
experimental ones for the electron effective mass
m = 0.04 me (this value was determined by the
comparison and adjustment of the calculated and
measured spectra; it is in good agreement with the
known data [17]).
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Fig. 7. Experimental spectra of TR from an n-InAs plate (Ne =
1.4×1018 cm−3, l = 100 µm, T = 375 K) without magnetic
field (a) and in the magnetic field B = 1.3 T. (b) – right-hand
circularly polarized mode; (c) – left-hand circularly polarized
mode; (d) – a sum of (c) and (d) spectra; (e) – the spectrum
recorded without analyzer.
It is worth to note that the reported effect appears in
the radiation of intrinsic unpolarized light and thus
differs in principle from the known magnetooptical
effects such as the Faraday and Voigt effects or
magnetoplasma reflection that describe the peculiarities
of the transmission (or reflection) of external polarized
light. It should be emphasized that the effect under
consideration is realized for TR with well-pronounced
coherent properties only.
3.5. Influence of uniaxial deformation on the TR
spectrum
When studying the TR and the transmission spectra of
uniaxially compressed plane-parallel plates, we used Si
with Nd – Na = 6×1017 cm−3 and 6×1016 cm−3,
respectively. The 5×8 mm crystals (the thickness
l = 350 µm) were compressed along their long face (i.e.,
along the [100] axis). We investigated the radiation
propagating along a normal to the direction of
compression, i.e., to the optical axis. The optical
surfaces of the sample coincided with the (100) plane. In
this case, the waves with the electric vector oriented
along and perpendicularly to the optic axis have different
velocities related to the refractive indices ||n and ⊥n ,
respectively. Therefore, the radiation leaving a
uniaxially strained crystal is a superposition of two
oscillating interference spectra, each of them
corresponding to ⊥n or ||n .
Fig. 8. Experimental transmission spectra of a uniaxially
strained plane-parallel Si sample. 1 – Р = 0, arbitrary direction
of the polarization plane of the detected radiation; 2 – Р =
6 kbar, the polarization plane is perpendicular to the direction
of compression; 3 – Р = 6 kbar, the polarization plane is
parallel to the direction of compression. Т = 355 K.
The samples were strained uniaxially in a special
facility; it ensured a mechanical stress that was uniform
over the sample cross section. The transmission spectra
were recorded when the sample was illuminated with
unpolarized radiation. Investigations of both TR and
transmission spectra were carried out in two ways,
namely, with and without an analyzer between the
sample under investigation and a spectrometer receptor.
The analyzer (a grating-polarizer) was placed so that its
axis was oriented either in parallel to the compression
direction or along a normal to it. The TR and
transmission spectra were measured in the 3–18-µm
wavelength range; the sample temperature was Т =
355 K.
In Fig. 8, we display the transmission spectra (in
the wavelength range 9.935–10.03 µm) of an unstrained
(curve 1) and strained (pressure Р = 6 kbar) samples
taken when the analyzer axis was perpendicular to the
compression direction (curve 2) and in parallel to it
(curve 3). (At Р = 0, the crystal is isotropic, so curve 1 is
repeated at any of the above analyzer orientations.) The
form of the spectra taken is typical of the transmission
interference spectra of plane-parallel samples. The
positions of maxima and minima in each spectrum are
determined by the refractive index n (curve 1), ⊥n (2),
and ||n (3), respectively. The spectral shifts of curves 2
and 3 relative to curve 1 determine birefringence
( ⊥n − ||n = 6×10−3 at Р = 6 kbar). We note that the TR
spectra are qualitatively similar to those of transmission
because the positions of maxima and minima in both
spectra of a plane-parallel plate are just the same if the
measurements are made at the same temperature.
The spectra of transmission of unpolarized light
and TR taken without an analyzer demonstrate the
presence of beats (low-frequency amplitude
modulation). Figure 9 presents the calculated emissivity
spectrum of an uniaxial deformed Si sample; the
parameters are as follows: l = 350 µm, n = 3.4,
⊥n = 3.4018, ||n = 3.3990 (P =3 kbar) and
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 94-102.
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Fig. 9. Calculated emissivity spectra of a uniaxially
compressed Si plane-parallel plate. Р = 3 kbar (a) and
4 kbar (b); Т = 355 K.
Fig. 10. Experimental transmission spectra of a uniaxially
compressed plane-parallel Si sample (no analyzer).
Р = 0 (a), 3 (b), and 4 kbar (c); Т = 355 K.
⊥n = 3.4025, ||n = 3.3986 (P = 4 kbar), the dependence
α(λ) was not taken into account. The values of ⊥n and
||n were obtained from an analysis of the experimental
transmission spectra. They correspond to those given by
Eq. (8) at γ = 7.3×10−7 bar−1 and σ = 0.3. One can see
that the beat node shifts towards a low-energy part of the
spectrum as the pressure P (and, with it, the difference
⊥n − ||n ) increases. The experimental transmission
spectrum is shown in Fig. 10 (the spectral bands near 9
and 16 µm are the regions of absorption induced by the
oxygen vibration absorption and the lattice absorption,
respectively). The position of the beat node, as well as
its spectral shift, is in close agreement with the
calculated data.
It should be noted that, as far as we know, the low-
frequency modulation in the transmission spectra of
anisotropic resonator objects has not been observed
before, and we seem to be the first to report it.
The investigation of the TR spectra under the
uniaxial compression, as well as under the action of an
external magnetic field, makes it possible to determine
the parameters of semiconductors at high temperatures.
4. Conclusions
In this paper, we have investigated the dependence of
coherent TR of semiconductor planar resonator
structures on their optical parameters, as well as the
spectrum modulation of coherent TR by an external
magnetic field and a uniaxial pressure.
Due to the multibeam interference, a smooth
spectral dependence of radiation is transformed into an
oscillating one with sharp peaks. In addition, the angular
dependence of TR does not obey Lambert’s law and
demonstrates a nonmonotonic character with clearly
pronounced lobes. We determined the optimal
conditions under which the intensity of TR of the planar
resonator structures may approach that of blackbody
radiation.
The results may be of use when developing new
thermal sources of IR radiation, whose operating spectral
band is given by the emitting element parameters. When
designing a controllable source or optical sensor
involving planar resonator structures as a radiating
element, one should take into account both the reflection
from interfaces and the absorption in emitting layers.
The optical parameters and hence the characteristics of
the radiating element can be modulated with the help of
the known methods of variation of the free charge carrier
concentration (e.g., by the contact injection or exclusion,
photogeneration, or the magnetoconcentration effect).
We have demonstrated a new magneto-optical
effect – the low-frequency modulation of a rapidly
oscillating spectrum of TR by an external magnetic field.
For the first time, we have investigated TR of
anisotropic crystals. We have shown that the low-
frequency modulation appears in the emission
(transmission) spectra of uniaxially deformed crystals.
The results obtained will serve to widen the
knowledge of emissive properties of different plane-
parallel objects (say, various films, layers, and coatings).
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They may be applied to the monitoring of layer
parameters and the proper account of heat exchange and
heat transfer as well.
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