Abstract

The propagation of light within a semiconductor Faraday-active Fabry–Perot resonator is investigated theoretically and experimentally. It is shown that an external magnetic field radically changes the angular and spectral characteristics of transmission, reflection, and emissivity of the resonator not only for polarized, but also for unpolarized, light. Suppression of interference patterns and phase inversion of the interference extrema were observed in both monochromatic and polychromatic light. The investigations were carried out for the plane-parallel plates of n-InAs in the spectral range of free charge carrier absorption. The results can be used to create new controllable optical and spectroscopic devices for investigation of Faraday-active material properties and for control of parameters of plane-parallel layers and structures.

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  1. M. Bass, J. M. Enoch, C. M. DeCusatis, V. Lakshminarayanan, G. Li, C. MacDonald, V. N. Mahajan, and E. Van Stryland, eds., Handbook of Optics, Vol. V of Atmospheric Optics, Modulators, Fiber Optics, X-Ray and Neutron Optics, 3rd ed. (McGraw-Hill, 2009).
  2. R. Rosenberg, C. B. Rubinstein, and D. R. Herriott, “Resonant optical Faraday rotation,” Appl. Opt. 3, 1079–1083 (1964).
    [CrossRef]
  3. V. A. Shamburov and E. A. Evdishchenko, “Exact Jones matrix for the plate from the natural gyrotropic nonmagnetic crystal,” Kristallografiya 38, 847–849 (1991) (in Russian).
  4. H. Y. Ling, “Theoretical investigation of transmission through a Faraday-active Fabry-Perot etalon,” J. Opt. Soc. Am. A 11754–758 (1994).
    [CrossRef]
  5. D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
    [CrossRef]
  6. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510(1972).
    [CrossRef]
  7. O. V. Ivanov and D. I. Sementsov, “Coherent and incoherent reflection and transmission of light in anisotropic layer structures,” Crystallogr. Rep. 45, 827–832 (2000).
    [CrossRef]
  8. I. V. Semchenko and V. E. Kaganovich, “Selective optical properties of a multilayered periodic gyrotropic structure at an arbitrary angle of incidence of waves,” Crystallogr. Rep. 49, 1032–1037 (2004).
    [CrossRef]
  9. D. G. Makarov, V. V. Danilov, and V. F. Kovalenko, “Multilayer structures with magnetically controlled light transmission,” Tech. Phys. 49, 598–602 (2004).
    [CrossRef]
  10. S. N. Kurilkina and A. L. Zykov, “Enhancement of the Faraday effect in finite three-layer periodic media,” Opt. Spectrosc. 98, 624–627 (2005).
    [CrossRef]
  11. O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
    [CrossRef]
  12. V. Morozhenko and O. G. Kollyukh, “Angular and spectral peculiarities of coherent thermal radiation of the magneto-optical Fabry-Perot resonator in magnetic field,” J. Opt. A 11, 085503 (2009).
    [CrossRef]
  13. O. Madelung, Semiconductors: Data Handbook (Springer-Verlag, 2004).
  14. K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
    [CrossRef]

2009

V. Morozhenko and O. G. Kollyukh, “Angular and spectral peculiarities of coherent thermal radiation of the magneto-optical Fabry-Perot resonator in magnetic field,” J. Opt. A 11, 085503 (2009).
[CrossRef]

2005

S. N. Kurilkina and A. L. Zykov, “Enhancement of the Faraday effect in finite three-layer periodic media,” Opt. Spectrosc. 98, 624–627 (2005).
[CrossRef]

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
[CrossRef]

2004

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

I. V. Semchenko and V. E. Kaganovich, “Selective optical properties of a multilayered periodic gyrotropic structure at an arbitrary angle of incidence of waves,” Crystallogr. Rep. 49, 1032–1037 (2004).
[CrossRef]

D. G. Makarov, V. V. Danilov, and V. F. Kovalenko, “Multilayer structures with magnetically controlled light transmission,” Tech. Phys. 49, 598–602 (2004).
[CrossRef]

2000

O. V. Ivanov and D. I. Sementsov, “Coherent and incoherent reflection and transmission of light in anisotropic layer structures,” Crystallogr. Rep. 45, 827–832 (2000).
[CrossRef]

1995

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

1994

1991

V. A. Shamburov and E. A. Evdishchenko, “Exact Jones matrix for the plate from the natural gyrotropic nonmagnetic crystal,” Kristallografiya 38, 847–849 (1991) (in Russian).

