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Manifestation of the Faraday effect in non-polarized light under optical resonance conditions
Optics Communications 426 (2018) 423–426
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Manifestation of the Faraday effect in non-polarized light under optical resonance conditions Vasyl Morozhenko *, Volodymyr Maslov, Nataliya Kachur V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 41 Pr. Nauky, Kyiv, 03028, Ukraine
ARTICLE
INFO
Keywords: Faraday effect Transmission Reflection Interference Resonator structure Magneto-controlled optical devices
ABSTRACT Effect of magnetic field on interaction of non-polarized light with a one-dimensional magneto-optic resonator has been investigated. Attention has been given to resonators with isotropic zero-field magneto-optic medium inside them. It has been shown that, in the presence of multi-beam interference, the Faraday effect is also observed in non-polarized light. Faraday rotation is manifested in significant changes of transmission and reflection spectral characteristics of the resonator in magnetic field. It has been ascertained that magnetic field substantially changes the interferential pattern for transmission and reflection of non-polarized light. The results of investigations show the possibility to use the magneto-optical resonators and resonator structures for controlling external nonpolarized light passing through them.
1. Introduction
2. Model and theory
As known, the Faraday effect is rotation of the light polarization plane in magnetic field, and when the selected plane of polarization is absent this effect does not appear. For this reason, all studies of Faraday rotation were carried out with the linearly polarized light. Numerous studies of the Faraday effect in magneto-optical resonators (MOR) are not an exception. Recently, investigations of Faraday rotation have been carried out in the Fabry–Perot cavity [1–4], 1D [5–12], 2D [12– 18] and 3D [12,19–21] magnetophotonic crystals as well as in the optical Tamm structures [22–24]. Synthesis of new magnetic materials, selection of their dimensions and location make it possible to create a new generation of magneto-optical devices for controlling coherent polarized light. In this paper, we have considered the possibility to use non-polarized light in magneto-optical phenomena observed in the resonator structures. Attention has been given to 1D resonators with isotropic zerofield magneto-optical medium inside. The presented here theoretical approach uses the multipath summation, taking into account both the phase difference and the difference in polarization directions. Unlike previous works, the experimental studies were carried out in the middle infrared (IR) spectral range. Interest in these studies is related with the possibility to create new methods for studying substances, as well as to develop new magnetically-operated IR devices for controlling external incoherent and non-polarized light passing through them.
Let us consider 1D-MOR that consists of a magneto-optical layer (MOL) sandwiched between two isotropic mirrors, as it is shown in Fig. 1. In general, the mirrors can be either single-layer or multi-layer Bragg structures. They are characterized by amplitudes of reflection 𝑟1 and 𝑟2 . This MOL is characterized by an isotropic zero-field complex refractive index 𝑛̃0 = 𝑛0 +𝑖𝜒0 (𝜒0 ≪ 𝑛0 ), Verdet constant 𝑉 and thickness 𝑑0 . The external magnetic field (H ) is directed along the normal to the surfaces of MOR. Let a non-polarized light with the wavelength 𝜆 falls at the angle 𝜃 on the mirror 𝑟1 . Since the incident light is non-polarized, it contains equal quantities of the linearly polarized components with any planes of polarization. Propagating inside the MOR, the light refracts and reflects back in its volume and splits into the series of coherent secondary waves 𝐸𝜉𝑟 and 𝐸𝜉𝑡 . Coherence of these waves is determined by coherence of their corresponding linearly polarized components. In magnetic field, the polarization directions of the corresponding components are different. When they are added at the output of resonator, the result depends both on the phase difference and on the magnitude of Faraday rotation. Since it happens with all the components, the total intensity of transmitted or reflected light is the sum of identical interference patterns. Calculations of optical characteristics inherent to MOR were carried out using the matrix method of multi-beam summation with account of Faraday rotation [25]. The matrix of electric field for the incident
*
Corresponding author. E-mail address: [email protected] (V. Morozhenko).
https://doi.org/10.1016/j.optcom.2018.05.062 Received 28 February 2018; Received in revised form 14 May 2018; Accepted 18 May 2018 0030-4018/© 2018 Published by Elsevier B.V.
V. Morozhenko et al.
Optics Communications 426 (2018) 423–426
Fig. 1. Scheme of the magneto-optical resonator.
non-polarized light can be written as: [ (𝑠) ] [ ] 𝐸 1 𝐄= = 𝐸 . 𝐸 (𝑝) 1
Fig. 2. Theoretical dependences of the magneto-optical resonator transmission on the phase difference under various values of the single-trip Faraday rotation angle: 1 – zero-field spectrum, 2 – 𝜑∕𝜋 = 0.07, 3 – 𝜑∕𝜋 = 0.25, 4 – 𝜑∕𝜋 = 0.43, 5 – 𝜑∕𝜋 = 0.5. The calculations were performed for normal incidence of nonpolarized light with the following resonator parameters: 𝑅1 = 0.7, 𝑅2 = 0.8, 𝜒 = 0.
