Abstract

We explore the optical properties of a Fabry-Perot resonator with an embedded Parity-Time (PT) symmetrical grating. This PT-symmetrical grating is non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a unidirectional reflective functionality. The incorporated PT-symmetrical grating forms a resonator with two embedded cavities. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. Depending on the resonator geometry these cavities can interact with different degrees of coherency: fully constructive interaction, partially constructive interaction, partially destructive interaction, and finally their interaction can be completely destructive. A number of very unusual (exotic) nonsymmetrical absorption and amplification behaviors are observed. The proposed structure also exhibits unusual lasing performance. Due to the PT-symmetrical grating, there is no chance of mode hopping; it can lase with only a single longitudinal mode for any distance between the distributed reflectors.

© 2014 Optical Society of America

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References

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  1. L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).
  2. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74(4), 043822 (2006).
    [Crossref]
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    [Crossref]
  4. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
    [Crossref] [PubMed]
  5. H.-P. Nolting, G. Sztefka, M. Grawert, and J. Ctyroky, “Wave Propagation in a Waveguide with a balance of Gain and Loss,” in Integrated Photonics Research ’96 (OSA, 1996), pp. 76–79.
  6. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
    [Crossref] [PubMed]
  7. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004).
    [Crossref] [PubMed]
  8. M. Kulishov, B. Kress, and R. Slavík, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21(8), 9473–9483 (2013).
    [Crossref] [PubMed]
  9. C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
    [Crossref]
  10. S. Longhi, “PT-symmetric Laser-absorber,” Phys. Rev. A 82(3), 031801 (2010).
    [Crossref]
  11. L. Feng, Z. J. Wong, R. Ma, Y. Wang, and X. Zhang, “Parity-Time Synthetic Laser,” arXiv:1405.2863.
  12. M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
    [Crossref] [PubMed]
  13. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT-symmetric microring lasers: Self-adapting broadband mode-selective resonators,” arXiv:1405.2103.
  14. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  15. L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
    [Crossref]
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    [Crossref]

2014 (3)

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

2013 (2)

2012 (2)

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

2011 (1)

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

2010 (1)

S. Longhi, “PT-symmetric Laser-absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

2006 (1)

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74(4), 043822 (2006).
[Crossref]

2005 (1)

2004 (1)

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-Symmetric Quantum Mechanics,” J. Math. Phys. 40(5), 2201 (1999).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Azaña, J.

Bélanger, N.

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-Symmetric Quantum Mechanics,” J. Math. Phys. 40(5), 2201 (1999).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-Symmetric Quantum Mechanics,” J. Math. Phys. 40(5), 2201 (1999).
[Crossref]

Cerjan, A.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

Chang, J.

Chong, Y. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74(4), 043822 (2006).
[Crossref]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Fan, S.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Gao, L.

Ge, L.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Greenberg, M.

Han, J. L.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Huang, C. Y.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Kress, B.

Kulishov, M.

Laniel, J. M.

Lei, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Liertzer, M.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

Lisyansky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Liu, T.

Long, G. L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Longhi, S.

S. Longhi, “PT-symmetric Laser-absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

Lv, G.

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-Symmetric Quantum Mechanics,” J. Math. Phys. 40(5), 2201 (1999).
[Crossref]

Monifi, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Ni, J.

Nori, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Orenstein, M.

Ozdemir, S. K.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Peng, B.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Peng, G.

Plant, D. V.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Pukhov, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Qi, H.

Rotter, S.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

Slavík, R.

Song, Z.

Stone, A. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74(4), 043822 (2006).
[Crossref]

Sun, Z.

Türeci, H. E.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74(4), 043822 (2006).
[Crossref]

Vinogradov, A. P.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Wang, C.

Wang, P.

Wang, Q.

Xu, J. Q.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Yang, L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Zhang, R.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Zhao, Y.

Zheng, J.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

Zyablovsky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-Symmetric Quantum Mechanics,” J. Math. Phys. 40(5), 2201 (1999).
[Crossref]

Nat. Phys. (1)

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (6)

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type II perfect absorption modes with controllable bandwidth in PT-symmetric/Traditional Bragg grating combined structures,” Phys. Rev. A 89, 023842 (2014).
[Crossref]

S. Longhi, “PT-symmetric Laser-absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

L. Ge, Y. D. Chong, S. Rotter, H. E. Türeci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84(2), 023820 (2011).

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74(4), 043822 (2006).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108(17), 173901 (2012).
[Crossref] [PubMed]

Other (3)

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT-symmetric microring lasers: Self-adapting broadband mode-selective resonators,” arXiv:1405.2103.

