Abstract

We explore a new class of Distributed Feedback (DFB) and Distributed Bragg Reflector (DBR) structures that employ the recently-developed concept of Parity-Time (PT) symmetry in optics. The approach is based on using so-called unidirectional Bragg gratings that are non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a uni-directional Bragg functionality. Such unusual property is achieved through diffraction through a grating having periodic variations in both, phase and amplitude. DFB and DBR structures traditionally consist of a gain medium and reflector(s) made via periodic variation of the (gain media) refractive index in the direction of propagation. As such structures are produced in a gain material. It becomes just possible to add periodic amplitude modulation in order to produce the unidirectional Bragg functionality. We propose here new and unique DFB and DBR structures by concatenating two such unidirectional Bragg gratings with their nonreflective ends oriented outwards the cavity. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. One of the unique characteristics of the structure is that it inherently supports only one lasing mode.

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References

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  1. H. Kogelnik and C. V. Shank, “Simulated emission in a periodic structure,” Appl. Phys. Lett.18(4), 152–154 (1971).
    [CrossRef]
  2. H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972).
    [CrossRef]
  3. E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasres,” IEEE J. Quantum Electron.18(1), 66–71 (1982).
    [CrossRef]
  4. D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
    [CrossRef]
  5. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(3), 2963–2975 (1996).
    [CrossRef] [PubMed]
  6. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express13(8), 3068–3078 (2005).
    [CrossRef] [PubMed]
  7. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
    [CrossRef] [PubMed]
  8. C. M. Bender and S. Boettcher, “Real spectra in non- Hermitian Hamiltinian having PT Symmetry,” Phys. Rev. Lett.80(24), 5243–5246 (1998).
    [CrossRef]
  9. K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
    [CrossRef]
  10. K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett.21(1), 68–70 (1996).
    [CrossRef] [PubMed]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
[CrossRef]

2005 (1)

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non- Hermitian Hamiltinian having PT Symmetry,” Phys. Rev. Lett.80(24), 5243–5246 (1998).
[CrossRef]

1996 (2)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett.21(1), 68–70 (1996).
[CrossRef] [PubMed]

1995 (1)

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
[CrossRef]

1982 (1)

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasres,” IEEE J. Quantum Electron.18(1), 66–71 (1982).
[CrossRef]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972).
[CrossRef]

1971 (1)

H. Kogelnik and C. V. Shank, “Simulated emission in a periodic structure,” Appl. Phys. Lett.18(4), 152–154 (1971).
[CrossRef]

Azaña, J.

Bélanger, N.

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non- Hermitian Hamiltinian having PT Symmetry,” Phys. Rev. Lett.80(24), 5243–5246 (1998).
[CrossRef]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non- Hermitian Hamiltinian having PT Symmetry,” Phys. Rev. Lett.80(24), 5243–5246 (1998).
[CrossRef]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

Cardimona, D. A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
[CrossRef]

Christodoulides, D. N.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
[CrossRef]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

El-Ganainy, R.

K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
[CrossRef]

Gavrielides, A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
[CrossRef]

Hardy, A.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasres,” IEEE J. Quantum Electron.18(1), 66–71 (1982).
[CrossRef]

Kapon, E.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasres,” IEEE J. Quantum Electron.18(1), 66–71 (1982).
[CrossRef]

Katzir, A.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasres,” IEEE J. Quantum Electron.18(1), 66–71 (1982).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, “Simulated emission in a periodic structure,” Appl. Phys. Lett.18(4), 152–154 (1971).
[CrossRef]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

Kovanis, V.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
[CrossRef]

Kulishov, M.

Laniel, J. M.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

Markis, K. G.

K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
[CrossRef]

Musslimani, Z. H.

K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
[CrossRef]

Nakazawa, M.

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. Topics54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, “Simulated emission in a periodic structure,” Appl. Phys. Lett.18(4), 152–154 (1971).
[CrossRef]

Sharma, M. P.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
[CrossRef]

Tamura, K.

Appl. Phys. Lett. (1)

H. Kogelnik and C. V. Shank, “Simulated emission in a periodic structure,” Appl. Phys. Lett.18(4), 152–154 (1971).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasres,” IEEE J. Quantum Electron.18(1), 66–71 (1982).
[CrossRef]

IEEE Quantum Electron. (1)

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasres,” IEEE Quantum Electron.31(1), 60–66 (1995).
[CrossRef]

Int. J. Theor. Phys. (1)

K. G. Markis, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodical optical potentials,” Int. J. Theor. Phys.50(4), 1019–1041 (2011).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys.43(5), 2327–2335 (1972).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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. Topics54(3), 2963–2975 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett.106(21), 213901 (2011).
[CrossRef] [PubMed]

C. M. Bender and S. Boettcher, “Real spectra in non- Hermitian Hamiltinian having PT Symmetry,” Phys. Rev. Lett.80(24), 5243–5246 (1998).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry for DBR/DFB Fabry-Perot structure formed by two gratings, M1 and M2, with PT-symmetry and non-reflective ends placed outwards the cavity.

Fig. 2
Fig. 2

Reflection and transmission spectra of the pair of unidirectional Bragg gratings L = 1 mm oriented as shown in Fig. 1 and separated by d = Λ/2 (red solid curves) and d = 0 (blue, dashed), with Λ = 0.5 µm. The spectra are shown for different values of the grating strength: below the threshold (a, b) and at the threshold (c, d).

