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Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms

Cristian Neipp, Inmaculada Pascual, and Augusto Beléndez

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Abstract

The Coupled Wave Theory of Kogelnik has given a well-established basis for the comprehension of how light propagates inside a hologram. This theory gives an accurate approximation for the diffraction efficiency of volume phase holograms and volume absorption holograms as well. Mixed holograms (phase and absorption) have been also treated from the point of view of this theory. For instance, Guibelalde theoretically described the diffraction efficiency of out of phase mixed volume gratings. In this work we will show that when using fixation-free rehalogenating bleaches, out of phase mixed volume gratings can be recorded on the hologram at high exposures. This is due to the oxidation products of the developer and the bleaching agent. The effects described theoretically for out of phase mixed volume hologram gratings are experimentally observed.

©2002 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Transmission efficiency as a function of the angle for different values of the phase difference, φ. n1 = 0.05, d = 8 μm, α0 = 0.03 μm-1 and α1 = 0.03 μm-1.

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Fig. 2.
Fig. 2. Diffraction efficiency as a function of the phase difference, φ, between the refractive index and the absorption constant for different values of the absorption constant modulation, α1 , n1 = 0.030 and α0 = 0.030 μm-1.

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Fig. 3.
Fig. 3. Experimental set-up.

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Fig. 4.
Fig. 4. Unslanted mixed diffraction gratings.

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Fig. 5.
Fig. 5. - Transmittance as a function of the angle for a mixed diffraction grating. Parameters: n1 = 0.085, d = 6.8 μm, α0 = 0.017 μm-1, α1 = 0.012 μm-1, αs = 0.018 μm-1, φ = 0.25 rad for θ∈ [-45°,0°], φ = 4.3l rad for θ∈ [0°,45°].

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Fig. 6.
Fig. 6. - Transmittance as a function of the angle for a mixed diffraction grating. Parameters: n1 = 0.091, d = 6.8 μm, α0 = 0.019 μm-1, α1 = 0.014 μm-1, αs = 0.014 μm-1, φ = 4.19 rad for θ∈ [-45°,0°], φ = 0.13 rad for θ∈ [0°,45°].

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Tables (2)

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Table I. Schedule procedure

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Table II. Bleach bath composition (modified version of R-10)

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Equations (20)

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Q = 2 πλd n Λ 2
n = n 0 + n 1 cos ( Kr )
α = α 0 + α 1 cos ( Kr + φ )
R ( z ) = − 1 c r ( γ 1 − γ 2 ) [ ( c r γ 2 + α 0 ) exp ( γ 1 z ) + ( c r γ 1 + α 0 ) exp ( γ 2 z ) ]
S ( z ) = − j χ 2 c s ( γ 1 − γ 2 ) [ exp ( γ 1 z ) − exp ( γ 2 z ) ]
γ 1,2 = − 1 2 [ α 0 c r + α 0 c s + j ϑ c s ] ± 1 2 [ ( α 0 c r − α 0 c s − j ϑ c s ) 2 − 4 χ 1 χ 2 c r c s ] 1 2
χ 1 = π n 1 λ − j exp ( − j φ ) α 1 2
χ 2 = π n 1 λ − j exp ( j φ ) α 1 2
η = ∣ c s ∣ c r S ( d ) S ( d ) *
τ = R ( d ) R ( d ) *
τ = exp ( − α s d ) · RR *
S ( d ) = − j exp ( − α 0 d c r ) ( a − j e jφ b ) sin [ ( a − j e jφ b ) · ( a − bj e − jφ ) ] ( a − bj e jφ ) · ( a − bj e − jφ )
a = dπ n 1 λc r
b = α 1 d 2 c r
S ( d ) = − j ( c r c s ) 1 2 exp ( − α 0 d c r ) sin ⁡ ( a − jb )
η = exp ( − 2 α 0 d c r ) · [ sin ⁡ 2 ( a ) + sinh ⁡ 2 ( b ) ]
n r = n 0 + n r 1 cos ( Kr + π )
n h = n 0 + n h 1 cos ( Kr )
α = α 0 + α 1 cos ( Kr )
n = n 0 + n 1 cos ( Kr + φ )
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