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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 10 — May. 10, 2010
  • pp: 10805–10812
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Real-time measurement of space-variant polarizations

Moti Fridman, Micha Nixon, Eran Grinvald, Nir Davidson, and Asher A. Friesem  »View Author Affiliations


Optics Express, Vol. 18, Issue 10, pp. 10805-10812 (2010)
http://dx.doi.org/10.1364/OE.18.010805


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Abstract

A novel configuration for real-time measurement of space-variant polarizations is presented. The experimental results reveal that the full state of polarization at each location within the beam can be accurately obtained every 10msec, limited only by the detection camera frame rate. We also present a more compact configuration which can be modified to determine the real-time wavelength variant polarization measurements.

© 2010 Optical Society of America

1. Introduction

In recent years there has been considerable activity in applying optical polarimetry to a range of applications such as ellipsometry, bioimaging, and imaging polarimetry. A commonly used method is to measure the time-dependent signal of a beam emerging from a rotating quarter-wave plate followed by an analyzer and then to determine the polarization state by Fourier analysis [1

1. G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16, 1168–1174 (1999). [CrossRef]

]. Increasing demands for faster and simpler methods has led to the development of the four-channel polarimeter [2–4

2. M. Mujat and A. Dogariu, “Real-time measurement of the polarization transfer function,” Appl. Opt. 40, 34–44 (2001). [CrossRef]

]. In this method a beam is split into four channels and each one is analyzed by use of different polarization optics. While this is being carried out, the realtime polarization state is calculated from the measured intensities. Unfortunately, this method requires multiple components and can only measure beams with uniform polarization [5

5. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 112, 455–546 (2004).

]. Recently, a new approach to polarimetry that makes use of a polarization grating composed of a space-variant polarizer was proposed [6

6. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]

], studied theoretically and experimentally demonstrated by use of sub-wavelength dielectric gratings [7–9

7. Y. Gorodetski, G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Space-variant polarization manipulation for far-field polarimetry by use of subwavelength dielectric gratings,” Opt. Lett. 30, 2245–2247 (2005). [CrossRef] [PubMed]

]. Another technique involves the use of a calcite crystal for splitting the polarization into two orthogonal components each of which is directed to a different detector, so as to obtain real-time measurement [10–12

10. J. Kim and D. E. Kim, “Measurement of the degree of polarization of the spectra from laser produced recombining Al plasmas,” Phys. Rev. E. 66, 017401 (2002). [CrossRef]

]. However, in this technique the full state of polarization is not measured. Accordingly, these and present commercial polarization measurement systems are limited to polarizations that are uniform in space and/or change very slowly in time. They are based on several sequential measurements with mechanical rotation of polarization elements in between measurements, and are hence expensive, complicated and slow [13–16

13. J. B. Kortright, M. Rice, Z. Hussain, H. A. Padmore, A. Adamson, W. R. A. Huff, A. T. Young, E. J. Moler, S. A. Kellar, R. X. Ynzunza, F. J. Palomares, H. Daimon, E. D. Tober, and C. S. Fadley, “Polarization measurement and vertical aperture optimization for obtaining circularly polarized bend-magnet radiation,” Rev. Sci. Instrum. 67, 3363–3363 (1996). [CrossRef]

].

There has been a continuous and increasing interest in measuring the space-variant polarization (i.e. polarization that can change from point to point in an arbitrary way) of more complex light beams in real time. Examples for such beams are either radially and azimuthally polarized beams that are useful for microscopy [17

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

], material processing [18

18. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D: Appl. Phys. 33, 1817–1822 (2000). [CrossRef]

, 19

19. K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603–2607 (2006). [CrossRef]

], trapping and acceleration of particles [20

20. M. O. Scully, “A simple laser linac,” Appl. Phys. B 51, 238–241 (1990). [CrossRef]

, 21

21. E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46, 6640–6653 (1992). [CrossRef] [PubMed]

] and laser light amplifications [22

22. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28, 807–809 (2003). [CrossRef] [PubMed]

], or randomly polarized beams that are exploited for polarization encryption [23]. Unfortunately, until now no simple and accurate solutions to realize such measurements were presented.

