Vortex retarders produced from photo-aligned liquid crystal polymers

Scott C. McEldowney; David M. Shemo; Russell A. Chipman

doi:10.1364/OE.16.007295

1. Introduction

There are many applications of polarization vortex beams. The usefulness of vortex fields stems from two primary factors. The first is that the polarization orientation, in the case of azimuthally or radially polarized light, can be aligned perpendicular to a lens (S-polarized) or parallel to a lens (P-polarization) or a focusing beam. When a high numerical aperture (NA) lens is used, the resulting field can be oriented purely transverse with respect to the propagation axis of an optical system in the case of the azimuthal field, or substantially along the direction of the axis of an optical system (longitudinal) in the case of a radial field. The second factor is that if a propagation invariant or Bessel beam is formed with a polarization vortex, a propagation invariant vector field is created

As a result of the aforementioned properties, many applications have emerged. Of particular interest are applications of radial polarized beams in conjunction with high NA lenses. Dorn, et al., and Youngsworth, et al., have predicted that in the case of non-paraxial imaging, the ratio between the transverse and longitudinal component of the intensity point spread function can be manipulated. For very high NA lenses and for a center obscured pupil, it was predicted that very high longitudinal components were possible [1, 2]. This has been also observed experimentally by Novotny [3] and by Dorn [4]. For lithography systems, it has been shown that light polarized normal to a photosensitive material will create better contrast [5]. Light which is purely transverse to the photosensitive material will couple more efficiently [5]. More general polarization vortices of single or higher orders can be used to create vectorial Bessel Beams [6].

One emerging application of Bessel beams is in enlarging the particle trapping region and the kinematics of optical tweezers [6]. Propagation invariant fields formed using polarization vortices have been demonstrated by Hasmin, et al., [6]. The unique characteristic of these fields are the propeller shaped intensity pattern formed for which the number of propellers and the rotation of the propeller with analyzer angle depends on the mode of the polarization vortex used in creating these fields. Example applications are optical tweezers or transport and guiding of microspheres [7].

Azimuthally and radially polarized fields were first formulated in the context of dipole radiation [8]. Radial polarized beams were first generated by Mushiake, et al., using a conical element in a laser cavity [9]. Hall demonstrated the creation of these fields using distributed feedback lasers designed to produce azimuthally polarized output [10, 11]. Tidwell and coworkers demonstrated that both radial and azimuthal polarization could be produced using coherent interference of different transverse electromagnetic (TEM) modes [12]. More recently, laser resonator cavities involving complex interferometric configurations have been developed to create azimuthal and radial polarized output [13].

Monolithic components which convert a polarization state to the desired polarization vortex have the advantage that they can be placed close to the exit pupil of the system. The first such component was demonstrated in 1989 when Yamaguchi and coworkers demonstrated the use of liquid crystals to create azimuthal and radial polarized modes [14]. For the case of passive elements, the approaches taken involve creating a special class of retarders which have a uniform retardance but have a fast axis which rotates around its own center; termed a vortex retarder. Some key characteristics of these components are the wavelength region over which they can operate, how well the fast axis can approximate a continuous variation, what output polarization vortex modes can be created, and the complexity of the manufacturing process.

A series of crystals can be assembled to make a vortex retarder [15,16]. However, the fast axis cannot be made continuous, therefore causing diffraction at the boundaries of the different crystals forcing the use of additional components to create a pure polarization vortex. The assembly process is also complex making the components expensive. Vortex retarders made from nanostructures [6] have the potential to create stable retarders with a continuous fast axis and, in principle, can be made over broad wavelength ranges. However, the feature size required to make visible vortex retarders is beyond the state-of-the-art of feature generation and thus, has only been demonstrated at infrared wavelengths. Vortex retarders made using twisted neumatic (TN) liquid crystal [17] are limited in that thus far, multiple components are required if polarization vortices other than order 2 are required.

The used of photo-aligned Liquid Crystal Polymers (LCP) to make vortex retarders has been reported by McEldowney, et al., [18]. LCP are materials which combine the birefringent properties of liquid crystals with the mechanical properties of polymers. The orientation of LCP is achieved through the use of an alignment layer [19]. Photo-alignment is a recent advancement that enables non-contact alignment which produces high quality alignment with low defects as compared to rubbing alignment techniques [20]. Photo-alignment also allows for a high degree of customization in producing birefringent components. Once the alignment has been established in the alignment layer, the LCP precursor is applied and crosslinked to make its alignment permanent.