Fig. 3
Fig. 3

Reflection and transmission spectra of the pair of unidirectional Bragg gratings L = 1 mm oriented as shown in Fig. 1and separated by d = 20000Λ (blue dashed curves) and d = 20000.5Λ (red, solid) with Λ = 0.5 um.

Fig. 4
Fig. 4

Temporal response of the pair of unidirectional Bragg gratings L = 1 mm oriented as it is shown in Fig. 1 and separated by d = 20000Λ in transmission (a) and reflection (b) (solid red curves). The input signal is depicted by the dashed blue curves.

Fig. 5
Fig. 5

Temporal response of the pair of unidirectional Bragg gratings L = 1 mm oriented as it is shown in Fig. 1 and separated by d = 20000Λ in reflection below the threshold (a) and at the threshold (b) κL = 1 (red curves). The input signal is depicted by the blue curves.

Fig. 6
Fig. 6

Temporal response of the pair of unidirectional Bragg gratings L = 1 mm oriented as shown in Fig. 1 and separated by d = 20000Λ in reflection at the threshold (κL = 1) (a) for the first seven signal replicas and (b) for the 63rd to 69th signal replicas. The black (dash) curves represent 8 ps FWHM input pulse, red (solid) curves – 9 ps FWHM input pulse and the blue (dot) curves is responsible for 10 ps FWHM input pulse. Magenta curve depicts F(t) = 0.085(1 + cos(Ωt)) with Ω = 55.3 GHz

Fig. 7
Fig. 7

Temporal response to 8-bit pulse sequence ((11011111) with FWHM 5 ps for each pulse in the train and 40 ps delay between each pulse) of the pair of unidirectional Bragg gratings L = 1 mm oriented as it is shown in Fig. 1 and separated by d = 200000Λ = 100 mm in reflection at the threshold (κL = 1): 1st and 10th replica are shown in detail.

Equations (22)

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[ E A (z+L) E B (z+L) ]=[ M 11 M 12 M 21 M 22 ][ E A (z) E B (z) ]
M 11 (L,z)=[ cosh( γL )+j σ γ sinh( γL ) ] е j( σ β ˜ )L , M 12 (L,z)=j κ Re κ Im γ sinh( γL ) е j( σ β ˜ )(L+2z) , M 21 (L,z)=j κ Re + κ Im γ sinh( γL ) е j( σ β ˜ )(L+2z) , M 22 (L,z)=[ cosh( γL )j σ γ sinh( γL ) ] е j( σ β ˜ )L ,
t (±) = exp(±j(σ β ˜ )L) cosh(γL)j(σ/γ)sinh(γL)
r (±) = j(( κ Re ± κ Im )/γ)sinh(γL) cosh(γL)j(σ/γ)sinh(γL)
t (Σ) = t 1 (+) t 2 (+) exp(j β ˜ d) n=0 ( r 1 () r 2 (+) exp(j2 β ˜ d) ) n .
r (Σ) = r 1 (+) + r 2 (+) t 1 (+) t 2 () exp(j β ˜ d) n=0 ( r 1 () r 2 (+) exp(j2 β ˜ d) ) n .
t (Σ) = t 1 (+) t 2 (+) exp(j β ˜ d) 1 r 1 () r 2 (+) exp(j2 β ˜ d) ,
r (Σ) = r 1 (+) + r 2 (+) t 1 (+) t 2 () exp(2j β ˜ d) 1 r 1 () r 2 (+) exp(j2 β ˜ d) ,
n=0 x n = 1 1x
t (Σ) =exp(2j( β B +σ)L)exp(j β ˜ d) n=0 ( jκL sin(σL) (σL) exp(j(σL+ β ˜ d) ) 2n
r (Σ) =jκL sin(σL) (σL) exp(j(3σL+2 β ˜ d)) n=0 ( jκL sin(σL) (σL) exp(j(σL+ β ˜ d) ) 2n .
t (Σ) = exp(j β ˜ (2L+d)) κ 2 L 2 sin 2 (σL)/ (σL) 2 exp(2j(σL+ β ˜ d)+1 ,
r (Σ) = jκLsin(σL)/(σL)exp(j( β ˜ d+3σL)) κ 2 L 2 sin 2 (σL)/ (σL) 2 exp(2j(σL+ β ˜ d)+1 .
M 11 (Σ) =exp[ j β ˜ ( 2L+d ) ], M 12 (Σ) =jκL sin(σL) σL exp(j(σ2 β ˜ )L)exp(j β ˜ d), M 21 (Σ) =jκL sin(σL) σL exp(j(3σ2 β ˜ )L)exp(j β ˜ d), M 22 (Σ) =( κ 2 L 2 κ 2 sin 2 (σL) σ 2 L 2 exp(2j(σL+ β ˜ d))+1 )exp(j β ˜ (2L+d)).
κL sin(σL) σL exp( jσ(L+d)+j πd Λ )=±j.
d/Λ=±( 2m+1 )/2
κL=1
κL sin(σL) σL exp( jσ(L+d) )=±j,
κL sin(σL) σL cos( σ(L+d) )=0 κL sin(σL) σL sin( σ(L+d) )=±1
κL=( m+ 1 2 ) πL (L+d) sin( ( m+ 1 2 ) πL (L+d) )
| r 1 (ω)|| r 2 (ω)|exp(j( φ 1 (ω)+ φ 2 (ω))exp(2d( g 0 (ω) α 0 (ω)))exp(2jβ(ω)d)=1
g 0 ( ω і ) α 0 ( ω і )= 1 2d ln( 1 | r 1 ( ω і )|| r 2 ( ω і )| ),

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