(1)
The indices (s) and (p) determine the direction of polarization perpendicular to the plane of incidence and parallel to it, respectively. The time dependence of 𝐄 was omitted. Propagation of light inside MOR is described by the following matrices: ⎡𝑟(𝑠) 0 ⎤ ⎡𝑡(𝑠) 0 ⎤ 1,2 ⎥ , 𝐏1,2 = ⎢ 1,2 ⎥, 𝐑1,2 = ⎢ (𝑝) ⎢0 ⎢0 ⎥ −𝑟1,2 ⎥⎦ 𝑡(𝑝) ⎣ ⎣ 1,2 ⎦ (2) [ ] cos(𝜑) sin(𝜑) 𝑖𝑘 𝑑 𝐅= e 𝑧 0, − sin(𝜑) cos(𝜑) √ where 𝑘𝑧 = (2𝜋∕𝜆) 𝑛̃0 2 − sin2 (𝜃), 𝜑 = 𝑉 𝑑0 𝐻 is the single-trip Faraday
where
𝜉 = 0, 1, …. The matrix of the total transmitted light matrix series ) (∞ ∑ (𝐑2 𝐅𝐑1 𝐅)𝜉 𝐏2 . 𝐄𝐭 = 𝐄𝐏1 𝐅
√ 𝑅1 − 2𝜂 𝑅1 𝑅2 cos(2𝛥± ) + 𝜂 2 𝑅2 𝑅 = . √ √ (1 − 𝜂 𝑅1 𝑅2 )2 + 4𝜂 𝑅1 𝑅2 sin2 (𝛥 ± 𝜑)
(11)
±
2 Here,𝑅1,2 = ||𝑟1,2 || , 𝜂 = exp(−2𝜋𝜒0 𝑑0 ∕𝜆), 𝛥 = 𝛿0 + (𝛿1 + 𝛿2 )∕2, 𝛿0 = 2𝜋𝑛0 𝑑0 ∕𝜆, and 𝛿1,2 are the phase shift for reflection from the mirrors, which can be present in the case of metal or multilayer Bragg mirrors. As it is seen, the spectral distributions of T and R are the sums of two independent interference patterns that are shifted relatively to each other by the value 2𝜑. Thus, depending on the magnitude of the magnetic field they can be in phase, antiphase or in-between states. They can also be in the same state relatively to the zero-field distributions. The divergence of their maxima is observed in the complete distribution as a splitting of the interferential line.
(3) 𝐄𝐭
(10)
𝑇
(𝜈) (𝜈 = 𝑠, 𝑝) are the reflection and transmission and 𝑡1,2 rotation angle, 𝑟(𝜈) 1,2 amplitudes, respectively, for the front and back mirrors. The matrix for transmitted waves is as follows:
𝐄𝐭𝜉 = 𝐄𝐏1 𝐅(𝐑2 𝐅𝐑1 𝐅)𝜉 𝐏2 ,
( )( ) 1 − 𝑅1 1 − 𝑅2 𝜂 = , √ √ (1 − 𝜂 𝑅1 𝑅2 )2 + 4𝜂 𝑅1 𝑅2 sin2 (𝛥 ± 𝜑)
±
is the sum of
(4)
𝜉=0
3. Results and discussions
It is easy to determine that for the eigenvalue 𝜇 of matrix 𝐅𝐑1 𝐅𝐑2 the condition |𝜇| < 1 is always true. Hence, the sum in Eq. (4) can be replaced with the expression [26]: ( )−1 𝐄𝐭 = 𝐄𝐏1 𝐅 𝐈 − 𝐑2 𝐅𝐑1 𝐅 𝐏2 ,
3.1. Theoretical results Figs. 2 and 3 show the theoretical dependences of transmission and reflection of MOR on the single-trip Faraday angle for non-polarized light at normal incidence. To generalize this description, the phase difference 𝛥∕𝜋 was plotted along the abscissa axis. It is seen that for 𝜑 = 0 (𝐻 = 0) the distributions of transmission and reflection have a narrow-band character with high contrast. In magnetic field, the interference line of transmission splits into two secondary lines, which diverge and decrease in their amplitude with increasing the field. For 𝜑 = 𝜋∕4, amplitudes of the transmission lines reach a minimal value that is equal to half of the zero-field one. With increasing 𝜑, the secondary lines begin to merge pairwise with the neighboring one, and when 𝜑 = 𝜋∕2 the distribution of transmission takes a pronounced narrow-band character again. However, there is an inversion of the interference extremes. Positions of the lines correspond to the zero-field minima. With further increasing 𝜑, this process repeats, and for 𝜑 = 𝜋 transmission takes the zero-field form. Contrary to that of transmission, in case of reflection the interference minimum splits (see Fig. 3). Its change with increasing 𝜑 is similar to the change of transmission line: lines of minimum reflection diverge along the 𝛥-axis simultaneously changing in their amplitude. For 𝜑 = 𝜋∕2, two contrast lines of the minimum reflection are observed. They are located at the places of the zero-field maxima. For 𝜑 = 𝜋, the reflection also takes the zero-field form.
(5)
where I is the unity matrix. The transmission of MOR is: 𝑇 = ||𝐸 𝑡 || ∕|𝐸|2 = 2
1 ∑ | |2 𝑢 , 2 𝑙,𝑘 | 𝑙𝑘 |
(6)
( )−1 where ||𝑢𝑙𝑘 || are the elements of a matrix 𝐏1 𝐅 𝐈 − 𝐑2 𝐅𝐑1 𝐅 𝐏2 . By doing the foregoing operations, it is possible to obtain an equation for reflection R: 1 ∑ | |2 𝑅 = |𝐸 𝑟 |2 ∕|𝐸|2 = (7) |𝑤𝑙𝑘 | 2 𝑙,𝑘
( ( )−𝟏 ) Here, 𝑤𝑙𝑘 are elements of the matrix 𝐑1 − 𝐏1 𝐅𝐑2 𝐅 𝐈 − 𝐅𝐑1 𝐅𝐑2 𝐏1 . At normal incidence of light, it is possible to obtain from Eqs. (6) and (7) the following sufficiently compact analytical expressions for the transmission and reflection of MOR in magnetic field: 𝑇 =
] 1[ + 𝑇 + 𝑇− 2
(8)
] 1[ + 𝑅 + 𝑅− , 2
(9)
and 𝑅=
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Optics Communications 426 (2018) 423–426
Fig. 3. Theoretical dependences of the magneto-optical resonator reflection on the phase difference under various values of the single-trip Faraday rotation angle: 1 – zero-field spectrum, 2 – 𝜑∕𝜋 = 0.07, 3 – 𝜑∕𝜋 = 0.25, 4 – 𝜑∕𝜋 = 0.43, 5 – 𝜑∕𝜋 = 0.5. The calculations were performed for normal incidence of nonpolarized light with the following resonator parameters: 𝑅1 = 0.7, 𝑅2 = 0.8, 𝜒 = 0.
