Abstract

Waveguide structures with an antisymmetric gain/loss profile were studied more than a decade ago as benchmark tests for beam propagation methods. These structures attracted renewed interest, recently e.g. as photonic analogues of quantum mechanical structures with parity-time symmetry breaking. In this paper, properties of both weakly and strongly guiding two-mode waveguides and directional couplers with balanced loss and gain are described. Rather unusual power transmission in such structures is demonstrated by using numerical methods. We found that the interface between media with balanced loss and gain supports propagation of confined unattenuated TM polarized surface wave and we have shown that its properties are consistent with the prediction of a simple analytical model.

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  1. M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12(17), 4013–4018 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-12-17-4013 .
    [CrossRef] [PubMed]
  2. M. Greenberg and M. Orenstein, “Optical Unidirectional Devices by Complex Spatial Single Sideband Perturbation,” IEEE J. Quantum Electron. 41(7), 1013–1023 (2005).
    [CrossRef]
  3. M. Kulishov, J. M. Laniel, N. Bélanger, and D. V. Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-9-3567 .
    [CrossRef] [PubMed]
  4. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
    [CrossRef] [PubMed]
  5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
    [CrossRef] [PubMed]
  6. C. E. Rüter and K. G. Makris, “R. E-Ganainy, D. N. Christoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [CrossRef]
  7. H.-P. Nolting, G. Sztefka, M. Grawert, and J. Čtyroký, “Wave Propagation in a Waveguide with a Balance of Gain and Loss,” in Integrated Photonics Research '96, (OSA, 1996), 76–79.
  8. G. Guekos, ed., Photonic Devices for telecommunications: how to model and measure (Springer, Berlin, 1998).
  9. J. Čtyroký, “Efficient Boundary Conditions for Bidirectional Propagation Algorithm Based on Fourier Series,” J. Lightwave Technol. 27(14), 2575–2582 (2009).
    [CrossRef]
  10. H. Raether, Surface plasmons, Springer tracts in modern physics (Springer, Berlin, 1988), Vol. 111.
  11. J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
    [CrossRef]

2010 (1)

C. E. Rüter and K. G. Makris, “R. E-Ganainy, D. N. Christoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

2009 (1)

2008 (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

2007 (1)

2005 (2)

2004 (1)

1907 (1)

J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
[CrossRef]

Bélanger, N.

Christodoulides, D. N.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Ctyroký, J.

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Greenberg, M.

M. Greenberg and M. Orenstein, “Optical Unidirectional Devices by Complex Spatial Single Sideband Perturbation,” IEEE J. Quantum Electron. 41(7), 1013–1023 (2005).
[CrossRef]

M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12(17), 4013–4018 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-12-17-4013 .
[CrossRef] [PubMed]

Kulishov, M.

Laniel, J. M.

Makris, K. G.

C. E. Rüter and K. G. Makris, “R. E-Ganainy, D. N. Christoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[CrossRef] [PubMed]

Orenstein, M.

M. Greenberg and M. Orenstein, “Optical Unidirectional Devices by Complex Spatial Single Sideband Perturbation,” IEEE J. Quantum Electron. 41(7), 1013–1023 (2005).
[CrossRef]

M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12(17), 4013–4018 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-12-17-4013 .
[CrossRef] [PubMed]

Plant, D. V.

Rüter, C. E.

C. E. Rüter and K. G. Makris, “R. E-Ganainy, D. N. Christoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

Zenneck, J.

J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
[CrossRef]

Annalen der Physik (1)

J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Greenberg and M. Orenstein, “Optical Unidirectional Devices by Complex Spatial Single Sideband Perturbation,” IEEE J. Quantum Electron. 41(7), 1013–1023 (2005).
[CrossRef]

J. Lightwave Technol. (1)

Nat. Phys. (1)

C. E. Rüter and K. G. Makris, “R. E-Ganainy, D. N. Christoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[CrossRef] [PubMed]

Other (3)

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

G. Guekos, ed., Photonic Devices for telecommunications: how to model and measure (Springer, Berlin, 1998).

H. Raether, Surface plasmons, Springer tracts in modern physics (Springer, Berlin, 1988), Vol. 111.

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Figures (12)

Fig. 1
Fig. 1

Strongly guiding waveguide with antisymmetric gain/loss profile.

