Abstract

This work presents the use of longitudinal refractive index modulation (apodization) in photosensitive glass for improved sidelobe suppression in volume holographic optical elements. We develop a numerical model for both uniform and apodized volume holograms based on rigorous coupled-wave analysis. We validate the model by comparison with a transmissive 1.55-µm uniform volume grating in photothermorefractive glass. We then apply our numerical model to calculate the spectral response of apodized gratings. The numerical results demonstrate that apodization of the refractive index modulation envelope improves spectral selectivity and reduces first and second-order side-lobe peaks by up to 33 and 65 dB, respectively. We suggest a method for creating apodization in volume holograms with approximately Gaussian spatial refractive index profile.

© 2004 Optical Society of America

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References

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  1. A.D. Cohen, M.C. Parker, and R.J. Mears, “100-GHz-resolution dynamic holographic channel management for WDM,” IEEE Phot. Tech. Lett.,  11, 851–3 (1999).
    [CrossRef]
  2. D.C. O’Brien, R.J. Mears, T.D. Wilkinson, and W.A. Crossland, “Dynamic holographic inter-connects that use ferroelectric liquid-crystal spatial light modulators”, Appl. Opt. 33, 2795–2803 (1994).
    [CrossRef]
  3. A. Marrakchi and K. Rastani, “Free-space holographic grating interconnects”, Photonics Switching and Interconnects, A. Marrakchi (ed.), Marcel Dekker, New York, 249–321 (1994).
  4. P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
    [CrossRef]
  5. O.M. Efimov, L.B. Glebov, L.N. Glebova, K.C. Richardson, and V.I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass”, Appl. Opt. 38, 619–27 (1999).
    [CrossRef]
  6. I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
    [CrossRef]
  7. S. Tao and G.W. Burr, “Performance optimization of volume gratings with finite size through numerical simulation”, CLEO/IQEC and PhAST Technical Digest (Optical Society of America, Washington, DC, 2004), CTuE5.
  8. T. K. Gaylord and M.G. Moharam, “Planar Dielectric Grating Diffraction Theories”, Appl. Phys. B 28, 1–14 (1982).
    [CrossRef]
  9. L.B. Glebov, “Kinetics modeling in photosensitive glass,” Optical Materials 25, 413–418 (2004).
    [CrossRef]
  10. M.G. Moharam and T.K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  11. T.K. Gaylord and M.G. Moharam, “Analysis of optical diffraction by gratings”, Proc. of the IEEE 73, 894–937 (1985).
    [CrossRef]
  12. K. Radhakrishnan and A.C. Hindmarsh, “Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations”, NASA reference publication 1327 (1993).
  13. G.D. Byrne and A.C. Hindmarsh, “Stiff ODE solvers: A review of current and coming attractions”, J. Comp. Phys. 70, 1–62 (1987).
    [CrossRef]

2004 (1)

L.B. Glebov, “Kinetics modeling in photosensitive glass,” Optical Materials 25, 413–418 (2004).
[CrossRef]

2003 (1)

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

1999 (2)

A.D. Cohen, M.C. Parker, and R.J. Mears, “100-GHz-resolution dynamic holographic channel management for WDM,” IEEE Phot. Tech. Lett.,  11, 851–3 (1999).
[CrossRef]

O.M. Efimov, L.B. Glebov, L.N. Glebova, K.C. Richardson, and V.I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass”, Appl. Opt. 38, 619–27 (1999).
[CrossRef]

1996 (1)

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

1994 (1)

1987 (1)

G.D. Byrne and A.C. Hindmarsh, “Stiff ODE solvers: A review of current and coming attractions”, J. Comp. Phys. 70, 1–62 (1987).
[CrossRef]

1985 (1)

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

1982 (1)

T. K. Gaylord and M.G. Moharam, “Planar Dielectric Grating Diffraction Theories”, Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

1981 (1)

Burr, G.W.

S. Tao and G.W. Burr, “Performance optimization of volume gratings with finite size through numerical simulation”, CLEO/IQEC and PhAST Technical Digest (Optical Society of America, Washington, DC, 2004), CTuE5.

Byrne, G.D.

G.D. Byrne and A.C. Hindmarsh, “Stiff ODE solvers: A review of current and coming attractions”, J. Comp. Phys. 70, 1–62 (1987).
[CrossRef]

Ciapurin, I.V.

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

Cohen, A.D.

A.D. Cohen, M.C. Parker, and R.J. Mears, “100-GHz-resolution dynamic holographic channel management for WDM,” IEEE Phot. Tech. Lett.,  11, 851–3 (1999).
[CrossRef]

Corkum, D.L.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Crossland, W.A.

Dorschner, T.A.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Efimov, O.M.

Friedman, L.J.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord and M.G. Moharam, “Planar Dielectric Grating Diffraction Theories”, Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

Gaylord, T.K.

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

M.G. Moharam and T.K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

Glebov, L.B.