Fig. 5. Experimental transmission spectra of the free n-InAs plane-parallel plate under normal incidence of non-polarized light. 1 – 𝐻 = 0; 2 – 𝐻 = 19 kOe. 𝑁𝑞 = 1.4 ⋅ 1018 cm−3 , 𝑑0 = 100 μm.
Figs. 5 and 6 demonstrate the experimental spectra of transmission and reflection in the InAs resonators without magnetic field as well as in its presence. The curves 1 correspond to the zero-field spectra, and the curves 2 — to the spectra in the presence of magnetic field. As it is seen, the zero-field spectra have an oscillating character and consist of interference maxima and minima. This is a consequence of multibeam interference in the used resonator. As a result of the absorption dispersion, contrast of the zero-field interference patterns decrease with increasing the wavelength. The strong dependence of free carrier Faraday rotation on the light wavelength (𝜑 ∼ 𝜆2 ) [27] makes it possible to trace the changes in the interference pattern with increasing 𝜑. In magnetic field, with increasing 𝜆 and 𝜑, contrast of the spectra is reduced, and the waist is visible, followed by an increase in contrast and an inversion of the extremes in the transmission spectrum within the waist region, the splitting of interference maxima is observed. In the reflection spectrum, the appearance and growth of interference maxima in the place of the minima are clearly pronounced. As it follows from the presented theory, the position of the waist in the spectra corresponds to the single-trip Faraday angle 𝜋∕4. Substituting this wavelength into the expression for the Faraday angle, it is possible to determine parameters of semiconductor. The free-carrier Faraday angle is given by the known expression
3.2. Experimental results In experimental studies, heavy doped InAs semiconductor with a free electron concentration 𝑁𝑞 = 1.4 ⋅ 1018 cm−3 was used as the magnetooptical medium. This semiconductor is optically isotropic. Its small value of the electron effective mass causes significant Faraday rotation provided by free current carriers in the infrared range [27]. In the investigations of transmission, a free plane-parallel InAs plate was used. The broad free faces were the resonator mirrors. To investigate reflection, the structure composed of the InAs plate with an applied layer of aluminum on the back surface was used. The plates were cut from a single crystal bar, then ground and subsequently polished on the broad faces. The 10 × 6 mm2 samples had the thickness 0.1 and 0.08 mm. The deviation from plane parallelism was no higher than a few seconds of arc. Experimental setups to investigate transmission and reflection spectra in magnetic field are shown in Fig. 4a and 4b, respectively. The resonator plate was placed between the magnet poles in such a manner that the magnetic field was directed along the normal to broad faces of the sample. Non-polarized light from a thermal emitter was sent to the sample and input to the FTIR spectrometer by using a mirror system. The angle of incidence on the broad face of the sample did not exceed 5◦ . The spectra were recorded with the spectral resolution 2 cm−1 .
𝜑=
𝜋 𝑑 (𝑛 − 𝑛− ), 𝜆 0 +
Fig. 4. Experimental setups for investigating (a) the transmission and (b) reflection spectra of the resonator plate in magnetic field. 425
(12)
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Optics Communications 426 (2018) 423–426 [4] H.Y. Ling, Theoretical investigation of transmission through a Faraday-active Fabry– Perot étalon, J. Opt. Soc. Am. A. 11 (1994) 754–758. [5] M. Zamani, A. Hocini, Giant magneto-optical kerr rotation, quality factor and figure of merit in cobalt-ferrite magnetic nanoparticles doped in silica matrix as the only defect layer embedded in magnetophotonic crystals, J. Magn. Magn. Mater. 449 (2018) 435–439. [6] T.V. Mikhailova, V.N. Berzhansky, A.N. Shaposhnikov, A.V. Karavainikov, A.R. Prokopov, Y.M. Kharchenko, I.M. Lukienko, O.V. Miloslavskaya, M.F. Kharchenko, Optimization of one-dimensional photonic crystals with double layer magneto-active defect, Opt. Mater. (Amst). 78 (2018) 521–530. [7] A.H. Gevorgyan, S.S. Golik, Band structure peculiarities of magnetic photonic crystals, J. Magn. Magn. Mater. 439 (2017) 320–327. [8] D.O. Ignatyeva, G.A. Knyazev, P.O. Kapralov, G. Dietler, S.K. Sekatskii, V.I. Belotelov, Magneto-optical plasmonic heterostructure with ultranarrow resonance for sensing applications, Sci. Rep. 6 (2016). [9] D. Jahani, A. Soltani-Vala, J. Barvestani, H. Hajian, Magneto-tunable one-dimensional graphene-based photonic crystal, J. Appl. Phys. 115 (2014) 153101. [10] H. Da, G. Liang, Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene, Appl. Phys. Lett. 98 (2011). [11] K.H. Chung, T. Kato, S. Mito, H. Takagi, M. Inoue, Fabrication and characteristics of one-dimensional magnetophotonic crystals for magneto-optic spatial light phase modulators, J. Appl. Phys. 107 (2010) 09A930–09A930–2. [12] M. Inoue, A.V. Baryshev, A.B. Khanikaev, M.E. Dokukin, K. Chung, J. Heo, H. Takagi, H. Uchida, P.B. Lim, J. Kim, Magnetophotonic materials and their applications, IEICE Trans. Electron. E91–C (2008) 1630–1638. [13] Q. Li, L. Hu, Q. Mao, H. Jiang, Z. Hu, K. Xie, Z. Wei, Light trapping and circularly polarization at a Dirac point in 2D plasma photonic crystals, Opt. Commun. 