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Jun 1, 2014 - Jose Luis Vilas,1,* Eusebio Bernabeu,1 Luis Miguel Sanchez-Brea,1 and. Rafael Espinosa-Luna2 ... 37150 León, Guanajuato, Mexico.
Circularly polarized light with high degree of circularity and low azimuthal error sensitivity Jose Luis Vilas,1,* Eusebio Bernabeu,1 Luis Miguel Sanchez-Brea,1 and Rafael Espinosa-Luna2 1

Optics Department, Applied Optics Complutense Group, Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Ciudad Universitaria s.n., 28040 Madrid, Spain

2

Centro de Investigaciones en Óptica, A.C., Loma del Bosque 115, Colonia Lomas del Campestre, 37150 León, Guanajuato, Mexico *Corresponding author: [email protected] Received 30 January 2014; revised 15 April 2014; accepted 15 April 2014; posted 15 April 2014 (Doc. ID 205744); published 22 May 2014

The generation of circularly polarized light with a high circularity degree and low azimuthal error sensitivity was analyzed using a system composed by two waveplates. It is shown how the high circularity degree is achieved using a combination of a half- λ∕2 and a quarter- λ∕4 waveplate λ∕2  λ∕4 configuration. However, the lowest azimuthal sensitivity under small variations in the azimuths of the waveplates is obtained by employing a λ∕4  λ∕2 configuration. Analytical calculus particularized for quartz and MgF2 waveplates is presented. © 2014 Optical Society of America OCIS codes: (220.4830) Systems design; (260.5430) Polarization; (260.1440) Birefringence; (160.1190) Anisotropic optical materials. http://dx.doi.org/10.1364/AO.53.003393

1. Introduction

There are many applications in which light with a high circularity degree is required. In particular, circularly polarized light has special relevance in applications such as ellipsometry-based tomography [1], characterization of biological tissues [2], optical information processing [3], or optical-quantum communications [4]. There are several techniques to obtain circularly polarized light. The most widely used include a linear polarizer followed by a quarter-waveplate, or via total internal reflection using prisms, such as Fresnel Rhomb [5]. Another possibility is the use of a system composed by a linear polarizer and two waveplates. The use of several waveplates has its origin in the Pancharatnam works. Pancharatnam proposed a combination of waveplates to achieve a retarder system with wide spectral range [6]. Thus different configurations with achromatic retarders 1559-128X/14/163393-06$15.00/0 © 2014 Optical Society of America

have been developed; for example, Violino suggested the use of two waveplates [7], and by using this technique, circularly polarized light was obtained by Corbalan and Bernabeu [8]. Moreover, Bernabeu and Aporta analyzed the obtainment of circularly polarized light with a high degree of circularity [9]. Other works have been carried out with two waveplates [10–15]. For example, in quantum communications, the configuration of a half-waveplate and a quarterwaveplate (λ∕2  λ∕4) is widely applied to code and decode states [16,17]. Nevertheless, we do not have any knowledge about an analysis of the azimuthal error sensitivity for this configuration. In addition, the sensitivity represents a critical point in some experiments that can be perturbed due to vibrations, temperature, illumination spectra, or incidence angle [18]. In this article, we analyze how to obtain circularly polarized light with different combinations of waveplates using several configurations: λ∕2  λ∕4, λ∕4  λ∕2, and λ∕4  λ∕4. Using the Jones formalism, two approaches are employed: in the first, the behavior 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS

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of the waveplates considers the transmittance as one, and in the second, the transmittance effect of the waveplates is taken into account. Due to the birefringence of the waveplates, the intensity transmitted is not the same along each axis. Thus, a few differences appear with respect to the formalism in which the transmittance takes on the unitary value. The eccentricity of the polarization ellipse obtained by each approach is used as a merit function. In this way, the configuration that produces the highest degree of circularity can be easily identified. In addition, we analyze the azimuthal error sensitivity of these systems under small variations in the azimuths of the waveplates. It is shown that the configuration λ∕4  λ∕2 exhibits the lowest azimuthal error sensitivity. However, the highest degree of circularity is obtained by using a configuration λ∕2  λ∕4. Therefore, it is demonstrated that a configuration λ∕4  λ∕2 can be used for obtaining a system with low azimuthal sensitivity. However, to obtain a state with a high degree of circularity, the suitable configuration is λ∕2  λ∕4. 2. Theoretical Frame A.

