Distortion cancellation of frequency converted pulses with simple linear signal processing and application to frequency modulation to amplitude modulation conversion in high power lasers

Sébastien Vidal; Jacques Luce; Steve Hocquet; Claude Gouédard; Pierre Calvet; Denis Penninckx

doi:10.1364/AO.51.005818

Applied Optics
Vol. 51,
Issue 24,
pp. 5818-5825
(2012)
•https://doi.org/10.1364/AO.51.005818

Distortion cancellation of frequency converted pulses with simple linear signal processing and application to frequency modulation to amplitude modulation conversion in high power lasers

Sébastien Vidal, Jacques Luce, Steve Hocquet, Claude Gouédard, Pierre Calvet, and Denis Penninckx

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Sébastien Vidal, Jacques Luce, Steve Hocquet, Claude Gouédard, Pierre Calvet, and Denis Penninckx, "Distortion cancellation of frequency converted pulses with simple linear signal processing and application to frequency modulation to amplitude modulation conversion in high power lasers," Appl. Opt. 51, 5818-5825 (2012)

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- Nonlinear Optics
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About this Article
History
- Original Manuscript: April 16, 2012
- Revised Manuscript: June 11, 2012
- Manuscript Accepted: July 12, 2012
- Published: August 13, 2012

Abstract

It is known that a linear filter may be easily compensated with its inverse transfer function. However, it was shown that this approach could also be valid even for such a complex nonlinear system as frequency conversion. As a matter of fact, it is possible to at least partly precompensate for distortions occurring within, or even downstream from, frequency conversion crystals with a simple linear optical filter set upstream. In this paper, we give the theoretical background and derive the optimum precompensation filter from simple analytical formulas even in the case of saturation. We first show the relevance of our approach for Gaussian pulses: the pulse may be short or not and chirped or not, and the same linear precompensation filter may be used as long as saturation is not reached. We then study the case of phase-modulated pulses, as can be found on high power lasers such as lasers for fusion. We show that previous experimental results are in perfect agreement with these calculations. Finally, justified by our simple analytical formulas, we present a rigorous parametrical study giving the distortion reduction for any second and third harmonic generation system in the case of phase-modulated pulses.

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	$β$	$γ$ (GHz)	$α_{in}$ (%)	$γ_{comp}$ (GHz)	$α_{comp}$ (%)	$β \cdot γ^{2} / γ_{comp}^{2}$
$I_{1} = I_{c} / 10$	2	80	34	56.5	$\approx 0$	$4.0 \pm 0.2$
$I_{1} = I_{c}$	1.7	77	38	49	$\approx 0$	$4.2 \pm 0.2$
$I_{1} = 2 \cdot I_{c}$	1.45	71	41	43	$\approx 2$	$4.0 \pm 0.2$
$I_{1} = 5 \cdot I_{c}$	1.05	59	55	29	$\approx 16$	$4.3 \pm 0.2$

	$β$	$φ_{2 ω}^{''} (\times 10^{8} {fs}^{2})$	$α_{in}$ (%)	$φ_{comp}^{''} (\times 10^{8} {fs}^{2})$	$α_{comp}$ (%)	$- φ_{comp}^{''} β / φ_{2 ω}^{''}$
$I_{1} = I_{c} / 10$	2	$- 0.8$	35	$+ 0.8$	$\approx 0$	$2.0 \pm 0.1$
$I_{1} = I_{c}$	1.7	$- 0.8$	35	$+ 0.94$	$\approx 0$	$2.0 \pm 0.1$
$I_{1} = 2 \cdot I_{c}$	1.45	$- 0.8$	35	$+ 1.1$	$\approx 1$	$2.0 \pm 0.1$
$I_{1} = 5 \cdot I_{c}$	1.05	$- 0.8$	35	$+ 1.5$	$\approx 14$	$2.0 \pm 0.1$