410 (2018) 431–437. [14] W. Zhou, H. ming Chen, K. Ji, Y. Zhuang, Vertically magnetic-controlled THz modulator based on 2-D magnetized plasma photonic crystal, Photonics NanostructuresFundam. Appl. 23 (2017) 28–35. [15] R. Deghdak, M. Bouchemat, M. Lahoubi, S. Pu, T. Bouchemat, H. Otmani, Sensitive magnetic field sensor using 2D magnetic photonic crystal slab waveguide based on BIG/GGG structure, J. Comput. Electron. 16 (2017) 392–400. [16] S. Baek, A.V. Baryshev, M. Inoue, Multiple diffraction in two-dimensional magnetophotonic crystals fabricated by the autocloning method, J. Appl. Phys. 109 (2011) 07B701–07B701–3. [17] M.E. Dokukin, A.V. Baryshev, A.B. Khanikaev, M. Inoue, Reverse and enhanced magneto-optics of opal-garnet heterostructures, Opt. Express 17 (2009) 9062–9070. [18] Z. Wang, S. Fan, Optical circulators in two-dimensional magneto-optical photonic crystals, Opt. Lett. 30 (2005) 1989–1991. [19] A. Hocini, R. Moukhtari, D. Khedrouche, A. Kahlouche, M. Zamani, Magnetophotonic crystal microcavities based on magnetic nanoparticles embedded in silica matrix, Opt. Commun. 384 (2017) 111–117. [20] V.V. Pavlov, P.A. Usachev, R.V. Pisarev, D.A. Kurdyukov, S.F. Kaplan, A.V. Kimel, A. Kirilyuk, T. Rasing, Optical study of three-dimensional magnetic photonic crystals opal/Fe3 O4 , J. Magn. Magn. Mater. 321 (2009) 840–842. [21] R. Fujikawa, A.V. Baryshev, A.B. Khanikaev, J. Kim, H. Uchida, M. Inoue, Enhancement of Faraday rotation in 3D/Bi:YIG/1D photonic heterostructures, J. Mater. Sci., Mater. Electron. 20 (2009) 493–497. [22] J. Li, N. Tang, T. Li, J. Luo, L. Yao, Highly sensitive sensors of fluid detection based on magneto-optical Tamm state, Sensors Actuators B 265 (2018) 644–651. [23] Y.H. Wu, F. Cheng, Y.C. Shen, G.Q. Lu, L.L. Li, One-way transmission through merging of magnetic defect state and optical Tamm states, Optik (Stuttg) 127 (2016) 3740–3744. [24] A.B. Khanikaev, A.V. Baryshev, M. Inoue, Y.S. Kivshar, One-way electromagnetic Tamm states in magnetophotonic structures, Appl. Phys. Lett. 95 (2009) 011101– 011101-3. [25] V. Morozhenko, O.G. Kollyukh, Angular and spectral peculiarities of the coherent thermal radiation of the magneto-optical Fabry–Perot resonator in a magnetic field, J. Opt. A Pure Appl. Opt. 11 (2009) 085503–085503-6. [26] P. Lancaster, Theory of Matrices, Academic Press, New York-London, 1969. [27] Y. Ukhanov, Magnetooptic Faraday effect in semiconductors, Sov. Phys.-Usp. 16 (1973) 236–250. [28] O. Madelung, Semiconductors: Data Handbook, third ed., Springer-Verlag, Berlin Heidelberg, 2004. [29] M. Cardona, Electron effective masses of InAs and GaAs as a function of temperature and doping, Phys. Rev. 121 (1961) 752–758.
Fig. 6. Experimental reflection spectra for the n-InAs/Al plane-parallel structure under normal incidence of non-polarized light. 1 – 𝐻 = 0; 2 – 𝐻 = 18 kOe. 𝑁𝑞 = 1.4 ⋅ 1018 cm−3 , 𝑑0 = 80 μm.
where −1 𝑛2± = 𝑛20 𝜆2 𝜆𝑐 𝜆−2 𝑝 (𝜆𝑐 ± 𝜆) .
𝜆2𝑝
𝑐 2 𝑛20 𝑚∗ ∕𝑞 2 𝑁𝑞 ,
(13) 2𝜋𝑚∗ 𝑐 2 ∕𝑞𝐻,
𝜆𝑐 = 𝑞 is the electron charge, Here, = 𝑚∗ – carrier effective mass, 𝑐 – speed of light in vacuum. The determined in this way electron effective mass of used n-InAs was 𝑚∗ = 0.043𝑚𝑒 , where 𝑚𝑒 is the electron mass. This result agrees with the known data [28,29]. 4. Conclusion In summary, the results of investigations of transmission and reflection of a one-dimensional magneto-optical resonator in an external magnetic field have been presented for non-polarized light. It has been shown that, despite the stereotyped view, magneto-optical rotation of non-polarized light clearly manifests itself in the above optical characteristics of MORs. It has been ascertained that a magnetic field substantially changes the interference pattern of transmission and reflection for non-polarized light. It has been observed the inversion of interference extremes in the applied magnetic field: maxima are transformed into the minima and vice versa. The results of investigations have great practical importance. The MORs can be used as narrow-band optical modulators, switches and controllable dispersion elements for non-polarized light. These magnetooptical devices can be applied in instruments with different functions. These results can be used to develop new methods for studying the parameters of substances as well. References [1] H. Sun, Y. Lei, S. Fan, Q. Zhang, H. Guo, Cavity-enhanced room-temperature high sensitivity optical Faraday magnetometry, Phys. Lett. Sect. A Gen. At. Solid State Phys. 381 (2017) 129–135. [2] K. Sycz, W. Gawlik, J. Zachorowski, Resonant Faraday effect in a Fabry–Perot cavity, Opt. Appl. XL (2010) 633–639. [3] E. Taskova, S. Gateva, E. Alipieva, K. Kowalski, M. Glódź, J. Szonert, Nonlinear Faraday rotation for optical limitation, Appl. Opt. 43 (2004) 4178–4181.
