Vortex retarders produced from photo-aligned
liquid crystal polymers

doi:10.1364/OE.16.007295

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Vortex retarders produced from photo-aligned liquid crystal polymers

Scott C. McEldowney, David M. Shemo, and Russell A. Chipman »View Author Affiliations

Optics Express, Vol. 16, Issue 10, pp. 7295-7308 (2008)
http://dx.doi.org/10.1364/OE.16.007295

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Abstract

We present developments using photo-aligned liquid crystal polymers for creating vortex retarders, halfwave retarders with a continuously variable fast axis. Polarization properties of components designed to create different polarization vortex modes are presented. We assess the viability of these components using the theoretical and experimental point spread functions and optical transfer functions in Mueller matrix format, point spread matrix (PSM) and optical transfer matrix (OTM). The measured PSM and OTM of these components in an optical system is very close to the theoretically predicted values thus showing that these components should provide excellent performance in applications utilizing polarized optical vortices. The impact of aberrations and of vortex retarder misalignment on the PSM and OTM are presented.

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

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

, 2

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

]. This has been also observed experimentally by Novotny [3

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 85, 5251–5253 (2001). [CrossRef]

] and by Dorn [4

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for radially polarized light beam,” Phys. Rev. Lett. 91, 23 (2003). [CrossRef]

]. For lithography systems, it has been shown that light polarized normal to a photosensitive material will create better contrast [5

Totzeck, et al., “Polarizer device for generating a defined spatial distribution of polarization states,” US 2006/0028706, Feb. 9, 2006.

]. Light which is purely transverse to the photosensitive material will couple more efficiently [5

Totzeck, et al., “Polarizer device for generating a defined spatial distribution of polarization states,” US 2006/0028706, Feb. 9, 2006.

]. More general polarization vortices of single or higher orders can be used to create vectorial Bessel Beams [6

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements,” Opt Lett 29, 238 (2004). [CrossRef] [PubMed]

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

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002). [CrossRef] [PubMed]

Azimuthally and radially polarized fields were first formulated in the context of dipole radiation [8

M. Born and E. Wolf, Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light , (Cambridge University Press, seventh ed., 1999). [PubMed]

]. Radial polarized beams were first generated by Mushiake, et al., using a conical element in a laser cavity [9

Y. Mushiake, K. Matzumurra, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972). [CrossRef]

]. Hall demonstrated the creation of these fields using distributed feedback lasers designed to produce azimuthally polarized output [10

T. Erdogan, K. G. Sullivan, and D. G. Hall, “Enhancement and inhibition of radiation in cylindrically symmetric, periodic structures,” J. Opt. Soc. Am. B 10, 391–398 (1993). [CrossRef]

, 11

T. Erdogan, O. King, W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992). [CrossRef]

]. Tidwell and coworkers demonstrated that both radial and azimuthal polarization could be produced using coherent interference of different transverse electromagnetic (TEM) modes [12

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990). [CrossRef] [PubMed]

]. More recently, laser resonator cavities involving complex interferometric configurations have been developed to create azimuthal and radial polarized output [13

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phy Lett. 77, 21 (2000) [CrossRef]

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

R. Yamaguchi, T. Nose, and S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. 28, 1730 (1989). [CrossRef]

]. 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

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007). [CrossRef]

,16

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 11 (2007). [CrossRef]

]. 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

M. Stalder and M. Schadt, “Linear polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt Lett. 21, 23 (1996). [CrossRef]

] 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

S. C. McEldowney, D. M. Shemo, R. A. Chipman, and P. K. Smith, “Creating vortex retarders using photoaligned liquid crystal polymers,” Opt. Lett. 33, 134–136 (2008). [CrossRef] [PubMed]

]. 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

M. Schadt, H. Seiberle, A. Schuster, and S. M. Kelly “Photo-induced alignment and patterning of hybrid liquid crystalline polymer films on single substrates,” Jpn. J. Appl. Phys. 34, L764–L767 (1995). [CrossRef]

]. 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

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996). [CrossRef] [PubMed]

]. 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

S. C. McEldowney, D. M. Shemo, R. A. Chipman, and P. K. Smith, “Creating vortex retarders using photoaligned liquid crystal polymers,” Opt. Lett. 33, 134–136 (2008). [CrossRef] [PubMed]

]. 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

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996). [CrossRef] [PubMed]

] and the non-paraxial case [24–26

C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999). [CrossRef]

]. 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

J. P. McGuire, Jr. and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations I: Formulation and example,” J. Opt. Soc. Am A. 7, 9, 1614–1626 (1990). [CrossRef]

]. 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);

(1)

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});

(2)

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}

(3)

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}]

(4)

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})] .