Here we present novel configurations that can indeed perform space-variant polarization measurements in real time. They are based on splitting the input beam into several parallel beams, each with a different polarization direction, using a single beam displacing crystal. These beams are then simultaneously detected and measured with a single CCD camera and the space variant polarization field is determined in real time, thus allowing measurements of polarizations which fluctuate in time. Moreover, these new configurations can be adapted to also measure wavelength-variant polarization in real time.

2. Experimental configuration and principles of operation

Our configuration for real-time space-variant polarization measurements is schematically presented in Fig. 1. It is comprised of a partially reflecting mirror (PR), a high reflecting mirror (HR), a quarter wave plate (QWP) and a calcite beam displacer. The input beam is split vertically into three parallel beams using the partially reflecting mirror and the high reflecting mirror, both oriented at a small angle with respect to the input beam in order to minimize variations of polarizations due to the reflective dielectric layers of the mirrors. The first beam, denoted as I, has the same polarization as the input beam. The second beam, denoted as II, propagates through the QWP that is oriented at 22.5° from the main axes of the calcite beam-displacer. The third beam, denoted as III, propagates three times through the QWP. These three beams then propagate through the calcite beam-displacer which horizontality splits them into their ordinary and extraordinary polarization components, to obtain six output parallel beams with two in each row that are detected by a CCD camera. Lenses can be added after the polarization elements to adjust the size of each beam to that of the CCD camera pixels and to compensate for diffraction due to the difference in propagation distances between the different beams.

Fig. 1. Configuration for real-time space variant polarization measurements. HR - high reflection mirror, QWP - quarter wave plate and PR - partial reflection mirror.

The six different beams are detected within each frame of the CCD camera, from which we can determine the full polarization state of each point in the beam. Specifically, we obtain the four Stokes parameters [24

24. E. Hecht, “Optics” (Addison Wesley, San Francisco, California) 2002, Chap. 8.

] at each point in the beam, as

S0(x,y)=I1(x,y)+I2(x,y),
(1)
S1(x,y)=I1(x,y)I2(x,y),
(2)
S2(x,y)=2(aI3(x,y)+bI5(x,y))2S0(x,y)S1(x,y),
(3)
S3(x,y)=2(aI3(x,y)bI5(x,y)),
(4)

where I1(x,y) and I2(x,y) are the intensity distributions of the two beams in the top row I, I3(x,y) and I4(x,y) the intensity distributions of the beams in the middle row II, and I5(x,y) and I6(x,y) the intensity distributions of the beams in the lower row III. The terms a and b are normalization constants obtained from

I1(x,y)+I2(x,y)=a(I3(x,y)+I4(x,y))=b(I5(x,y)+I6(x,y)).
(5)

The normalization constants can also be spatially dependent to account for spatial non-uniformities in the optical element and in the CCD sensitivity. The parameter S0(x,y) denotes the incident input intensity distribution, while S1(x,y), S2(x,y) and S3(x,y) specify the state of polarization at each point. Note that measurement of all four Stokes parameters fully characterize also partially polarized and unpolarized light [24

24. E. Hecht, “Optics” (Addison Wesley, San Francisco, California) 2002, Chap. 8.

].

Equations (1)–(4) are valid under the following conditions. First,the locations of corresponding points in the six intensity distributions must be accurately known. Second, the sum of the intensities at every point in the two intensity distributions in each row should be normalized to be the same using the normalization factors a and b. Finally, inaccuracies in the QWP and undesired polarization changes due to dielectric layers of mirrors, misalignments, etc., can be calibrated with input beams of known polarizations, and compensated for, so the total errors can be reduced to less than 1%.