	$I_{1} = I_{c} / 10$	$I_{1} = I_{c}$	$I_{1} = 2 \cdot I_{c}$	$I_{1} = 5 \cdot I_{c}$
$γ (I_{1}) = 3 Δ F_{2 ω}$	$α_{in} = 5 %$ $α_{comp} = 0 %$	$α_{in} = 5 %$ $α_{comp} = 0 %$	$α_{in} = 5 %$ $α_{comp} = 0 %$	$α_{in} = 5 %$ $α_{comp} = 0 %$
$γ (I_{1}) = 2 Δ F_{2 ω}$	$α_{in} = 13 %$ $α_{comp} = 0 %$	$α_{in} = 13 %$ $α_{comp} = 0 %$	$α_{in} = 13 %$ $α_{comp} = 0 %$	$α_{in} = 13 %$ $α_{comp} = 0 %$
$γ (I_{1}) = 1.5 Δ F_{2 ω}$	$α_{in} = 23 %$ $α_{comp} = 1.5 %$	$α_{in} = 23 %$ $α_{comp} = 0.3 %$	$α_{in} = 23 %$ $α_{comp} = 1 %$	$α_{in} = 23 %$ $α_{comp} = 3 %$
$γ (I_{1}) = 1 Δ F_{2 ω}$	$α_{in} = 55 %$ $α_{comp} = 1.5 %$	$α_{in} = 55 %$ $α_{comp} = 3 %$	$α_{in} = 55 %$ $α_{comp} = 6 %$	$α_{in} = 53 %$ $α_{comp} = 12 %$
$γ (I_{1}) = 0.75 Δ F_{2 ω}$	$α_{in} = 95 %$ $α_{comp} = 4 %$	$α_{in} = 95 %$ $α_{comp} = 7 %$	$α_{in} = 92 %$ $α_{comp} = 18 %$	$α_{in} = 88 %$ $α_{comp} = 36 %$

	$I_{1} = I_{c} / 10$	$I_{1} = I_{c}$	$I_{1} = 1.5 \cdot I_{c}$	$I_{1} = 2.5 I_{c}$
$γ (I_{1}) = 3 Δ F_{3 ω}$	$α_{in} = 5 %$ $α_{comp} = 0 %$	$α_{in} = 5 %$ $α_{comp} = 0 %$	$α_{in} = 5 %$ $α_{comp} = 0.7 %$	$α_{in} = 5 %$ $α_{comp} = 1 %$
$γ (I_{1}) = 2 Δ F_{3 ω}$	$α_{in} = 12 %$ $α_{comp} = 0 %$	$α_{in} = 12 %$ $α_{comp} = 0 %$	$α_{in} = 12 %$ $α_{comp} = 1 %$	$α_{in} = 12 %$ $α_{comp} = 3 %$
$γ (I_{1}) = 1.5 Δ F_{3 ω}$	$α_{in} = 23 %$ $α_{comp} = 0.6 %$	$α_{in} = 23 %$ $α_{comp} = 1 %$	$α_{in} = 23 %$ $α_{comp} = 2 %$	$α_{in} = 23 %$ $α_{comp} = 6 %$
$γ (I_{1}) = 1 Δ F_{3 ω}$	$α_{in} = 57 %$ $α_{comp} = 2.3 %$	$α_{in} = 57 %$ $α_{comp} = 7 %$	$α_{in} = 57 %$ $α_{comp} = 16 %$	$α_{in} = 57 %$ $α_{comp} = 34 %$
$γ (I_{1}) = 0.75 Δ F_{3 ω}$	$α_{in} = 105 %$ $α_{comp} = 6.5 %$	$α_{in} = 105 %$ $α_{comp} = 20 %$	$α_{in} = 105 %$ $α_{comp} = 48 %$	$α_{in} = 102 %$ $α_{comp} = 65 %$

	$β$	$γ$ (GHz)	$α_{in}$ (%)	$γ_{comp}$ (GHz)	$α_{comp}$ (%)	$β \cdot γ^{2} / γ_{comp}^{2}$
$I_{1} = I_{c} / 10$	2	80	34	56.5	$\approx 0$	$4.0 \pm 0.2$
$I_{1} = I_{c}$	1.7	77	38	49	$\approx 0$	$4.2 \pm 0.2$
$I_{1} = 2 \cdot I_{c}$	1.45	71	41	43	$\approx 2$	$4.0 \pm 0.2$
$I_{1} = 5 \cdot I_{c}$	1.05	59	55	29	$\approx 16$	$4.3 \pm 0.2$

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Applied Optics