(5)

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}}

(6)

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)

(7)

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}

(8)

where

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

(9)

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} .

(10)

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

(11)

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

Information on Axometrics polarimeter from http://www.axometrics.com/

]. 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.

Fig. 2. m=2.0: (a) linear retardance map (in nm) (b) fast axis orientation map, (c) transmission between crossed polarizers.

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

Figure 4 shows the fast axis of the m=3 component with the center expanded so that only the center 6mm×6mm region is presented. This result demonstrates the orientation changes continuously over the azimuth of the component. (The pixilation in the image is due to the mapping resolution).

Fig. 4. Expanded view of measured fast axis orientation showing a 6mm×6mm central region of the m=3 vortex retarder

4. Measurement of PSM and OTM

This section presents the measured PSM and OTM of an optical system utilizing a vortex retarder to create a polarization vortex at its exit pupil. Note that the measured OTM is calculated from the measured PSM as described in the section 2.

The experimental setup for measuring the PSM can be found in [29

J. L. Pezzanitti and R. A. Chipman, “Mueller matrix imaging polarimeter,” Opt. Eng. 34, 6 (1995). [CrossRef]

]. A simplified instrument schematic is shown in Fig. 5. The measurement consists of a light source, a polarization state generator (PSG) consisting of a stationary polarizer followed by a rotating retarder, a microscope objective, a polarization state analyzer (PSA) consisting of a rotating retarder and a stationary polarizer, and a CCD detector. The principle of the measurement is to take Q different images acquired at different orientations of the linear retarder and linear polarizer in both PSG and PSA. The camera was focused on an illumination point source. The illumination source was an Oriel monochrometer with an arc lamp and approximately 10nm spectral bandwidth and spectral range from 400–900nm enabling measurements to be performed over the entire visible wavelength range. Each sample was measured at the wavelength where the vortex retarder was a half-wave, approximately 550nm. A 10µm pinhole was illuminated by a monochrometer. A collimating lens forms an approximately uniform plane wave at the vortex retarder. An aperture stop after the vortex retarder defines the stop of the system thus defining the exit pupil. After setting up the system, small adjustments were made to minimize the aberrations in the point spread function measurement. To further minimize the aberrations, the aperture stop was reduced to approximately 2.5mm which provided a nearly aberration free image. The final illumination region on the vortex retarder was approximately 3.0mm yielding an f-150 beam in the image plane of the system; well within the paraxial approximation used in the calculations.

Fig. 5. Schematic of the PSM measurement

Figures 6, 7, and 8 show the predicted and measured PSM of the m=1, 2, and 3 vortex retarders respectively. These figures demonstrate the measured PSM is very close to the predicted PSM. Some observations of the measurements are that (a) the m₀₀ and m₃₃ elements of the PSM show a “donut” shaped PSF components; (b) the PSF components of m₁₁, m₁₂, m₂₁, m₂₂ show a “propeller” type response indicating a rotation of the linear polarized Stokes vectors (S₁ and S₂) contribution; (c) The m₀₃ element is equal and opposite to m₀₀ indicating the system will convert a circular polarized input to an orthogonal circular polarized output. This response, in combination with the propeller PSF of m₁₁, m₁₂, m₂₁, m₂₂, indicates each pixel in the PSF is acting like a half wave plate with variable orientation; and (d) the non-zero m₃₀ and m₀₃ elements suggest small circular diattenuation and circular polarizance in the PSM.

Fig. 6. Comparison of the predicted PSM (left) and measured PSM (right) for the m=1 vortex retarder.

Fig. 7. Comparison of the predicted PSM (left) and measured PSM (right) for the m=2 vortex retarder.

Fig. 8. Comparison of the predicted PSM (left) and measured PSM (right) for the m=3 vortex retarder.

Figure 9 compares the intensity PSF for an m=2 vortex retarder in a system with a horizontal linear input with no analyzer, a horizonal linear analyzer, and a vertical linear analyzer. The results show the measured performance of the photo-aligned vortex retarder match the predicted performance.

Fig. 9. Comparison between predicted intensity PSF with (a) no analyzer, (b) horizontal linear analyzer, (c) vertical linear analyzer and measured intensity PSF with (d) no analyzer, (e) horizontal linear analyzer, (f) vertical linear analyzer.