1972

1964

Berreman, D. W.

Bretenaker, F.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

Danilov, V. V.

D. G. Makarov, V. V. Danilov, and V. F. Kovalenko, “Multilayer structures with magnetically controlled light transmission,” Tech. Phys. 49, 598–602 (2004).
[CrossRef]

Evdishchenko, E. A.

V. A. Shamburov and E. A. Evdishchenko, “Exact Jones matrix for the plate from the natural gyrotropic nonmagnetic crystal,” Kristallografiya 38, 847–849 (1991) (in Russian).

Guga, K. Yu.

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

Herriott, D. R.

Ivanov, O. V.

O. V. Ivanov and D. I. Sementsov, “Coherent and incoherent reflection and transmission of light in anisotropic layer structures,” Crystallogr. Rep. 45, 827–832 (2000).
[CrossRef]

Jacob, D.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

Kaganovich, V. E.

I. V. Semchenko and V. E. Kaganovich, “Selective optical properties of a multilayered periodic gyrotropic structure at an arbitrary angle of incidence of waves,” Crystallogr. Rep. 49, 1032–1037 (2004).
[CrossRef]

Kollyukh, O. G.

V. Morozhenko and O. G. Kollyukh, “Angular and spectral peculiarities of coherent thermal radiation of the magneto-optical Fabry-Perot resonator in magnetic field,” J. Opt. A 11, 085503 (2009).
[CrossRef]

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
[CrossRef]

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

Kovalenko, V. F.

D. G. Makarov, V. V. Danilov, and V. F. Kovalenko, “Multilayer structures with magnetically controlled light transmission,” Tech. Phys. 49, 598–602 (2004).
[CrossRef]

Kurilkina, S. N.

S. N. Kurilkina and A. L. Zykov, “Enhancement of the Faraday effect in finite three-layer periodic media,” Opt. Spectrosc. 98, 624–627 (2005).
[CrossRef]

Le Floch, A.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

Le Naur, R.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

Ling, H. Y.

Liptuga, A. I.

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
[CrossRef]

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

Madelung, O.

O. Madelung, Semiconductors: Data Handbook (Springer-Verlag, 2004).

Makarov, D. G.

D. G. Makarov, V. V. Danilov, and V. F. Kovalenko, “Multilayer structures with magnetically controlled light transmission,” Tech. Phys. 49, 598–602 (2004).
[CrossRef]

Morozhenko, V.

V. Morozhenko and O. G. Kollyukh, “Angular and spectral peculiarities of coherent thermal radiation of the magneto-optical Fabry-Perot resonator in magnetic field,” J. Opt. A 11, 085503 (2009).
[CrossRef]

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
[CrossRef]

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

Pipa, V. I.

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
[CrossRef]

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

Rosenberg, R.

Rubinstein, C. B.

Semchenko, I. V.

I. V. Semchenko and V. E. Kaganovich, “Selective optical properties of a multilayered periodic gyrotropic structure at an arbitrary angle of incidence of waves,” Crystallogr. Rep. 49, 1032–1037 (2004).
[CrossRef]

Sementsov, D. I.

O. V. Ivanov and D. I. Sementsov, “Coherent and incoherent reflection and transmission of light in anisotropic layer structures,” Crystallogr. Rep. 45, 827–832 (2000).
[CrossRef]

Shamburov, V. A.

V. A. Shamburov and E. A. Evdishchenko, “Exact Jones matrix for the plate from the natural gyrotropic nonmagnetic crystal,” Kristallografiya 38, 847–849 (1991) (in Russian).

Vallet, M.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

Zykov, A. L.

S. N. Kurilkina and A. L. Zykov, “Enhancement of the Faraday effect in finite three-layer periodic media,” Opt. Spectrosc. 98, 624–627 (2005).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and R. Le Naur, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef]

Crystallogr. Rep.