426
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Manifestation of the Faraday effect in non-polarized light under optical resonance conditions Vasyl Morozhenko *, Volodymyr Maslov, Nataliya Kachur V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 41 Pr. Nauky, Kyiv, 03028, Ukraine
ARTICLE
INFO
Keywords: Faraday effect Transmission Reflection Interference Resonator structure Magneto-controlled optical devices
ABSTRACT Effect of magnetic field on interaction of non-polarized light with a one-dimensional magneto-optic resonator has been investigated. Attention has been given to resonators with isotropic zero-field magneto-optic medium inside them. It has been shown that, in the presence of multi-beam interference, the Faraday effect is also observed in non-polarized light. Faraday rotation is manifested in significant changes of transmission and reflection spectral characteristics of the resonator in magnetic field. It has been ascertained that magnetic field substantially changes the interferential pattern for transmission and reflection of non-polarized light. The results of investigations show the possibility to use the magneto-optical resonators and resonator structures for controlling external nonpolarized light passing through them.
1. Introduction
2. Model and theory
As known, the Faraday effect is rotation of the light polarization plane in magnetic field, and when the selected plane of polarization is absent this effect does not appear. For this reason, all studies of Faraday rotation were carried out with the linearly polarized light. Numerous studies of the Faraday effect in magneto-optical resonators (MOR) are not an exception. Recently, investigations of Faraday rotation have been carried out in the Fabry–Perot cavity [1–4], 1D [5–12], 2D [12– 18] and 3D [12,19–21] magnetophotonic crystals as well as in the optical Tamm structures [22–24]. Synthesis of new magnetic materials, selection of their dimensions and location make it possible to create a new generation of magneto-optical devices for controlling coherent polarized light. In this paper, we have considered the possibility to use non-polarized light in magneto-optical phenomena observed in the resonator structures. Attention has been given to 1D resonators with isotropic zerofield magneto-optical medium inside. The presented here theoretical approach uses the multipath summation, taking into account both the phase difference and the difference in polarization directions. Unlike previous works, the experimental studies were carried out in the middle infrared (IR) spectral range. Interest in these studies is related with the possibility to create new methods for studying substances, as well as to develop new magnetically-operated IR devices for controlling external incoherent and non-polarized light passing through them.
Let us consider 1D-MOR that consists of a magneto-optical layer (MOL) sandwiched between two isotropic mirrors, as it is shown in Fig. 1. In general, the mirrors can be either single-layer or multi-layer Bragg structures. They are characterized by amplitudes of reflection 𝑟1 and 𝑟2 . This MOL is characterized by an isotropic zero-field complex refractive index 𝑛̃0 = 𝑛0 +𝑖𝜒0 (𝜒0 ≪ 𝑛0 ), Verdet constant 𝑉 and thickness 𝑑0 . The external magnetic field (H ) is directed along the normal to the surfaces of MOR. Let a non-polarized light with the wavelength 𝜆 falls at the angle 𝜃 on the mirror 𝑟1 . Since the incident light is non-polarized, it contains equal quantities of the linearly polarized components with any planes of polarization. Propagating inside the MOR, the light refracts and reflects back in its volume and splits into the series of coherent secondary waves 𝐸𝜉𝑟 and 𝐸𝜉𝑡 . Coherence of these waves is determined by coherence of their corresponding linearly polarized components. In magnetic field, the polarization directions of the corresponding components are different. When they are added at the output of resonator, the result depends both on the phase difference and on the magnitude of Faraday rotation. Since it happens with all the components, the total intensity of transmitted or reflected light is the sum of identical interference patterns. Calculations of optical characteristics inherent to MOR were carried out using the matrix method of multi-beam summation with account of Faraday rotation [25]. The matrix of electric field for the incident
*
Corresponding author. E-mail address: [email protected] (V. Morozhenko).
https://doi.org/10.1016/j.optcom.2018.05.062 Received 28 February 2018; Received in revised form 14 May 2018; Accepted 18 May 2018 0030-4018/© 2018 Published by Elsevier B.V.
V. Morozhenko et al.
Optics Communications 426 (2018) 423–426
Fig. 1. Scheme of the magneto-optical resonator.
non-polarized light can be written as: [ (𝑠) ] [ ] 𝐸 1 𝐄= = 𝐸 . 𝐸 (𝑝) 1
Fig. 2. Theoretical dependences of the magneto-optical resonator transmission on the phase difference under various values of the single-trip Faraday rotation angle: 1 – zero-field spectrum, 2 – 𝜑∕𝜋 = 0.07, 3 – 𝜑∕𝜋 = 0.25, 4 – 𝜑∕𝜋 = 0.43, 5 – 𝜑∕𝜋 = 0.5. The calculations were performed for normal incidence of nonpolarized light with the following resonator parameters: 𝑅1 = 0.7, 𝑅2 = 0.8, 𝜒 = 0.