Figure 10 gives an example of the calculated magnitude OTM and phase OTM for an m=2 vortex retarder. Figures 11, 12, and 13 compare the modulation transfer function (MTF) and phase transfer function (PTF) along the ν_v=0 axis of the OTM for horizontal polarized input and no analyzer. For all the vortex retarders, the measured and predicted MTF are in excellent agreement for the lower spatial frequencies and in very good qualitative agreement at the higher spatial frequencies. The measured PTF of all the vortex retarders have relatively good agreement with the predicted values particularly at the lower spatial frequencies. The location of phase discontinuities for the PTF is shifted as the spatial frequency increases. Additionally, there is significant structure in the measured PTF in regions where the theoretical PTF is relatively constant possibly suggesting the presence of small aberrations in the PSM measurements.

Fig. 10. Predicted magnitude OTM (left) and phase OTM (right)

Fig. 11. Comparison of predicted vs. measured MTF (left) and PTF (right) for an m=1 vortex retarder for the case of horizontal polarized input and no analyzer.

Fig. 12. Comparison of predicted vs. measured MTF (left) and PTF (right) for an m=2 vortex retarder for the case of horizontal polarized input and no analyzer.

Fig. 13. Comparison of predicted vs. measured MTF (left) and PTF (right) for an m=3 vortex retarder for the case of horizontal polarized input and no analyzer.

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

To understand the impact of optical system errors on the performance of the vortex retarders, this section analyzes some possible deviations of the measured vortex PSM and OTM from the predictions. To analyze the differences, aberrations and misalignment of the optics were added to the predictions. Previously, a theoretical discussion of aberrations in focused radial and azimuthal beams was developed by Biss and Brown [30]. We explore deviations of misalignment and aberrations for low NA systems by calculating the effect of various errors and comparing these to the measurements.

Figures 6, 7, and 8 compare the predicted PSM of an ideal vortex retarder to the measured PSM showing qualitative agreement but small differences, particularly the non-zero m03 and m30 elements. The measured PSMs are blurred compared to the predicted results; this is seen best in the m₁₁, m₁₂, m₂₁, m₂₂, elements. The comparison between predicted and measured MTF and PTF shows excellent qualitative agreement at low spatial frequencies. At high spatial frequencies, there is good qualitative agreement, but varies quantitatively.

In order to analyze the differences between measured and predicted results, we improved the calculations incorporating the ability to model aberrations in the optical system and the ability to misalign the center of the vortex retarder from the center of the optical axis of the system. Aberrations were included in the simulation by multiplying the PAM by a wavefront aberration identity matrix prior to taking the Fourier Transform of the PAM. The PSM matrix was then calculated as previously described using the modified PAM. The misalignment was simulated by shifting the center of the pupil function.

To evaluate the effect of aberrations on the PSM, we calculated the PSM including 0.2 waves of wavefront aberration for the coma and astigmatism. Figures 14 and 15 show the PSM for the case where there are no aberrations compared to the case where there is 0.2 waves of wavefront distortion.

Fig. 14. PSM for m=2 vortex retarder for ideal case (left) and with 0.2 waves of coma (right).

Fig. 15. PSM for m=2 vortex retarder for ideal case (left) and with 0.2 waves of astigmatism (right).

The impact of aberrations on the MTF and the PTF was investigated. Figure 16 shows the MTF and PTF for an m=2 vortex retarder with 0.1 waves of coma and astigmatism. Also shown in the figure are the values for the measured result and the aberration-free prediction. Figure 16 indicates that the 0.1 waves of coma and astigmatism have negligible impact on the MTF. However, 0.1 waves of coma has a significant impact on the PTF. The results suggest that adding 0.1 waves of coma partially explain the measurement results.

Fig. 16. MTF (top) and PTF(bottom) comparison between aberration free predicted, measured, predicted with 0.1 waves of astigmatism, and predicted with 0.1 waves of coma for an m=2 vortex retarder with horizontal linear input and no analyzer.

The use of vortex retarders requires careful alignment of the center of the vortex retarder to the optical axis of the system since misalignment reduces performance. The PSM was calculated for an m=2 vortex retarder with the center of the vortex shifted from the center of the optical axis of the system by 1% and 5% of the pupil radius and the results are shown in Fig. 17.

Figure 17 suggests that even a small amount of misalignment (1%) introduces measurable circular diattenuation. As the misalignment increases, the changes increase. At 5% misalignment, there is significant blurring of the PSF of all the elements, in addition to the circular polarization present.

