The Quantum Well of One-Dimensional Photonic Crystals

Abstract

We have studied the transmissivity of one-dimensional photonic crystals quantum well (QW) with quantum theory approach. By calculation, we find that there are photon bound states in the QW structure (BA[superscript])6[/superscript] (BBABB[superscript])n[/superscript] (AB[superscript])6[/superscript] , and the numbers of the bound states are equal to n+1. We have found that there are some new features in the QW, which can be used to design optic amplifier, attenuator, and optic filter of multiple channel.

Xiao-Jing Liu 1, 2 and Ji Ma 1 and Xiang-Dong Meng 1 and Hai-Bo Li 1 and Jing-Bin Lu 2 and Hong Li 1 and Wan-Jin Chen 1 and Xiang-Yao Wu 1 and Si-Qi Zhang 2 and Yi-Heng Wu 3

Academic Editor:Ashok Chatterjee

1, Institute of Physics, Jilin Normal University, Siping 136000, China
2, Institute of Physics, Jilin University, Changchun 130012, China
3, School of Physics and Electronic Engineering, Anqing Normal University, Anqing 246133, China

Received 6 January 2015; Accepted 8 April 2015; 18 June 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Photonic crystals (PCs) are artificial structures with a periodic dielectric constant in one, two, or three dimensions [1, 2]. They are characterized by photonic band structures owing to the multiple Bragg scatterings [3, 4]. Between photonic bands there may exist a photonic band gap (PBG), in which the propagation of electromagnetic waves or photons is strongly inhibited [5]. This facilitates the manipulation and control of the flow of electromagnetic waves or photons as well as the design of high-performance optoelectric devices [6, 7].

The concept of super lattice and quantum well (QW) stemmed from the pioneering work of Shimuzu and Ishihara [8]. It is well known that there are many interesting and new phenomena for electrons in semiconductor QW structures [9]. The QW structures and super lattices can be used to tailor the electronic band structures of semiconductors [9-13]. Similar to the idea of semiconductor QW structures, one can use different PCs to construct photonic QW structures, provided that the PBG of the constituent PCs are aligned properly. The constituents can be one-dimensional (1D), two-dimensional (2D), or three-dimensional (3D) PCs. It has been shown by the authors [14] that the transmission properties of the 1D and 2D PCs can be tailored by using QW structures. The nontransmission frequency range can be enlarged as desired by using QW. The use of QW exciton embedded in high-finesse semiconductor microcavities of the Fabry-Perot type has allowed observing a modification of spontaneous emission (weak coupling regime) [15-21] as well as the occurrence of a vacuum Rabi splitting (strong coupling regime) [22-25]. The latter effect arises when the radiation-matter coupling energy overcomes the damping rates of QW exciton and microcavities photons.

In [26, 27], we have studied the quantum transmission characteristic of 1D PCs with quantum theory approach and given the quantum transform matrix, quantum transmissivity, and reflectivity. In this paper, we use the quantum method to research the QW transmissivity of 1D PCs. It is found that there are some new features in the QW structure [figure omitted; refer to PDF] , which can be used to design optic amplifier, attenuator, and optic filter of multiple channel.

2. Quantum Transform Matrix and Transmissivity of QW

The QW structures consisted of two different 1D PCs. The first and second 1D PCs structures are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, where [figure omitted; refer to PDF] are the numbers of the second PCs layers. The two 1D PCs can consist of 1D PCs of QW structure, which is [figure omitted; refer to PDF] .

