Abstract

A theoretical analysis of asymmetrical diffraction in Raman-Nath, intermediate and Bragg diffraction regimes is presented. The asymmetry is achieved by combining matched periodic modulations of the phase and of the loss/gain of the material, which enables the breakdown of optical symmetry and redirects all resulting optical energy in only positive or only negative diffraction orders, depending on the quarter period shift directions between the phase and the loss/gain modulations. Analytic expressions for the amplitudes of the diffraction orders are derived based on rigorous multimode coupled mode equations in slowly varying amplitude approximation.

©2012 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  2. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
    [Crossref]
  3. C. Neipp, I. Pascual, and A. Belendez, “Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms,” Opt. Express 10(23), 1374–1383 (2002).
    [Crossref] [PubMed]
  4. M. Fally, M. Ellabban, and I. Drevensek-Olenik, “Out-of-phase holographic gratings: a quantitative analysis,” Opt. Express 16(9), 6528–6536 (2008).
    [Crossref] [PubMed]
  5. M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
    [Crossref]
  6. M. Chang and N. George, “Holographic dielectric Grating: theory and practice,” Appl. Opt. 9(3), 713–719 (1970).
    [Crossref] [PubMed]
  7. G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
    [Crossref]
  8. H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
    [Crossref]
  9. 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]
  10. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
    [Crossref]
  11. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004).
    [Crossref] [PubMed]
  12. 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]
  13. T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981).
    [Crossref] [PubMed]
  14. R. Birabassov, A. Yesayan, and T. V. Galstyan, “Nonreciprocal diffraction by spatial modulation of absorption and refraction,” Opt. Lett. 24(23), 1669 (1999).
    [Crossref]
  15. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
    [Crossref]
  16. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
    [Crossref]
  17. L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
    [Crossref]

2009 (1)

M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
[Crossref]

2008 (2)

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

M. Fally, M. Ellabban, and I. Drevensek-Olenik, “Out-of-phase holographic gratings: a quantitative analysis,” Opt. Express 16(9), 6528–6536 (2008).
[Crossref] [PubMed]

2007 (1)

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

2005 (1)

2004 (1)

2002 (1)

1999 (1)

1998 (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[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]

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

1984 (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
[Crossref]

1981 (1)

1980 (1)

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[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]

1970 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Azana,

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Azaña, J.

Bélanger, N.

Belendez, A.

Berry, M. V.

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

Birabassov, R.

Chang, M.

Clausen, A. T.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Drevensek-Olenik, I.

Ellabban, M.

Fally, M.

Galili, M.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Galstyan, A. V.

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

Galstyan, T. V.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981).
[Crossref] [PubMed]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

George, N.

Greenberg, M.

Guibelalde, E.

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
[Crossref]

Jeppesen, P.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[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, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kulishov, M.

Laniel, J. M.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981).
[Crossref] [PubMed]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

Mulvad, H. C. H.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Neipp, C.

Orenstein, M.

Oxenlowe, L. K.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Pascual, I.

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]

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]

Slavik, R.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Vasnetsov, M. V.

M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
[Crossref]

Yesayan, A.

Yongwoo Park, J.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Zakharyan, G. G.

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE J. Sel. Top. Quantum Electron. (1)

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[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]

J. Phys. Math. Gen. (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

Opt. Commun. (2)

M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
[Crossref]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
[Crossref]

Opto-Electron. Rev. (1)

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[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]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Geometry of asymmetrical diffraction on the combined gratings of refractive index and gain/loss modulations. When these two modulations are balanced, π n 1 /λ= α 1 /2 , only diffraction into non-negative or non- positive orders occurs depending on the shift between the gratings.

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Fig. 2
Fig. 2 Diffraction efficiencies in zero and higher diffraction orders by the index grating (a), the gain/loss grating with zero average gain/loss value (b), and the perfectly asymmetrical grating (c) also with zero average gain/loss value.

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Fig. 3
Fig. 3 Diffraction efficiencies in the second (a) and third (b) diffraction orders for different values of the Nath factor (ρ) at Bragg angles. The diffraction efficiency for the third order is magnified 10x, 100x and 1000x for ρ=3 , ρ=6 ,and ρ=12 respectively.