H.-P. Nolting, G. Sztefka, M. Grawert, and J. Ctyroky, “Wave Propagation in a Waveguide with a balance of Gain and Loss,” in Integrated Photonics Research ’96 (OSA, 1996), pp. 76–79.

L. Feng, Z. J. Wong, R. Ma, Y. Wang, and X. Zhang, “Parity-Time Synthetic Laser,” arXiv:1405.2863.

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Figures (8)

Fig. 1
Fig. 1 Geometry for a FP structure formed by two traditional Bragg gratings, L1 and L2, with a PT-SBG, LPT, placed inside the cavity.
Fig. 2
Fig. 2 Fully constructive interference: left reflection (a, c) and transmission (b, d) spectra of the DBG FP resonator with the PT-SBG grating inside the cavity and of the DBR FP resonator without PT-SBG (b, d) and with an amplifying waveguide (α = −12 µm−1). The structure parameters are: Λ = 0.5 µm; L1 = L2 = 2000Λ = 1 mm; κ1 = κ2 = 1257 m−1; LPT = 20000Λ = 10000 µm; κPT = 32.67 m−1; d1 = 100Λ = 50 µm ; d2 = 1000.5Λ = 500.25 µm; The insert in (a) shows the reflection spectra of the left (blue), right (red) and PT-symmetric Bragg grating (magenta).
Fig. 3
Fig. 3 Fully constructive interference: reflection from the left (a,d) and right (c, f) and transmission (b,e) spectra of the DBG FP resonator with the PT-SBG grating inside the cavity. (The spacings Λ = 0.5µm, L1 = L2 = 2000Λ = 1 mm, LPT = 20000Λ = 10000 µm, d1 = 100Λ = 50 µm, d2 = 1000.5Λ = 500.25 µm, are common to all panels. In the left panels, with parameters κ1 = 942.5 m−1, κ2 = 3142 m−1, κPT = 26.74 m−1, reflection from the right Bragg reflector is close to 100% at the resonance wavelength. In the right panels, with parameters κ1 = 3142 m−1, κ2 = 942.5 m−1, κPT = 36.12 m−1, reflection from the left Bragg reflector is close to 100% at resonance. The inserts in (a) and (c) show the reflection spectra of the left (blue), right (red) and PT-symmetric Bragg grating (magenta).
Fig. 4
Fig. 4 Fully constructive interference: reflection from the left (a,d) and right (c, f) and transmission (b,e) spectra of the FP resonator with the PT-SBG grating inside the cavity The spacings Λ = 0.5 µm, L1 = L2 = 2000Λ = 1 mm, LPT = 10000Λ = 5000 µm, d1 = 100Λ = 50 µm, d2 = 1000.5Λ = 500.25 µm, are common to all panels. For the left panels, with parameters κ1 = 1571 m−1; κ2 = 942.5 m−1; κPT = 53.54 m−1, reflection from the left is completely suppressed at the resonance wavelength, while for the right panels, with parameters κ1 = 942.5 m−1, κ2 = 1571 m−1, κPT = 36 m−1, reflection from the right is completely suppressed at resonance.
Fig. 5
Fig. 5 Partially constructive interference: spectra of the DBG FP resonator with the PT-SBG inside the cavity with the following spacings Λ = 0.5 µm; L1 = L2 = 2000Λ = 1 mm; LPT = 20000Λ = 10000 µm, d1 = 100.5Λ = 50.25 µm ; d2 = 1000.5Λ = 500.25 µm. In the upper panels, with parameters κ1 = κ2 = 1257 m−1 and κPT = 202.63 m−1, the reflectivities of the left and right DBR are equal, while in the lower panels, with parameters κ1 = 0; κ2 = 3770 m−1 and κPT = 100 m−1, the reflectivities are 0 and 100% respectively. The inserts in (a) and (c) show the reflection spectra of the left (red), right (blue) and PT-symmetric Bragg grating (magenta).
Fig. 6
Fig. 6 Partially constructive interference with right-side reflection suppressed at resonance: left-side reflection spectra (a, d, g), transmission spectra (b, e, h) and right-side reflection spectra (c, f, i) of the DBG FP resonator with the PT-SBG inside the cavity with the following structure parameters: Λ = 0.5 µm; L1 = L2 = 2000Λ = 1 mm; LPT = 20000Λ = 10000 µm, d1 = 100.5 Λ = 50.25 µm ; d2 = 1000.5Λ = 500.25 µm; (a), (b) and (c) κ1 = 3142 m−1; κ2 = 942.5 m−1; κPT = 236.18 m−1; (d), (e) and (f) κ1 = κ2 = 1287 m−1; κPT = 236.18 m−1; (g), (h) and (i) κ1 = 942.5 m−1; κ2 = 3142 m−1; κPT = 236.18 m−1.
Fig. 7
Fig. 7 Partially constructive interference with left-side reflection suppressed at resonance: left-side reflection spectra (a, d, g), transmission spectra (b, e, h) and right-side reflection spectra (c, f, i) of the DBG FP resonator with the PT-SBG inside the cavity with the following structure parameters: Λ = 0.5 µm; L1 = L2 = 2000Λ = 1 mm; LPT = 20000Λ = 10000 µm, d1 = 100.5Λ = 50.25 µm ; d2 = 1000.5Λ = 500 µm; (a), (b) and (c) κ1 = 3142 m−1; κ2 = 942.5 m−1; κPT = 173.26 m−1; (d), (e) and (f) κ1 = κ2 = 1287 m−1; κPT = 173.26 m−1; (g), (h) and (i) κ1 = 942.5 m−1; κ2 = 3142 m−1; κPT = 173.26 m−1.
Fig. 8
Fig. 8 Partially destructive interference: Left-side (a) and (d), right-side (c) and (f) reflection and transmission (b) and (e) spectra of the DBG FP resonator with the PT-SBG grating inside the cavity. The spacings Λ = 0.5 µm; L1 = L2 = 2000Λ = 1 mm; LPT = 20000Λ = 10000 µm, d1 = 100.5 Λ = 50.25 µm ; d2 = 1000Λ = 500 µm are common to all panels. In the left panels, with parameters κ1 = 942.48 m−1; κ2 = 3142 m−1; κPT = 35.42 m−1, left-side reflection is suppressed at resonance, while in the right panels, with parameters κ1 = 3142 m−1; κ2 = 942.48 m−1; κPT = 32.61 m−1, right-side reflection is suppressed at resonance.