Previously, we demonstrated photo-aligned LCP vortex retarders creating m=1, 2, and 3 vortex modes at 550 nm [18]. We analyzed these elements experimentally by measuring the space variant Mueller matrix of each component. We characterized these components by determining the Mueller matrix of the point spread function (PSF) or point spread matrix (PSM); the combination of these allowing us to fully understand the optical vortex created by the component and its impact in an optical system. In this paper, we further develop the theoretical and measured imaging properties in Mueller Matrix format of vortex retarders produced using Photo-aligned LCP. From the PSM previously calculated and measured, we present a theoretical and experimental analysis to determine the optical transfer matrix (OTM) to assess the spatial frequency transfer characteristics of the vortex retarders. Additionally, the impact of aberrations and misalignments are assessed by adding these factors to the theoretical calculations and comparing these to the measured PSM and OTM.

2. PSM and OTM of a Vortex Retarder

This section develops a model for the point spread function (PSF) and optical transfer function (OTF) of an optical system with a polarization vortex at the exit pupil plane. The formalism for the PSF in the presence of polarization aberrations has been well developed for both the paraxial case [20–23] and the non-paraxial case [24–26]. The description of the PSF and OTF in Mueller Matrix format as a point spread matrix (PSM) and optical transfer matrix (OTM) is well developed [27]. However, the application of the PSM and OTM to a pupil plane polarization vortex is unique.

The paraxial PSM and the OTM have been used to assess the applicability of the photoaligned LCP vortex retarder. The closer the performance is to ideal, the greater the applicability. If significant undesired polarization aberrations, non-uniformity, or stray light are present, the device is less useful.

The first step is determining the pupil function as a Jones matrix of pupil coordinates. This pupil function includes the space variant polarization element, the pupil aperture, and wavefront aberrations. This function, termed the polarization aberration matrix (PAM), is defined as [27]:

PAM (h, ρ, λ) = P (h, ρ) \exp (j k W (h, ρ, λ)) J_{VR} (h, ρ, λ);

where P is the pupil function, W is the wavefront aberration function, and J_VR is a Jones Matrix describing the vortex retarder. h, ρ and λ are the object coordinate, pupil coordinate and wavelength of light respectively. This function relates the Jones vector at the entrance pupil to that at the exit pupil providing a detailed complete description a polarization aberrated imaging system. In the aberration free case where the retardance is a halfwave and the fast axis orientation is $θ (x, y) = \frac{1}{2} m ϕ$ , the PAM is given by

PAM (m ϕ) = (\begin{matrix} \cos (m ϕ) & \sin (m ϕ) \\ \sin (m ϕ) & - \cos (m ϕ) \end{matrix});

where ϕ is the azimuthal angle in polar coordinates and m is the order of the polarization vortex created by the vortex retarder.

The amplitude distribution in the image space is equal to the convolution of the amplitude response matrix (ARM) and the amplitude distribution in the object space.

U_{i} = h * U_{o}

The detailed derivation of the ARM is given in [27]. The key assumptions are (a) Kirchhoff’s principle applies in that the extent of the stop does not perturb the field at the exit pupil; (b) the exit pupil can be related to the field incident in the entrance pupil by a Jones Matrix; and (c) scalar diffraction theory is applicable. The assumption that scalar diffraction theory applies restricts the NA of the incoming and outgoing beams to less than 0.1.

Given the object space Cartesian coordinates (ξ,η), the pupil coordinates (x,y), and the image space coordinates (u,v), the ARM is given as [27]:

h = C \exp [j k Ψ (ξ, η; u, v) [\begin{matrix} ℱ {j_{11}} & ℱ {j_{12}} \\ ℱ {j_{21}} & ℱ {j_{22}} \end{matrix}]

where ℱ{j_ij} is the Fourier Transform of the ij^th element of the PAM. The constant C is given by

C = \frac{- A^{2}}{4 λ^{2} z_{1} z_{2}} \exp [j k (z_{1} + z_{2})] .

