Abstract

The use of a complex short-period (Bragg) grating which combines matched periodic modulations of refractive index and loss/gain allows asymmetrical mode coupling within a contra-directional waveguide coupler. Such a complex Bragg grating exhibits a different behavior (e.g. in terms of the reflection and transmission spectra) when probed from opposite ends. More specifically, the grating has a single reflection peak when used from one end, but it is transparent (zero reflection) when used from the opposite end. In this paper, we conduct a systematic analytical and numerical analysis of this new class of Bragg gratings. The spectral performance of these, so-called nonreciprocal gratings, is first investigated in detail and the influence of device parameters on the transmission spectra of these devices is also analyzed. Our studies reveal that in addition to the nonreciprocal behavior, a nonreciprocal Bragg grating exhibits a strong amplification at the resonance wavelength (even with zero net-gain level in the waveguide) while simultaneously providing higher wavelength selectivity than the equivalent index Bragg grating. However, it is also shown that in order to achieve nonreciprocity in the device, a very careful adjustment of the parameters corresponding to the index and gain/loss gratings is required.

© 2005 Optical Society of America

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References

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  1. X. Daxhelet, M. Kulishov, �??Theory and practice of long-period gratings: when a loss becomes a gain,�?? Opt. Lett. 28, 686-688 (2003).
    [CrossRef] [PubMed]
  2. M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, D.V. Plant, �??Tunable waveguide transmission gratings based on active gain control,�?? IEEE J. Quantum Electron. 40, 1715-1724 (2004).
    [CrossRef]
  3. L. Poladian, �??Rresonance mode expansions and exact solutions for nonuniform gratings,�?? Physical Review E 54, 2963-2975 (1996).
    [CrossRef]
  4. M. Greenberg, M. Orenstein, �??Irreversible coupling by use of dissipative optics,�?? Opt. Lett. 29, 451-453 (2004).
    [CrossRef] [PubMed]
  5. M. Greenberg, M. Orenstein, �??Unidirectional complex gratings assisted couplers,�?? Opt. Express 12, 4013-4018 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4013.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-4013.</a>
    [CrossRef] [PubMed]
  6. R. Kashyap, Fiber Bragg Gratings (SanDiego, CA: Academic, 1999, ch.4).
  7. T. Erdogan, �??Fiber grating spectra,�?? J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  8. H. Kogelnik, C.V. Shank, �??Coupled mode theory of distributed feedback lasers,�?? J.Appl.Phys. 43, 2327-2335 (1972).
    [CrossRef]
  9. D.R. Zimmerman, L.H. Spiekman, �??Amplifiers for the masses: EDFA, EDWA, and SOA Amplets for metro and access applications,�?? IEEE J. Lightwave Technol. 22, 63-71 (2004).
    [CrossRef]

IEEE J. Lightwave Technol. (1)

D.R. Zimmerman, L.H. Spiekman, �??Amplifiers for the masses: EDFA, EDWA, and SOA Amplets for metro and access applications,�?? IEEE J. Lightwave Technol. 22, 63-71 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Kulishov, V. Grubsky, J. Schwartz, X. Daxhelet, D.V. Plant, �??Tunable waveguide transmission gratings based on active gain control,�?? IEEE J. Quantum Electron. 40, 1715-1724 (2004).
[CrossRef]

J. Lightwave Technol. (1)

T. Erdogan, �??Fiber grating spectra,�?? J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

J.Appl.Phys. (1)

H. Kogelnik, C.V. Shank, �??Coupled mode theory of distributed feedback lasers,�?? J.Appl.Phys. 43, 2327-2335 (1972).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Physical Review E (1)

L. Poladian, �??Rresonance mode expansions and exact solutions for nonuniform gratings,�?? Physical Review E 54, 2963-2975 (1996).
[CrossRef]

Other (1)

R. Kashyap, Fiber Bragg Gratings (SanDiego, CA: Academic, 1999, ch.4).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic for the interacting modes, A(z) and B(z), in the Bragg grating.

Fig. 2.
Fig. 2.

Transmission (blue, dash) and reflection (red, solid) spectra of a IBG (κnL=π/2, κα =0) (a) and an ideal NRBG (Δz =0 and κn =κa , so that κ21L=(κα +κn )L=π) (b). The group delay (red, solid) of the reflected light is shown in (c) and (d) with respect to (a) and (b) respectively. The length of the grating is 5 mm.