In quantum theory approach [26, 27], we consider the photon travels along with the [figure omitted; refer to PDF] -axis, and the QW structure and quantum wave functions distribution are shown in Figure 1. The thicknesses and refractive indexes of layers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively. The [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the photon wave functions of the first period media [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The photon wave functions of incident, reflection, and transmission are [26, 27] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the wave vector of photon in vacuum, mediums [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the wave function amplitudes of incident, reflection, and transmission wave. By calculation, similarly as [26, 27], we can directly give the wave functions of photon in arbitrary layers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . For medium [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of the [figure omitted; refer to PDF] th ( [figure omitted; refer to PDF] th) layer, the photon wave function can be written as [figure omitted; refer to PDF] before the [figure omitted; refer to PDF] th layer medium [figure omitted; refer to PDF] , there are [figure omitted; refer to PDF] layers medium [figure omitted; refer to PDF] and [figure omitted; refer to PDF] layers medium [figure omitted; refer to PDF] , and before the [figure omitted; refer to PDF] th layer medium [figure omitted; refer to PDF] , there are [figure omitted; refer to PDF] layers medium [figure omitted; refer to PDF] and [figure omitted; refer to PDF] layers medium [figure omitted; refer to PDF] . The constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the wave function amplitudes. By the condition of wave function and its derivative continuation at the interface of two mediums, we can obtain the quantum transfer matrix of [figure omitted; refer to PDF] th medium layer; it is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is the wave vector of photon in the [figure omitted; refer to PDF] th ( [figure omitted; refer to PDF] th) layer medium and [figure omitted; refer to PDF] is the thickness of [figure omitted; refer to PDF] th layer medium. For the QW structure [figure omitted; refer to PDF] , its total quantum transfer matrix is [figure omitted; refer to PDF] and the quantum transmissivity [figure omitted; refer to PDF] is [26, 27] [figure omitted; refer to PDF]

Figure 1: The QW structure of 1D PCs.

[figure omitted; refer to PDF]

3. Numerical Result

In this section, we report our numerical results of the QW quantum transmissivity. The refractive indexes and the thicknesses of medium [figure omitted; refer to PDF] and medium [figure omitted; refer to PDF] are as follows: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] nm, [figure omitted; refer to PDF] nm. The quantum transmissivity of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is shown in Figures 2(a) and 2(b). In Figures 3 and 4, we only change the refractive index of medium [figure omitted; refer to PDF] in relation to Figure 2; they are [figure omitted; refer to PDF] (active medium) and [figure omitted; refer to PDF] (absorbing medium). From Figures 2 to 4, we can obtain results as follows. (1) In Figures 2(b), 3(b), and 4(b), when the ratio [figure omitted; refer to PDF] , the quantum transmission peaks are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for the 1D PCs [figure omitted; refer to PDF] ; that is, the quantum transmission peaks are gained and attenuated when the medium [figure omitted; refer to PDF] is active medium [figure omitted; refer to PDF] and absorbing medium [figure omitted; refer to PDF] . (2) The forbidden band of 1D PCs [figure omitted; refer to PDF] is in the range of [figure omitted; refer to PDF] , and the conduction band of 1D PCs [figure omitted; refer to PDF] is inside the forbidden band. So, the PCs [figure omitted; refer to PDF] play a role similar to a barrier to PCs [figure omitted; refer to PDF] , and the PCs [figure omitted; refer to PDF] act as a well in the forbidden band. We put the 1D PCs [figure omitted; refer to PDF] and [figure omitted; refer to PDF] together to constitute the 1D PCs of QW structure [figure omitted; refer to PDF] , which are shown in Figure 1. In Figures 5, 6, and 7, we should study the quantum transmissivity of the QW structure [figure omitted; refer to PDF] . As mentioned above, the conduction band of PCs [figure omitted; refer to PDF] is inside the forbidden band of PCs [figure omitted; refer to PDF] ; that is, the PCs [figure omitted; refer to PDF] prohibit the propagation of photon in its forbidden band; then the photon will be confined in the PCs [figure omitted; refer to PDF] . Because of the quantum effect of photon in QW of 1D PCs, the photon should form the bound state in the QW, which is analogous to the bound state of electron in semiconductor QW. The photon can pass the QW by the resonance perforation way and form the very sharp peaks of quantum transmissivity within the forbidden band of the PCs [figure omitted; refer to PDF] , which are shown in Figures 5, 6, and 7. In Figures 5, 6, and 7, (a), (b), and (c) are corresponding to [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for the QW structure [figure omitted; refer to PDF] . In Figures 5, 6, and 7 refractive indexes of medium [figure omitted; refer to PDF] are real number, [figure omitted; refer to PDF] (convention medium), and complex numbers [figure omitted; refer to PDF] (active medium) and [figure omitted; refer to PDF] (absorbing medium), respectively. From Figures 5 to 7, we can obtain some results. (1) The numbers of the sharp peaks (bound states) are equal to [figure omitted; refer to PDF] ; that is, when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , the numbers of the sharp peaks are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . (2) In Figure 5, the quantum transmissivity of the sharp peaks [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , which can be designed optic filter of multiple channel. (3) In Figure 6, the quantum transmissivity of the sharp peaks [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] increase, the sharp peaks value [figure omitted; refer to PDF] increase, which can be used to design optic amplifier and optic filter of multiple channel. (4) In Figure 7, the quantum transmissivity of the sharp peaks [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; when [figure omitted; refer to PDF] increase, the sharp peaks value [figure omitted; refer to PDF] decrease, which can be used to design optic attenuator.

