Birefringence measurements of diamond space-variant subwavelength gratings

Pierre Piron; Ernesto Vargas Catalan; Olivier Absil; Mikael Karlsson

doi:10.1364/AO.57.004909

1. INTRODUCTION

Subwavelength gratings are, as their name suggests, gratings with periods $(Λ)$ smaller than the incident wavelength $(λ)$ . These gratings only allow the transmission of the zeroth order. The zero-order condition [1–3] is

\frac{Λ}{λ} \leq \frac{1}{n_{I} \sin ω_{i} + \max (n_{I}, n_{II})},

where

n_{I}

is the refractive index of the incident medium,

n_{II}

is the refractive index of the substrate, and

ω_{i}

is the incident angle.

These gratings act as birefringent plates and they can be described by four optical parameters, namely, the phase shift induced between the TE and TM components of the incident beam $(ϕ)$ , the local fast axis orientation $(α)$ , and the transmission efficiency of the TE and TM components ( $η_{TE}^{2}$ and $η_{TM}^{2}$ ). These parameters depend on the grating parameters, namely, grating width on top $(w)$ , depth ( $d$ ), side wall angles ( $β$ ), and grating orientation (see Fig. 1). Using the effective medium theory and the rigorous coupled-wave analysis (RCWA), one can design the grating parameters to achieve a specific fast axis orientation and a desired phase shift over a specified bandwidth [2–6]. Another advantage of these gratings is the possibility to create space-variant elements, i.e., elements with a variation of their fast axis orientation: $α = α (x, y)$ .

These elements have several applications such as in coronagraphy [1,2,7], polarization analysis [3,8], and in the generation of radially and azimuthally polarized beams [9]. Finally, diamond has advantages of high transmission in the infrared region, robustness to external conditions, chemical resistance, low thermal expansion, and high thermal conductivity.

The efficiency of these components for the applications mentioned previously are affected by a deviation from the ideal fast axis orientation and by a variation of the phase shift with the wavelength; although this variation is mitigated by design [6], it still impacts performance [2,3,10]. For a proper spectral analysis of the optical parameters, the infrared source is needed to fulfill two requirements. It must possess a high intensity over a broadband domain to improve the signal-to-noise ratio and a fine spectral resolution to only acquire data over a small spectral range. Those requirements were achieved with a tunable quantum cascade laser (QCL). Using the measurements of the four optical parameters, one can compute the performance of the element. The paper will present the validation of an experimental method to measure $α$ , $ϕ$ , and the efficiency ratio $q = \frac{η_{TE}^{2}}{η_{TM}^{2}}$ for one kind of subwavelength grating, namely, the annular groove phase mask (AGPM). AGPMs are constituted by concentric circular subwavelength gratings and they are used for coronagraphy [2,11–13]; they can be seen as half-wave plates with a radial orientation of their fast axis.

In this paper, first, the experimental method, the optical components, and the mathematical model will be described. Then, numerical simulations of the measurement process will be performed for three AGPM cases of increased complexity. Those numerical simulations aim to validate the principle of the measurement and to verify that the birefringent parameters can be extracted by fitting the intensity ratio curves. Therefore, the parameters obtained through the fitting will be compared with the parameters introduced to produce the intensity curves. Afterward, the components of the optical setup will be presented and the measurement results of one component will be shown and compared with RCWA simulations. Finally, the principle of the method and its results will be summarized.

2. DESCRIPTION OF THE EXPERIMENTAL METHOD

The experimental method is inspired by the one used by Yang and Yeh [14] to measure the phase shift and local fast axis orientation of birefringent elements. Our experimental setup is presented in Fig. 2. The QCL emits a collimated beam, which is transmitted through a linear polarizer $P_{0}$ and a quarter-wave plate $λ / 4$ to achieve a circular polarization. Then, the beam is transmitted through a computer-controlled linear polarizer $P_{1}$ , the sample $S$ (AGPM), and a second computer-controlled linear polarizer $P_{2}$ . Finally, the beam intensity is recorded with an imaging detector. The method principle is as follows: the transmission axis of $P_{1}$ will perform a full rotation in increments of 10°. For each orientation of $P_{1}$ with a transmission axis orientation of $θ$ , two intensities, $I_{∥} (x, y, θ)$ and $I_{⊥} (x, y, θ)$ , will be recorded. First, $P_{2}$ rotates to achieve parallel orientation of both transmission axes and $I_{∥} (x, y, θ)$ is recorded with the detector. Then $P_{2}$ is rotated to achieve perpendicular orientation of transmission axes and $I_{⊥} (x, y, θ)$ is recorded. After a complete revolution of $P_{1}$ , the intensity ratio $Γ (x, y, θ)$ will be computed by dividing the intensities by each other for each orientation of $P_{1}$ and each pixel: $Γ (x, y, θ) = \frac{I_{⊥} (x, y, θ)}{I_{∥} (x, y, θ)}$ (see Fig. 3 for the $Γ$ curve for one pixel). Afterward, $Γ (x, y, θ)$ will be fitted as a function of $θ$ to extract $ϕ$ , $α$ , and $q$ for each pixel.

