Nonlinear refractive index measurement by SPM-induced phase regression

Piotr Kabaciński; Tomasz M. Kardaś; Yuriy Stepanenko; Czesław Radzewicz

doi:10.1364/OE.27.011018

Optics Express
Vol. 27,
Issue 8,
pp. 11018-11028
(2019)
•https://doi.org/10.1364/OE.27.011018

Nonlinear refractive index measurement by SPM-induced phase regression

Piotr Kabaciński, Tomasz M. Kardaś, Yuriy Stepanenko, and Czesław Radzewicz

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Piotr Kabaciński, Tomasz M. Kardaś, Yuriy Stepanenko, and Czesław Radzewicz, "Nonlinear refractive index measurement by SPM-induced phase regression," Opt. Express 27, 11018-11028 (2019)

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- Nonlinear Optics
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About this Article
History
- Original Manuscript: January 28, 2019
- Revised Manuscript: March 9, 2019
- Manuscript Accepted: March 17, 2019
- Published: April 4, 2019

Abstract

Herewith, we describe how intensity and phase of the ultrashort pulse retrieved with second-harmonic frequency-resolved optical gating (SHG FROG) can be utilized for measurement of the nonlinear refractive index (n $_{2}$ ). Through comparison with available literature, we show that our method surpasses Z-scan in terms of precision by a factor of two, and thus, constitutes an interesting alternative. We present results for various materials: fused silica, calcite, YVO $_{4}$ , BiBO, CaF $_{2}$ , and YAG at 1030 nm. Unlike the Z-scan, the use of this method is not restricted to free-space geometry, but due to its characteristics, it can be used in integrated waveguides or photonic crystal fibers as well.

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Fig. 1 Schematic description of the method of nonlinear index of refraction measurement. (a) First, the known ultrashort pulse is propagated through the sample to be measured, which modifies its phase via self-phase modulation, whilst having little effect on the intensity envelope. (b) Subtraction of the phase of the initial pulse from the phase measured after propagation through the sample yields SPM-induced phase component. (c) Intensity and the SPM-induced phase are fitted together and according to Eq. (1) the value of n

_{2}

is obtained.

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Fig. 2 Measurement setup used in the experiment. Femtosecond pulses are weakly focused onto the sample and after attenuation of the beam are characterized via the FROG apparatus.

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Fig. 3 Procedure of fitting SPM-induced phase to intensity profile for measuring n

_{2}

in fused silica for 160 μJ. (a) Visual comparison of intensity profiles for a pulse propagated through the sample and a reference pulse. (b) Subtraction of the reference phase from the phase measured after propagation through the sample. Their difference is the SPM-induced component of the phase. (c) Phase predicted from intensity profile is fitted to the measured SPM-induced phase. Best fit line is described by Eq. (3). Slope of the purple line presented here is equal to n

_{2}

value for measured material. (d) Intensity profile scaled according to Eq. (3) is compared to the measured SPM-induced phase, showing almost perfect agreement.

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Fig. 4 SPM-induced phases (dashed red) fitted to intensity profiles (blue) for different incident pulse energies for fused silica, CaF

_{2}

, YAG and BiBO.

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Fig. 5 SPM-induced phases (dashed red) fitted to intensity profiles (blue) for different incident pulse energies for birefringent materials: Calcite and YVO

_{4}

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Fig. 6 All measured n₂ values for varying pulse energy for different samples. Straight lines are weighted averages of values for given sample.

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Tables (1)

Table 1 Nonlinear Refractive Index Measured in This Work at 1030 nm Compared with Values Known from Literature^a

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Equations (9)

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Δ ϕ (t) = n_{2} \frac{2 π d}{λ} I (t),

Δ ϕ (t) = ϕ_{s} (t) - ϕ_{i} (t)

Δ ϕ (t) = n_{2} \frac{2 π d}{λ} I (t) + a t + b,

ϕ_{t} = \frac{2 π d}{λ} I_{s} = \frac{2 π d}{λ} \frac{E}{π w_{0}^{2}} \frac{I_{s}}{\int I_{s} d t}

ϕ_{t}^{e r r} = \sqrt{\sum_{p} {(\frac{d ϕ_{t}}{d p} p^{e r r})}^{2}} = ϕ_{t} \sqrt{{(\frac{I_{s}^{e r r}}{I_{s}})}^{2} + {(\frac{λ^{e r r}}{λ})}^{2} + {(\frac{d^{e r r}}{d})}^{2} + {(\frac{E^{e r r}}{E})}^{2} + 4 {(\frac{w_{0}^{e r r}}{w_{0}})}^{2}}

W_{1} = \frac{1}{{(ϕ_{S P M}^{e r r})}^{2}}

W_{j} = \frac{1}{{(ϕ_{S P M}^{e r r})}^{2} + {(n_{2, j - 1} ϕ_{t}^{e r r})}^{2}}

{\bar{n}}_{2} = \frac{\sum_{i}^{k} n_{2, i} / n_{2, i}^{e r r}}{\sum_{i}^{k} 1 / n_{2, i}^{e r r}}, {\bar{n}}_{2}^{e r r} = \sqrt{max (σ_{i n t}^{2}, σ_{e x t}^{2})}

σ_{i n t}^{2} = \frac{1}{\sum_{i}^{k} 1 / n_{2, i}^{e r r}} and σ_{e x t}^{2} = \frac{σ_{i n t}^{2}}{k - 1} \sum_{i}^{k} {(\frac{n_{2, i} - {\bar{n}}_{2}}{n_{2, i}^{e r r}})}^{2}

Material	This work	Literature
Fused silica	$2.19 \pm 0.05$	$2.23 \pm 0.12$ at 1030 nm [21]
		$2.74 \pm 0.17$ at 1053 nm [33]
		$2.46$ at 1064 nm [17]
$C a F_{2}$	$1.71 \pm 0.09$	$1.9 \pm 0.26$ at 1064 nm [34]
		$1.26$ at 1064 nm [17]
YAG	$6.13 \pm 0.07$	$6.25$ at 1064 nm [17]
		$7.23 \pm 1.4$ at 1064 nm [35]
BiBO $E \| \| X$	$7.65 \pm 0.09$	$7.39, 8.00$ at 1064 nm [36]
Calcite n_o	$3.22 \pm 0.09$	$2.83$ at 1064 nm [17]
Calcite n_e	$2.12 \pm 0.07$	$2.35$ at 1064 nm [17]
$Y V O_{4} n_{o}$	$15.6 \pm 0.7$	19 at 1080 nm in Yb:YVO $_{4}$ [37]
$Y V O_{4} n_{e}$	$14.9 \pm 0.8$	15 at 1080 nm in Yb:YVO $_{4}$ [37]

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