O. V. Ivanov and D. I. Sementsov, “Coherent and incoherent reflection and transmission of light in anisotropic layer structures,” Crystallogr. Rep. 45, 827–832 (2000).
[CrossRef]

I. V. Semchenko and V. E. Kaganovich, “Selective optical properties of a multilayered periodic gyrotropic structure at an arbitrary angle of incidence of waves,” Crystallogr. Rep. 49, 1032–1037 (2004).
[CrossRef]

J. Opt. A

V. Morozhenko and O. G. Kollyukh, “Angular and spectral peculiarities of coherent thermal radiation of the magneto-optical Fabry-Perot resonator in magnetic field,” J. Opt. A 11, 085503 (2009).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Kristallografiya

V. A. Shamburov and E. A. Evdishchenko, “Exact Jones matrix for the plate from the natural gyrotropic nonmagnetic crystal,” Kristallografiya 38, 847–849 (1991) (in Russian).

Opt. Spectrosc.

S. N. Kurilkina and A. L. Zykov, “Enhancement of the Faraday effect in finite three-layer periodic media,” Opt. Spectrosc. 98, 624–627 (2005).
[CrossRef]

Phys. Rev. B

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Magnetic-field modulation of the spectrum of coherent thermal radiation of semiconductor layers,” Phys. Rev. B 71, 073306 (2005).
[CrossRef]

Semiconductors

K. Yu. Guga, O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, and V. I. Pipa, “Features of thermal radiation of plane-parallel semiconductor wafers,” Semiconductors 38, 507–511(2004).
[CrossRef]

Tech. Phys.

D. G. Makarov, V. V. Danilov, and V. F. Kovalenko, “Multilayer structures with magnetically controlled light transmission,” Tech. Phys. 49, 598–602 (2004).
[CrossRef]

Other

M. Bass, J. M. Enoch, C. M. DeCusatis, V. Lakshminarayanan, G. Li, C. MacDonald, V. N. Mahajan, and E. Van Stryland, eds., Handbook of Optics, Vol. V of Atmospheric Optics, Modulators, Fiber Optics, X-Ray and Neutron Optics, 3rd ed. (McGraw-Hill, 2009).

O. Madelung, Semiconductors: Data Handbook (Springer-Verlag, 2004).

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Figures (7)

Fig. 1.
Fig. 1.

Experimental setup for investigating (a) transmission and (b) reflection of a plane-parallel n-InAs plate in a magnetic field. The Fourier transform interferometer (FTIR) measures the resultant spectrum. The theoretical analysis is also for this FAFR–magnetic field arrangement.

Fig. 2.
Fig. 2.

Contour plots of the dependence of the angular distribution of the FAFR transmission on magnetic field. Incident light is (a)  s -polarized, (b)  p -polarized, and (c) unpolarized. λ = 7.96 μm . In a magnetic field, the interference maxima split into doublets, which change their positions with increasing H . They combine with neighboring maxima into a phase reversed single maxima. This effect exists in polarized as well as unpolarized light.

Fig. 3.
Fig. 3.

Contour plots of the dependence of the angular distribution of the FAFR reflection on magnetic field. Incident light is (a)  s -polarized, (b)  p -polarized, and (c) unpolarized. λ = 7.96 μm . In a magnetic field, the interference minima split into doublets, which change their positions with increasing H . They combine with neighboring minima into phase reversed single minima. This effect exists in polarized as well as unpolarized light.

Fig. 4.
Fig. 4.

Fringes of constant inclination of the FAFR for an n-InAs resonator of thickness l = 0.08 mm . The incident light is unpolarized. λ = 7.849 7.974 μm . (a)  H = 0 , (b)  H = 21 kG , and (с)  H = 42 kG . The wavelength scale is shown on top. The magnetic field changes the dispersion characteristics of the resonator. Near H = 21 kG the resonator ceases to be a dispersive element and no fringe pattern is observed.

Fig. 5.
Fig. 5.

Angular dependencies of the FAFR emissivity in magnetic field. λ = 7.96 μm . (a)  H = 0 ; (b)  H = 21 kG , χ = 0.25 ; (с)  H = 42 kG , χ = 0.5 . Angular dependence of the emissivity changes with the magnetic field, thus, changing the directional pattern of the FAFR thermal emission.

