Open Access
Add to BibSonomyAbstract
We have made an ultra-thin (~2.26 µm) f/2.1 lens based on the Pancharatnam phase effect using the polarization holography alignment technique. This lens exhibits a continuous phase profile, high efficiency (>97%), and is switchable from having a positive focal length to a negative one by changing the handedness of input circularly polarized light. We analyzed its optical performance and simulated it as a gradient index lens for further comparison, and to discuss its bandwidth limitation. The conditions required for improving the performance and its low-cost fabrication method is discussed. Because of the nature of Pancharatnam devices and the demonstrated fabrication method, these results are applicable to a wide size range.
© 2015 Optical Society of America
1. Introduction
Pancharatnam or geometric phase refers to the phase shift acquired by an electromagnetic wave that undergoes a continuous sequence of polarization state transformations following a closed path in the space of polarization states [1, 2]. Unlike conventional phase or amplitude gratings, Pancharatnam phase devices operate by locally modifying the polarization state of light waves passing through them. Their unique optical properties have been utilized recently in applications including wide-angle non-mechanical beam-steerers, imaging spectro-polarimeters, and polarization independent displays [3–5]. A particularly interesting application is an optical lens, which can be made by modulating the Pancharatnam phase to provide an appropriate profile across an aperture. This is accomplished by fabricating a half-wave retarder that has its optics axis in the plane of the film with the azimuthal angle (β), which is spatially varying along the radial direction from the concentric center. Specifically, the phase of circularly polarized light exiting a Pancharatnam phase device is related to the angle β between the optic axis of a half-wave retarder with respect to a fixed lab axis. If β changes in a continuous manner, the phase Γ changes continuously with Γ = 2β. A Pancharatnam lens can be fabricated by meeting the condition Γ = π r2/ λ f, which defines a parabolic phase profile for a lens with radius r and focal length f at a designed wavelength λ. A lens fabricated in this way can change its sign (e.g. from a ‘positive’ lens to a ‘negative’ lens) when the handedness of the circularly polarized light incident on it is changed.
Previous investigators have reported fabricating these lenses from different methods, for example a micro-rubbing technique [6], a space-variant subwavelength dielectric grating [7, 8], and a holographic alignment technique to create the desired phase profile [9–11]. Oh and Escuti [12–14] showed their lenses have high efficiency and a continuous phase profile. In some of these references, lenses based on Pancharatnam phase are referred to as Fresnel lenses, and their phase profile is described to be similar to a diffractive lens. However, as shown above, a Pancharatnam lens provides a continuous phase profile that does not have the same limitations as a Fresnel lens or diffractive lens [15]. A key feature of this type of lens, due to its continuous phase profile, is that its quality is not reduced by diffraction effects as the phase gradient becomes high. Therefore, ultra-thin lenses with exceptional quality and easy fabrication are possible over a wide range of aperture dimensions.
In this paper, we provide the details of the construction of a Pancharatnam lens with 5 mm diameter and a measured 8.5 mm focal length, as well as analysis of its optical performance. We characterize its point spread function (PSF) and modulation transfer function (MTF) and show images taken with different apertures. Also, we simulate this lens as a gradient index (GRIN) lens [16] in ZEMAX®, and compare it with the experimental results. This ultra-thin lens exhibits diffraction limited performance with high image quality when the aperture is limited to 2 mm. The effect of larger apertures is also quantified and analyzed. Finally, the lens phase profile and spectral bandwidth of illumination are discussed in terms of possible future improvements.
2. Lens fabrication
Our Pancharatnam lenses are fabricated using a method similar to that used by Oh and Escuti [12]. It involves using a photo-alignment layer that defines the orientation of local optic axis of a subsequently fabricated half-wave retarder using reactive mesogen (RM), a type of liquid crystal (LC) monomers.
In the first step, a photo-alignment layer is applied to a thin-glass substrate and exposed to provide the desired half-wave retarder orientation. In this work, we choose brilliant yellow (BY) as the alignment layer material. To prepare photo-alignment films, BY material was dissolved in DMF solution with fixed concentration of 1.5% by weight. The solution was spin-coated onto the substrate at 3000 rpm for 30 seconds. Then the films were dried at 120°C for 30 minutes in the oven. This layer is a thin photo-sensitive film showing strong orientational response to the local polarization axis of light.