Fig. 1. Representation of the grating parameters for a beam with an incident angle of $ω_{i}$ .

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Fig. 2. Representation of the optical setup: a fixed linear polarizer $P_{0}$ and a quarter-wave plate $λ / 4$ aim to create a circular polarization to avoid intensity modulation on the sample for different orientations of $P_{1}$ . $P_{1}$ and $P_{2}$ are computer-controlled linear polarizers, $S$ represents the component to be measured, and $D$ stands for the detector. The polarization states at different places of the optical setup are represented by blue arrows and the beam is shown in red.

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Fig. 3. Representation of $Γ$ as a function of $θ$ with a sampling step of 10°, for a half-wave plate component with perfect transmission of TE and TM components, and a local orientation of 49° of its fast axis.

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To develop the mathematical model, the Jones formalism [2,15] was chosen. After the quarter-wave plate, the beam is circularly polarized and the electric field is given by $J_{in}$ :

J_{in} = \frac{\sqrt{2}}{2} (\begin{matrix} 1 \\ - i \end{matrix}) .

The first polarizer

P_{1}

is expressed as

P_{1} = (\begin{matrix} \cos^{2} (θ) & 0.5 \sin (2 θ) \\ 0.5 \sin (2 θ) & \sin^{2} (θ) \end{matrix}),

where

θ

is the orientation of the transmissive axis of the polarizer. The sample

S

is given by Eq. (2) [2]:

S = (\begin{matrix} η_{TM} \cos^{2} (α) + η_{TE} \sin^{2} (α) e^{i ϕ} & \frac{1}{2} \sin (2 α) (η_{TM} - η_{TE} e^{i ϕ}) \\ \frac{1}{2} \sin (2 α) (η_{TM} - η_{TE} e^{i ϕ}) & η_{TM} \sin^{2} (α) + η_{TE} \cos^{2} (α) e^{i ϕ} \end{matrix}),

where

α

is the local orientation of the local fast axis,

ϕ

is the phase shift between the TE and TM components, and

η_{TE}^{2}

and

η_{TM}^{2}

are, respectively, the transmission efficiencies for the TE and TM components of the incident polarization. Finally, the second polarizer

P_{2}

is expressed as

P_{2} = (\begin{matrix} \cos^{2} (γ) & 0.5 \sin (2 γ) \\ 0.5 \sin (2 γ) & \sin^{2} (γ) \end{matrix}),

where

γ

is the angle formed by the transmissive axis of

P_{2}

and the horizontal,

γ = θ

for the computation of

I_{∥}

, and

γ = 90 + θ

for

I_{⊥}

. At the detector, the Jones vector

J_{out}

is simply the result of the multiplications of the matrices:

J_{out ∥} = P_{2 ∥} \times S \times P_{1} \times J_{in} J_{out ⊥} = P_{2 ⊥} \times S \times P_{1} \times J_{in} .

The intensities at the detector are given by Eqs. (3) and (4):

I_{∥} = \frac{1}{16} (η_{TM}^{2} + η_{TE}^{2}) \cos (4 α - 4 θ) + \frac{1}{4} (η_{TM}^{2} - η_{TE}^{2}) \cos (2 α - 2 θ) + \frac{3}{16} (η_{TM}^{2} + η_{TE}^{2}) - \frac{1}{8} η_{TM} η_{TE} \cos (4 α - 4 θ) \cos (ϕ) + \frac{1}{8} η_{TM} η_{TE} \cos (ϕ);

I_{⊥} = \frac{1}{16} (η_{TM}^{2} + η_{TE}^{2}) - \frac{1}{16} (η_{TM}^{2} + η_{TE}^{2}) \cos (4 α - 4 θ) + \frac{1}{8} η_{TM} η_{TE} \cos (4 α - 4 θ) \cos (ϕ) - \frac{1}{8} η_{TM} η_{TE} \cos (ϕ) .