System Description and Waveplate Representation

Consider a device composed by a linear polarizer, P, followed by two waveplates, C1 and C2 , with retardations δ1 and δ2 , respectively. A scheme of the system is shown in Fig. 1. The fast axis of the first plate will define the reference axis, θ represents the angle between the linear polarizer axis and the reference axis, and ϕ is the relative angle between the fast axes of the two waveplates. The linear polarizer produces linearly polarized light with azimuth θ, whose state is ji θ. The system is immersed in air and illuminated with a plane wave normal to the optical elements, at the wavelength λ. Under this configuration, we will analyze how circularly polarized light with a high degree of circularity can be obtained. Using the Jones formalism, a linear optical element is described by a 2 × 2 complex matrix. The linear response of an optical system, represented by the matrix M, to an incident polarization state, ji , is Mji . Therefore the polarization state at the output of our system, jo , will be the product

Fig. 1. System under study. A linear polarizer Pθ and two phase plates C1 and C2 with retardations δ1 , δ2 and azimuths 0 and ϕ, respectively. 3394

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jo  C2 C1 ji θ;

(1)

where C1 and C2 are the Jones representation matrices for the respective waveplates. In order to analyze the system we will consider two mathematical approaches for the waveplates: unitary transmittance and nonunitary transmittance. The first representation considers the matrices of linear retarders that have unit transmittance, and the retardation is given by δ  2πne − no d∕λ, where λ represents the wavelength, d the thickness, and ne , no the extraordinary and ordinary refractive indices of the material, respectively. Thus, the operator of these retarders is unitary. Under this hypothesis the following matrix is represented by the birefringent waveplate:  Cδ; ϕ 

 cos 2δ  i sin 2δ cos 2ϕ i sin 2δ sin 2ϕ : cos 2δ − i sin 2δ cos 2ϕ i sin 2δ sin 2ϕ (2)

The proposed system is composed by two waveplates, and it is described by the product M  C2 δ2 ; ϕ C1 δ1 ; 0 where the fast axis of the first waveplate has been fixed as a reference axis for convenience. Since the matrices C1 and C2 are unitary, their product has the form   A B M ; (3) −B A where the symbol  denotes the complex conjugate operation, A  eiδ1 ∕2 cosδ2 ∕2  i sinδ2 ∕2 cos2ϕ and B  ie−iδ1 ∕2 sinδ2 ∕2 sin2ϕ. The overall retardation of the system, Δ, is obtained from the matrix M, Eq. (3), as follows [19]: tan2

Δ ImA2  ImB2  : 2 ReA2  ReB2

(4)

Introducing the expressions for A and B and operating, Eq. (4) can be expressed as tan2

  Δ sin2 δ21 − δ22  12 sin δ1 sin δ2 1  cos 2ϕ  : (5)   2 cos2 δ21 − δ22 − 12 sin δ1 sin δ2 1  cos 2ϕ

It is remarkable that using this treatment into Eq. (5) for the waveplates, their order does not affect the overall retardation. This means that, when we permute δ1 with δ2 in Eq. (5), the overall retardation is not affected. In order to obtain a high degree of circularity, we consider a second approach consisting of taking into account the effects of the transmittance of the waveplates’ interfaces. This fact entails losses in the transmitted intensity, which is expressed in a nonunitary matrix of the waveplate. The derivation of the characteristic matrix in this approach is obtained using the inclusion of matrices that model the interfaces effect with the Fresnel transmission coefficients, and finally a rotation in order to consider

any azimuth of the fast axis. Thus, the matrix of a waveplate in transmittance representation is given by the product Cr  RϕT2 Ci δ; 0T1 R−ϕ;

(6)

where T1 , T2 represents the transmittance matrices  T1 

2 no 1

0

0 2 ne 1

 ;

 T2 

2no no 1

0

0 2ne ne 1

 ;

(7)

no and ne are the ordinary and extraordinary refractive indices of the birefringent waveplate, and   cos ϕ − sin ϕ Rϕ  ; (8) sin ϕ cos ϕ represents an azimuthal rotation with angle ϕ. Thus, the retarder matrix is described by  Cr δ; ϕ; tx ; ty  

