## Abstract

By a combination of quarter-wave plates made of different birefringent materials it is possible to produce achromatic quarter-wave plates whose degree of achromatism is dependant on the dispersions of birefringence and on the thicknesses of the individual quarter-wave plates. These waveplates are widely used in optical instrumentation and the residual errors associated with these devices can be very important in high resolution spectro-polarimetry measurements. The misalignment of optic axis in a double crystal waveplate is one of the main source of error and leads to elliptical eigenpolarization modes in the retarder and the oscillation of its orientation according to the wavelength. This paper will discuss, first, how the characteristics of a quartz-MgF_{2} quarter-wave plate is affected by such a misalignment. A correlation with the experiment is then achieved in order to highlight the interest of taking a possible tilt error into consideration when doing polarimetric measurements.

©2001 Optical Society of America

## 1. Introduction

Retardation plates are generally made of a single piece of material, although when the thickness required for a plate is too small, two thicker pieces may be used with the fast axis of one aligned parallel to the slow axis of the other to cancel out all but the desired retardation. Those devices are called zero-order waveplates. Achromatic retardation plates which have almost the same retardation over a given range of wavelengths can be made from two or more different materials [1, 2] or two or more plates of the same material whose axes are oriented at the appropriate angle [3, 4]. In the first case, the best choice of materials would be a pair of positive and negative crystals [5]. In this respect the best combination would be colorless sapphire (negative crystal) and magnesium fluoride (positive crystal) [6]. Unfortunately, the refractive indices of these materials differ considerably and incident natural light would become polarized by the reflection from their cemented interface. For these reasons it seems that the most convenient pair is quartz with magnesium fluoride with the fast axis of one aligned parallel with the slow axis of the other [7]. These waveplates are widely used in instrumentation and the residual errors associated with these devices can be of high importance in spectropolarimetry measurements [8]. The present paper describes how a misalignment inside a quartz-MgF_{2} achromatic quarter-wave plate can affect those measurements.

## 2. Quart-MgF2 achromatic quarter-wave plate

#### 2.1 Optical system

A reasonably achromatic retardation plate can be constructed from pairs of birefringent materials. Difference between the birefringence of quartz and those of MgF_{2} depends little of the wavelength on the visible spectral range [9]. As a matter of fact, both these materials are used with the fast axis of one aligned parallel to the slow axis of the other to make a zero-order achromatic quarter-wave plate on the range 400–700 nm. Assume that quartz and MgF_{2} have thicknesses d_{q} and d_{m} and a birefringence Δn_{q} and Δn_{m} respectively, and that is to be achromatized at wavelengths λ_{1} and λ_{2}. We obtain the relations:

Where Δn’s are the values of birefringence for the materials at the wavelength specified. If the quarter-wave plate is made of quartz and MgF_{2} with the thicknesses d_{q} and d_{n} respectively, the retardance will be 90° for λ_{1} and λ_{2} and the retardance variation according to the wavelength will be less than few percent. We have tested a quartz-MgF_{2} quarter-wave plate (Optique de precision J. Fichou) whose thicknesses for the quartz and the MgF2 are 239.1 µm and 197.1 µm respectively (commercial specifications). As a consequence of equations 1, the retardance is exactly equal to 90° at the wavelengths 432.2 nm and 614.5 nm(figure 1). The birefringence values are extracted from an article of J. M. Beckers [7].

#### 2.2 Misalignment error

Atilt error between the two plates has been introduced in our calculations, as schematized in figure 2, in order to describe the consequences on the optical characteristics of the retarder.

The Jones matrix [M] of the quartz-MgF_{2} quarter-wave plate with a tilt error θ between the two plates can be calculated as follows:

With [M_{m}] the Jones matrix of the MgF_{2} plate, [M_{q}] those of the quartz plate and [R(θ)] the matrix of a rotational change of basis.

δ_{q} and δ_{m} are the retardance of the quartz plate and the MgF_{2} plate respectively.