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Fig. 4
Fig. 4 Diffraction efficiencies of the first (red), second (blue) and third (green) diffraction orders as a function of the mismatch factor μ for different values of the Nath factor (ρ) and the dimensionless grating thickness ξ . The proportional contribution of the second and the third diffraction orders at μ=1 is shown in percentage in each plot.

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Fig. 5
Fig. 5 Comparative diffraction on the index grating (a, b, c, d) and the perfectly asymmetrical grating (e, f, g, h) for different incidence directions and different incidence angles.

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Fig. 6
Fig. 6 Diffraction efficiencies of the first (red), second (blue) and third (green) orders for normal light incidence ( θ=0 ) as a function of the grating strength ξ for different values of the Nath factor (ρ).

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Equations (14)

Equations on this page are rendered with MathJax. Learn more.

n(x)= n 0 + n 1 cos( Kx );
α(x)= α 0 ± α 1 sin( Kx )
∇ 2 E y (x,z)+k (x) 2 E y (x,z)=0
E y (x,z)= ∑ m=−∞ +∞ S m (z)exp{ −j[ ( 2π n 0 λ sinθ−mK )x+ 2π n 0 λ cosθz ] }
cosθ d S m (z) dz +j( πλm Λ 2 n 0 (μ−m)+ α 0 ) S m + j 2 ( κ + S m+1 (z)+ κ − S m−1 (z) )=0
κ + =π n 1 /λ+ α 1 /2; κ − =π n 1 /λ− α 1 /2
2 d T m (ξ) dξ + j 2 ( η + T m+1 + η − T m−1 )=jm(m−μ)ρ T m
2 d T m (ξ) dξ +j T m−1 =jm( m−μ )ρ T m
T m (ξ)= R m (ξ)exp(+jξm(m−μ)ρ/2)
R m (ξ)=− j 2 ∫ 0 ξ R m−1 ( ξ ′ ) exp(−j ξ ′ (2m−μ−1)ρ/2)
R 0 (ξ)=1; R 1 (ξ)=−2j sin((μ−1)ξρ/4) ρ(μ−1) exp(−j(μ−1)ξρ/4); R 2 (ξ)= j2 ρ 2 ( sin((μ−2)ξρ/2) 2(μ−1)(μ−2) exp(j(μ−2)ξρ/2)− sin((μ−3)ξρ/4) (μ−1)(μ−3) exp(j(μ−3)ξρ/4) ); R 3 (ξ)= −j ρ 3 ( sin(3(μ−3)ξρ/4) 3(μ−1)(μ−2)(μ−3) exp(j3(μ−3)ξρ/4) − − sin((μ−4)ξρ/2) (μ−1)(μ−3)(μ−4) exp(j(μ−4)ξρ/2)+ sin((μ−5)ξρ/4) (μ−2)(μ−3)(μ−5) exp(j(μ−5)ξρ/4) )
T m (ξ)= R m (ξ)= (ξ) m 2 m m! ; for m=0,+1,...+N
R 0 (ξ)=1; R 1 (ξ)=−j ξ 2 ; at μ=1 R 2 (ξ)= 2j ρ ( ξ 4 − sin(ξρ/4) ρ exp(−jξρ/4 ) ) at μ=2 R 3 (ξ)= −j ρ 2 [ ξ 8 +( ξ 8 cos(ρξ/2)− 1 2ρ sin(ρξ/2) )exp(−jξρ/2) ] at μ=3
R 0 (ξ)=1; R 1 (ξ)=−2j sin(ξρ/4) ρ exp(jξρ/4); R 2 (ξ)= −2j ρ 2 ( sin(ξρ) 4 exp(−jξρ)− sin(3ξρ/4) 3 exp(−j3ξρ/4) ); R 3 (ξ)= −j ρ 3 ( sin(9ξρ/4) 9 exp(−j9ξρ/4) − sin(2ξρ) 6 exp(−j2ξρ)+ sin(5ξρ/4) 15 exp(−j5ξρ/4) )

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