After introducing the transmission efficiency ratio

q = \frac{η_{TE}^{2}}{η_{TM}^{2}}

and simplification, the intensity ratio

Γ = \frac{I_{⊥}}{I_{∥}}

is expressed by Eq. (5):

Γ (α, ϕ, θ, q) = \frac{(1 + q) - (1 + q) \cos (4 α - 4 θ) + 2 q^{0.5} \cos (4 α - 4 θ) \cos (ϕ) - 2 q^{0.5} \cos (ϕ)}{(1 + q) \cos (4 α - 4 θ) + (1 - q) \cos (2 α - 2 θ) + 3 (1 + q) - 2 q^{0.5} \cos (4 α - 4 θ) \cos (ϕ) + 2 q^{0.5} \cos (ϕ)} .

3. NUMERICAL SIMULATIONS

The numerical simulations will focus on the extraction of $α$ , $ϕ$ , and $q$ for various AGPM cases. In these simulations, the intensity at the detector will be computed using the matrix multiplication presented in Section 2. Then the parameters $φ$ , $α$ , and $q$ will be computed for each pixel by fitting the $Γ$ curves using Eq. (5) with the Matlab curve fitting tool for each pixel. For each AGPM case, three initial values and acceptable variation domains will be given to the fitting process of $Γ$ , $ϕ_{0} \in [ϕ_{\min}, ϕ_{\max}]$ , $q_{0} \in [q_{\min}, q_{\max}]$ , and $α_{0} \in [α_{\min}, α_{\max}]$ . For $ϕ$ and $q$ , the initial values and computation domains are derived from the corresponding RCWA design simulations, whereas for $α$ , the initial values come from the visual interpretation of the transmitted intensities.

A. Ideal Case

As expressed before, the ideal element exhibits a constant phase shift of 180°, a perfect radial orientation of its fast axis, and a perfect transmission of both TE and TM components: $ϕ = 180 °$ , $α = arc \tan (y / x)$ , $η_{TE} = 1$ , and $η_{TM} = 1$ . The parameters introduced in the fitting routine are $ϕ_{0} = 180 ° \in [135, 225] °$ , $q_{0} = 1 \in [0.5, 1.5]$ , and $α_{0} = \arctan (y / x) \in [- 180, 180] °$ . Figure 4 presents $I_{∥}$ and $I_{⊥}$ for several values of $θ$ . Table 1 presents the results of the fitting process with $ϕ_{in}$ , the introduced phase shift; $⟨ ϕ ⟩$ , the spatially averaged computed phase shift; $σ_{ϕ}$ , the spatial standard deviation of the computed phase shift; $q_{in}$ , the introduced efficiency ratio; $⟨ q ⟩$ , the spatially averaged computed $q$ ; $σ_{q}$ , the spatial standard deviation of the computed $q$ ; and $⟨ ε ⟩$ , the spatially averaged error between the introduced fast axis orientation and the computed one. Figure 5 presents the fast axis orientation computed for this component.

Fig. 4. Representation of the simulated intensities for the ideal case, for three values of $θ$ , namely, 0°, 30°, and 60°. Panels (a–c) present $I_{∥}$ and panels (d–f) stand for $I_{⊥}$ .

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Table 1. Table of the Results Obtained with the Fitting of $Γ$ for the Ideal Case

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Fig. 5. Representation of the fast axis orientation for the ideal case. The fast axis rotates around the center of the component from $- 180 °$ to 180°. Only the introduced fast axis is shown since there is no observable difference between the introduced one and the computed one.

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By observing the results, it can be seen that the fitting of the $Γ$ curve leads to accurate computation of the element parameters. This good agreement allows validating the fitting principle and to proceed with a more complex case.

B. Chromatic Phase Shift Case

For the second case, the phase shift depends on the incident wavelength, $ϕ = ϕ (λ)$ . For this case, the phase shift values were obtained through RCWA. Those values $[φ_{RCWA} (λ)]$ correspond to grating parameters designed to achieve a phase shift close to 180° in the L-band region (3.5–4.1 μm) and were calculated in the 3.5–11.5 μm region for our analysis. The grating parameters are $w_{t} = 0.59 μm$ , $d = 4.42 μm$ , and $α = 2.45 °$ . They correspond to the expected grating parameters of AGPM L13r presented by Vargas et al. [11] along with the manufacturing and tuning processes. Afterward, the intensities and $Γ$ curves were computed for each pixel and each wavelength. Afterward, the extraction process for $α$ , $ϕ$ , and $q$ was performed as before. The parameters introduced in the fitting routine are $ϕ_{0} (λ) = ϕ_{RCWA} (λ) \in [ϕ_{RCWA} (λ) - 45, ϕ_{RCWA} (λ) + 45]$ , $q_{0} = 1 \in [0.5, 1.5]$ , and $α_{0} = \arctan (y / x) \in [- 180, 180] °$ . The phase domain extremes are derived from RCWA simulations. Those simulations were performed with a deviation of 20% from the ideal set of gratings parameters: $w \in [0.8 w_{i d}, 1.2 w_{i d}]$ , $d \in [0.8 d_{i d}, 1.2 d_{i d}]$ , and $β \in [0.8 β_{i d}, 1.2 β_{i d}]$ . Figure 6 presents $I_{∥}$ and $I_{⊥}$ at 5.7 μm for the same values of $θ$ as in Fig. 4. Table 2 presents some results for different wavelengths and Fig. 7 shows the comparison between $⟨ ϕ ⟩$ over the range of 3.5–11.5 μm and $ϕ_{in}$ .