Equations (42)

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Δ n ( z ) = a cos ( k z ) + j b sin ( k z )
M P T = [ exp ( j β L P T ) m 12 P T exp ( j ( σ β ) L P T ) 0 exp ( j β L P T ) ] ,
m 12 P T = j ( κ P T / σ ) sin ( σ L P T ) ,
M і = [ m 11 ( i ) exp ( j ( σ β ) L i ) m 12 ( i ) exp ( j ( σ β ) ( L і + 2 z i ) m 21 ( i ) exp ( j ( σ β ) ( L і + 2 z i ) m 22 ( i ) exp ( j ( σ β ) L і ) ]
m 11 ( і ) = cos h ( γ i L i ) + j σ γ i sin h ( γ i L i ) ; m 12 ( і ) = j κ i γ i sin h ( γ i L i ) ; m 21 ( і ) = j κ i γ i sin h ( γ i L i ) ; m 22 ( і ) = cos h ( γ i L i ) j σ γ i sin h ( γ i L i ) ; ,
M Σ = M 2 I ( d 2 ) M P T I ( d 1 ) M 1 ,
І і = [ exp ( j β d i ) 0 0 exp ( j β d і ) ]
m 11 Σ = [ m 11 ( 1 ) m 11 ( 2 ) exp ( 2 j β ( d 1 + d 2 + L P T ) + m 21 ( 1 ) m 11 ( 2 ) m 12 ( P T ) exp ( 2 j β ( d 2 + L P T ) ) exp ( j σ L P T ) + + m 21 ( 1 ) m 12 ( 2 ) ] exp ( j Φ )
m 12 Σ = [ m 11 ( 2 ) m 12 ( 1 ) exp ( 2 j β ( d 1 + d 2 + L P T ) ) + m 11 ( 2 ) m 22 ( 1 ) m 12 ( P T ) exp ( 2 j β ( d 2 + L P T ) ) exp ( j σ L P T ) + + m 12 ( 2 ) m 22 ( 1 ) ] exp ( j Φ )
m 21 Σ = [ m 21 ( 2 ) m 11 ( 1 ) exp ( 2 j β ( d 1 + d 2 + L P T ) ) + m 21 ( 1 ) m 21 ( 2 ) m 12 ( P T ) exp ( 2 j β ( d 2 + L P T ) exp ( j σ L P T ) + + m 21 ( 1 ) m 22 ( 2 ) ] exp ( j Φ )
m 22 Σ = [ m 21 ( 2 ) m 12 ( 1 ) exp ( 2 j β ( d 1 + d 2 + L P T ) ) + m 21 ( 2 ) m 22 ( 1 ) m 12 ( P T ) exp ( 2 j β ( d 2 + L P T ) ) exp ( j σ L P T ) + + m 22 ( 1 ) m 22 ( 2 ) ] exp ( j Φ ) ,
Т Σ ( L ) = | t Σ ( L ) | 2 = | m 11 Σ m 22 Σ m 12 Σ m 21 Σ m 22 Σ | 2 ; R Σ ( L ) = | r Σ ( L ) | 2 = | m 21 Σ m 22 Σ | 2
Т Σ ( R ) = | t Σ ( R ) | 2 = | 1 m 22 Σ | 2 ; R Σ ( R ) = | r Σ ( R ) | 2 = | m 12 Σ m 22 Σ | 2
| T Σ 1 | = R Σ ( R ) R Σ ( L )
m 21 ( 2 ) m 22 ( 2 ) [ m 12 ( 1 ) m 22 ( 1 ) exp ( 2 j β ( d 1 + d 2 + L P T ) ) + m 12 ( P T ) exp ( 2 j β ( d 2 + L P T ) ) exp ( j σ L P T ) ] + 1 = 0
r 2 ( L ) ( r 1 ( R ) exp ( 2 j β ( d 1 + d 2 + L P T ) ) + r P T ( R ) exp ( 2 j β ( d 2 + L P T ) ) exp ( j σ L P T ) ) + 1 = 0
r 1 ( R ) r 2 ( L ) exp ( 2 j β ( d 1 + d 2 + L A ) + 1 = 0 ,
