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Laser-written wave plates inside the silicon enabled by stress-induced birefringence

Alperen Saltik and Onur Tokel

Open Access Open Access

Abstract

Laser writing enables optical functionality by altering the optical properties of materials. To achieve this goal, efforts generally focus on laser-written regions. It has also been shown that birefringence surrounding the modified regions can be exploited for achieving functionality. The effect has been used to fabricate wave plates in glass, with significant potential for other materials. Here, we establish analogous stress control and birefringence engineering inside silicon. We first develop a robust analytical model enabling the prediction of birefringence maps from arbitrary laser-written patterns. Then, we tailor three-dimensional laser lithography to create the first, to the best of our knowledge, polarization-control optics inside silicon.

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Supplementary Material (1)

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Data can be obtained from the authors upon reasonable request.

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Figures (4)
Fig. 1.
Fig. 1. Compressive stress resulting from a simple laser-written structure on its surroundings is shown in (a), with the associated transmission microscope image in (b) and experimental retardance map in (c). The orange contour in (c) and the green contour in (f) are plotted in (d), along with the cos$(2\theta )$ function. The computational model of the basic micro-structure is shown in (e), along with the associated retardance map in (f), evaluated with the developed model.

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Fig. 2.
Fig. 2. (a) Transmission microscope image of a laser-written star pattern. (b) Unknown transfer function from written pattern to retardance. (c) Experimental retardance map. (d) Computational recreation of the star shape. (e) Transfer function in our model. (f) Predicted retardance map from the model. The borders of laser writing is indicated with black lines in (c) and (f).

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Fig. 3.
Fig. 3. (a) Architectural design for the wave plates. (b) Retardance is averaged over 1 $\times$ 1 mm2 of the active area and is shown for different thicknesses and number of levels. The experimental results are shown with empty circles, and the analytical model predictions are given with crosses. The model is calibrated with the 0.25-mm-thick experimental data points. The retardance values close to HWP and QWP operation are encircled at 3.25 rad and 1.60 rad. The experimental retardance map for the 0.75-mm-thick, single-level data, and the corresponding model prediction are given in (c) and (d), respectively. The averaged 1 $\times$ 1 mm2 area is indicated in (c) and (d) with black squares. Color bar applies to both (c) and (d).

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Fig. 4.
Fig. 4. (a) HWP characterization scheme based on Malus’s law. (b) Transmitted intensity as a function of input polarization passing through the active region is compared with control experiment. (c) Phase difference between these curves defines the retardance, found as 3.23 rad. This is in strong agreement with phase microscopy results (3.25 rad) acquired from the retardance map. (d) and (e) Intensity images for maximum and minimum transmissions, respectively. Data in (b) are averages over the 1 $\times$ 1 mm2 active area, indicated with green squares in (d) and (e).

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Equations (10)

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σ r , θ = [ σ r = α r 2 σ r θ = 0 σ θ r = 0 σ θ = − α r 2 ] .
n r − n θ = C ( θ ) ( σ r − σ θ ) = C ( θ ) 2 α r 2 ,
T ≡ e i 2 π n θ d / λ [ e i Δ φ 0 0 1 ] r , θ .
Δ φ = 2 π d λ ( n r − n θ ) = 2 π d λ C ( θ ) 2 α r 2 ,
T ≡ [ e i Δ φ cos 2 ⁡ ( θ ) + sin 2 ⁡ ( θ ) ( e i Δ φ − 1 ) sin ⁡ ( θ ) cos ⁡ ( θ ) ( e i Δ φ − 1 ) sin ⁡ ( θ ) cos ⁡ ( θ ) e i Δ φ sin 2 ⁡ ( θ ) + cos 2 ⁡ ( θ ) ] x , y .
B 0 = φ y − φ x = arg ⁡ ( T 22 ) − arg ⁡ ( T 11 ) .
B 0 ( r , θ ) = β r − 2 cos ⁡ ( 2 θ ) ,
B 0 ( x , y ) = β ( x 2 + y 2 ) − 1 cos ⁡ ( 2 tan − 1 ⁡ ( y x ) ) .
B ( x , y ) = LW ( x , y ) ⊛ B 0 ( x , y ) .
B ( x , y ) = FT − 1 [ FT [ LW ( x , y ) ] . FT [ B 0 ( x , y ) ] ] .
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