Fig. 6. Representation of the simulated intensities normalized by the maximum values of the six pictures, for the chromatic phase shift case at 5.7 μm ( $ϕ_{in} = 139.7 °$ ) and perfect transmission of both TE and TM components, for three values of $θ$ , namely, 0°, 30°, and 60°. Panels (a–c) present $I_{∥}$ and panels (d–f) stand for $I_{⊥}$ .

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Table 2. Table of the Results Obtained with the Fitting of $Γ$ for the Component for Several Wavelengths

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Fig. 7. Representation of $ϕ$ obtained after the fitting routine in the chromatic $ϕ$ case. Error bars are not visible on the graph due to their small values. The blue dots represent the averaged fitted phase shift $⟨ ϕ ⟩$ and the black line represents the phase shift obtained due to the RCWA computations $ϕ_{RCWA}$ and used to compute the $Γ$ curves for each pixel.

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From these results, it appears that the fitting process is also reliable for the chromatic phase shift case. The next case, which combines the wavelength-dependent phase shift and transmission ratio, can therefore be studied.

C. Chromatic Phase Shift and Chromatic Efficiency Ratio Case

Finally, a more realistic component combines the wavelength-dependent phase shift and efficiency ratio, both obtained through RCWA using the same grating parameters as in case B. The fitting principle is the same as in the previous case, only the $q$ values are changed: $q_{0} = q_{RCWA} \in [q_{RCWA} - 0.5, q_{RCWA} + 0.5]$ . As in case B, the $q$ domain extremes are also derived from RCWA in the 20% error combination of the grating parameters. Figure 8 presents $I_{∥}$ and $I_{⊥}$ at 5.7 μm. Table 3 presents the fitting results for the same wavelength as before and Fig. 9 presents the results for the computation of $q$ in the 3.5–11.5 μm domain.

Fig. 8. Representation of the simulated intensities normalized by the maximum values of the six pictures, for the chromatic phase shift and efficiency ratio case at 5.7 μm ( $ϕ_{in} = 139.7 °$ ) and $q_{in} = 0.9348$ for three values of $θ$ , namely, 0°, 30°, and 60°. Panels (a–c) present $I_{∥}$ and panels (d–f) stand for $I_{⊥}$ .

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Table 3. Table of the Results Obtained with the Fitting of $Γ$ for the Component for Several Wavelengths

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Fig. 9. Representation of $q$ obtained after the fitting routine in the chromatic $ϕ$ and $q$ case. Error bars are not visible on the graph due to their small values. The blue dots represent the averaged fitted efficiency ratio $⟨ q ⟩$ and the black line represents the efficiency ratio obtained due to the RCWA computations $q_{RCWA}$ and used to compute the $Γ$ curves for each pixel.

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Comparing Figs. 8 and 6, the effect of $q$ can be viewed on sub-image (a) in both figures, where the pixels on the horizontal axis are characterized by a higher intensity than the ones on $y$ axis in Fig. 8.

From these results, it appears that the fitting routine is able to compute both the chromatic phase shift and efficiency ratio. This corresponds to the expected component. The experimental measurement of a real component is presented in the next section.

4. EXPERIMENTAL RESULTS

Since the fitting process was validated by numerical simulations, the experimental setup was built based on the design presented in Fig. 2. Figure 10 presents the mounted optical setup.

Fig. 10. Picture of the experimental setup (QCL not present).

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Our components are:

• A tunable QCL (Daylight Solutions, MIRcat-QT) from 5.2 to 11 μm, with a pulse width of 400 ns and a duty cycle of 10 μs.
• Three linear ${BaF}_{2}$ polarizers with transmission higher than 80% and an extinction ratio of 150 each for our bandwidth ( $P_{0}$ , $P_{1}$ , and $P_{2}$ ).
• Two computer-controlled rotators each with an accuracy of 5 arcmin and a repeatability of 10 arcsec.
• A quarter-wave plate ( $λ / 4$ ) at 6.55 μm with a phase shift between 65° and 110° in our measurement domain.
• A pyrocam ( $D$ ; Ophir Photonics, Pyrocam IIIHR) sensitive from 1.1 to 30 μm, with a pixel size of $75 μm \times 75 μm$ and a sensor size of $1.6 {cm}^{2}$ .
• An AGPM optimized for the 3.5–4.1 μm wavelength domain (component L13r from Vargas et al. [11]).

