Pulse compression beyond the Fourier-transform limit

Haitao Liu; Guoguang Mu; Lie Lin; Zhongwei Fan

doi:10.1364/JOSAA.23.000848

Journal of the Optical Society of America A
Vol. 23,
Issue 4,
pp. 848-857
(2006)
•https://doi.org/10.1364/JOSAA.23.000848

Pulse compression beyond the Fourier-transform limit

Haitao Liu, Guoguang Mu, Lie Lin, and Zhongwei Fan

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Haitao Liu, Guoguang Mu, Lie Lin, and Zhongwei Fan, "Pulse compression beyond the Fourier-transform limit," J. Opt. Soc. Am. A 23, 848-857 (2006)

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Abstract

Theories of a technique for compression of an ultrashort femtosecond laser pulse beyond its Fourier-transform limit based on the pulse shaping technique are proposed. The technique is called superresolution in time domain (STD). Global optimization theories for the design of a mask to modulate the spectrum of an input pulse to obtain a STD output pulse are proposed. Several design examples illustrate the feasibility of STD. Some fundamental limits of STD are also provided.

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Variable	Location of definition
G, S, M	Paragraph 1 of Section 2
$e (t)$ , t, ν, $ν_{0}$ , $Δ ν$ , $h (ν)$ , $u (ν)$ , $φ_{s} (ν)$ , $T_{m} (ν)$ , $φ_{m} (ν)$	Before and following Eq. (1)
$E (τ)$ , τ, μ, $H (μ)$ , $U (μ)$ , $I (τ)$ , $I_{L} (τ)$ , $τ_{STD}^{+}$ , $τ_{STD}^{-}$ , $τ_{FTL}^{+}$ , $τ_{FTL}^{-}$ , $τ_{ML}^{+}$ , $τ_{ML}^{-}$	Following Eq. (2)
$τ_{1}$ , $τ_{2}$ , …, $τ_{N}$ , $τ_{ϵ}$	Following problem (3)
$A (μ)$ , $B (μ)$	Before problem (4)
P, $γ (p)$ , p, m, n	Following relation (6)
$A_{k}$ , $B_{k}$ , $μ_{k}$ $(k = 1, 2, \dots, K)$	Before problem (7)
$μ_{j}^{b}$ $(j = 1, 2, \dots, N_{b})$	Following problem (7)
$w_{0}$	2nd paragraph of Section 3; following Eq. (8)
$μ_{j, δ}^{b}$ , δ, $G_{δ}$ , $S_{δ}$ , $M_{δ}$	Before and following Eq. (9)
$Δ μ_{RMS}$ , $Δ τ_{RMS}$ , $Δ τ_{RMS}^{L}$ , $Δ τ_{FWHM}$ , $Δ τ_{FWHM}^{L}$	Before and following relations (10) and (11)
$S^{e u} (G)$	Paragraph 1 of Section 4
$τ_{0.5}$	Following problem (12)

Parameter	Example
Parameter	1	2	3	4
ϵ	0.1	0.1	0.2	0.1
$τ_{ϵ}$	1.467	1.467	1.230	1.126
$τ_{1} ∕ τ_{ϵ}$	0.67	0.75	0.694	0.75
$I (\pm τ_{1}) ∕ I (0)$	0.1	0.1	0.2	0.1
$τ_{N}$	40	10	20	30
G	0.7293	0.8075	0.7297	0.7836
S	0.2429	0.4456	0.2818	0.2264
$S^{e u} (G)$	0.3202	0.4702	0.3209	0.4041
M	0.1044	0.1006	0.2031	0.1079
$μ_{j}^{b}$	$- 0.125$ , $- 0.093$ , $- 0.068$ , $- 0.056$ , $- 0.036$ , $- 0.027$ , $- 0.012$ , $- 0.005$ , 0.005, 0.012, 0.027, 0.036, 0.056, 0.068, 0.093, 0.125	$- 0.065$ , $- 0.033$ , $- 0.004$ , 0.004, 0.033, 0.065	$- 0.090$ , $- 0.047$ , $- 0.016$ , $- 0.006$ , 0.006, 0.016, 0.047, 0.090	$- 0.233$ , $- 0.201$ , $- 0.160$ , $- 0.144$ , $- 0.120$ , $- 0.108$ , $- 0.085$ , $- 0.076$ , $- 0.053$ , $- 0.043$ , $- 0.022$ , $- 0.012$ , 0.001, 0.009, 0.025, 0.039, 0.064, 0.077, 0.108, 0.120, 0.152, 0.188
δ	0.002	0.003	0.003	0.001
$G_{δ} - G$	$- 0.0041$	0.0053	$- 0.0089$	$- 0.0055$
$S_{δ} - S$	$- 0.0051$	0.0135	$- 0.0116$	$- 0.0045$
$M_{δ} - M$	0.0351	$- 0.0034$	0.0325	0.0284
$Δ τ_{FWHM}$	0.5913	0.6547	0.5916	0.5121
$Δ τ_{FWHM}^{L}$	0.8108	0.8108	0.8108	0.6535
$Δ τ_{RMS}$	30.45	19.92	22.01	27.53
$Δ τ_{RMS}^{L}$	0.6857	0.6857	0.6857	0.7878
$Δ μ_{RMS}$	0.1165	0.1165	0.1165	0.1683
$4 π Δ τ_{RMS} Δ μ_{RMS}$	44.56	29.16	32.22	58.22
$4 π Δ τ_{RMS}^{L} Δ μ_{RMS}$	1.004	1.004	1.004	1.666