4no ; no  12

ty 

4ne : ne  12

(10)

A polarization state is characterized by the polarization ellipse x2 y2 2xy  − 2 2 cos δ  sin2 δ; 2 2 Ex Ey Ex Ey

1 iδ∕2 − ty e−iδ∕2  2 sin 2ϕtx e 2 iδ∕2 tx sin ϕe  ty cos2 ϕe−iδ∕2

 ;

(9)

    s   1 1 1 1 4 1 1 2 2  cos δ − 1∕b  − : 2 sin2 δ E2x E2y E2x E2y E2x E2y (14)

Eccentricity of the Polarization State

(11)

where Ex , Ey represent the orthogonal amplitudes of the electric field and δ is the phase retardation, obtained from the Jones vector, j  Ex ; Ey eiδ . The polarization state is determined by the eccentricity of this ellipse and can be described by b2 ; (12) a2 where a and b are the semi-major and semi-minor axes of the polarization ellipse, respectively. Thus, when the eccentricity is zero, the light is circularly polarized. Using the nonunitary transmittance approach of the waveplates implies a slight, but significant, difference between this representation and the unitary transmittance approach. Using this representation, the configurations can be classified according e2  1 −

(13)

2

Since tx ≠ ty , the transmittance through the fast and the slow axes is different. Thus, when trying to obtain circularly polarized light with a linear polarizer and a λ∕4 waveplate, the polarization at the output is quasi-circular, because the transmittance in each axis is different. It will be shown in Section 3 how a combination of two waveplates can compensate this, usually, unwanted effect. B.

    s   1 1 1 1 4 1 1 2 2 1∕a   cos δ 2− 2  ; 2 sin2 δ E2x E2y E2x E2y Ex Ey 2

tx cos2 ϕeiδ∕2  ty sin2 ϕe−iδ∕2 1 iδ∕2 − ty e−iδ∕2  2 sin 2ϕtx e

where tx 

to their circularity degree. That is, by considering the transmittances, the output state will be close to the pure state circular. Therefore the eccentricity is a good parameter to measure these differences. In general, Eq. (11) represents a rotated ellipse. Consider a nonrotated arbitrary ellipse, with semimajor axis, b, and semi-minor axis, a, and 90° retardation phase, described by the equation x2 ∕a2  y2 ∕ b2  1. Then, an affine transformation over this arbitrary ellipse can be performed, x0  x cos α− y sin α, y0  x sin α  y cos α. Finally, identifying terms with Eq. (11), semi-major and semi-minor axes are obtained:

An elliptical polarization state is given by j  cos α; sin αeiγ , with tan α  Ey ∕Ex and γ the retardation. Then, using Eqs. (13) and (14), Eq. (12) can be written in terms of α and γ, p 2 cos2 γ sin2 2α  cos2 2α p : e  1  cos2 γ sin2 2α  cos2 2α 2

(15)

Therefore, Eq. (15) represents a family of solutions for different polarization states. For example, circular polarization light is obtained when cos2 γ sin2 2α cos2 2α  0, which occurs when γ  π∕2 and α  π∕4 simultaneously. 3. Results

Different possibilities to obtain circularly polarized light using two waveplates have been analyzed according to the setup given in Fig. 1. In particular, the configurations λ∕2  λ∕4, λ∕4  λ∕2, and λ∕4  λ∕4 have been studied. Also, a single λ∕4 waveplate is analyzed for comparison purposes. It will be shown how these configurations produce different polarization states, and the sensitivity under small variation in the azimuth of the waveplates. As a consequence, they are not equivalent to each other, because the transmittance in any fast or slow axis of the waveplates is not the same. Numerical results can be obtained particularizing to the case of quartz and MgF2 waveplates, considering that a combination of two waveplates of different material can form an achromatic retarder. We have chosen these materials 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS

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because they are used broadly [20–23]. The dependence of the refractive indices with the wavelength is given by the Sellmeier relations, which for the case of MgF2 are

n2e − 1 

0.48755108λ2 0.39875031λ2  2 2 − 0.04338408 λ − 0.094614422 2.3120353λ2  2 ; (16) λ − 23.7936042 λ2