The measurements are acquired automatically using Labview scripts (National Instruments V 2015). For image processing reasons, a measurement cycle includes the acquisition of two darks when the laser is not lasing ( $D_{a}$ and $D_{b}$ ), one after the acquisition of $I_{∥}$ , and one after the acquisition of $I_{⊥}$ . The measurement cycle is divided into the following six steps:

1. $P_{1}$ rotates to achieve position $θ_{i}$ and $P_{2}$ rotates accordingly to achieve parallel orientation in relation to $P_{1}$ .
2. $N$ frames are acquired with the pyrocam while the laser is active $\Rightarrow I_{∥} (θ_{i})$ .
3. $N$ frames are acquired with the pyrocam while the laser is passive $\Rightarrow D_{a} (θ_{i})$ .
4. $P_{2}$ rotates to achieve a crossed configuration where the transmission axis of $P_{2}$ is perpendicular to $P_{1}$ axis.
5. $N$ frames are acquired with the pyrocam while the laser is active $\Rightarrow I_{⊥} (θ_{i})$ .
6. $N$ frames are acquired with the pyrocam while the laser is passive $\Rightarrow D_{b} (θ_{i})$ .

Then,

P_{1}

moves to reach the next position

θ_{i + 1}

and the cycle is repeated until a turn of 360° in 10° steps is completed. Afterward, the frames are used to compute

Γ (θ_{i})

as

Γ (θ_{i}) = \frac{I_{⊥} (θ_{i}) - 0.5 [D_{a} (θ_{i}) + D_{b} (θ_{i})]}{I_{∥} (θ_{i}) - 0.5 [D_{a} (θ_{i}) + D_{b} (θ_{i - 1})]} .

However, for

i = 1

,

D_{b} (θ_{0})

does not exist; it is taken as the initial dark before the acquisition of

I_{∥} (θ_{1})

. It is important to remark that the number of frames

N

and the camera parameters (gain, exposure time, averaged number of frames) are constant during one measurement, but can be changed from one wavelength to another. The reason being the variation of our source intensity and a compromise between the signal-to-noise ratio and the measurement speed. Figure 11 shows the intensities measured at 5.7 μm.

Fig. 11. Representation of the measured intensities normalized by the maximum values of the six pictures, for the chromatic case at 5.7 μm ( $ϕ_{RCWA} = 139.7 °$ ) and $q_{RCWA} = 0.935$ for three values of $θ$ , namely, 0°, 30°, and 60°. Panels (a–c) present $I_{∥}$ and panels (d–f) stand for $I_{⊥}$ . Contrast and luminosity were modified compared with the previous pictures to enhance visibility.

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From these images, it can be observed that the transmitted intensity at the center of the component is very low. This is due to the singularity at the center. Therefore, pixels with low intensity, which cannot be separated from the background noise, will be omitted by the fitting algorithm. Consequently, an intensity criterion, which is based on the sum of the intensity of all $P_{1}$ positions, will be designed:

\sum_{i} I_{T} (θ_{i}, x, y) \geq 0.02 \times \max_{[x, y]} [\sum_{i} I_{T} (θ_{i}, x, y)],

where

I_{T} (θ_{i}, x, y) = I_{∥} (θ_{i}, x, y) + I_{⊥} (θ_{i}, x, y)

. Moreover, pixels close to the center represent circular gratings with small radii, leading to the merging of several orientations of the fast axis into the same camera pixel, thus inducing errors in the extraction of

α

,

ϕ

, and

q

. To define a distance to the center criterion, binning simulations were performed. The intensities

I_{⊥} (θ_{i}, x, y)

and

I_{∥} (θ_{i}, x, y)

were computed for an AGPM with a pixel size of two by two periods (

2.84 μm \times 2.84 μm

) and those intensities were combined into camera pixels (

75 μm \times 75 μm

):

I_{⊥} (θ_{i}, X, Y) = \sum_{k = 1}^{27} \sum_{l = 1}^{27} I_{⊥} (θ_{i}, x_{k}, y_{l}), I_{∥} (θ_{i}, X, Y) = \sum_{k = 1}^{27} \sum_{l = 1}^{27} I_{∥} (θ_{i}, x_{k}, y_{l}) .