Fig. 6.
Fig. 6.

Experimental transmission spectra of the plane-parallel n-InAs plate at H = 0 (dotted curve) and H = 24 kG (solid curve) at a normal incidence of light. The incident light is unpolarized. As λ increases, the interference maxima split, initially minimizing intensity contrast and then combining to form maxima of inverted phase.

Fig. 7.
Fig. 7.

Experimental reflection spectra of the plane-parallel n-InAs plate at H = 0 (dotted curve) and H = 24 kG (solid curve). The incidence angle is near the Brewster angle ϑ = ( 71.5 ± 1 ) ° . The incident light is (a) unpolarized and (b)  p -polarized. (c) Theoretical reflection spectrum of the p -polarized light in the absence of absorption. (a) In a magnetic field, unpolarized light shows a change in interference contrast with increasing λ . (b) and (c) For p -polarized light at Brewster angle incidence, the magnetic field leads to the appearance of reflection as well as to interference of the reflected waves .

Equations (16)

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k 0 2 ε E k 2 E + k ( kE ) = 0 .
ξ = q / k 0 , s 1 = ( 1 ξ 2 ) 1 / 2 , s 2 = ( ε z z ξ 2 ) 1 / 2 .
k z ± = k 0 { ( 2 ε z z ) 1 [ 2 ε x x ε z z ξ 2 ( ε z z + ε x x ) ± f ] } 1 / 2 ,
E x ( z ) = A + exp ( i k z + z ) + B + exp ( i k z + z ) + A exp ( i k z z ) + B exp ( i k z z ) .
a 11 = exp ( i k z + l ) , a 22 = exp ( i k z l ) , a 33 = exp ( i k z + l ) , a 44 = exp ( i k z l ) ,
E s = C 1 S E t s ,
E p = C 1 P E t p ,
R ν = α | r ν α | 2 , T ν = α | t ν α | 2 .
b ν = ν 42 / ν 41 , d ν = ( b ν ν 11 + ν 12 ) 1 , F ν = | b ν ν 21 + ν 22 | 2 , G ν = | b ν ν 31 + ν 32 | 2 ,
R s = | d s | 2 ( 4 F s + G s ) , R p = | d p | 2 ( F p + 4 G p cos 2 ϑ ) ,
T ν = 4 | C d ν | 2 ( | b ν | 2 + cos 2 ϑ ) .
R = ( R s + R p ) / 2 , T = ( T s + T p ) / 2 .
A = 1 R T .
m 11 = C + ( s 1 + n z + ) , m 12 = C ( s 1 + n z ) , m 13 = C + ( s 1 n z + ) , m 14 = C ( s 1 n z ) , m 21 = m 22 = m 23 = m 24 = 1 , m 31 = m 13 , m 32 = m 14 , m 33 = m 11 , m 34 = m 12 , m 41 = s 2 2 + s 1 ε z z n z + , m 42 = s 2 2 + s 1 ε z z n z , m 43 = s 2 2 s 1 ε z z n z + , m 44 = s 2 2 s 1 ε z z n z ,
c 11 = C ( n z + + V ) / n z + , c 12 = ( s 1 + n z + ) / n z + , c 13 = c 14 = 0 , c 21 = C + ( n z + V ) / n z , c 22 = ( s 1 + n z ) / n z , c 23 = c 24 = 0 , c 31 = C ( n z + V ) / n z + , c 32 = ( s 1 n z + ) / n z + , c 33 = c 34 = 0 , c 41 = C + ( n z V ) / n z , c 42 = ( s 1 n z ) / n z , c 43 = c 44 = 0.
n 11 = m 41 / s 2 2 , n 12 = m 42 / s 2 2 , n 13 = m 43 / s 2 2 , n 14 = m 44 / s 2 2 , n 21 = m 43 / s 2 2 , n 22 = m 44 / s 2 2 , n 23 = m 41 / s 2 2 , n 24 = m 42 / s 2 2 , n 31 = C + , n 32 = C , n 33 = C + , n 34 = C , n 41 = m 11 , n 42 = m 12 , n 43 = m 13 , n 44 = m 14 .

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