Fig. 17. PSM for an m=2 vortex retarder shifted from the center by 1% (left), 5% (right).

6. Conclusions

This report addresses the creation of polarization vortex beams. The goal was to develop a simple method for producing vortex retarders for visible wavelengths, with a continuous fast axis, and for multiple vortex modes. The approach was to use photo-aligned liquid crystal polymers (LCP). The target was a half-wave retardance for 550nm wavelength. Elements produced were analyzed by measuring the space variant Mueller Matrix of each component. Our measurements demonstrated that the vortex retarders were half-wave plates with a continuous fast axis orientation. The viability of these components was assessed by determining the point spread matrix (PSM) and the optical transfer matrix (OTM) and comparing these to theoretical calculations. The agreement between the measured and predicted PSM was excellent. The major difference was the non-zero response in the m03 and m30 elements, indicating circular diattenuation. The OTM comparison between measurement and prediction demonstrated an excellent quantitative match at lower spatial frequencies and a good qualitative match at higher spatial frequencies. Measured results confirm that vortex retarders produced using photo-aligned LCP produce near theoretical performance in an optical system.

We also analyzed the impact of aberrations and vortex retarder misalignment. Our analysis showed that the introduction of 0.1 waves of astigmatism can cause circular diattenuation, and the introduction of the same amount of coma can partially explain the PTF observed in our measured results. Additionally, misalignments of the vortex retarder as small as 1% can cause measurable circular diattenuation.

References and links

1.	S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
2.	K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]
3.	L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 85, 5251–5253 (2001). [CrossRef]
4.	R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for radially polarized light beam,” Phys. Rev. Lett. 91, 23 (2003). [CrossRef]
5.	Totzeck, et al., “Polarizer device for generating a defined spatial distribution of polarization states,” US 2006/0028706, Feb. 9, 2006.
6.	A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements,” Opt Lett 29, 238 (2004). [CrossRef] [PubMed]
7.	V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002). [CrossRef] [PubMed]
8.	M. Born and E. Wolf, Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light , (Cambridge University Press, seventh ed., 1999). [PubMed]
9.	Y. Mushiake, K. Matzumurra, and N. Nakajima, “Generation of radially polarized optical beam mode by laser oscillation,” Proc. IEEE 60, 1107–1109 (1972). [CrossRef]
10.	T. Erdogan, K. G. Sullivan, and D. G. Hall, “Enhancement and inhibition of radiation in cylindrically symmetric, periodic structures,” J. Opt. Soc. Am. B 10, 391–398 (1993). [CrossRef]
11.	T. Erdogan, O. King, W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992). [CrossRef]
12.	S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990). [CrossRef] [PubMed]
13.	R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phy Lett. 77, 21 (2000) [CrossRef]
14.	R. Yamaguchi, T. Nose, and S. Sato, “Liquid crystal polarizers with axially symmetrical properties,” Jpn. J. Appl. Phys. 28, 1730 (1989). [CrossRef]
15.	A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007). [CrossRef]
16.	G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 11 (2007). [CrossRef]
17.	M. Stalder and M. Schadt, “Linear polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt Lett. 21, 23 (1996). [CrossRef]
18.	S. C. McEldowney, D. M. Shemo, R. A. Chipman, and P. K. Smith, “Creating vortex retarders using photoaligned liquid crystal polymers,” Opt. Lett. 33, 134–136 (2008). [CrossRef] [PubMed]
19.	M. Schadt, H. Seiberle, A. Schuster, and S. M. Kelly “Photo-induced alignment and patterning of hybrid liquid crystalline polymer films on single substrates,” Jpn. J. Appl. Phys. 34, L764–L767 (1995). [CrossRef]
20.	D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996). [CrossRef] [PubMed]
21.	R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: The azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994). [CrossRef] [PubMed]
22.	P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-Gauss beam,” J. Opt. Soc.Am. A 13, 962–966 (1996). [CrossRef]
23.	P. L. Greene and D. G. Hall, “Properties and diffraction of vector Bessel-Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998). [CrossRef]
24.	C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999). [CrossRef]
25.	K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express. 7, 77–87, (2000). [CrossRef] [PubMed]
26.	S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
27.	J. P. McGuire, Jr. and R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations I: Formulation and example,” J. Opt. Soc. Am A. 7, 9, 1614–1626 (1990). [CrossRef]
28.	Information on Axometrics polarimeter from http://www.axometrics.com/
29.	J. L. Pezzanitti and R. A. Chipman, “Mueller matrix imaging polarimeter,” Opt. Eng. 34, 6 (1995). [CrossRef]

