Synthesis of 2-dimensional inhomogeneous polarized beams

Based on the interferometrical setup, an idea is proposed that arbitrary state of polarization can be obtained in spatial plane. The following sections include background information of mathematics and the analytical description of synthesis process. The first section focuses on the Jones calculus which is a useful tool to analyze the state of polarization.

4.1.1 Jones calculus

Jones Calculus which includes Jones matrix and Jones vector is a quantitatively mathematical description of polarized light. The operation of optical systems can be described by a cascade multiplication based on 2 x 2 matrices. Each component of the system has associated Jones matrix, and the analysis of the system as a whole is performed by multiplication of the 2 x 2 component matrices. Moreover, from the final vector of multiplication the state of polarization is clearly known at each stage of the calculation. The polarization information can be reduced to a vector, called the Jones vector. The field was described by the Jones vector whose vector components

are including Ex and Ey. From Eqs. Ex Ae and Ey Be ^∆ , the Jones vector of the electric field is

E E

E e A

Be ^∆ where A and B are amplitude, and ∆ is the phase retardation of the field. For convenience of calculation, the time-dependent term is omitted and then we denote

tors in their normalized forms. For instance, the linearly polarized beam the Jones vec

Ae

0 can be normalized to 10 .

When putting an optical component as a converter of the state of polarization from E E into E E , the function of the optical component is represented by Jones matrix.

where J is the incident beam, J’ is the emergent beam, and M is the Jones matrix.

The Jones calculus also describes the optical functionality of components. This method is useful to describe the propagation of polarized light through optical elements such as polarizer and wave-plates. A polarizer is an optical element that passes a state of polarization by absorbing or reflecting the orthogonal mode of polarization. A wave-plate is an optical element that consists of an anisotropic material, which changes the phase of transmitted light depending on its polarization state. The following table summarizes the Jones vectors for common optical components.

Optical element Jones matrix Linear polarized with axis of

transmission at angle

Cos Cos sin

Cos sin Sin Half-wave plate with fast axis in

x-direction

0 0 Quarter-wave plate with fast axis in

x-direction E 1 0

0 Quarter-wave plate with fast axis at

angle 45°

1

√2

1 j

j 1 Quarter-wave plate with fast axis at

angle -45°

1

√2

1 j

j 1

Table 4-1. The reference between optical element and Jones matrix.

Common optical elements are described by a 2×2 Jones matrix, enabling the description of the change of polarization states by multiplication of the Jones vector with the Jones matrix. Table 4-2 shows some examples of Jones vectors and Jones matrices of the optical elements.

Polarization Jones ector v Illustration

Linear in x-direction 1

Left circular 1

Table 4-2 The reference between optical element and Jones vector.

4.1.2 Mathematics of synthesis

The synthesis of the inhomogeneous beams is based on the interferometrical circular approach. The key issue is to find the relationship between the phase retardation and polarization. Moreover, the derived formula can be used to modulate the spatial light modulator in our experiment. The associated instrument will be introduced in the chapter 3.

Consideration of letting the beam pass through the polarizer and a quarter wave plate can result in turning the x-polarized beam to left-handed circularly polarized beams. Jones mat x ri em gent from quarter-waveplate with fast axiser at angle -45° is

√

Similarly, the x-polarized beam can be turned to right-handed circularly polarized beam. Jones matrix emergent from quarter-waveplate w h fast axis at angle 45° is it

√

The terms of phase retardation of light emergent from reflected SLM are e ^∆ and e^∆ respectively. As combining these beam, we can get the final vector which describes the state of polarization.

where Jl is Jones vector of the emergent beam at local point and the relative phase difference is the retardance ∆. Additionally, it is easy to control the state of polarization from SLM by referring to Eq. (4-2). When ∆ is 0°, the polarization is linear x-polarized beam. We put the retardance and Jones vector in order, as showed in Table 4-3. From this table, it is convenient for us to design the pattern on the SLM.

Table 4-3 The relationship of retardance and Jones vector.

In order to check feasibility of this method, we create a eight-zones polarization pupil to simulate the property of the inhomogeneous beams, as shown as showed in Fig. 4-1. Two cases of examples are radial-like and azimuthal-like beams.

When taking radial-like polarization as an example, we can use the phase information with respective circular polarization as illustrated in Fig. 4-1 (b), (c). By using the software Diffract, we can simulate the propagation of radial-like beam passing

through the polarizer. Fig. 4-2 describes the simulation results depending on the transmission axis of polarizer. The result was brought into harmony with our prediction.

(a) (b) (c) (d) (e) (f)

Fig. 4-1 Two examples of synthesizing the inhomogeneous beams. (a) Radial-like polarization with specific phase (b), (c) and (d) azimuthal-like beams polarization with specific phase (e), (f).

(a) (b) (c) (d) Fig. 4-2 The simulation of passing the radial-like beam through the transmission axis at the angle (a) 0°, (b) 45°, (c) 90°, (d) -45° of polarizer.