Figure 2: The quantum transmissivity of 1D PCs with [figure omitted; refer to PDF] . (a) The structure is [figure omitted; refer to PDF] ; (b) the structure is [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 3: The quantum transmissivity of 1D PCs with [figure omitted; refer to PDF] . (a) The structure is [figure omitted; refer to PDF] ; (b) the structure is [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 4: The quantum transmissivity of 1D PCs with [figure omitted; refer to PDF] . (a) The structure is [figure omitted; refer to PDF] ; (b) the structure is [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 5: The quantum transmissivity for QW structure [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . (a) [figure omitted; refer to PDF] , (b) [figure omitted; refer to PDF] , and (c) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Figure 6: The quantum transmissivity for QW structure [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . (a) [figure omitted; refer to PDF] , (b) [figure omitted; refer to PDF] , and (c) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Figure 7: The quantum transmissivity for QW structure [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . (a) [figure omitted; refer to PDF] , (b) [figure omitted; refer to PDF] , and (c) [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

4. Conclusion

In summary, we have studied the quantum transmissivity of the QW of 1D PCs with quantum theory approach. By calculation, we find that there are photon bound states in QW structure [figure omitted; refer to PDF] , and the numbers of the bound states are equal to [figure omitted; refer to PDF] , which are formed by the quantum effect of photon in QW. We also find that the QW [figure omitted; refer to PDF] can be used to design optic amplifier, attenuator, and optic filter of multiple channel.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61275047), the Research Project of Chinese Ministry of Education (no. 213009A), and Scientific and Technological Development Foundation of Jilin Province (no. 20130101031JC).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] C. M. Bowden, J. P. Dowling, H. O. Everitt, "Development and applications of materials exhibiting photonic band gaps," Journal of the Optical Society of America B , vol. 10, pp. 280-282, 1993.

[2] A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, A. Imamo[gcaron]lu, "Strongly correlated photons on a chip," Nature Photonics , vol. 6, no. 2, pp. 93-96, 2012.

[3] J.-T. Liu, T.-B. Wang, X.-J. Li, N.-H. Liu, "Enhanced absorption of monolayer MoS₂ with resonant back reflector," Journal of Applied Physics , vol. 115, 2014.

[4] C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, "Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity," Physical Review Letters , vol. 69, article 3314, 1992.

[5] D. Ballarini, M. de Giorgi, E. Cancellieri, R. Houdre, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, D. Sanvitto, "All-optical polariton transistor," Nature Communications , vol. 4, article 1778, 2013.

[6] G. Kurizki, J. W. Haus, "Special issue on photonic band structures," Journal of Modern Optics , vol. 41, no. 2, pp. 171-172, 1994.

[7] N. M. R. Peres, Y. V. Bludov, "Enhancing the absorption of graphene in the terahertz range," EPL , vol. 101, no. 5, 2013.

[8] M. Shimuzu, T. Ishihara, "Subpicosecond transmission change in semiconductor-embedded photonic crystal slab: toward ultrafast optical switching," Applied Physics Letters , vol. 80, no. 16, pp. 2836-2838, 2002.