2

0.41344023λ 0.50497499λ  2 2 − 0.03684262 λ − 0.090761622 2 2.4904862λ  2 ; (17) λ − 23.7719952

0.7

0.5 0.4 0.3 0.2 0.1

180 0

45

90 φ (degrees)

135

180

(b)

0.665721λ:2 0.503511λ2 0.214792λ2   λ2 − 0.06002 λ2 − 0.10602 λ2 − 0.11902 0.539173λ2 1.8076613λ2  2  2 : (19) λ − 8.7922 λ − 19.702

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(c) 0

1 0.9 0.8

45

0.7 θ (degrees)

We assume the system is illuminated at λ  638.2 nm. The use of this specific wavelength obeys the quartz waveplates with d  369.87 μm being a retarder λ∕2 at λ  638.1 nm, and the waveplate of MgF2 with d  298.12 μm exhibits a λ∕4 behavior at λ  638.3 nm. Essentially, both kinds of waveplates work at the same wavelength. Thus, if the wavelength is λ  638.2 nm, it is possible to obtain the configurations λ∕4  λ∕2 or λ∕2  λ∕4 with a quartz waveplate and a MgF2 waveplate. In order to get a high degree of circularity for a circular polarization state, the eccentricity of the polarization ellipse for all the possible azimuths θ; ϕ has been computed and mapped from the output Jones vector from Eq. (12) and Eqs. (13) and (14). The eccentricity minimum values are associated to circular polarization states, in which the polarization is circular. In Fig. 2, the eccentricity is shown for the nonunitary transmittance representation when the system is illuminated at λ  638.2 nm. It is noteworthy that using the unitary transmittance representation formalism, the minimum eccentricity is zero. In particular, performing a simple calculus for the unitary transmittance case, the azimuth solutions are θ  π∕4, ϕ is arbitrary for λ∕4  λ∕2, and the relation between θ and ϕ for the λ∕2  λ∕4 configuration is described by θ  −ϕ  π∕4 modπ∕2. Therefore, the calculus using the transmittance effect treatment achieves in discriminating the degree of circularity obtained. Thus, the system λ∕2  λ∕4 3396

0.6 90

135

0.663044λ 0.517852λ2 0.175912λ2 −1 2  2  2 2 2 λ − 0.1060 λ − 0.11902 λ − 0.0600 0.565380λ2 1.675299λ2  2  2 ; (18) 2 λ − 8.844 λ − 20.7422

n2e − 1 

0.8

45

and for the quartz n2o

1 0.9

2

λ2

(a)

0

θ (degrees)

n2o − 1 

exhibits the highest circular polarization with an eccentricity e  9.94 · 10−4, for θ  171.3° and ϕ  143.7°. The other configurations have eccentricities around 10−2 , in particular, e  1.49 · 10−2 for λ∕4  λ∕2 at θ  45.0° and ϕ  90.0°, e  2.74 · 10−2 for

0.6 90

0.5 0.4 0.3

135

0.2 0.1 180 0

45

90 φ (degrees)

135

180

Fig. 2. Eccentricity of the polarization ellipse for a system illuminated at λ  638.2 nm and composed by (a) a linear polarizer and a configuration λ∕4  λ∕2 of quartz-MgF2 , (b) a linear polarizer and a configuration λ∕2  λ∕4 of quartz-MgF2 , and (c) a linear polarizer and a configuration λ∕4  λ∕4 of quartz-MgF2 .

(a)

(b)

0.06

0.06 λ/2+λ/4 λ/4+λ/2 λ/4+λ/4

0.04

0.02

−1

∂ e/∂ φ (degrees )

−1 ∂ e/∂ θ (degrees )

0.04

0

−0.02

−0.04

−0.06 −5

λ/4 λ/2+λ/4 λ/4+λ/2 λ/4+λ/4

0.02

0

−0.02

−0.04

−3

−1 0 1 θ (degrees)

3

5

−0.06 −5

−3

−1 0 1 φ (degrees)

3

5

Fig. 3. Curves of azimuthal error sensitivity for the configurations λ∕2  λ∕4, λ∕4  λ∕2, λ∕4  λ∕4, and λ∕4 under illumination at λ  638.2 nm: (a) ∂e∕∂ϕ versus ϕ; all the curves are essentially the same. (b) ∂e∕∂θ versus θ; the configuration λ∕4  λ∕2 exhibits better azimuthal error sensitivity.