r 2 ( L ) r P T ( R ) exp ( 2 j β ( d 2 + L P T ) ) exp ( j σ L P T ) + 1 = 0
exp ( 2 j β ( d 1 + d 2 + L P T ) = 1 exp ( 2 j β ( d 2 + L P T ) = 1 } d 1 = m Λ ; d 2 = ( p + 1 / 2 ) Λ ; L P T = q Λ
exp ( 2 j β ( d 1 + d 2 + L P T ) = 1 exp ( 2 j β ( d 2 + L P T ) = 1 } d 1 = ( m + 1 / 2 ) Λ ; d 2 = ( p + 1 / 2 ) Λ ; L P T = q Λ
exp ( 2 j β ( d 1 + d 2 + L P T ) = 1 exp ( 2 j β ( d 2 + L P T ) = 1 } d 1 = ( m + 1 / 2 ) Λ ; d 2 = p Λ ; L P T = q Λ
exp ( 2 j β ( d 1 + d 2 + L P T ) = 1 exp ( 2 j β ( d 2 + L P T ) = 1 } d 1 = m Λ ; d 2 = p Λ ; L P T = q Λ
( κ P T L P T ) 2 = coth ( κ 2 L 2 ) 2 + 2 cos ( θ ) coth ( κ 2 L 2 ) tan h ( κ 1 L 1 ) + tan h ( κ 1 L 1 ) 2
| r P T ( L ) | = | tan h ( κ 2 L 2 ) + κ P T L P T tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) tan h ( κ 1 L 1 ) 1 tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) κ P T L P T tan h ( κ 2 L 2 ) | ; | r P T ( R ) | = | tan h ( κ 1 L 1 ) + κ P T L P T tan h ( κ 2 L 2 ) 1 tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) κ P T L P T tan h ( κ 2 L 2 ) | ; | t P T | = | s e c h ( κ 1 L 1 ) s e c h ( κ 2 L 2 ) 1 tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) κ P T L P T tan h ( κ 2 L 2 ) | .
κ P T L P T = coth ( κ 2 L 2 ) tan h ( κ 1 L 1 )
( κ P T L P T ) 0 L = coth ( κ 2 L 2 ) coth ( κ 1 L 1 ) ,
| r P T ( R ) | = sin h ( κ 1 L 1 κ 2 L 2 ) cos h ( κ 1 L 1 + κ 2 L 2 ) sin h 2 ( κ 2 L 2 ) ; | t P T | = sin h ( κ 1 L 1 ) sin h ( κ 2 L 2 ) .
( κ P T L P T ) 0 R = tan h ( κ 2 L 2 ) tan h ( κ 1 L 1 ) ,
| r P T ( L ) | = sin h ( κ 2 L 2 κ 1 L 1 ) cos h ( κ 1 L 1 + κ 2 L 2 ) cos h 2 ( κ 1 L 1 ) ; | t P T | = cos h ( κ 2 L 2 ) cos h ( κ 1 L 1 ) ;
| r P T ( L ) | = | tan h ( κ 2 L 2 ) κ P T L P T tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) + tan h ( κ 1 L 1 ) 1 κ P T L P T tan h ( κ 2 L 2 ) + tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | | r P T ( R ) | = | tan h ( κ 1 L 1 ) κ P T L P T + tan h ( κ 2 L 2 ) 1 κ P T L P T tan h ( κ 2 L 2 ) + tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | | t P T | = | s e c h ( κ 1 L 1 ) sech ( κ 2 L 2 ) 1 κ P T L P T tan h ( κ 2 L 2 ) + tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) |