3. Experimental results

We started our experiments with input beams that have specific known polarizations - linear polarization at horizontal orientation, linear polarization at 45° orientation, and circular polarization. The linear polarization at horizontal orientation was obtain by passing an unpolarized doughnut shaped beam through a thin film polarizer, the linear polarization at 45° orientation by passing the beam through a thin film polarizer and half wave plate, and the circular polarization by passing the beam through a thin film polarizer and a quarter wave plate. Representative experimental intensity distributions of the six output beams for each input polarization are shown in Fig. 2. Figure 2(a) shows the output intensity distributions for a horizontal linearly polarized input beam, Fig. 2(b) for a 45° linearly polarized input beam, and Fig. 2(c) for a circularly polarized input beam. Using Eqs. (1)–(4), we determined the Stokes parameters for these three input beams. The results are presented in Fig. 3. As expected, for the horizontal linearly polarized input beam most of the light is indeed observed in S1, as shown in Fig. 3(a). For the 45° linearly polarized input beam most of the light is indeed observed in S2, as shown in Fig. 3(b). For the circularly polarized input beam most of the light is indeed observed in S3, as shown in Fig. 3(c). We determined that undesired residual intensities in the other Stokes parameters in our results were less than 1%, indicating that our configuration indeed operates effectively.

Fig. 2. Representative experimental intensity distributions of the six output beams for different polarized input beams. (a) For a horizontal linearly polarized input beam; (b) for a 45° linearly polarized input beam; (c) for a circularly polarized input beam.

To illustrate that our configuration can measure light distributions with spatial variations of polarization, we first transformed the linearly polarized beam into either radial or azimuthal polarization by means of a space variant polarization retarder [15

15. M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]

], and used it as the input beam. Then, we detected the six intensity distributions from which we determined the Stokes parameters for the entire light distribution of the beam simultaneously with our configuration. The calculated and experimental Stokes parameters as well as the corresponding state of polarization of the input beams are presented in Figs. 4 and 5. Figure 4 shows the results for the radially polarized input beam. Figure 4(a) the calculated Stokes parameters, Fig. 4(b) the experimental Stokes parameters, and Fig. 4(c) the full polarization state indicated by the arrows which represent the main axis of the local polarization ellipse. As evident from Fig. 4(c), the input beam is indeed radially polarized. The corresponding results for an azimuthally polarized input beam are shown in Fig. 5. These results clearly indicate that the input beam is indeed azimuthally polarized. There are some differences between the calculated and experimental Stokes parameters, and we attribute the differences to imperfections in the polarization distribution of the input beam [15

15. M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]

].

Fig. 3. Stokes parameters determined from the experimental results shown in Fig. 2. (a) For a horizontal linearly polarized input beam; (b) for a 45° linearly polarized input beam; (c) for a circularly polarized input beam.
Fig. 4. Calculated and experimental Stokes parameters and full polarization state at each point for a radially polarized input beam.(a) Calculated Stokes parameters;(b) experimental Stokes parameters; (c) arrows representing the resulting main axis of the local polarization ellipse.

Our configuration can also measure the space-variant polarization of an input beam as it varies with time. To illustrate this we lunched a radially polarized input beam into a large mode area (LMA) fiber amplifier with numerical aperture of 0.07, core diameter of 22μm and cladding diameter of 125μm. At time t = 0, the amplifier was turned on and we measured the Stokes parameters and the corresponding states of polarization at each point of the amplified beam as they varied with time due to thermal and stresses changes in the fiber amplifier. The measurements rate was limited by the simple CCD camera that was used to 10msec, but it can be greatly increased by resorting to faster cameras. The results of the state of polarization as a function of time for three representative points in the amplified beam are presented in Fig. 6(a) on a normalized Poincar sphere [24

24. E. Hecht, “Optics” (Addison Wesley, San Francisco, California) 2002, Chap. 8.

]. The full results for the four Stokes parameters at each point in the beam as a function of time can be seen in a movie online at the supplementary materials. A single frame from this movie is presented in Fig. 6(b).

Fig. 5. Calculated and experimental Stokes parameters and full polarization state at each point for a azimuthally polarized input beam.(a) Calculated Stokes parameters;(b) experimental Stokes parameters; (c) arrows representing the resulting main axis of the local polarization ellipse.
Fig. 6. (Media 1) The trajectory of the state of polarization on the Poincar sphere. (a) The trajectory of three representative points (marked by crosses in the inset) in a radially polarized amplified beam as a function of time after the amplifier is turned on. For each trajectory 200 equally spaced measurements taken over 25sec are plotted. (b) Single frame from a movie presents all four Stokes parameters and the full polarization state on the the Poincar sphere as a function of time.