[9] H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, U. Gösele, "Nonlinear optical tuning of a two-dimensional silicon photonic crystal," Physical Review B , vol. 70, no. 20, 2004.

[10] F. D. M. Haldane, S. Raghu, "Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry," Physical Review Letters , vol. 100, no. 1, 2008.

[11] S. Raghu, F. D. M. Haldane, "Analogs of quantum-Hall-effect edge states in photonic crystals," Physical Review A , vol. 78, 2008.

[12] Z. Wang, Y. Chong, J. D. Joannopoulos, M. Soljacic, "Observation of unidirectional backscattering-immune topological electromagnetic states," Nature , vol. 461, no. 7265, pp. 772-775, 2009.

[13] Y. Poo, R.-X. Wu, Z. Lin, Y. Yang, C. T. Chan, "Experimental realization of self-guiding unidirectional electromagnetic edge states," Phys. Rev. Lett , vol. 106, no. 9, 2011.

[14] J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, A. Knorr, "Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells," Applied Physics Letters , vol. 81, no. 23, pp. 4332-4334, 2002.

[15] G. Björk, S. MacHida, Y. Yamamoto, K. Igeta, "Modification of spontaneous emission rate in planar dielectric microcavity structures," Physical Review A , vol. 44, no. 1, pp. 669-681, 1991.

[16] K. Fang, Z. Yu, S. Fan, "Realizing effective magnetic field for photons by controlling the phase of dynamic modulation," Nature Photonics , vol. 6, no. 11, pp. 782-787, 2012.

[17] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, A. Szameit, "Photonic Floquet topological insulators," Nature , vol. 496, no. 7444, pp. 196-200, 2013.

[18] J. Cho, D. G. Angelakis, S. Bose, "Fractional quantum Hall state in coupled cavities," Physical Review Letters , vol. 101, 2008.

[19] J. Koch, A. A. Houck, K. L. Hur, S. M. Girvin, "Time-reversal-symmetry breaking in circuit-QED-based photon lattices," Physical Review A , vol. 82, 2010.

[20] R. O. Umucalilar, I. Carusotto, "Artificial gauge field for photons in coupled cavity arrays," Physical Review A , vol. 84, no. 4, 2011.

[21] M. Hafezi, E. A. Demler, M. D. Lukin, J. M. Taylor, "Robust optical delay lines with topological protection," Nature Physics , vol. 7, no. 11, pp. 907-912, 2011.

[22] M. Minkov, V. Savona, "Long-distance radiative excitation transfer between quantum dots in disordered photonic crystal waveguides," Physical Review B , vol. 88, 2013.

[23] R. Houdre, C. Weisbuch, R. P. Stanley, U. Oesterle, P. Pellandini, M. Ilegems, "Measurement of cavity-polariton dispersion curve from angle-resolved photoluminescence experiments," Physical Review Letters , vol. 73, no. 15, pp. 2043-2046, 1994.

[24] M. Hafezi, S. Mittal, J. Fan, A. Migdall, J. M. Taylor, "Imaging topological edge states in silicon photonics," Nature Photonics , vol. 7, no. 12, pp. 1001-1005, 2013.

[25] G. Q. Liang, Y. D. Chong, "Optical resonator analog of a two-dimensional topological insulator," Physical Review Letters , vol. 110, 2013.

[26] X.-Y. Wu, J. Ma, X.-J. Liu, J.-H. Yang, H. Li, S.-Q. Zhang, H.-X. Gao, X.-G. Yin, S. Chen, "Quantum theory of photonic crystals," Physica E: Low-Dimensional Systems and Nanostructures , vol. 59, pp. 174-180, 2014.

[27] X.-Y. Wu, J. Ma, X.-J. Liu, J.-H. Yang, H. Li, S.-Q. Zhang, H.-X. Gao, H.-M. Li, H.-C. Yuan, "Study photonic crystals defect model property with quantum theory," Optics Communications , vol. 321, pp. 211-218, 2014.