L.B. Glebov, “Kinetics modeling in photosensitive glass,” Optical Materials 25, 413–418 (2004).
[CrossRef]

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

O.M. Efimov, L.B. Glebov, L.N. Glebova, K.C. Richardson, and V.I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass”, Appl. Opt. 38, 619–27 (1999).
[CrossRef]

Glebova, L.N.

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

O.M. Efimov, L.B. Glebov, L.N. Glebova, K.C. Richardson, and V.I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass”, Appl. Opt. 38, 619–27 (1999).
[CrossRef]

Hindmarsh, A.C.

G.D. Byrne and A.C. Hindmarsh, “Stiff ODE solvers: A review of current and coming attractions”, J. Comp. Phys. 70, 1–62 (1987).
[CrossRef]

K. Radhakrishnan and A.C. Hindmarsh, “Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations”, NASA reference publication 1327 (1993).

Hobbs, D.S.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Holz, M.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Liberman, S.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Marrakchi, A.

A. Marrakchi and K. Rastani, “Free-space holographic grating interconnects”, Photonics Switching and Interconnects, A. Marrakchi (ed.), Marcel Dekker, New York, 249–321 (1994).

McManamon, P.F.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Mears, R.J.

A.D. Cohen, M.C. Parker, and R.J. Mears, “100-GHz-resolution dynamic holographic channel management for WDM,” IEEE Phot. Tech. Lett.,  11, 851–3 (1999).
[CrossRef]

D.C. O’Brien, R.J. Mears, T.D. Wilkinson, and W.A. Crossland, “Dynamic holographic inter-connects that use ferroelectric liquid-crystal spatial light modulators”, Appl. Opt. 33, 2795–2803 (1994).
[CrossRef]

Moharam, M.G.

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

T. K. Gaylord and M.G. Moharam, “Planar Dielectric Grating Diffraction Theories”, Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

M.G. Moharam and T.K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

Nguyen, H.Q.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

O’Brien, D.C.

Parker, M.C.

A.D. Cohen, M.C. Parker, and R.J. Mears, “100-GHz-resolution dynamic holographic channel management for WDM,” IEEE Phot. Tech. Lett.,  11, 851–3 (1999).
[CrossRef]

Radhakrishnan, K.

K. Radhakrishnan and A.C. Hindmarsh, “Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations”, NASA reference publication 1327 (1993).

Rastani, K.

A. Marrakchi and K. Rastani, “Free-space holographic grating interconnects”, Photonics Switching and Interconnects, A. Marrakchi (ed.), Marcel Dekker, New York, 249–321 (1994).

Resler, D.P.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Richardson, K.C.

Rotari, E.V.

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

Sharp, R.C.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Smirnov, V.I.

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

O.M. Efimov, L.B. Glebov, L.N. Glebova, K.C. Richardson, and V.I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass”, Appl. Opt. 38, 619–27 (1999).
[CrossRef]

Tao, S.

S. Tao and G.W. Burr, “Performance optimization of volume gratings with finite size through numerical simulation”, CLEO/IQEC and PhAST Technical Digest (Optical Society of America, Washington, DC, 2004), CTuE5.

Watson, E.A.

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

Wilkinson, T.D.

Appl. Opt. (2)

Appl. Phys. B (1)

T. K. Gaylord and M.G. Moharam, “Planar Dielectric Grating Diffraction Theories”, Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

IEEE Phot. Tech. Lett. (1)

A.D. Cohen, M.C. Parker, and R.J. Mears, “100-GHz-resolution dynamic holographic channel management for WDM,” IEEE Phot. Tech. Lett.,  11, 851–3 (1999).
[CrossRef]

J. Comp. Phys. (1)

G.D. Byrne and A.C. Hindmarsh, “Stiff ODE solvers: A review of current and coming attractions”, J. Comp. Phys. 70, 1–62 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Optical Materials (1)

L.B. Glebov, “Kinetics modeling in photosensitive glass,” Optical Materials 25, 413–418 (2004).
[CrossRef]

Proc. of the IEEE (2)

P.F. McManamon, T.A. Dorschner, D.L. Corkum, L.J. Friedman, D.S. Hobbs, M. Holz, S. Liberman, H.Q. Nguyen, D.P. Resler, R.C. Sharp, and E.A. Watson, “Optical phased array technology”, Proc. of the IEEE 84, 268–298 (1996).
[CrossRef]

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

Proceedings of SPIE (1)

I.V. Ciapurin, L.B. Glebov, L.N. Glebova, V.I. Smirnov, and E.V. Rotari, “Incoherent combining of 100-W Yb-fiber laser beams by PTR Bragg grating”, In Advances in Fiber Devices, L. N. Durvasula, Editor, Proceedings of SPIE 4974, 209–219 (2003).
[CrossRef]

Other (3)

S. Tao and G.W. Burr, “Performance optimization of volume gratings with finite size through numerical simulation”, CLEO/IQEC and PhAST Technical Digest (Optical Society of America, Washington, DC, 2004), CTuE5.

A. Marrakchi and K. Rastani, “Free-space holographic grating interconnects”, Photonics Switching and Interconnects, A. Marrakchi (ed.), Marcel Dekker, New York, 249–321 (1994).