The polarization holography setup, based on a modified Mach-Zehnder interferometer, to expose the photo-alignment layer is shown in Fig. 1(a). The 457 nm laser beam was firstly expanded to 5mm in diameter after passing the first beam expander (BE1), and then split into two optical paths by the first beam splitter (BS1). The beam in Path 1 is circularly polarized by a quarter-wave plate (QWP) but it is still in 5mm diameter. The beam in Path 2 was expanded to ~41.7 mm in diameter after passing the second beam expander (BE2) and circularly polarized by QWP. It is then focused by the ‘template’ lens and interferes with the reference beam from Path 1 after a beam combiner (BS2). The resulting interference pattern provides linearly polarized light whose polarization axis is a function of the radial distance from the center of the pattern with gradually decreased grating pitch to the edge [5]. To visualize the resulting interference pattern, a linear polarizer is placed in front of a CCD camera with a 20x objective lens that is focused on the plane as shown in Fig. 1(b). The substrate with the photo-alignment layer is then placed at the position where two interfered beams have both 5 mm diameter (with ND filter, CCD, and LP2 removed in Fig. 1(a)), and exposed ~10 minutes. The total power after BS2 is about 5.38 mW with 2.70 mW from Path 1 and 2.68 mW from Path 2.
Fig. 1 (a) The schematic of full optical setup for holographic exposure. (ND: neutral density; BE: beam expander; M: magnification; LP: linear polarizer; BS: beam splitter; QWP: quarter-wave plate;) (b) Direct observation of interference pattern from the CCD coupled with 20x objective lens.
In choosing the ‘template’ lens, there are two considerations that need to be taken into account. One is that the focal length of the template lens must be long enough so that the glass substrate coated by photo-alignment layer can be placed between BS2 and the focal plane of the ‘template’ lens, at a distance (L) from the focal plane of the ‘template’ lens, where L = fp*λd/λexp (fp is the desired focal length of the resulting Pancharatnam lens in the wavelength of light λd used in imaging system, and λexp is the wavelength of light used in the exposure system). Also, the diameter (Dt) of the ‘template’ lens needs to be met with the condition Dt > Dp*ft /L (where Dp is the diameter of the desired Pancharatnam lens, and ft is the focal length of the ‘template’ lens). With the goal of fabricating a lens that has 5 mm diameter and 8.5 mm focal length, when illuminated by 632 nm light, we used an aspherical lens with 50 mm diameter and 100 mm focal length purchased from Thorlabs, Inc. (AL50100-A) as the ‘template’ lens.
Next in the process, reactive mesogens are coated on the BY layer, and aligned by BY molecules. For high efficiency, the retardation of the RM film must be equal to half of testing wavelength. However, it is not possible to deposit a single layer thick enough to provide the required retardation because the monomers tend to align out of the plane of the film to reduce the high elastic energy [17]. To overcome this, we coat multiple thin RM layers. Each layer is aligned by the one beneath it and then polymerized with the desired orientation. In this work, we use RM257 (Δn = 0.14) from Merck, dissolved in Toluene solution with the concentration of 10% by weight. The photo-initiator (Iragcure 651) was also added so that its concentration was 5% of the RM by weight. The solution was then spin-coated on the BY layer at 3000 rpm for 30 seconds, immediately after the holographic exposure. With exposure to UV lamp (spectral peak at 365 nm and power density in 2 mW/cm2) ~7 minutes in N2 rich environment, these LC monomers then resulted in a cross-linked polymer. This layer-by-layer process is continued until the desired half-wave retardation is reached as determined by the following test, in which we set up the red laser (632 nm wavelength, and 500 µW power) with a circular polarizer and let this circularly polarized light pass through our sample. Depending on the handedness of the circular polarizer, the sample will show optical property of a ‘negative’ lens or ‘positive’ lens. As a ‘negative’ lens, after the first layer of RM257 is deposited, the retardation is much less than a half-wave and the intensity of the zero-order spot was high. We are also able to measure the intensity of the zero-order spot after each RM257 layer is coated, which will be the minimum when the sample reaches the effect of half-wave plate. With our spin conditions, we found this occurs when the 8th layer of RM257 was coated, and the diffraction efficiency can be >97%. Using the birefringence of RM257, this thickness should be about 2.26 microns.