Afterward, a binned intensity ratio

Ξ (X, Y)

was computed as

Ξ = \frac{I_{⊥}}{I_{∥}}

and the fitting algorithm was performed to extract

ϕ

and

Q

. Finally, the differences between the computed parameters and the one introduced in the fine resolution model (

ϕ_{in}

and

q_{in}

) were computed. Figures 12 and 13 present the results at 5.7 μm as function of the distance to the component singularity in the binned configuration,

r

.

Fig. 12. Absolute value of the difference between the binned phase retard $ϕ$ and the introduced one $ϕ_{in}$ as a function of the radius of the component for the binned case.

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Fig. 13. Absolute value of the difference between the binned phase retard $Q$ and the introduced one $q_{in}$ as a function of the radius of the component for the binned case.

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From those results, we decided that only pixels at a distance larger than 20 camera pixels (1.5 mm) from the component center would be processed by the fitting algorithm. Afterward, the fitting process was performed for pixels fulfilling those two criteria. As in case C, the parameters introduced into the fitting algorithm are $ϕ_{0} = ϕ_{RCWA} (λ) \in [ϕ_{RCWA} - 45, ϕ_{RCWA} + 45]$ , $[q_{RCWA} - 0.5, q_{RCWA} + 0.5]$ , and $α_{0} = \arctan (y / x) \in [0, 360] °$ . The results of the measurements are presented in Table 4. Figures 14 and 15 compare the measured parameters with the parameters obtained through RCWA for the ideal grating parameters and a realistic error on them of 2.5% expected from previously cracked similar components [11]. Finally, Fig. 16 shows a comparison of $α_{0}$ and $α$ .

Table 4. Table of the Results Obtained with the Fitting of $Γ$ for the Real Component

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Fig. 14. Measured values of $ϕ$ . The error bars represent $σ_{ϕ}$ . The green values correspond to the maximum and minimum $ϕ$ obtained using RCWA with a maximum error of 2.5% on the grating parameters.

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Fig. 15. Measured values of $q$ . The error bars represent $σ_{q}$ . The green values correspond to the maximum and minimum $q$ obtained using RCWA with a maximum error of 2.5% on the grating parameters.

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Fig. 16. Representation of $α$ . Panel (a) shows the expected values resulting from the intensity visual interpretation. Panel (b) shows the values obtained by the measurement process. The ring form originates from our intensity and binning criteria, where only the pixels present in the ring are taken into account in the fitting computation.

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From the table and graphics, it can be observed that the measurement parameters are in the same range as the expected ones. However, the spatial deviations are important $σ_{ϕ} \approx 0.1 ⟨ ϕ ⟩$ and $σ_{q} \approx 0.17 ⟨ q ⟩$ . To determine the origins of these variations, $ϕ$ and $q$ were presented as functions of the distance to the singularity (Figs. 17 and 18, respectively).

Fig. 17. Scatter plot of $ϕ$ as a function of distance to the component center at 5.7 μm.

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Fig. 18. Scatter plot of $q$ as function of distance to the component center at 5.7 μm.

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By observing those graphs for every wavelength, emerging trends were noticed: $ϕ$ and $q$ show radial variations and their spatial deviation at constant radius is smaller than the global one. To explain the radial variations and dispersions at constant radius, we propose two origins: an imperfect wavefront and a radial variation of the grating geometry. For a non-planar wavefront at non-normal incidence, the incident angle and the optical path in the component are not uniform; radial variations could be observed for a diverging beam and dispersion at constant radius could originate from astigmatism. Moreover, our RCWA simulations considered an ideal anti-reflective grating (ARG) [11,16] located on the back-side of the component. Those gratings are 2D symmetrical subwavelength gratings, whose purpose is to reduce the internal reflection at the back-side of the component, to increase the intensity at the exit of the component and its performance as a coronagraph [13]. However, the ARG present on the component was designed for wavelengths smaller than the measured range, and so its maximum efficiency is not reached. Therefore, a fraction of the beam exiting the component experiences internal reflections and impacts the measured $ϕ$ and $q$ . Also, small radial variations of the grating geometry have been observed on previously cracked components [11]; variations such as alterations of the grating shape or fluctuations of the grating parameters can also induce a change in the measured $ϕ$ and $q$ .

To reduce the spatial deviation, statistical smoothing was implemented. Our method is based on the computation of the lower and upper quartiles, which excludes values that are separated by more than 1.5 times the interquartile range from the lower or upper quartile. First, the $ϕ$ and $q$ values belonging to the pixels located inside our area of interest are grouped into vectors corresponding to rings with a radius of one pixel. Then, the tests are performed on those sub-vectors and only the elements passing the tests for their $ϕ$ and $q$ are preserved. Afterward, the sub-vectors are merged together to reform a global vector and the tests are performed once again to exclude the remaining outliers. Finally, the averaged values and spatial deviations are computed; they are presented in Table 5 and compared with unfiltered data.