The constant C is made up of several constants that describe the system and a phase factor related to the position of the object and the image. For an object on axis, the transformation coordinates are $f_{u} = \frac{u NA}{λ}$ and $f_{v} = \frac{v NA}{λ}$ . The phase factor is given as:

Ψ (ξ, η; u, v) = \frac{ξ^{2} + η^{2}}{2 z_{1}} + \frac{u^{2} + v^{2}}{2 z_{2}}

For our cases, Ψ is very small and is ignored.

The PSM represents image intensities using a 4×4 matrix that relate the object-plane Stokes vector to the image-plane Stokes vector. The formalism is valid for partially coherent light propagating through a polarizing optical system.

The PSM is determined by taking the direct product of the ARM with it’s complex conjugate [27];

P_{c} (ξ_{1}, η_{1}; ξ_{2}, η_{2}; u, v) = h (ξ_{1}, η_{1}; u, v) \otimes h^{*} (ξ_{2}, η_{2}; u, v)

where P_c is the PSM relating the mutual coherence vector at the object plane to the mutual coherence vector at the image plane and ⊗ is the direct (or Kronecker) product [27]. The PSM in the Stokes basis is found by rotating the matrix P_c

P = S \cdot P_{C} \cdot S^{- 1}

where

S = [\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & - 1 \\ 0 & 1 & 1 & 0 \\ 0 & - j & - j & 0 \end{matrix}]

is the rotation matrix for converting the PSM to the Stokes basis when the coherence vectors are in Cartesian coordinates.

The OTM is the polarization generalization of the optical transfer function (OTF). It describes the polarization sensitive frequency transfer characteristics of the optical system. The OTM is determined by taking the Fourier Transform of PSM and normalizing to the zero frequency components.

H (v_{u}, v_{v}) = \frac{ℱ {P_{s}}}{N} .

N is given by:

N = \frac{1}{2} \iint_{\infty} ({∣ H_{11} (p, q) ∣}^{2} + {∣ H_{21} (p, q) ∣}^{2} + {∣ H_{12} (p, q) ∣}^{2} + {∣ H_{22} (p, q) ∣}^{2}) dpdq

3. Vortex Retarders using Photo-aligned LCP

The specific process used to make the samples follows. A photo-alignment layer (ROP108 from Rolic) was spin-coated onto the substrate, baked, and then the alignment was set through exposure to linear polarized UV (LPUV) light. The alignment layer was exposed through a narrow wedge shaped aperture located between the substrate and the polarizer. Both the polarizer and the substrate were continuously rotated during the exposure process in order to create a continuous variation in photo-alignment orientation with respect to azimuthal locations on the substrate. The variation in fast axis orientation with azimuthal angle was determined by the relative rotation speeds. The LCP precursor (ROF5104 from Rolic) was then spin-coated and subsequently polymerized using a UV curing processes. A post-bake stabilized the films.

The samples were fabricated on 2 inch squares of Corning 1737F glass, having a broadband visible AR on the back surface. The process targeted a half wave retardance for wavelengths in the range of 540~550nm. Three samples were produced with of mode m=1, 2, and 3. Each samples polarization properties were mapped on a commercially available Axoscan Mueller Matrix Spectro-Polarimeter available from Axometrics [28]. The samples were placed on an x-y stage and mapped at a 0.5×0.5 mm xy-resolution. For each sample summary maps of (a) linear retardance; (b) fast axis orientation; and (c) a photograph between crossed-polarizers are provided in Figs. 1 to 3.

The fast axis maps are given in degrees and with a range from -90° to 90°. The orientation in all cases varies continuously. Because the components are half-wave plates, a linear polarized input field will be converted to a linear polarized output field that rotates with respect to azimuth angle. This yields a periodic modulation of the intensity when analyzed by a linear analyzer, as seen in the photos in Figs. 1(c), 2(c), and 3(c).

Fig. 1. m=3: (a) linear retardance map (in nm) (b) fast axis orientation map, (c) transmission between crossed polarizers.

Abstract

1. Introduction

2. PSM and OTM of a Vortex Retarder

3. Vortex Retarders using Photo-aligned LCP

4. Measurement of PSM and OTM

5. Impact of aberrations, alignment errors on PSM/OTM

6. Conclusions

References and links

Cited By

Figures (17)

Equations (11)

Optics Express