3. Results and discussions
3.1. Characterization of phase profile and image quality
Figure 2 shows the resulting Pancharatnam lens between crossed polarizers when viewed with a microscope with 632 nm light. The black rings correspond to the regions where the spiraling optics axis is aligned with the polarizer or analyzer axis. Therefore, the director angle β changes by 90 degrees between each dark ring. We define the grating pitch, to correspond to the director angle changing by 180 degrees. Some point defects are observed in Fig. 2(a), which may come from the dust in the spin coating process or particles not being totally filtered from the RM solution. Even though the exposure of the BY was done on an optical table with vibration damping, some slight air flow can cause uneven fringes of polarization grating. Line pairing is sometimes observed, depending on the exact method of focusing the camera image. Also, microscope pictures at higher magnification around in different radius are shown in Fig. 2. At the outmost edge of the lens, the pitch is measured to be ~2.2 µm.
Fig. 2 (a) The Pancharatnam Lens observed under an interference filter using polarized microscope, with magnified pictures (b) around the center; (c) at 0.5 mm radius (d) at 1 mm radius (e) at 1.5 mm radius (f) at 2 mm radius; (g) close to the edge. Figures (b) and (g) are from the areas designated by boxes in figure (a)
A half-wave retarder between crossed polarizers will have a transmitted intensity proportional to sin2 (2β), where β is the azimuthal angle of the optic axis in the plane of the film. For a Pancharatnam phase device, the phase of circularly polarized light exiting the film is Γ = 2β, so we can characterize the phase profile of our lens by obtaining β as a function of r from the intensity of the device between crossed polarizers as a function of r by using the expression:
where Γο and Io are the initial phase and the maximum intensity of light, respectively.By using the 2nd-order polynomial fit to the effective optical path difference (which will be designated as OPD' to differentiate from a physical OPD) of our Pancharatnam lens measured from Fig. 2(a), we found out it has parabolic shape as shown in Fig. 3, which can be written as:
The measured phase profile can be compared with that of the ‘template’ lens whose ZEMAX® file can be directly obtained from Thorlabs Inc., also shown in Fig. 3. It is key to realize here that the phase profile of the Pancharatnam lens would be the same as that of any ‘template’ phase profile used in the holographic exposure of the photo-alignment layer.Fig. 3 The comparison of OPD of aspherical lens used in holographic exposure and the Pancharatnam lens with polynomial fit shown in red dash line.
In order to obtain the measured PSF, a laser beam with 632 nm wavelength was expanded, and passed through the ND filter, circular polarizer, a 2 mm diameter pinhole, and the Pancharatnam lens, sequentially. A CCD (Aptina Imaging Corp., MT9J001) coupled with 20x objective lens was used for recording the PSF as shown in the insert of Fig. 4(a). We took three different cross sections from each of three measurements of the PSF resulting in 9 curves. We normalized these measured curves in one figure so that the area below each curve is 1, and then obtained an average PSF curve from them as shown in Fig. 4(a). The bars represent the range of values obtained from the 9 curves. The corresponding MTF, shown in Fig. 4(b) is obtained from the average PSF curve, and the bars show the range of values obtained from the 9 PSF curves, which were obtained by performing a Fourier transform of the measured PSF curve normalized to a maximum contrast of 1 in a simple Matlab® program. To compare the measured PSF of our lens with that simulated assuming a parabolic phase profile, we use the surface type of Gradient 1 of GRIN lens in ZEMAX®, with the refractive index expressed by:
where no, nr1, nr2 are the coefficient of constant, linear and quadratic term, respectively.Fig. 4 (a) Characterization of PSF and (b) MTF of Pancharatnam lens under 2 mm aperture and 632 nm wavelength. The bars show the range of data points acquired from three cross sections of each of 3 PSF measurements.