Table 5. Comparison between the Old Results $O$ and the Filtered Ones $F$

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As a result of the statistical smoothing, the spatial dispersions were reduced. However, the radial variations of $ϕ$ and $q$ remained, confirming our track on the impact of an aberrated incident wavefront and of the radial variation of the grating parameters. Moreover, the first results for an AGPM designed to be used in the 11–13.2 μm region and measured at 11 μm ( $ϕ_{RCWA} = 188.0 °$ , $q_{RCWA} = 0.9275$ , $⟨ ϕ ⟩ = 201.5 °$ and $⟨ q ⟩ = 0.9275$ ) presented smaller radial variations, confirming our hypothesis on the role of the ARG in reducing the radial variations. To achieve a better understanding of those radial variations and to improve the measurement accuracy, our future plan is to change our measurement protocol and to improve our performance prediction tool to investigate the effect of aberrated wavefronts. Our new procedure will include the measurement of the wavefront before the beam hits the component, to assess its characteristics. The wavefront will be measured between $P_{1}$ and the component for each wavelength before running the rotation of $P_{1}$ . Besides, finite-difference time domain (FDTD) simulations will be used to investigate the origin of the radial variations of $ϕ$ and $q$ . Using the Comsol Multiphysics (COMSOL Inc., V5.2) electric field propagation module, several cases of incident beam and component geometry will be studied to determine their effects on $ϕ$ and $q$ . In those simulations, realistic aberrated wavefronts will be coupled to components with ARGs and more advanced simulations will tackle the radial variations of the grating parameters.

5. CONCLUSIONS AND PERSPECTIVES

In this paper, an experimental method to measure the birefringent parameters of space-variant subwavelength gratings was described. The method principle and its mathematical model using the Jones formalism were presented. Then, the results of numerical simulations representing increasing component complexity were displayed and discussed. The simulations of the measurement process showed that the experimental method was suited to optically evaluate space-variant subwavelength gratings. Afterward, the optical setup and measurement process were detailed. Finally, the measurement results for one AGPM were summarized and compared with the parameters expected from its design. It was observed that the measured parameters were in the same range as the expected ones. However, both $ϕ$ and $q$ exhibited large spatial dispersions when they were supposed to be constant over the whole component. Statistical smoothing reduced those dispersions but radial variations remained. The effect of an aberrated wavefront combined with the impact of an ARG and the radial variations of the gratings were proposed to explain those behaviors.

In the future, the experimental procedure will include a measurement of the incident wavefront and FDTD simulations will be performed to correlate its effect on the measured parameters. Moreover, due to the improved accuracy on $ϕ$ and $q$ , the identification of the grating parameters within a significant domain would be possible and performance prediction would be a reality, making this method an excellent complement to the already existing measurement methods.

Funding

Vetenskapsrådet (VR) (621-2014-5959); Carl Tryggers Stiftelse för Vetenskaplig Forskning (CTS 15:259); FP7 Ideas: European Research Council (IDEAS-ERC) (337569).

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$ϕ_{in}$ (°)	$⟨ ϕ ⟩$ (°)	$σ_{ϕ}$ (°)	$q_{in}$	$⟨ q ⟩$	$σ_{q}$	$⟨ ε ⟩$ (°)
180	180	$1 e - 6$	1	1	$2 e - 15$	$2 e - 7$

$λ$ (μm)	$ϕ_{RCWA}$ (°)	$⟨ ϕ ⟩$ (°)	$σ_{ϕ}$ (°)	$q_{in}$	$⟨ q ⟩$	$σ_{q}$	$⟨ ε ⟩$ (°)
3.5	171.0	171.0	$1 e - 11$	1	1	0	$2 e - 7$
3.9	176.3	176.3	$1 e - 11$	1	1	0	$2 e - 7$
4.1	178.9	178.9	$7 e - 12$	1	1	0	$2 e - 7$
4.9	157.6	157.6	$1 e - 11$	1	1	0	$2 e - 7$
5.3	147.6	147.6	$4 e - 12$	1	1	0	$2 e - 7$
5.7	139.7	139.7	$1 e - 12$	1	1	0	$2 e - 7$
6.3	127.4	127.4	$1 e - 13$	1	1	0	$2 e - 7$
6.6	120.8	120.8	$9 e - 12$	1	1	0	$2 e - 7$
7.3	107.7	107.7	$1 e - 11$	1	1	0	$2 e - 7$
7.7	102.6	102.6	$6 e - 12$	1	1	0	$2 e - 7$