κ P T L P T = c o t h ( κ 2 L 2 ) + tan h ( κ 1 L 1 )
( κ P T L P T ) 0 ( L ) = c o t h ( κ 1 L 1 ) + coth ( κ 2 L 2 ) ,
| r P T ( R ) | = cos h ( κ 1 L 1 κ 2 L 2 ) sin h ( κ 1 L 1 + κ 2 L 2 ) sin h 2 ( κ 2 L 2 ) ; | t P T | = sin h ( κ 1 L 1 ) sin h ( κ 2 L 2 )
( κ P T L P T ) 0 ( R ) = tan h ( κ 1 L 1 ) + tan h ( κ 2 L 2 )
| r P T ( L ) | = cos h ( κ 1 L 1 κ 2 L 2 ) sin h ( κ 1 L 1 + κ 2 L 2 ) cos h 2 ( κ 1 L 1 ) ; | t P T | = cos h ( κ 2 L 2 ) cos h ( κ 1 L 1 ) ;
| r P T ( L ) | = | tan h ( κ 1 L 1 ) + κ P T L P T tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) tan h ( κ 2 L 2 ) 1 + κ P T L P T tan h ( κ 2 L 2 ) tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | ; | r P T ( R ) | = | tan h ( κ 2 L 2 ) + κ P T L P T tan h ( κ 1 L 1 ) 1 + κ P T L P T tan h ( κ 2 L 2 ) tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | ; | t P T | = s e c h ( κ 1 L 1 ) s e c h ( κ 2 L 2 ) | 1 + κ P T L P T tan h ( κ 2 L 2 ) tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | .
κ P T L P T = tan h ( κ 2 L 2 ) coth ( κ 1 L 1 ) ,
( κ P T L P T ) 0 ( L ) = coth ( κ 1 L 1 ) coth ( κ 2 L 2 ) ,
| r P T ( R ) | = cos h ( κ 1 L 1 + κ 2 L 2 ) sin h ( κ 2 L 2 κ 1 L 1 ) sin h 2 ( κ 2 L 2 ) ; | t P T | = sin h ( κ 1 L 1 ) sin h ( κ 2 L 2 ) .
( κ P T L P T ) 0 ( R ) = tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) ,
| r P T ( L ) | = cos h ( κ 1 L 1 + κ 2 L 2 ) sin h ( κ 1 L 1 κ 2 L 2 ) cos h 2 ( κ 2 L 2 ) ; | t P T | = cos h ( κ 1 L 1 ) cos h ( κ 2 L 2 ) .
| r P T ( L ) | = tan h ( κ 2 L 2 ) + κ P T L P T tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) + tan h ( κ 1 L 1 ) 1 + κ P T L P T tan h ( κ 1 L 1 ) + tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | r P T ( R ) | = tan h ( κ 1 L 1 ) + κ P T L P T + tan h ( κ 2 L 2 ) 1 + κ P T L P T tan h ( κ 1 L 1 ) + tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 ) | t P T | = s e c h ( κ 1 L 1 ) s e c h ( κ 2 L 2 ) 1 + κ P T L P T tan h ( κ 1 L 1 ) + tan h ( κ 1 L 1 ) tan h ( κ 2 L 2 )

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