Then the effective OPD of the Pancharatnam lens can be associated with the refractive index of the GRIN lens by using its definition:
where m is the number of spiral turns of the optic axis in the plane of the Pancharatnam lens; n and d are the index of refraction and thickness of the GRIN lens, respectively. The constant term of refractive index of GRIN lens can be calculated from R2/2df, where r< = R = 2.5mm in this work. We set the thickness of the GRIN lens to be equal to that of our Pancharatnam lens (d~2.26 µm). Here we used Eq. (3) to specify n(r) with the linear term dropped and the quadratic term obtained from Eqs. (2) and (4).For the simulated PSF, we also normalized the curve so that the area under the curve is equal to 1 as shown in Fig. 4 (a). For a 2 mm aperture (f/4.3) under 632 nm wavelength, the airy radius in theory is expected to be 3.3 µm from 1.22*λ* f/#. The value from simulation and experimental result is 3.3 and ~3.6 µm, respectively. The difference of spatial frequency where MTF is 50% is ~20 cycles/mm between simulation and experimental result. The real lens is comparable to the simulated one at the diffraction limit, and therefore can provide a good image quality. It is seen that a parabolic phase profile works well for the 2 mm aperture. For a 4mm aperture (f/2.1), the PSF curves for the parabolic phase profile of GRIN lens simulated in ZEMAX®, and from the average of our measured curves are shown in Fig. 5(a). The airy radius in theory is expected to be 1.6 µm. The value from simulation and experimental result is 1.7 and ~2.1 µm, respectively. However, the peak height of both simulation and experimental result are 30% less than that of diffraction-limited one in Fig. 5(a). The corresponding MTF curves are shown in Fig. 5(b). Although the average of measured data has a similar MTF to the simulated one (as expected, since our lens has the parabolic phase profile) as shown in Fig. 3, it can be seen that both curves are far away from that expected for a diffraction-limited lens.
Fig. 5 (a) Characterization of PSF and (b) MTF of Pancharatnam lens under 4 mm aperture and 632 nm wavelength. The bars show the range of data points acquired from three cross sections of each of 3 PSF measurements.
This same effect of observing excellent image quality for an f/4.3 lens, which is degraded as the f/# is lowered, can be seen in Fig. 6. These images were acquired with our Pancharatnam lens from a back illuminating 1951 USAF resolution test chart using a white LED coupled with a 632 nm interference filter which has a 11.5 nm full width at half maximum (FWHM). Because we have shown the phase profile of our Pancharatnam lens accurately follows the phase profile of the ‘template’ lens, the image degradation shown in Fig. 6 is related to the parabolic phase profile of the ‘template’ lens. If a ‘template’ lens was available with a more accurate phase profile of a diffraction-limited lens, the Pancharatnam lens would provide the diffraction-limited MTF shown in Fig. 5(b) and improved image quality over that shown in Fig. 6.
Fig. 6 This Pancharatnam lens was attached to CCD (a) with 2 mm aperture; (b) with 3 mm aperture; (c) with 4 mm aperture; (d) with 5 mm aperture for imaging tests using 11.5 nm FWHM of interference filter.
3.2. Bandwidth limitation of Pancharatnam lenses
Unlike a diffractive lens, the effect of wavelengths for a Pancharatnam lens, different from the design value, is not to diffract the light to undesired orders, but to change the focal length of the lens. Therefore, there is a strong wavelength dependence on the focal length of the lens. Assuming a parabolic phase profile, we can combine Eqs. (2) and (4) while considering Γ = 2β to obtain the focal length of a Pancharatnam lens to be:
Because of this effect of the wavelength of light on the focal length, the image quality is expected to be strongly affected by the spectral bandwidth of the light being imaged by a Pancharatnam lens.To find a metric of this degradation, we first found the spatial frequency corresponding to an MTF value of 0.5 for a diffraction-limited lens at its design wavelength; and then changed the wavelength away from the design wavelength to the value that caused the spatial frequency corresponding to the MTF value of 0.5 to be halved. We did this for changes in the wavelength greater than and also less than the design wavelength and took the difference between these two wavelengths as our “MTF/2 bandwidth limit”. The MTF/2 bandwidth limit (in nm) was calculated for the specific cases of a focal length of 8.5 mm, and for diameters of 1, 2, and 4mm, and for a design wavelength of 632 nm. Because the image quality of a Pancharatnam lens is expected to be related to maximum deflection angle at the lens periphery [18], these results can be generalized to being representative of lens with f/8.5, f/4.3, and f/2.1, respectively, and are shown in Fig. 7.