wavelength, despite the fact that the proposed systems formed by two waveplates are achromatic. Indeed, using Eq. (5) the overall retardation of the system composed by two waveplates can be plotted in terms of the wavelength. It is important to take into account that Eq. (5) allows us to permute δ1 and δ2 without changing the overall retardation. In Fig. 4 the overall retardation has been plotted for a combination 180 160 140 Retardation (degrees)

λ∕4  λ∕4 at θ  0.0°, and ϕ  45.1° and e  3.12 · 10−2 for a single plate λ∕4 with θ  45.0°. Hence, to get circularly polarized light with a high polarization degree, the system λ∕2  λ∕4 should be used. It is remarkable that in the configuration λ∕4  λ∕2 the light is already circular once it has passed the λ∕4 waveplate. The λ∕2 waveplate has a double function: first, the slight lack of circularity after the first waveplate due to different transmittance in the fast and slow axes is compensated with the transmittances of any axis of the second waveplate. Thus, the circularity is improved using the second waveplate. The second benefit of using a second waveplate is that overall retardation is quasi-constant and equal to π∕2. Therefore, the system will be achromatic, as will be shown below. However, the configuration λ∕2  λ∕4 has a higher azimuthal error sensitivity than the λ∕4  λ∕2 configuration, as shown below. The azimuthal error sensitivity has been studied analyzing the first derivative of the eccentricity with respect to θ or ϕ in an interval centered around the minimum angular radius 5° for each configuration. If the variations in that interval are small, the system will be low azimuthally sensible; otherwise the system will be highly azimuthally sensible. Thus, the azimuthal sensitivity can be affected by the parameter θ; ϕ. The sensitivity curves ∂e∕∂ϕ versus ϕ and ∂e∕∂θ versus θ can be observed in Fig 3. Thereby, looking at these curves, the configuration λ∕4  λ∕2 presents the lowest azimuthal error sensitivity, because it is very stable for any azimuth θ of the polarizer. The other configurations composed by two waveplates present more sensible behavior under azimuthal variations. Then the azimuthal sensitivity with one waveplate λ∕4 and the system λ∕4  λ∕2 is essentially the same. However, a single waveplate is designed for a specific

120 100 80 60 40 20 0 500

550

600 Wavelength (nm)

650

700

Fig. 4. Overall retardation for plane spectrum in the bandwidth [500, 700] nm: (continuous) configuration λ∕2  λ∕4 at 638.2 nm of quartz-MgF2 with ϕ  π∕2, d1  369.87 μm, and d2  298.12 μm or equivalently the same curve is obtained with a combination λ∕4  λ∕2 at the same wavelength of MgF2 -quartz with d1  298.12 μm and d2  369.87 μm. (Dashed) single quartz waveplate λ∕4 with thickness d  369.87 μm. λ∕4  λ∕2 at the same wavelength of MgF2 -quartz with d1  298.12 μm and d2  369.87 μm, respectively. 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS

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λ∕2  λ∕4 at 638.2 nm of quartz-MgF2 with d1  369.87 μm and d2  298.12 μm, respectively. The retardation curve is the same considering a combination λ∕4  λ∕2 at the same wavelength of MgF2 -quartz with d1  298.12 μm and d2  369.87 μm, respectively. Thereby, the spectral behavior of the two-waveplate system is spectrally stable for the retardation of the system, despite the fact that a single λ∕4 is unstable as can be observed in Fig. 4. Therefore, a system composed with two waveplates in formats of half- and quarter-wave achieves a better circularity than a single quarter-waveplate. In particular, the configuration λ∕2  λ∕4 provides the best circularity degree. However, in terms of azimuthal error sensitivity, the configuration λ∕4  λ∕2 exhibits better behavior despite the fact that its circularity is lower than the case λ∕2  λ∕4. 4. Conclusions