(1)
The indices (s) and (p) determine the direction of polarization perpendicular to the plane of incidence and parallel to it, respectively. The time dependence of 𝐄 was omitted. Propagation of light inside MOR is described by the following matrices: ⎡𝑟(𝑠) 0 ⎤ ⎡𝑡(𝑠) 0 ⎤ 1,2 ⎥ , 𝐏1,2 = ⎢ 1,2 ⎥, 𝐑1,2 = ⎢ (𝑝) ⎢0 ⎢0 ⎥ −𝑟1,2 ⎥⎦ 𝑡(𝑝) ⎣ ⎣ 1,2 ⎦ (2) [ ] cos(𝜑) sin(𝜑) 𝑖𝑘 𝑑 𝐅= e 𝑧 0, − sin(𝜑) cos(𝜑) √ where 𝑘𝑧 = (2𝜋∕𝜆) 𝑛̃0 2 − sin2 (𝜃), 𝜑 = 𝑉 𝑑0 𝐻 is the single-trip Faraday
where
𝜉 = 0, 1, …. The matrix of the total transmitted light matrix series ) (∞ ∑ (𝐑2 𝐅𝐑1 𝐅)𝜉 𝐏2 . 𝐄𝐭 = 𝐄𝐏1 𝐅
√ 𝑅1 − 2𝜂 𝑅1 𝑅2 cos(2𝛥± ) + 𝜂 2 𝑅2 𝑅 = . √ √ (1 − 𝜂 𝑅1 𝑅2 )2 + 4𝜂 𝑅1 𝑅2 sin2 (𝛥 ± 𝜑)
(11)
±
2 Here,𝑅1,2 = ||𝑟1,2 || , 𝜂 = exp(−2𝜋𝜒0 𝑑0 ∕𝜆), 𝛥 = 𝛿0 + (𝛿1 + 𝛿2 )∕2, 𝛿0 = 2𝜋𝑛0 𝑑0 ∕𝜆, and 𝛿1,2 are the phase shift for reflection from the mirrors, which can be present in the case of metal or multilayer Bragg mirrors. As it is seen, the spectral distributions of T and R are the sums of two independent interference patterns that are shifted relatively to each other by the value 2𝜑. Thus, depending on the magnitude of the magnetic field they can be in phase, antiphase or in-between states. They can also be in the same state relatively to the zero-field distributions. The divergence of their maxima is observed in the complete distribution as a splitting of the interferential line.
(3) 𝐄𝐭
(10)
𝑇
(𝜈) (𝜈 = 𝑠, 𝑝) are the reflection and transmission and 𝑡1,2 rotation angle, 𝑟(𝜈) 1,2 amplitudes, respectively, for the front and back mirrors. The matrix for transmitted waves is as follows:
𝐄𝐭𝜉 = 𝐄𝐏1 𝐅(𝐑2 𝐅𝐑1 𝐅)𝜉 𝐏2 ,
( )( ) 1 − 𝑅1 1 − 𝑅2 𝜂 = , √ √ (1 − 𝜂 𝑅1 𝑅2 )2 + 4𝜂 𝑅1 𝑅2 sin2 (𝛥 ± 𝜑)
±
is the sum of
(4)
𝜉=0
3. Results and discussions
It is easy to determine that for the eigenvalue 𝜇 of matrix 𝐅𝐑1 𝐅𝐑2 the condition |𝜇| < 1 is always true. Hence, the sum in Eq. (4) can be replaced with the expression [26]: ( )−1 𝐄𝐭 = 𝐄𝐏1 𝐅 𝐈 − 𝐑2 𝐅𝐑1 𝐅 𝐏2 ,
3.1. Theoretical results Figs. 2 and 3 show the theoretical dependences of transmission and reflection of MOR on the single-trip Faraday angle for non-polarized light at normal incidence. To generalize this description, the phase difference 𝛥∕𝜋 was plotted along the abscissa axis. It is seen that for 𝜑 = 0 (𝐻 = 0) the distributions of transmission and reflection have a narrow-band character with high contrast. In magnetic field, the interference line of transmission splits into two secondary lines, which diverge and decrease in their amplitude with increasing the field. For 𝜑 = 𝜋∕4, amplitudes of the transmission lines reach a minimal value that is equal to half of the zero-field one. With increasing 𝜑, the secondary lines begin to merge pairwise with the neighboring one, and when 𝜑 = 𝜋∕2 the distribution of transmission takes a pronounced narrow-band character again. However, there is an inversion of the interference extremes. Positions of the lines correspond to the zero-field minima. With further increasing 𝜑, this process repeats, and for 𝜑 = 𝜋 transmission takes the zero-field form. Contrary to that of transmission, in case of reflection the interference minimum splits (see Fig. 3). Its change with increasing 𝜑 is similar to the change of transmission line: lines of minimum reflection diverge along the 𝛥-axis simultaneously changing in their amplitude. For 𝜑 = 𝜋∕2, two contrast lines of the minimum reflection are observed. They are located at the places of the zero-field maxima. For 𝜑 = 𝜋, the reflection also takes the zero-field form.
(5)
where I is the unity matrix. The transmission of MOR is: 𝑇 = ||𝐸 𝑡 || ∕|𝐸|2 = 2
1 ∑ | |2 𝑢 , 2 𝑙,𝑘 | 𝑙𝑘 |
(6)
( )−1 where ||𝑢𝑙𝑘 || are the elements of a matrix 𝐏1 𝐅 𝐈 − 𝐑2 𝐅𝐑1 𝐅 𝐏2 . By doing the foregoing operations, it is possible to obtain an equation for reflection R: 1 ∑ | |2 𝑅 = |𝐸 𝑟 |2 ∕|𝐸|2 = (7) |𝑤𝑙𝑘 | 2 𝑙,𝑘
( ( )−𝟏 ) Here, 𝑤𝑙𝑘 are elements of the matrix 𝐑1 − 𝐏1 𝐅𝐑2 𝐅 𝐈 − 𝐅𝐑1 𝐅𝐑2 𝐏1 . At normal incidence of light, it is possible to obtain from Eqs. (6) and (7) the following sufficiently compact analytical expressions for the transmission and reflection of MOR in magnetic field: 𝑇 =
] 1[ + 𝑇 + 𝑇− 2
(8)
] 1[ + 𝑅 + 𝑅− , 2
(9)
and 𝑅=
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Fig. 3. Theoretical dependences of the magneto-optical resonator reflection on the phase difference under various values of the single-trip Faraday rotation angle: 1 – zero-field spectrum, 2 – 𝜑∕𝜋 = 0.07, 3 – 𝜑∕𝜋 = 0.25, 4 – 𝜑∕𝜋 = 0.43, 5 – 𝜑∕𝜋 = 0.5. The calculations were performed for normal incidence of nonpolarized light with the following resonator parameters: 𝑅1 = 0.7, 𝑅2 = 0.8, 𝜒 = 0.