OCIS Codes
(230.3720) Optical devices : Liquid-crystal devices
(230.5440) Optical devices : Polarization-selective devices
(260.1440) Physical optics : Birefringence
(260.5430) Physical optics : Polarization

ToC Category:
Optical Devices

History
Original Manuscript: March 21, 2008
Revised Manuscript: May 1, 2008
Manuscript Accepted: May 1, 2008
Published: May 6, 2008

Citation
Scott C. McEldowney, David M. Shemo, and Russell A. Chipman, "Vortex retarders produced from photo-aligned liquid crystal polymers," Opt. Express 16, 7295-7308 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7295

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References

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical vector beams," Opt. Express 7, 77-87 (2000). [CrossRef] [PubMed]
L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 85, 5251-5253 (2001). [CrossRef]
R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for radially polarized light beam," Phys. Rev. Lett. 91, 23 (2003). [CrossRef]
Totzeck, et al., "Polarizer device for generating a defined spatial distribution of polarization states," US 2006/0028706, Feb. 9, 2006.
A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements," Opt Lett 29, 238 (2004). [CrossRef] [PubMed]
V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002). [CrossRef] [PubMed]
M. Born and E. Wolf, Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, (Cambridge University Press, seventh ed., 1999). [PubMed]
Y. Mushiake, K. Matzumurra, and N. Nakajima, "Generation of radially polarized optical beam mode by laser oscillation," Proc. IEEE 60, 1107-1109 (1972). [CrossRef]
T. Erdogan, K. G. Sullivan, and D. G. Hall, "Enhancement and inhibition of radiation in cylindrically symmetric, periodic structures," J. Opt. Soc. Am. B 10, 391-398 (1993). [CrossRef]
T. Erdogan, O. King, W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992). [CrossRef]
S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990). [CrossRef] [PubMed]
R. Oron, S. Blit, N. Davidson, and A. A. Friesem, Z. Bomzon and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phy Lett. 77, 21 (2000). [CrossRef]
R. Yamaguchi, T. Nose, and S. Sato, "Liquid crystal polarizers with axially symmetrical properties," Jpn. J. Appl. Phys. 28, 1730 (1989). [CrossRef]
A. K. Spilman and T. G. Brown, "Stress birefringent, space-variant wave plates for vortex illumination," Appl. Opt. 46, 61-66 (2007). [CrossRef]
G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, "Efficient extracavity generation of radially and azimuthally polarized beams," Opt. Lett. 32, 11 (2007). [CrossRef]
M. Stalder and M. Schadt, "Linear polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt Lett. 21, 23 (1996). [CrossRef]
S. C. McEldowney, D. M. Shemo, R. A. Chipman, and P. K. Smith, "Creating vortex retarders using photoaligned liquid crystal polymers," Opt. Lett. 33, 134-136 (2008). [CrossRef] [PubMed]
M. Schadt, H. Seiberle, A. Schuster, and S. M. Kelly "Photo-induced alignment and patterning of hybrid liquid crystalline polymer films on single substrates," Jpn. J. Appl. Phys. 34, L764-L767 (1995). [CrossRef]
D. G. Hall, "Vector-beam solutions of Maxwell�??s wave equation," Opt. Lett. 21, 9-11 (1996). [CrossRef] [PubMed]
R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave equation: The azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994). [CrossRef] [PubMed]
P. L. Greene and D. G. Hall, "Diffraction characteristics of the azimuthal Bessel-Gauss beam," J. Opt. Soc.Am. A 13, 962-966 (1996). [CrossRef]
P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998). [CrossRef]
C. J. R. Sheppard and S. Saghafi, "Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation," Opt. Lett. 24, 1543-1545 (1999). [CrossRef]
K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical vector beams," Opt. Express. 7, 77-87, (2000). [CrossRef] [PubMed]
S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
J. P. McGuire, Jr. and R. A. Chipman, "Diffraction image formation in optical systems with polarization aberrations I: Formulation and example," J. Opt. Soc. Am A. 7, 9, 1614-1626 (1990). [CrossRef]
Information on Axometrics polarimeter from http://www.axometrics.com/
J. L. Pezzanitti and R. A. Chipman, "Mueller matrix imaging polarimeter," Opt. Eng. 34, 6 (1995). [CrossRef]

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