M1C.4.pdf
International Conference on Fiber Optics and Photonics © OSA 2012
Controlled Polarization Engineering by Holography Amit Kumar Singh*, Rakesh Kumar Singh Department of Physics, Indian Institute of Space Science and Technology (IIST) Thiruvananthapuram, Kerala, India-695547 *e-mail:
[email protected]
Abstract: We have demonstrated use of holography for controlled generation of polarized light. This permits easy generation of desired spatial polarization structure, of course under limitation of holography, in comparison to conventional polarization optics method. OCIS codes: (260.5430) Polarization; (090.1760) Computer holography
1. Introduction Polarization, which traditionally refers transverse nature of light, is subject of great importance due to various practical and theoretical applications [1]. Characterization of polarized light has drawn attention of researchers cutting across boundaries of disciplines. This characterization is performed by state of polarization (SOP) mapping with help of polarization ellipse or Poincare sphere methods either using complex fields or Stokes parameters [1]. Majority of investigations on polarization are limited to analysis part in spite of growing demand of engineered polarization in several practical applications ranging from microscopy to nano-photonics. Manipulation of polarization of light is performed using traditional polarization elements such as retarders, analyzers made of crystals. Liquid crystals devices are also being used for this purpose but these elements are sensitive to wavelength, temperature in addition to being fragile. With increasing applications of polarization in different fields of science, holography can be used as an effective technique for polarization engineering. Potential of holography for this goal lies with its capability to record and reconstruct complex field using intensity modulation [2-4]. This unique property of holography is exploited for controlled polarization engineering in this paper. This task is performed by recording a vectorial object in terms of intensity modulations for the orthogonal polarization components, and their subsequent reconstruction. 2. Theory Let us consider an off- axis vectorial object Ei ( ro ) , and a reference point Ri ( ro ) at the source plane, as show in Fig. 1. Interference of object and reference at transverse plane r, i.e. at focal plane, is given as [5]:
ESi (r) Ei (r) R i (r)
Fig.1: Schematic diagram for recording and reconstruction of polarization holograms
(1)
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International Conference on Fiber Optics and Photonics © OSA 2012
Here E is ( r ) is the resultant field of i th polarization component with i x , y and * stands for conjugate field. Terms Ei ( r ) and Ri ( r ) represent object and reference field at the focal plane r. Intensity modulation is given as [2-5]: 2
2
IiH (r) Ei (r) R i (r) Ei* (r )R i (r) Ei (r )R i* (r )
(2)
Here I iH represents intensity hologram of i th polarization component. Complex amplitude of the object encoded into intensity hologram is retrieved by illuminating the holograms with coherent light as shown in Fig. 1, and this field is given as [3]: IiR (r) R i IiH (r) (3) Note that we have removed central un-modulated term (DC) from the reconstruction as shown in Fig.1. This is performed to highlight the object under high value of DC. This becomes possible with help of digital processing. Spatial distribution of SOP is demonstrated using reconstructed complex amplitude of orthogonal components,
given by E( r ) Eox ei x Eoy e
i y
, where E
ox and
E0 y are amplitudes of orthogonal components. Terms x and
y stands for respective phase factors. The Stokes parameters (SP) is defined using reconstructed complex fields and mathematically given as: S0 (r ) E 02x (r ) E 02 y (r) S1 (r ) E 02x (r ) E 02 y (r) S2 (r ) 2E 0 x (r )E 0 y (r ) cos (r) S3 (r ) 2E 0 x (r )E 0 y (r ) sin (r )
(4) (5) (6) (7)
Here S 0 ( r ), S1 ( r ), S 2 ( r ) and S3 ( r ) are Stokes parameters at spatial position r. These Stokes parameters play important role in characterization of polarized light by denoting the SOP of light on the Poincare sphere. Other method of SOP mapping is based on the polarization ellipse, and its mathematical form is given as: E 2x (r) E 2 y ( r ) 2E x ( r ) E y ( r ) 2 cos sin 2 , where y x (8) 2 E ( r ) E ( r ) E 0 x (r ) E 0 y (r ) 0x 0y 3. Results and Discussions To demonstrate usefulness of our technique, we have considered a star shaped polarized object as shown in Fig. 1. The amplitude transmittance of the x and y polarized component of the object is assumed to be same with phase delay of δ=π/4 and π/2 for two cases. The CGHs for vectorial objects are made digitally using scheme shown in Fig. 1. The coherent illuminations of intensity holograms of the orthogonal polarization components reconstruct the vectorial field. Reconstructed amplitude distribution of a orthogonal polarization components and its corresponding SOP are shown in Fig. 2 with the help of polarization ellipse and Poincare sphere for / 4 . Amplitude distribution of one of the orthogonal polarization component is shown in Fig. 2a. Similar amplitude structure also exists for other polarization component. Result in Fig. 2(b) describes SOP, for / 4 which is elliptically polarized. The SOP on the Poincare sphere is represented in Fig. 2c by a pink dot on the Poincare sphere. Other case is demonstrated in Fig. 3 for circular polarization with / 2 . Result in Fig. 3(a) represents amplitude distribution of the object and Figs. 3b & 3c represent its SOP with help of polarization ellipse and Poincare sphere. Fig. 3(b) is result of / 2 for circularly polarized and its SOP is shown by pink dots on one of the pole of Poincare sphere. Note that we have demonstrated SOP of one part of the reconstructed field after removal of DC term from the reconstructed field, and SOP of conjugate object which remains to be same as the original object is not presented.
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(a)
(b)
International Conference on Fiber Optics and Photonics © OSA 2012
(c)
Fig. 2: (a) Reconstructed amplitude distribution; SOP mapping using (b) polarization ellipse, (c) Poincare sphere
(a)
(b)
(c)
Fig. 3: (a) Reconstructed amplitude distribution; SOP mapping using (b) polarization ellipse, (c) Poincare sphere
4. Conclusions In this paper, we have demonstrated application of holography for controlled generation of desired polarization structure, and results are presented for two difference cases. This technique is useful to generate polarization structure in any form of the object with limitation of holography, which is not the case with traditional polarization elements. Acknowledgement: Part of this work is carried out under IIST-Fast track research project scheme. 5. References [1] D. H. Goldstein, Polarized light (CRC Press, 2010). [2] L. Nikolova and P. S. Ramanujjam, Polarization Holography (Cambridge University Press, New York, 2009). [3] R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Vectorial coherence holography” Opt. Express 19, 11558-11567 (2011). [4] R. K. Singh, D. N. Naik, H. Itou, Y. Miyamoto, and M. Takeda, “Stokes holography” Opt. Lett.37 , 966-968 (2012). [5] Joseph W. Goodman, Introduction to Fourier Optics (Roberts & Company 2007)