Fig. 2
Fig. 2

Weakly guiding mutually coupled waveguides with gain/loss sections.

Fig. 3
Fig. 3

Dependence of the effective indices of guided modes of the loss/gain waveguide plotted in Fig. 1 on the loss/gain coefficient α. a) real part, b) imaginary part. Red solid lines: TE modes, blue dots: TM modes.

Fig. 4
Fig. 4

Dependence of the effective indices of guided modes of the loss/gain waveguide plotted in Fig. 2 on the loss/gain coefficient α. a) real part, b) imaginary part. Red solid lines: TE modes, blue dots: TM modes.

Fig. 5
Fig. 5

Magnetic field distribution of TM-polarized guided modes in the gain/loss waveguide in Fig. 1. Solid lines – magnitude, dashed – phase in radians. a) loss/gain coefficient α = 0.5 µm–1, i.e. below the critical value, b) α = 0.7 µm–1, i.e. above the critical value.

Fig. 6
Fig. 6

Magnetic field distribution of two TM-polarized guided modes in the gain/loss directional coupler structure in Fig. 2. Notation same as in Fig. 5. a) – loss/gain coefficient α = 30 cm–1 (below the critical value), b) α = 120 cm–1.(above the critical value).

Fig. 7
Fig. 7

Waveguide configurations investigated numerically. Gray – lossless waveguide, light red – waveguide with gain, light blue – lossy waveguide. Numbers and numbers primed denote the input and output waveguides, respectively.

Fig. 8
Fig. 8

Magnetic field distribution | H y ( x , z ) | 2 in waveguide structures c) and f) in Fig. 7 for α = 60  cm 1 . The length of the gain/loss section is 4000 µm. a) excitation into the lossy waveguide 1 (configuration c), b) excitation into the waveguide 2 with gain (configuration f).

Fig. 9
Fig. 9

Magnetic field distribution | H y ( x , z ) | 2 excited in structures c) and f) in Fig. 7 for α = 100  cm 1 . The length of the gain/loss section is 250 µm. a) excitation into the lossy waveguide 1 (configuration c), b) excitation into the waveguide 2 (with gain – configuration f). Note the difference between the color scales in figs. a) and b); the excitation amplitude is in both cases identical.

Fig. 10
Fig. 10

Power transmitted by the directional coupler section with balanced loss and gain versus the coupler length L c . a) α = 60  cm 1 (below the critical value), b) α = 100  cm 1 (above the critical value).

Fig. 11
Fig. 11

Magnetic field distribution of the TM polarized surface mode in the gain/loss waveguide in Fig. 1 for α = 12.5  µm 1 . Solid line – magnitude, dashed – phase (in radians). a) transversal field distribution, b) 2D field distribution Re { H SW ( x , z ) } .

Fig. 12
Fig. 12

Dependences of the effective index N SW and transversal propagation constants γ G and γ L on the imaginary part of the refractive index n of the gain/loss media.

Equations (11)

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d 2 f ( x ) d x 2 + k 0 2 [ n 2 ( x ) N 2 ] f ( x ) = 0 ,
n ( x ) = n * ( x ) ,
f ( x ) = ± f * ( x ) ,
f 1 ( x ) f 2 ( x ) d x = 0    (TE),    n 2 ( x ) f 1 ( x ) f 2 ( x ) d x = 0    (TM) .
E 1 y ( x ) H 2 x ( x ) d x = 0    (TE),     E 1 x ( x ) H 2 y ( x ) d x = 0   (TM) .
H SW ( x , z ) = { H 0 exp [ i k 0 ( γ L x + N SW z ) ] ,    x < 0 , H 0 exp [ i k 0 ( γ G x + N SW z ) ] ,      x > 0 ,
γ G = n G 2 N SW 2 ,    γ L = n L 2 N SW 2 , n G = n w i n ,         n L = n w + i n .
γ G n G 2 = γ L n L 2 .
N SW = n L 2 n G 2 n L 2 + n G 2 = n w 2 + ( n ) 2 2 [ n w 2 ( n ) 2 ] ,
γ G = γ + i γ ,    γ L = γ + i γ ,   γ > 0 ,   γ > 0 ;
H SW ( x , z ) = H 0 exp ( k 0 γ   | x |   ) exp [ i k 0 ( γ x + N SW z ) ] .

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