Fig. 5. Experimental transmission spectra of the free n-InAs plane-parallel plate under normal incidence of non-polarized light. 1 – 𝐻 = 0; 2 – 𝐻 = 19 kOe. 𝑁𝑞 = 1.4 ⋅ 1018 cm−3 , 𝑑0 = 100 μm.
Figs. 5 and 6 demonstrate the experimental spectra of transmission and reflection in the InAs resonators without magnetic field as well as in its presence. The curves 1 correspond to the zero-field spectra, and the curves 2 — to the spectra in the presence of magnetic field. As it is seen, the zero-field spectra have an oscillating character and consist of interference maxima and minima. This is a consequence of multibeam interference in the used resonator. As a result of the absorption dispersion, contrast of the zero-field interference patterns decrease with increasing the wavelength. The strong dependence of free carrier Faraday rotation on the light wavelength (𝜑 ∼ 𝜆2 ) [27] makes it possible to trace the changes in the interference pattern with increasing 𝜑. In magnetic field, with increasing 𝜆 and 𝜑, contrast of the spectra is reduced, and the waist is visible, followed by an increase in contrast and an inversion of the extremes in the transmission spectrum within the waist region, the splitting of interference maxima is observed. In the reflection spectrum, the appearance and growth of interference maxima in the place of the minima are clearly pronounced. As it follows from the presented theory, the position of the waist in the spectra corresponds to the single-trip Faraday angle 𝜋∕4. Substituting this wavelength into the expression for the Faraday angle, it is possible to determine parameters of semiconductor. The free-carrier Faraday angle is given by the known expression
3.2. Experimental results In experimental studies, heavy doped InAs semiconductor with a free electron concentration 𝑁𝑞 = 1.4 ⋅ 1018 cm−3 was used as the magnetooptical medium. This semiconductor is optically isotropic. Its small value of the electron effective mass causes significant Faraday rotation provided by free current carriers in the infrared range [27]. In the investigations of transmission, a free plane-parallel InAs plate was used. The broad free faces were the resonator mirrors. To investigate reflection, the structure composed of the InAs plate with an applied layer of aluminum on the back surface was used. The plates were cut from a single crystal bar, then ground and subsequently polished on the broad faces. The 10 × 6 mm2 samples had the thickness 0.1 and 0.08 mm. The deviation from plane parallelism was no higher than a few seconds of arc. Experimental setups to investigate transmission and reflection spectra in magnetic field are shown in Fig. 4a and 4b, respectively. The resonator plate was placed between the magnet poles in such a manner that the magnetic field was directed along the normal to broad faces of the sample. Non-polarized light from a thermal emitter was sent to the sample and input to the FTIR spectrometer by using a mirror system. The angle of incidence on the broad face of the sample did not exceed 5◦ . The spectra were recorded with the spectral resolution 2 cm−1 .
𝜑=
𝜋 𝑑 (𝑛 − 𝑛− ), 𝜆 0 +
Fig. 4. Experimental setups for investigating (a) the transmission and (b) reflection spectra of the resonator plate in magnetic field. 425
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Optics Communications 426 (2018) 423–426 [4] H.Y. Ling, Theoretical investigation of transmission through a Faraday-active Fabry– Perot étalon, J. Opt. Soc. Am. A. 11 (1994) 754–758. [5] M. Zamani, A. Hocini, Giant magneto-optical kerr rotation, quality factor and figure of merit in cobalt-ferrite magnetic nanoparticles doped in silica matrix as the only defect layer embedded in magnetophotonic crystals, J. Magn. Magn. Mater. 449 (2018) 435–439. [6] T.V. Mikhailova, V.N. Berzhansky, A.N. Shaposhnikov, A.V. Karavainikov, A.R. Prokopov, Y.M. Kharchenko, I.M. Lukienko, O.V. Miloslavskaya, M.F. Kharchenko, Optimization of one-dimensional photonic crystals with double layer magneto-active defect, Opt. Mater. (Amst). 78 (2018) 521–530. [7] A.H. Gevorgyan, S.S. Golik, Band structure peculiarities of magnetic photonic crystals, J. Magn. Magn. Mater. 439 (2017) 320–327. [8] D.O. Ignatyeva, G.A. Knyazev, P.O. Kapralov, G. Dietler, S.K. Sekatskii, V.I. Belotelov, Magneto-optical plasmonic heterostructure with ultranarrow resonance for sensing applications, Sci. Rep. 6 (2016). [9] D. Jahani, A. Soltani-Vala, J. Barvestani, H. Hajian, Magneto-tunable one-dimensional graphene-based photonic crystal, J. Appl. Phys. 115 (2014) 153101. [10] H. Da, G. Liang, Enhanced Faraday rotation in magnetophotonic crystal infiltrated with graphene, Appl. Phys. Lett. 98 (2011). [11] K.H. Chung, T. Kato, S. Mito, H. Takagi, M. Inoue, Fabrication and characteristics of one-dimensional magnetophotonic crystals for magneto-optic spatial light phase modulators, J. Appl. Phys. 107 (2010) 09A930–09A930–2. [12] M. Inoue, A.V. Baryshev, A.B. Khanikaev, M.E. Dokukin, K. Chung, J. Heo, H. Takagi, H. Uchida, P.B. Lim, J. Kim, Magnetophotonic materials and their applications, IEICE Trans. Electron. E91–C (2008) 1630–1638. [13] Q. Li, L. Hu, Q. Mao, H. Jiang, Z. Hu, K. Xie, Z. Wei, Light trapping and circularly polarization at a Dirac point in 2D plasma photonic crystals, Opt. Commun. 