The obtainment of circularly polarized light with a high degree of circularity and low azimuthal error sensitivity has been analyzed using a system composed by two waveplates in formats of half- and quarterwaveplate representation. The Fresnel transmission coefficients have been considered in the waveplate matrix representation, allowing the classification of systems according to the degree of circularity achieved by them. Thus, it has been shown that a configuration λ∕2  λ∕4 produces the highest circularity degree, and in addition the use of two waveplates improves the circularity of the output polarization state compared to the use of a single waveplate λ∕4. The reason for such behavior is the birefringence of the waveplates. The existence of two refractive indices generally implies two orthogonal polarizations transmitted with different intensities. The slight differences of intensity of the first waveplate are compensated with the second waveplate, and then a high degree of circularity is obtained. The azimuthal sensitivity under a small variation in the azimuths of the waveplates has been studied for the same configurations. A combination of two waveplates can provide an achromatic configuration, as has been shown. The azimuthal sensitivity analysis concludes that the lowest-sensitivity system is the configuration λ∕4  λ∕2. This system has the same sensibility as a single λ∕4, but the advantage of using λ∕4  λ∕2 is in the overall retardation, Δ. A single waveplate is designed at a specific wavelength. Therefore, when the system requires a high degree of circularity, a combination of two waveplates should be employed. This combination is achromatic. The highest circularity is obtained with a combination λ∕2  λ∕4. However, when the lowest azimuthal error sensitivity is required, the suitable configuration is λ∕4  λ∕2. The authors express their gratitude to Jorge Jimenez de la Morena and Yanalte de Haro for their advice and interest in this research. This work was

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supported by the Ministry of Science and Innovation of Spain by project DPI2011-27851 and by the project “Photonic Transceiver for Secure Communications Space” of the European Space Agency with Tecnológica Ingeniería, Calidad y Ensayos. References 1. C. J. Yu, C. E. Lin, L. P. Yu, and C. Chou, “Paired circularly polarized heterodyne ellipsometer,” Appl. Opt. 48, 758–764 (2009). 2. V. Samkaran, “Comparison of polarized-light propagation in biological tissue and phantoms,” Opt. Lett. 24, 1044–1046 (1999). 3. C. Wagenknecht, C. M. Li, A. Reingruber, X. H. Bao, A. Goebel, Y. A. Chen, Q. Zhang, K. Chen, and J. W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photonics 4, 549–552 (2010). 4. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006). 5. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987). 6. S. Pancharatnam, “Achromatic combinations of birefringent plates, Part II. An achromatic quarter-wave plate,” Proc. Indian Acad. Sci. 41A, 137–144 (1955). 7. P. Violino, “Polariseur circulaire réglable sur une large domaine de longueurs d’onde,” Rev. Optique 44, 109–114 (1965). 8. R. Corbalan and E. Bernabeu, “On the obtaining circular polarization for each of the optical doublet Caesium lines with available commercial components,” Opt. Pura Apl. 5, 80–84 (1972). 9. E. Bernabeu and J. Aporta, “On obtaining circularly polarized light,” Atti della Fondazione Giorgio Ronchi 2, 351–355 (1975). 10. P. Hariharan, “Achromatic retarders using quartz and mica,” Meas. Sci. Technol. 6, 1078–1079 (1995). 11. P. Hariharan and D. Malacara, “A simple achromatic halfwave retarder,” J. Mod. Opt. 41, 15–18 (1994). 12. A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51, 1976–1980 (2012). 13. J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Optimal achromatic wave retarders using two birefringent wave plates,” Appl. Opt. 52, 7078–7080 (2013). 14. X. Zhang, “Optimal achromatic wave retarders using two birefringent wave plates: comment,” Appl. Opt. 52, 7078– 7080 (2013). 15. J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Optimal achromatic wave retarders using two birefringent wave plates: reply,” Appl. Opt. 52, 7081–7082 (2013). 16. X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C.-Z. Peng, “Experimental quantum error detection,” Sci. Rep. 2, 626 (2012). 17. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996). 18. P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (wave plates),” Appl. Opt. 27, 5146–5153 (1988). 19. J. B. Masson and G. Gallot, “Terahertz achromatic quarterwave plate,” Opt. Lett. 31, 265–267 (2006). 20. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23, 1980–1985 (1984). 21. M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. Van Stryland, in Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 4. 22. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163, 95–102 (1999). 23. S. Chandrasekhar, “The dispersion and thermo-optic behaviour of vitreous silica,” Proc. Indian Acad. Sci. 34A, 275–282 (1951).