Variable	Location of definition
G, S, M	Paragraph 1 of Section 2
$e (t)$ , t, ν, $ν_{0}$ , $Δ ν$ , $h (ν)$ , $u (ν)$ , $φ_{s} (ν)$ , $T_{m} (ν)$ , $φ_{m} (ν)$	Before and following Eq. (1)
$E (τ)$ , τ, μ, $H (μ)$ , $U (μ)$ , $I (τ)$ , $I_{L} (τ)$ , $τ_{STD}^{+}$ , $τ_{STD}^{-}$ , $τ_{FTL}^{+}$ , $τ_{FTL}^{-}$ , $τ_{ML}^{+}$ , $τ_{ML}^{-}$	Following Eq. (2)
$τ_{1}$ , $τ_{2}$ , …, $τ_{N}$ , $τ_{ϵ}$	Following problem (3)
$A (μ)$ , $B (μ)$	Before problem (4)
P, $γ (p)$ , p, m, n	Following relation (6)
$A_{k}$ , $B_{k}$ , $μ_{k}$ $(k = 1, 2, \dots, K)$	Before problem (7)
$μ_{j}^{b}$ $(j = 1, 2, \dots, N_{b})$	Following problem (7)
$w_{0}$	2nd paragraph of Section 3; following Eq. (8)
$μ_{j, δ}^{b}$ , δ, $G_{δ}$ , $S_{δ}$ , $M_{δ}$	Before and following Eq. (9)
$Δ μ_{RMS}$ , $Δ τ_{RMS}$ , $Δ τ_{RMS}^{L}$ , $Δ τ_{FWHM}$ , $Δ τ_{FWHM}^{L}$	Before and following relations (10) and (11)
$S^{e u} (G)$	Paragraph 1 of Section 4
$τ_{0.5}$	Following problem (12)

Parameter	Example
Parameter	1	2	3	4
ϵ	0.1	0.1	0.2	0.1
$τ_{ϵ}$	1.467	1.467	1.230	1.126
$τ_{1} ∕ τ_{ϵ}$	0.67	0.75	0.694	0.75
$I (\pm τ_{1}) ∕ I (0)$	0.1	0.1	0.2	0.1
$τ_{N}$	40	10	20	30
G	0.7293	0.8075	0.7297	0.7836
S	0.2429	0.4456	0.2818	0.2264
$S^{e u} (G)$	0.3202	0.4702	0.3209	0.4041
M	0.1044	0.1006	0.2031	0.1079
$μ_{j}^{b}$	$- 0.125$ , $- 0.093$ , $- 0.068$ , $- 0.056$ , $- 0.036$ , $- 0.027$ , $- 0.012$ , $- 0.005$ , 0.005, 0.012, 0.027, 0.036, 0.056, 0.068, 0.093, 0.125	$- 0.065$ , $- 0.033$ , $- 0.004$ , 0.004, 0.033, 0.065	$- 0.090$ , $- 0.047$ , $- 0.016$ , $- 0.006$ , 0.006, 0.016, 0.047, 0.090	$- 0.233$ , $- 0.201$ , $- 0.160$ , $- 0.144$ , $- 0.120$ , $- 0.108$ , $- 0.085$ , $- 0.076$ , $- 0.053$ , $- 0.043$ , $- 0.022$ , $- 0.012$ , 0.001, 0.009, 0.025, 0.039, 0.064, 0.077, 0.108, 0.120, 0.152, 0.188
δ	0.002	0.003	0.003	0.001
$G_{δ} - G$	$- 0.0041$	0.0053	$- 0.0089$	$- 0.0055$
$S_{δ} - S$	$- 0.0051$	0.0135	$- 0.0116$	$- 0.0045$
$M_{δ} - M$	0.0351	$- 0.0034$	0.0325	0.0284
$Δ τ_{FWHM}$	0.5913	0.6547	0.5916	0.5121
$Δ τ_{FWHM}^{L}$	0.8108	0.8108	0.8108	0.6535
$Δ τ_{RMS}$	30.45	19.92	22.01	27.53
$Δ τ_{RMS}^{L}$	0.6857	0.6857	0.6857	0.7878
$Δ μ_{RMS}$	0.1165	0.1165	0.1165	0.1683
$4 π Δ τ_{RMS} Δ μ_{RMS}$	44.56	29.16	32.22	58.22
$4 π Δ τ_{RMS}^{L} Δ μ_{RMS}$	1.004	1.004	1.004	1.666

Abstract

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Equations (61)

Journal of the Optical Society of America A