410 (2018) 431–437. [14] W. Zhou, H. ming Chen, K. Ji, Y. Zhuang, Vertically magnetic-controlled THz modulator based on 2-D magnetized plasma photonic crystal, Photonics NanostructuresFundam. Appl. 23 (2017) 28–35. [15] R. Deghdak, M. Bouchemat, M. Lahoubi, S. Pu, T. Bouchemat, H. Otmani, Sensitive magnetic field sensor using 2D magnetic photonic crystal slab waveguide based on BIG/GGG structure, J. Comput. Electron. 16 (2017) 392–400. [16] S. Baek, A.V. Baryshev, M. Inoue, Multiple diffraction in two-dimensional magnetophotonic crystals fabricated by the autocloning method, J. Appl. Phys. 109 (2011) 07B701–07B701–3. [17] M.E. Dokukin, A.V. Baryshev, A.B. Khanikaev, M. Inoue, Reverse and enhanced magneto-optics of opal-garnet heterostructures, Opt. Express 17 (2009) 9062–9070. [18] Z. Wang, S. Fan, Optical circulators in two-dimensional magneto-optical photonic crystals, Opt. Lett. 30 (2005) 1989–1991. [19] A. Hocini, R. Moukhtari, D. Khedrouche, A. Kahlouche, M. Zamani, Magnetophotonic crystal microcavities based on magnetic nanoparticles embedded in silica matrix, Opt. Commun. 384 (2017) 111–117. [20] V.V. Pavlov, P.A. Usachev, R.V. Pisarev, D.A. Kurdyukov, S.F. Kaplan, A.V. Kimel, A. Kirilyuk, T. Rasing, Optical study of three-dimensional magnetic photonic crystals opal/Fe3 O4 , J. Magn. Magn. Mater. 321 (2009) 840–842. [21] R. Fujikawa, A.V. Baryshev, A.B. Khanikaev, J. Kim, H. Uchida, M. Inoue, Enhancement of Faraday rotation in 3D/Bi:YIG/1D photonic heterostructures, J. Mater. Sci., Mater. Electron. 20 (2009) 493–497. [22] J. Li, N. Tang, T. Li, J. Luo, L. Yao, Highly sensitive sensors of fluid detection based on magneto-optical Tamm state, Sensors Actuators B 265 (2018) 644–651. [23] Y.H. Wu, F. Cheng, Y.C. Shen, G.Q. Lu, L.L. Li, One-way transmission through merging of magnetic defect state and optical Tamm states, Optik (Stuttg) 127 (2016) 3740–3744. [24] A.B. Khanikaev, A.V. Baryshev, M. Inoue, Y.S. Kivshar, One-way electromagnetic Tamm states in magnetophotonic structures, Appl. Phys. Lett. 95 (2009) 011101– 011101-3. [25] V. Morozhenko, O.G. Kollyukh, Angular and spectral peculiarities of the coherent thermal radiation of the magneto-optical Fabry–Perot resonator in a magnetic field, J. Opt. A Pure Appl. Opt. 11 (2009) 085503–085503-6. [26] P. Lancaster, Theory of Matrices, Academic Press, New York-London, 1969. [27] Y. Ukhanov, Magnetooptic Faraday effect in semiconductors, Sov. Phys.-Usp. 16 (1973) 236–250. [28] O. Madelung, Semiconductors: Data Handbook, third ed., Springer-Verlag, Berlin Heidelberg, 2004. [29] M. Cardona, Electron effective masses of InAs and GaAs as a function of temperature and doping, Phys. Rev. 121 (1961) 752–758.
Fig. 6. Experimental reflection spectra for the n-InAs/Al plane-parallel structure under normal incidence of non-polarized light. 1 – 𝐻 = 0; 2 – 𝐻 = 18 kOe. 𝑁𝑞 = 1.4 ⋅ 1018 cm−3 , 𝑑0 = 80 μm.
where −1 𝑛2± = 𝑛20 𝜆2 𝜆𝑐 𝜆−2 𝑝 (𝜆𝑐 ± 𝜆) .
𝜆2𝑝
𝑐 2 𝑛20 𝑚∗ ∕𝑞 2 𝑁𝑞 ,
(13) 2𝜋𝑚∗ 𝑐 2 ∕𝑞𝐻,
𝜆𝑐 = 𝑞 is the electron charge, Here, = 𝑚∗ – carrier effective mass, 𝑐 – speed of light in vacuum. The determined in this way electron effective mass of used n-InAs was 𝑚∗ = 0.043𝑚𝑒 , where 𝑚𝑒 is the electron mass. This result agrees with the known data [28,29]. 4. Conclusion In summary, the results of investigations of transmission and reflection of a one-dimensional magneto-optical resonator in an external magnetic field have been presented for non-polarized light. It has been shown that, despite the stereotyped view, magneto-optical rotation of non-polarized light clearly manifests itself in the above optical characteristics of MORs. It has been ascertained that a magnetic field substantially changes the interference pattern of transmission and reflection for non-polarized light. It has been observed the inversion of interference extremes in the applied magnetic field: maxima are transformed into the minima and vice versa. The results of investigations have great practical importance. The MORs can be used as narrow-band optical modulators, switches and controllable dispersion elements for non-polarized light. These magnetooptical devices can be applied in instruments with different functions. These results can be used to develop new methods for studying the parameters of substances as well. References [1] H. Sun, Y. Lei, S. Fan, Q. Zhang, H. Guo, Cavity-enhanced room-temperature high sensitivity optical Faraday magnetometry, Phys. Lett. Sect. A Gen. At. Solid State Phys. 381 (2017) 129–135. [2] K. Sycz, W. Gawlik, J. Zachorowski, Resonant Faraday effect in a Fabry–Perot cavity, Opt. Appl. XL (2010) 633–639. [3] E. Taskova, S. Gateva, E. Alipieva, K. Kowalski, M. Glódź, J. Szonert, Nonlinear Faraday rotation for optical limitation, Appl. Opt. 43 (2004) 4178–4181.
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