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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  2. D.C. O�??Brien, R.J. Mears, T.D. Wilkinson and W.A. Crossland, �??Dynamic holographic interconnects 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]

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]

CLEO/IQEC 2004 (1)

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.

IEEE Phot. Tech. Lett. (1)

A.D. Cohen, M.C. Parker, 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)

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

Optical Materials (1)

L.B. Glebov, �??Kinetics modeling in photosensitive glass,�?? Optical Materials 25, 413-418 (2004).
[CrossRef]

Photonics Switching and Interconnects (1)

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

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]

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 (1)

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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