Elsevier

Optik

Volume 125, Issue 16, August 2014, Pages 4534-4537
Optik

Achromatic half-wave combination of birefringent plates

https://doi.org/10.1016/j.ijleo.2014.02.012Get rights and content

Abstract

Half-wave retarders are primarily used in rotating the plane of polarization of polarized light. These retarders usually exhibit strong wavelength dependence. In this paper, the design and characteristics of an achromatic half-wave plate, formed by a cascaded system of birefringent plates, have been studied. Pancharatnam proposed a combination of three retarders in series and discussed the possibility of fabricating reasonably good achromatic quarter-wave plate with a suitable combination of their retardance. The system studied here is similar in construction to Pancharatnam configuration and behaves as an achromatic half-wave plate over a wide spectral range. The proposed configuration exhibits a maximum variation of only about ±1.4 degree over the entire wavelength range of 500–750 nm. In our analysis we have used Jones matrix formalism for the derivation of the general expression for the equivalent retardation and the azimuth of the combination.

Introduction

A half-wave plate is a useful polarization device and is required in many experimental layouts involving polarization optics. This device is ubiquitously used for rotating the plane of vibration or polarization ellipse by any pre-specified amount. For example, a λ/2-plate can convert an oblique ellipse to a symmetrical ellipse by adjusting the azimuth of the half-wave plate. We know that the retardance of a single plate birefringent retarder is highly chromatic, as is obvious from the expression for the retardation δ of a birefringent plate between the vibrations along two privileged directions of the plate.δ=2πλ0Δndwhere Δn is the birefringence of material of the plate. Since Δn of birefringent materials commonly used for making a retarder does not appreciably vary within the visible range, it is evident that the retardation varies almost inversely with the wavelength of light. If an experimental layout using polarized polychromatic light demands a use of a λ/2-plate, it must act as a λ/2-plate for the entire range of wavelengths constituting the polychromatic light used. Thus, an achromatic half-wave plate would be a useful polarizing device. Design of achromatic retarders, over at least the wavelength range of interest, therefore assumes significance. Different techniques have been used to attain such achromatism using both passive devices as well as active devices. In some of the cases the same birefringent material has been used to fabricate the retarder combination, whereas in other cases different materials are employed. Retarders with the best achromatic characteristics are based on the principle of unequal phase retardation between the fast and slow components of an incident beam on total internal reflection at the glass-air interface. An example of this is the Fresnel rhomb. This subject has been covered in detail by Bennett [1]. Several researchers worked in this field to design various types of achromatic retarders using this principle of total internal reflection in Fresnel rhomb [2], [3], [4], [5], [6], [7], [8]. However, in general, retarders based on the principle of the Fresnel rhomb are limited in aperture due to their geometry. Recently achromatic retarders are designed using liquid crystal cells [9], [10], [11], [12]. Achromatic phase retarders have also been designed using sub wavelength grating structures [13], [14], [15], [16] known as form birefringence. Composite plates, consisting of two or more plates of the same material whose axes are oriented at the appropriate angles, can also be used as achromatic circular polarizers or achromatic polarization rotators [17]. In fact Pancharatnam proposed one of the most successful combinations of three plates which were produced of the same material [18], with the extreme retardation plates having similar optical axes orientations and retardations while the central plate has a different retardation and orientation. Subsequently, achromatic quarter-wave and half-wave retarders have been produced using a combination of two or three plates of different birefringent materials [19], [20], [21] to minimize retardation dispersion across a broad wavelength range. Hariharan has devised a procedure to optimize the choice of materials for broadband retarders [22]. Hariharan and Malacara proposed a combination of two quarter-wave plates and one half-wave plate of the same material, which can be used as an achromatic half-wave retarder [23]. Hariharan has also showed that two plates of different birefringent materials, when suitably configured, exhibits achromatic behavior over a limited range of wavelengths [24]. A reconfigurable achromatic half-wave and quarter-wave retarder in near infrared has been proposed using three crystalline quartz plates [25]. A cascaded system of four plates has also been studied [26]. However, one of the early works in this domain was done by Destriau et al. [27] where the authors showed that an almost achromatic quarter wave retarder could be obtained by juxtaposing two retarders – one half-wave and the other quarter-wave plate. However, they used the Poincaré sphere method to design the system and the variation of azimuth is not shown in their work. Two different configurations for producing achromatic quarter-wave retardation using only two plates have also been studied recently where the problems of the works of Destriau has been removed [28]. However, in most of the cases, the retarders designed are either costly or show the performance over only a very limited range.

Section snippets

Formulation of the problem

The combination considered by us is a cascaded system of three birefringent plates, as shown in Fig. 1. For general analysis of such a system of birefringent plate, we assume δ1 to be the retardation of the two extreme plates and δ2 to be that of the central plate. The angle ϕ is the inclination between the fast axis of the central plate and the fast axis of the two extreme plates.

We now calculate the retardation of the achromatic combination using the well-known Jones matrix method. The Jones

The design procedure

The aim of this study is to achieve an achromatic half-wave plate over the wavelength range 500–750 nm. To achieve this, a combination of three half-wave plates, all designed for 600 nm, have been considered. We have calculated the retardation of the overall system for the design wavelength (i.e., 600 nm), the two extreme wavelengths (i.e., 500 nm and 750 nm), and two other intermediate wavelengths (i.e., 550 nm and 700 nm) for different orientations of the central plate. The retardation for the

Computational results and discussion

We show that the combination of these three birefringent plates of crystalline quartz behaves as a single achromatic half-wave retarder, within the 500–750 nm wavelength spectrum range, when the central plate is oriented at an angle of 58.27°. It is found that in this entire wavelength range the retardation varies maximum by ±1.4° (as illustrated in Fig. 3), which amounts to about only ±0.8% shown in Fig. 4. This region of spectrum may be shifted to some other wavelength region by a proper

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