4. Compact configuration

The configuration presented in Fig. 1 is relatively bulky and can be used over a limited wavelength range because of the relatively narrow spectral bandwidths of the QWP and dielectric mirrors. Such a limitation can be overcome by resorting to the configuration depicted in Fig. 7. Here, the mirror for splitting the input beam and the QWP are replaced by a single parallel plate that is partially coated with 100% reflecting tungsten layer (HR) on the front surface and with 90% reflecting tungsten layer (PR) on the other surface (approximately 30nm thick in the visible regime). The input light beam propagates through the un-coated area of the plate and bounces inside back-and-forth, resulting in a multiplicity of parallel beams at the output, where only the outputs from the odd reflections need to be used.

Using rigorous coupled wave analysis we calculated that for a fused silica plate and silver coatings, an angle of 33° will result in a phase difference of λ/16 between the S and P polarizations. Thus, after four reflections the difference between the S and P polarizations in the third beam relative to the first beam will be λ/4, corresponding to a QWP, and after eight reflections the difference in the fifth beam relative to the first beam will be λ/2, corresponding to a half wave plate. Thus, we can determine the full polarization state of each point in the input beam with the first, third and fifth beams that are depicted in Fig. 7. The calcite beam displacer should be oriented at 22.5° with respect to the plate, as similarly arranged in Fig. 1.

Fig. 7. Compact configuration for real-time space-variant polarization measurement.

Our calculations reveal that the chromatic dispersion in this compact configuration is less than 1% over a wavelength range of λ = 400nm to λ = 800nm, while in the configuration of Fig. 1 with the QWP, the chromatic dispersion is 1% for only ±10nm range. Lower reflectivity of the PR give better light efficiency but at the expense of somewhat larger chromatic dispersion. Alternatively, the parallel plate can be tilted at 88° where the phase difference between the S polarization and the P polarization at each reflection will be λ/8 so only four reflections are needed but at the expense of working close to the grazing angle. It is important to note that the compact configuration limits the size of the input beam.

The configuration of Fig. 7 can be modified to obtain real-time polarization measurements as a function of wavelength rather than space. This is achieved by adding a diffraction grating and an array of three lenses between the parallel plate and the calcite beam displacer, as shown in Fig 8. The CCD camera is located at the focal plane of the lenses so it will detect six spectra. From each CCD camera frame, the four Stokes parameters can be obtained for all wavelengths, following Eqs. (1)–(4). Accordingly, the full state of polarization for each wavelength can be determined as a function of time. The wavelength-variant polarization measurement configuration of Fig. 8 can be extend to include also space-variant polarization measurement along one dimension, by using a slit at the input. It is also possible to place the grating before the beam splitter at the expense of increasing the size of the beam splitter.

Fig. 8. Configuration for real-time wavelength-variant polarization measurement.

5. Concluding remarks

To conclude, we developed and demonstrated a novel configuration for measuring space variant polarization of an incident light beam in real-time. We showed that the full polarization state at each point of the input beam can be accurately obtained every 10ms, which in our experiment was limited by the speed of the CCD camera. We believe that the measurements rate could be significantly increase by resorting to faster cameras. We also presented a compact configuration that can be adapted for measuring wavelength-variant polarization in real-time. We believe that these configurations could lead to a number of applications that where hitherto not possible.

References and links

1.

G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16, 1168–1174 (1999). [CrossRef]

2.

M. Mujat and A. Dogariu, “Real-time measurement of the polarization transfer function,” Appl. Opt. 40, 34–44 (2001). [CrossRef]

3.

R. M. A. Azzam, “In-line light-saving photopolarimeter and its fiber-optic analog,” Opt. Lett. 12, 558–560 (1987). [CrossRef] [PubMed]

4.

B. L. Heffner, “Automated Measurement of Polarization Mode Dispersion Using Jones Matrix Eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]

5.

D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 112, 455–546 (2004).

6.

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]

7.

Y. Gorodetski, G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Space-variant polarization manipulation for far-field polarimetry by use of subwavelength dielectric gratings,” Opt. Lett. 30, 2245–2247 (2005). [CrossRef] [PubMed]

8.

E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Polarization dependent focusing lens by use of quantized PancharatnamBerry phase diffractive optics,” Appl. Phys. Lett. 82, 328–330 (2003). [CrossRef]

9.

