Single shot interferometric method for measuring the nonlinear refractive index

Ioan Dancus; Silviu T. Popescu; Adrian Petris

doi:10.1364/OE.21.031303

Optics Express
Vol. 21,
Issue 25,
pp. 31303-31308
(2013)
•https://doi.org/10.1364/OE.21.031303

Single shot interferometric method for measuring the nonlinear refractive index

Ioan Dancus, Silviu T. Popescu, and Adrian Petris

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Ioan Dancus, Silviu T. Popescu, and Adrian Petris, "Single shot interferometric method for measuring the nonlinear refractive index," Opt. Express 21, 31303-31308 (2013)

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Abstract

In this paper we are introducing a new single shot method for measuring the nonlinear refractive index of materials in a simple interferometric pump-probe configuration. The theoretical model proposed for extracting the nonlinear refractive index from the experimental fringe pattern and the experimental configuration are presented and discussed. The results obtained by this method are in good agreement with that obtained on the same sample using the conventional Z-scan method.

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Fig. 1 The experimental setup based on a Michelson interferometer (a) and an experimental fringe pattern obtained when the investigated sample is excited (b).

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Fig. 2 (a) 3D representation of the reconstructed phase (dots) fitted with Eq. (10) (surface). (b) The top view of the fitting surface. Inset: the reconstructed phase change corresponding to the tilted Gaussian excitation only.

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Equations (19)

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I (x, y, z = 0) = I_{0} \cdot e^{(- \frac{{((x - x_{0}) \cdot \cos (θ_{e, x}))}^{2} + {((y - y_{0}) \cdot \cos (θ_{e, y}))}^{2}}{w^{2}})} .

Δ n (x, y, z) = n_{2} \cdot I_{e x} (x, y, z),

I_{e x} (x, y, z) = I (x, y, z = 0) \cdot e^{- α_{0} \cdot z} .

Δ Φ_{n l} (x, y, z = L) = k \cdot \int_{0}^{L} Δ n (x, y, z) \cdot d z .

Δ Φ_{n l} (x, y, z = L) = k \cdot n_{2} \cdot I_{e x} (x, y, z = 0) \cdot \frac{1 - e^{- α_{0} L}}{α_{0}}

Δ Φ_{n l} (x, y) \equiv Δ Φ_{n l} (x, y, z > L) = Δ Φ_{n l} (x, y, z = L) .

Δ Φ_{n l} (x, y, z = L) = Δ Φ {(x, y, z)}^{(3)} + Δ Φ {(x, y, z)}^{(5)} + \dots

Δ Φ {(x, y, z = L)}_{}^{(3)} = k n_{2} I_{e x} (x, y, z = 0) [1 - \exp (- α_{0} L)] / α_{0},

Δ Φ {(x, y, z = L)}_{}^{(5)} = k n_{4} I_{e x}^{2} (x, y, z = 0) [1 - \exp (- 2 α_{0} L)] / 2 α_{0} .

Δ n (x, y, z) = \frac{n_{2} \cdot I_{e x} (x, y, z)}{1 + I_{e x} (x, y, z) / I_{s a t}} .

Δ Φ_{n l} (x, y, z = L) = \frac{k \cdot n_{2} \cdot I_{s a t}}{α_{0}} \cdot \ln [\frac{I_{s a t} + I_{e x} (x, y, z = 0)}{I_{s a t} + I_{e x} (x, y, z = 0) \cdot e^{- α_{0} \cdot L}}]

I (x, y) = I_{1} (x, y) + I_{2} (x, y) + 2 \sqrt{(I_{1} (x, y) \cdot I_{2} (x, y))} \cdot \cos (Δ Φ (x, y)) .

Δ Φ (x, y) \equiv Δ Φ_{t o t a l} (x, y) = 2 \cdot Δ Φ_{n l} (x, y) + Δ Φ_{t i l t} (x, y),

Δ Φ_{t i l t} (x, y) = k \cdot [x \cdot \sin (θ_{x}) + y \cdot \sin (θ_{y})],

I (x, y) = A (x, y) + B (x, y) \cdot \cos (2 \cdot Δ Φ_{n l} (x, y) + k \sin (θ_{x}) x),

A (x, y) = I_{1} (x, y) + I_{2} (x, y) and B (x, y) = 2 \sqrt{I_{1} (x, y) + I_{2} (x, y)} .

I (x, y) = A (x, y) + C (x, y) \cdot e^{i k x \sin (θ_{x})} + C^{*} (x, y) \cdot e^{- i k x \sin (θ_{x})},

C (x, y) = (1 / 2) \cdot B (x, y) \cdot e^{2 i Δ Φ_{n l} (x, y)}

\tilde{I} (u_{x}, u_{y}) = \tilde{A} (u_{x}, u_{y}) + \tilde{C} (u_{x} - k \sin (θ_{x}), u_{y}) + {\tilde{C}}^{*} (u_{x} + k \sin (θ_{x}), u_{y}),

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