#### 2.3 Consequences of an optic axis tilt error

If the misalignment between the two plates is small, its effect on the retardance has been calculated to be negligible [10]. On the other hand, we examined the consequences on the ellipticity of the eigenpolarization modes. Indeed, because of the tilt, the eigenpolarization modes, *i.e*. the states of polarization that propagate without transformation through the plate, are no more linear but elliptic. A general elliptical vector J is depicted in figure 3. A and B represent the values of the great and the small axes respectively, α is the azimuth angle of J, ε its ellipticity and υ the diagonal angle.

The calculation of the eigenvectors of matrix [M] gives two orthogonal elliptical vectors J_{1} and J_{2}:

With a, b and c real.

The calculation of the ellipticity ε of J_{1} was made with the help of the following expression 7 :

For the elliptical eigenvector J_{1}, ϕ and E_{y} are the argument and the absolute value of the complex number b+i c respectively. E_{x}=a and $\upsilon =\mathrm{arctan}\left[\frac{{E}_{y}}{{E}_{x}}\right]$.

Ellipticity ε can be calculated according to the wavelength on the visible spectral range for several values of the tilt error θ. Here again, the thicknesses of quartz and MgF_{2} are the commercial specifications *i.e*. 239.1 nm and 197.1 nm. The results are presented on figure 4.

From the calculation illustrated on figure 4, one can observe a linear dependence of the ellipticity with the tilt (ε≅1.38 θ). It oscillates three times over the 400–700 nm wavelength range and the period of modulation increases as the wavelength increases. This first result evidences the interest of taking ellipticity into account in spectropolarimetry measurements with imperfect zero-order achromatic quarter-wave plates.

We calculated as well the orientation of a quartz-MgF_{2} quarter-wave plate according to the wavelength. So as to calculate the orientation of the neutral axes of the retarder, we multiplied the eigenvector J_{1} by the matrix of a rotational change of basis R(-α). Indeed, when J=R(-α). J_{1} is the vector of an elliptical polarization state whose axes are aligned with the reference axes X and Y, then the phase difference ϕ occuring between the x and the y components of J is equal to $\frac{\pi}{2}$. This procedure allowed us to determine the orientation α of one of the neutral axes with the use of the method described as follows. If the x and y components of J are ${J}_{x}={E}_{x}{e}^{i\varphi}x$ and ${J}_{y}={E}_{y}{e}^{i\varphi}y$, then $\varphi ={\varphi}_{y}-{\varphi}_{x}=\frac{\pi}{2},\frac{{J}_{x}}{{J}_{y}}=\frac{{E}_{x}}{{E}_{y}}{e}^{-i\varphi}$, and finally the real part of $\frac{{J}_{x}}{{J}_{y}}$ equal to zero. Further to these explanations, the azimuth α of the eigenvector J_{1} is given by:

the azimuth α was calculated according to the wavelength on the visible spectral range for several values of θ. The results are presented on figure 5.

Figure 5 shows that, as for the ellipticity, the azimuth of the eigenpolarization modes oscillates three times over the visible spectral range. The modulation increases as the wavelength increases and the oscillation is linearly dependant of the tilt error. We experimentally observed an oscillation of the fast axis of almost 1° (see chapter 4 of the present paper) corresponding to a misalignment inside the retarder of θ=0.72°. Further to this observation we took this value in the calculations described bellow.

To understand these oscillations, we examined both the ellipticity of the eigenpolarization modes and the orientation of the retarder fast axis, as a function of the wavelength, for θ=0.72° (the thicknesses are the commercial specifications). We plotted as well the values δ/2π according to the wavelength for the quartz and MgF_{2} respectively for a best understanding of the physical effect occurring to the materials (Figure 6).

At λ_{1}=448 nm, (Δn_{q}×d_{q})/λ=5 (Figure 6) that means that the quartz plate displays no retardance. At the same wavelength, theMgF_{2} plate acts as a perfect quarter-wave plate ((Δn_{m}×d_{m})/λ=5.25). We considered the materials (quartz and MgF_{2} plates) as perfect devices in our calculations and especially we assumed that the quartz has no optical activity. Consequently, if the first plate is a wave-plate at the wavelength 448 nm, the ellipticity and the orientation of the fast axis of the retarder would be those of the second plate *i.e*. ε=0° and α=0.72° (figure 7). At λ_{2}=469 nm, (Δn_{q}×d_{q})/λ=4.75 and the quartz plate contributes the perfect quarter wave of retardance when the second plate has zero retardance. The retarder ellipticity and its orientation are thus those of the first plate: ε=0° (no optical activity) and α=0°. At λ_{3}=548 nm, theMgF_{2} plate acts as a perfect quarter-wave plate and the quartz one as a wave-plate (ε=0°, α=0.72°) and at λ_{4}=582 nm the optical functions of the materials are reversed. Optical characteristics of the retarder according to the wavelength are resumed in figure 8.

