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Femtosecond pulse synthesis by efficient second-harmonic generation in engineered quasi phase matching gratings

Usman K. Sapaev and Gaetano Assanto

Open Access Open Access

Abstract

We numerically design quasi-phase matched crystals with domains of arbitrary sizes for second harmonic generation by femtosecond pulses, taking into account both group velocity mismatch and dispersion. An efficient simulated-annealing algorithm is developed to design quasi-phase matching gratings which can yield the desired amplitude and phase of second-harmonic pulses in the presence of pump depletion. The method is illustrated with reference to single, double-hump and chirped fs Gaussian pulses in a lithium niobate crystal.

©2007 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. Scheme of the arbitrary QPM grating, with δ(z) the dimensionless sign-changing aperiodic function of amplitude ∣δ(z)∣ = 1. The grating is comprised of N inverted domains with individual lengths qm(1 ≤ m ≤ N).

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Fig. 2.
Fig. 2. Integration scheme. Here the number of steps in each domain is j=4.

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Fig. 3.
Fig. 3. (Left) Normalized SHG conversion and (Right) domain size variation versus normalized crystal length (units of LNL ), for various QPM conditions: black lines, uniform grating; blue lines, positively chirped; red lines, negatively chirped; green and magenta lines, randomly varied. Here ε=100*(qm-lo )/lo .

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Fig. 4.
Fig. 4. Illustration of the synthesis task. A fundamental frequency Gaussian shaped pulse (red profile) excites with duration τ=100 fs and peak intensity I0=5 GW/cm2 the QPM structure to be determined (blue box) in order to generate the desired SH target pulses (green profiles on the right).

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Fig. 5.
Fig. 5. Left: FF intensity distributions (green (ξ=0), magenta (ξ=ξo)), desired target profile (∙) and obtained SH profile (blue). Center: PG-FROG spectrograms of target (center left) and final (center right) SH pulses. Right: FF (red) and SH (blue) power versus propagation length. (a) case (i) with a flat phase Gaussian pulse of 100 fs; (b) case (ii) with a 150 fs pulse; (c) case (iii) with 200 fs duration.

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Fig. 6.
Fig. 6. Left: FF intensity distributions (green (ξ=0), magenta (ξ=ξo)), desired target profile (∙) and obtained SH profile (blue). Center: PG-FROG spectrograms of target (center left) and final (center right) SH pulses. Right: FF (red) and SH (blue) power versus propagation length. (a) Case (iv) with two equal width pulses; (b) case (v) with two unequal width pulses of 100 and 200 fs, respectively.

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Fig. 7.
Fig. 7. Left: FF intensity distributions (green (ξ=0), magenta (ξ=ξo)), desired target profile (∙) and obtained SH profile (blue). Center: PG-FROG spectrograms of target (center left) and final (center right) SH pulses. Right: FF (red) and SH (blue) power versus propagation length. (a) Case (vi) with positive chirp ∼150 fs2 on a 100 fs pulse; (b) case (vii) with negative chirp ∼-150 fs2.

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Fig. 8.
Fig. 8. Domain size distribution along the crystal and corresponding number of domains N for the seven target profiles: (i) 100 fs Gaussian, N=37; (ii) 150 fs Gaussian, N=75; (iii) 200 fs Gaussian, N=120; (iv) 100 fs twin-pulses with a separation of 300 fs, N=135; (v) 100 and 200fs pulses with a separation of 400 fs, N=185; (vi) a positively chirped 100 fs Gaussian pulse (φ″ ∼ 150 fs2), N=54; (vii) a negatively chirped 100 fs Gaussian pulse (φ″ ∼ −150 fs2), N=66.

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Fig. 9.
Fig. 9. Calculated SH pulse shape with different resolution in domain size: 100 nm (blue line), 500 nm (green line), 1 μm (red line).

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Equations (8)

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∂ A 1 ∂ z + 1 V 1 ∂ A 1 ∂ t − i α 1 2 ∂ 2 A 1 ∂ t 2 = − iγ 1 δ ( z ) ( A 1 ) * A 2 exp ( − i Δ kz )
∂ A 2 ∂ z + 1 V 2 ∂ A 2 ∂ t − i α 2 2 ∂ 2 A 2 ∂ t 2 = − iγ 2 δ ( z ) ( A 1 ) 2 exp ( i Δ kz )
A 1 ( z , t ) ∣ z = 0 = A 0 exp ( − 2 ln 2 ( 1 τ ) 2 + i φ 1 )
A 2 ( z , t ) ∣ z = 0 = 0
∂ a 1 ∂ ξ − i β 1 ∂ 2 a 1 ∂ μ 2 = − i δ ( ξ ) ( a 1 ) * a 2 exp ( − i Δ S ξ )
∂ a 2 ∂ ξ + ρ ∂ a 2 ∂ μ − i β 2 ∂ 2 a 2 ∂ μ 2 = − i δ ( ξ ) ( a 1 ) 2 exp ( i Δ S ξ )
a 1 ( ξ , μ ) ∣ ξ = 0 = exp ( − 2 ln 2 μ 2 + iφ 1 )
a 2 ( ξ , μ ) ∣ ξ = 0 = 0
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