Fig. 3.
Fig. 3.

Electric field intensity distribution (red, solid) inside the NRBG with respect to the imaginary grating (blue, dash) for (a) κL =π/2 and (b) κL=π. The negative segments of the imaginary grating correspond to the gain periods, whereas the positive ones correspond to the loss periods.

Fig. 4.
Fig. 4.

(a) Reflectivity and (c) the group delay of the IBG (blue, κnL=π; κα =0) and (b) reflectivity and (d) group delay of the ideal NRBG (red, Δz=0, κ21L=(κα +κn )L=2π, κ12 =0) for the gratings with zero net gain (solid, α=0) and nonzero net gain (dash, α=-0.5 cm-1 in the waveguide). All the gratings are assumed to be 5mm long.

Fig. 5.
Fig. 5.

Spectral characteristics of the NRBG for different values of the amplitude imbalance between the real κn and κα imaginary components. There is no grating position deviation (Δz =0): (a) reflection spectrum and (b) reflection group delay for light launched into the left side of the grating (z=0); (c) transmission spectrum and (d) transmission group delay (the transmission characteristics are the same regardless the light input direction, z=0 or L); (e) reflection spectrum and (f) reflection group delay for light injected into the right side of the grating (z=L).

Fig. 6.
Fig. 6.

Spectral characteristics of the NRBG for the different values of the grating position deviation Δz between the real and imaginary gratings. There is no grating amplitude imbalance (κn =κα ). (a) reflection spectrum and (b) reflection group delay for light launched into the left side of the grating (z=0); (c) transmission spectrum and (d) transmission group delay (transmission characteristics are the same regardless the light input direction z=0 or L); (e) reflection spectrum and (f) reflection group delay for light injected into the right side of the grating (z=L).

Equations (22)

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Δ n = Δ n DC + j Δ α DC k 0 + Δ n AC cos [ 2 π Λ z ] j Δ α AC k 0 sin [ 2 π Λ ( z + Δ z ) ] ,
dA dz = j σ ˜ A + j κ 12 B exp [ j 2 Δ β z ] ,
dB dz = j σ ˜ B j κ 21 A exp [ j 2 Δ β z ] ,
κ 12 = [ κ n κ α exp ( j 2 π Δ z Λ ) ] ,
κ 21 = [ κ n + κ α exp ( j 2 π Δ z Λ ) ] .
dR dz = j σ ̂ R + j κ 12 S ,
dS dz = j σ ̂ S j κ 21 R ,
R ( z ) = A ( z ) exp [ j Δ β z ]
S ( z ) = B ( z ) exp [ j Δ β z ]
κ 12 = ( κ n κ α ) ,
κ 21 = ( κ n + κ α ) .
[ A ( L ) B ( L ) ] = [ M 11 M 12 M 21 M 22 ] [ A ( 0 ) B ( 0 ) ] ,
M 11 = [ cosh ( γ L ) + j σ ̂ γ sinh ( γ L ) ] exp [ j ( σ ̂ β ˜ ) L ] ,
M 12 = j κ 12 γ sinh ( γ L ) exp [ j ( σ ̂ β ˜ ) L ] ,
M 21 = j κ 21 γ sinh ( γ L ) exp [ j ( σ ̂ β ˜ ) L ] ,
M 22 = [ cosh ( γ L ) j σ ̂ γ sinh ( γ L ) ] exp [ j ( σ ̂ β ˜ ) L ] ,
M = [ exp ( j β ˜ L ) 0 j κ 21 σ ̂ sin ( σ ̂ L ) exp [ j ( σ ̂ β ˜ ) L ] exp ( j β ˜ L ) ]
Δ n ( z ) = j Δ α AC k 0 sin [ 2 π z Λ ]
E ( z ) = E + ( z ) + E ( z ) = 1 2 [ A ( z ) exp ( + j π z Λ ) + B ( B ) exp ( j π z Λ ) ] + c . c .
I ( z ) = A 0 2 [ 1 + 4 κ 2 ( z L ) 2 4 κ ( z L ) sin ( 2 π z Λ ) ]
Δ λ nr λ = λ n eff L ,
Δ λ c λ = κ λ π n eff 1 + ( π κ L ) 2 ,

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