## 3. Experimental results

#### 3.1 Experimental methodology and measurement set-up

The azimuth of one of the eigenpolarization mode of the retarder was measured using a dichotomous method of null intensity. Indeed, when the plate rotates between two orthogonal polarizers (figure 9), the minimum of intensity is obtained when one of its neutral axes is aligned with those of the first polarizer. This situation remains valid even if the ellipticity of the retarder is not null. Figure 10 shows indeed that the minimum of intensity on the detector is constant whatever the ellipticity ε. This curve was plotted using the model of a dichroic elliptic object developed by Scierski and Ratajczyk [11].

Figure 11 displays the experimental set-up. The light source is a commercial optical parametric oscillator (Sunlite - Continnum) pumped by a frequency-tripled Nd:YAG laser with a 6-ns pulse width. The detection is made by a photodiode. The axis of the optical elements are aligned with a step-by-step motor (accuracy: 0.01°). Intensity variations are corrected by means of a reference signal (detector 1) and the estimated standard deviation on the corrected signal is less than 5‰. L_{1}, P_{2} and L_{2} are used in order to control the intensity on the detectors. An additional polarizer P_{1} has been added so as to eliminate a possible intensity noise effect when reducing the power [12].

#### 3.2 Experimental results

The dichotomous method gives access to the experimental value of the azimuth (figure 12). Reproducibility of the measurements showed an estimated accuracy to within 0.02°. We plotted as well the theoretical azimuth calculated with the thicknesses already used (commercial specifications) and a tilt error of 0.72°. The difference occurring between the experimental and the theoretical points is due to false values of the thicknesses and evidences the need of determining the real characteristics of the quarter-wave plate in order to obtain a reasonably good fit between the two curves. We consequently used a least-square analysis. This method allowed us to have an estimation of the characteristics of the retarder: d_{q}=242 µm, d_{m}=172 µm and θ=0.78° (figure 13).

One can observe on figure 13 that the angular amplitude of the calculated azimuth oscillations, in opposition to the experimentation, is not constant. The best fit between experimental data and theoretical calculation was obtained in the range 575–675 nm. The difference still remaining is probably due to the optical activity of the quartz plate [13] especially since this difference is higher for the low wavelengths [14]. The model used to describe the retarder is not consequently accurate and the calculated characteristics only an estimation of the real values. Fitting only the first part of the data (on the wavelength range 400–475 nm and 400–600 nm) and because of the optical activity, we obtained different values of the retarder characteristics. The thicknesses and tilt value corresponding to this two simulated curves (table 1) allowed us to know the accuracy of the retarder characteristics calculated by mean of this method.

Thicknesses calculated by fitting the experimental and the simulated azimuth of the retarder were thus precise to within 4–5 µm and the tilt error was obtained with an accuracy of 0.1°.

## 4. Conclusion

An experimental method to determine the characteristics of a quartz-MgF_{2} achromatic quarter-wave plate has been presented. The theoretical model of two plates with an optical axis tilt error gave a good correlation with the experiment and allowed us to obtain an estimation of the tilt error with an accuracy equal to 0.1°. Thicknesses of the two materials used to achieve the achromatic retarder have been calculated as well and do not correspond to those given by the constructor. A better accuracy could be reached by taking into account the optical activity of the quartz. Nevertheless, the results reported here evidences a non negligible misalignment inside the retarder. Simulations showed that the eigenpolarization modes of the retarder are consequently elliptic and that this ellipticity varies strongly with the wavelength. Azimuth which was also experimentally measured according to the wavelength oscillates on the whole visible spectral range and could lead to error measurement when using this device in order to encode the polarization information in a spectropolarimeter.

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