K. Radhakrishnan and A.C. Hindmarsh, “Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations”, NASA reference publication 1327 (1993).

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Figures (7)

Fig. 1.
Fig. 1.

Diffraction from a planar dielectric grating bounded by homogeneous media.

Fig. 2.
Fig. 2.

Dependence of refractive index increment for the second exposure of PTR glass to UV radiation at 325 nm. Kinetics parameters are Δnmax =0.7×10-3 and ��=0.5 J/cm2.

Fig. 3.
Fig. 3.

Distribution of refractive index increment in depth of PTR glass after two step exposure to UV radiation at 325 and 250 nm with dosages of 0.7 and 0.6 J/cm2, respectively. The red line is a Gaussian function with half-width of 0.2 cm.

Fig. 4.
Fig. 4.

|S 1|2 as function of d for f (z)=1 and dmax =1.15 mm.

Fig. 5.
Fig. 5.

The spectrum of |S 0(dmax )|2 and |S 1(dmax )|2 for f (z)=1.

Fig. 6.
Fig. 6.

|S 1|2 as function of d for f (z)=Gq , with q=1… 4.

Fig. 7.
Fig. 7.

The spectrum of |S 1(dmax )|2 for uniform and apodized gratings.

Equations (26)

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2 E ̅ + · ( E ̅ · ε ε ) μ ε 2 E ̅ t 2 = 0
2 E y ( x , z ) + k 2 ε ˜ ( x , z ) E y ( x , z ) = 0
ε ˜ 2 ( x , z ) = ε ˜ 20 + ε ˜ 21 ( z ) cos [ K ̅ · x ̅ ]
E 2 y ( x , z ) = m = + S m ( z ) exp ( j k ̅ 2 m · x ̅ )
S m + n a mn S n + n b mn S n = 0
a mn = { 2 j ( k ̅ 2 n · z ̂ ) if n = m 0 if n m
b mn = { [ k 2 2 ( k ̅ 2 n · x ̂ ) 2 ( k ̅ 2 n · z ̂ ) 2 ] n = m k 2 ε ˜ 21 ( z ) 2 n = m 1 or n = m + 1 0 otherwise
E 1 y = exp ( j k ̅ 1 · x ̅ ) + m = R m exp ( j k ̅ 1 m · x ̅ )
E 3 y = m = T m exp ( j k ̅ 3 m · ( x ̅ d z ̂ ) )
( k ̅ 1 m · x ̂ ) = ( k ̅ 2 m · x ̂ ) = ( k ̅ 3 m · x ̂ )
( k ̅ 1 m · z ̂ ) = [ k 1 2 ( k ̅ 2 m · x ̂ ) 2 ] 1 2
( k ̅ 3 m · z ̂ ) = + [ k 3 2 ( k ̅ 2 m · x ̂ ) 2 ] 1 2
( E ̅ ) t ( z = 0 ) : S m ( 0 ) δ 0 m = R m
( H ̅ ) t ( z = 0 ) : j [ k 1 2 ( k ̅ 2 m · x ̂ ) 2 ] 1 2 ( δ 0 m R m ) = S m ( 0 ) j ( k ̅ 2 m · z ̂ ) S m ( 0 )
( E ) ¯ t ( z = d ) : S m ( d ) exp [ j ( k ̅ 2 m · z ̂ ) d ] = T m
( H ̅ ) t ( z = d ) : j [ S m ( d ) j ( k ̅ 2 m · z ̂ ) S m ( d ) ] exp [ j ( k ̅ 2 m · z ̂ ) d ] = ( k ̅ 3 m · z ̂ ) T m
S m ( 0 ) j { ( k ̅ 2 m · z ̂ ) + [ k 1 2 ( k ̅ 2 m · x ̂ ) 2 ] 1 2 } S m ( 0 ) = 2 j [ k 1 2 ( k ̅ 2 m · x ̂ ) 2 ] 1 2 δ 0 m
S m ( d ) j [ ( k ̅ 2 m · z ̂ ) ( k ̅ 3 m · z ̂ ) ] S m ( d ) = 0
ε ˜ 20 + ε ˜ 21 ( z ) cos [ K ̅ · x ̅ ] n 20 2 + 2 n 20 n 21 f ( z ) cos [ K ̅ · x ̅ ]
G ( z ; α , σ q ) = exp [ ( z α ) 2 2 σ q 2 ]
cos ( ϕ θ 2 ) = k 2 k 2
S m ( z ) = i C i Φ im exp ( γ i z )
Δ n = Δ n max E 𝓔 + E ,
Δ n = Δ n max ( E b + E 𝓔 + E b + E E b 𝓔 + E b ) .
E b = E bi [ exp ( Az ) + exp ( A ( z d ) ) ] = E bi B ( z ) ,
Δ n = Δ n max ( E bi B ( z ) + E 𝓔 + E bi B ( z ) + E E bi B ( z ) 𝓔 + E bi B ( z ) )

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