Fig. 7 The MTF/2 bandwidth limit (as defined in the text) vs. f-number of a diffraction-limited Pancharatnam lens.
4. Conclusion
We have shown that a thin film Pancharatnam lens can be fabricated using a ‘template’ lens to define its phase profile. With this technique, we have made a low f/# lens with high quality which is only a few microns thick. We further have shown that diffraction-limited performance can be expected for low f/# lenses if an appropriate ‘template’ lens is used. Finally, we clarified the effect of bandwidth on the optical quality of these lenses.
Acknowledgment
We gratefully acknowledge financial support from Intel Corporation.
References and links
1. S. Pancharatnam, “Generalized theory of interference, and its application,” Proc. Ind. Acad. Sci. A 44(5), 247–262 (1956).
2. M. Martinelli and P. Vavassori, “A geometric (Pancharatnam) phase approach to the polarization and phase control in the coherent optics circuits,” Opt. Commun. 80(2), 166–176 (1990). [CrossRef]
3. H. H. Cheng, A. Bhowmik, and P. J. Bos, “Large angle image steering using a liquid crystal device,”SID Symp. Dig. Tech. Pap. 45(1), 739–742 (2014). [CrossRef]
4. M. J. Escuti, C. Oh, C. Sánchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207 (2006). [CrossRef]
5. M. J. Escuti and W. M. Jones, “Polarization independent switching with high contrast from a liquid crystal polarization grating,” SID Symp. Dig. Tech. Papers 37, 1443–1446 (2006). [CrossRef]
6. M. Honma and T. Nose, “Liquid-Crystal Fresnel Zone Plate Fabricated by Microrubbing,” Jpn. J. Appl. Phys. 44(1A), 287–290 (2005). [CrossRef]
7. E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Polarization dependent focusing lens by use of quantized Pancharatnam–Berry phase diffractive optics,” Appl. Phys. Lett. 82(3), 328–330 (2003). [CrossRef]
8. Y. Gorodetski, G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Optical properties of polarization-dependent geometric phase elements with partially polarized light,” Opt. Commun. 266(2), 365–375 (2006). [CrossRef]
9. T. Todorov, L. Nikolova, and N. Tomova, “Polarization holography. 2: Polarization holographic gratings in photoanisotropic materials with and without intrinsic birefringence,” Appl. Opt. 23(24), 4588–4591 (1984). [CrossRef] [PubMed]
10. G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98(12), 123102 (2005). [CrossRef]
11. H. Sarkissian, S. V. Serak, N. V. Tabiryan, L. B. Glebov, V. Rotar, and B. Y. Zeldovich, “Polarization-controlled switching between diffraction orders in transverse-periodically aligned nematic liquid crystals,” Opt. Lett. 31(15), 2248–2250 (2006). [CrossRef] [PubMed]
12. C. Oh, “Broadband Polarization Gratings for Efficient Liquid Crystal Display, Beam Steering, Spectropolarimetry, and Fresnel Zone Plate,” Ph. D. Thesis, North Carolina State University (2009).
13. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008). [CrossRef] [PubMed]
14. C. Oh and M. J. Escuti, “Achromatic polarization gratings as highly efficient thin-film polarizing beamsplitters for broadband light,” Proc. SPIE 6682, 668211 (2007). [CrossRef]
15. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13(11), 2219–2231 (1996). [CrossRef]
16. E. Hecht, Optics, 2nd ed. (Addison Wesley, 1987).
17. R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 021701 (2007). [CrossRef] [PubMed]
18. C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76(4), 043815 (2007). [CrossRef]