$λ$ (μm)	$ϕ_{RCWA}$ (°)	$⟨ ϕ ⟩$ (°)	$σ_{ϕ}$ (°)	$q_{RCWA}$	$⟨ q ⟩$	$σ_{q}$	$⟨ ε ⟩$ (°)
3.5	171.0	171.0	$1 e - 11$	0.949	0.949	0	$2 e - 7$
3.9	176.3	176.3	$1 e - 11$	0.930	0.930	0	$2 e - 7$
4.1	178.9	178.9	$7 e - 12$	0.948	0.948	0	$2 e - 7$
4.9	157.6	157.6	$1 e - 11$	0.960	0.960	0	$2 e - 7$
5.3	147.6	147.6	$4 e - 12$	0.929	0.929	0	$2 e - 7$
5.7	139.7	139.7	$1 e - 12$	0.935	0.935	0	$2 e - 7$
6.3	127.4	127.4	$1 e - 13$	1.013	1.013	0	$2 e - 7$
6.6	120.8	120.8	$9 e - 12$	1.029	1.029	0	$2 e - 7$
7.3	107.7	107.7	$1 e - 11$	0.968	0.968	0	$2 e - 7$
7.7	102.6	102.6	$6 e - 12$	0.923	0.923	0	$2 e - 7$

$λ$ (μm)	$ϕ_{RCWA}$ (°)	$⟨ ϕ ⟩$ (°)	$σ_{ϕ}$ (°)	$q_{RCWA}$	$⟨ q ⟩$	$σ_{q}$
5.2	149.8	138.9	19.4	0.937	0.901	0.297
5.3	147.6	141.9	13.1	0.929	0.923	0.242
5.4	145.5	139.3	15.5	0.924	0.891	0.256
5.5	143.6	140.0	9.5	0.924	0.955	0.221
5.6	141.6	134.7	9.7	0.927	0.924	0.210
5.7	139.7	134.1	8.1	0.935	0.935	0.210
5.8	137.8	139.0	10.8	0.946	0.892	0.222
5.9	135.8	126.9	10.3	0.859	0.778	0.191
6.3	127.4	124.4	11.8	1.013	0.656	0.120
6.6	120.8	120.0	10.0	1.029	0.903	0.226
6.7	118.6	111.3	9.3	1.030	0.829	0.217
7.3	107.7	110.4	11.1	0.968	0.769	0.221
7.7	102.6	104.0	11.3	0.923	0.890	0.270

$λ$ (μm)	${⟨ ϕ ⟩}_{O}$	$σ_{ϕ O}$ (°)	${⟨ q ⟩}_{O}$	$σ_{q O}$	${⟨ ϕ ⟩}_{F}$ (°)	$σ_{ϕ F}$ (°)	${⟨ q ⟩}_{F}$	$σ_{q F}$
5.2	138.9	19.4	0.901	0.297	139.5	19.0	0.891	0.294
5.3	141.9	13.1	0.923	0.242	140.8	8.3	0.916	0.228
5.4	139.3	15.5	0.891	0.256	142.7	8.3	0.904	0.251
5.5	140.0	9.5	0.955	0.221	139.6	7.0	0.969	0.210
5.6	134.7	9.7	0.924	0.210	134.5	7.7	0.938	0.200
5.7	134.1	8.1	0.935	0.210	133.9	7.8	0.946	0.206
5.8	139.0	10.8	0.892	0.222	138.7	8.8	0.883	0.213
5.9	126.9	10.3	0.778	0.191	126.4	9.3	0.757	0.170
6.3	124.4	11.8	0.656	0.120	126.4	9.3	0.758	0.115
6.6	120.0	10.0	0.903	0.226	119.9	9.3	0.893	0.223
6.7	111.3	9.3	0.829	0.217	110.9	9.3	0.810	0.203
7.3	110.4	11.1	0.769	0.221	109.9	10.8	0.778	0.219
7.7	104.0	11.3	0.890	0.270	100.6	11.3	0.943	0.258

Birefringence measurements of diamond space-variant subwavelength gratings

Abstract

1. INTRODUCTION

2. DESCRIPTION OF THE EXPERIMENTAL METHOD

3. NUMERICAL SIMULATIONS

A. Ideal Case

B. Chromatic Phase Shift Case

C. Chromatic Phase Shift and Chromatic Efficiency Ratio Case

4. EXPERIMENTAL RESULTS

5. CONCLUSIONS AND PERSPECTIVES

Funding

REFERENCES

Cited By

Figures (18)

Tables (5)

Equations (12)

Applied Optics