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of a discrete space-variant subwavelength grating,” J. Opt. Soc. Am. A 20, 1940–1948 (2003). [CrossRef]

10.

J. Kim and D. E. Kim, “Measurement of the degree of polarization of the spectra from laser produced recombining Al plasmas,” Phys. Rev. E. 66, 017401 (2002). [CrossRef]

11.

J. C. Kieffer, J. P. Matte, H. Pepin, M. Chaker, Y. Beaudoin, T. W. Johnston, C. Y. Chien, S. Coe, G. Mourou, and J. Dubau, “Electron Distribution Anistropy in Laser-Produced Plasmas from X-Ray Line Polarization Measurements,” Phys. Rev. Lett. 68, 480–483 (1992). [CrossRef] [PubMed]

12.

W. J. Bock and W. Urbanczyk, “Measurement of polarization mode dispersion and modal birefringence in highly birefringent fibers by means of electronically scanned shearing-type interferometry,” Appl. Opt. 32, 5841–5848 (1993). [CrossRef] [PubMed]

13.

J. B. Kortright, M. Rice, Z. Hussain, H. A. Padmore, A. Adamson, W. R. A. Huff, A. T. Young, E. J. Moler, S. A. Kellar, R. X. Ynzunza, F. J. Palomares, H. Daimon, E. D. Tober, and C. S. Fadley, “Polarization measurement and vertical aperture optimization for obtaining circularly polarized bend-magnet radiation,” Rev. Sci. Instrum. 67, 3363–3363 (1996). [CrossRef]

14.

N. Chinone and R. Ulrich, “Elasto-optic polarization measurement in optical fiber,” Opt. Lett. 6, 16–18 (1981). [CrossRef] [PubMed]

15.

M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]

16.

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. Phys. Lett. 77, 3322–3324 (2000). [CrossRef]

17.

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

18.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D: Appl. Phys. 33, 1817–1822 (2000). [CrossRef]

19.

K. Venkatakrishnan and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603–2607 (2006). [CrossRef]

20.

M. O. Scully, “A simple laser linac,” Appl. Phys. B 51, 238–241 (1990). [CrossRef]

21.

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46, 6640–6653 (1992). [CrossRef] [PubMed]

22.

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28, 807–809 (2003). [CrossRef] [PubMed]

23.

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Space-variant polarization scrambling for image encryption obtained with subwavelength gratings,” Opt. Commun. 261, 5–12 (2006) [CrossRef]

24.

E. Hecht, “Optics” (Addison Wesley, San Francisco, California) 2002, Chap. 8.

OCIS Codes
(230.5440) Optical devices : Polarization-selective devices
(260.5430) Physical optics : Polarization

History
Original Manuscript: January 29, 2010
Revised Manuscript: April 15, 2010
Manuscript Accepted: May 3, 2010
Published: May 10, 2010

Virtual Issues
Unconventional Polarization States of Light (2010) Optics Express

Citation
Moti Fridman, Micha Nixon, Eran Grinvald, Nir Davidson, and Asher A. Friesem, "Real-time measurement of unique space-variant polarizations," Opt. Express 18, 10805-10812 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-10805


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References

  1. G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16, 1168–1174 (1999). [CrossRef]
  2. M. Mujat, and A. Dogariu, “Real-time measurement of the polarization transfer function,” Appl. Opt. 40, 34–44 (2001). [CrossRef]
  3. R. M. A. Azzam, “In-line light-saving photopolarimeter and its fiber-optic analog,” Opt. Lett. 12, 558–560 (1987). [CrossRef] [PubMed]
  4. B. L. Heffner, “Automated Measurement of Polarization Mode Dispersion Using Jones Matrix Eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]
  5. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 112, 455–546 (2004).
  6. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999). [CrossRef]
  7. Y. Gorodetski, G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Space-variant polarization manipulation for farfield polarimetry by use of subwavelength dielectric gratings,” Opt. Lett. 30, 2245–2247 (2005). [CrossRef] [PubMed]
  8. 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, 328–330 (2003). [CrossRef]
  9. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of a discrete space-variant subwavelength grating,” J. Opt. Soc. Am. A 20, 1940–1948 (2003). [CrossRef]
  10. J. Kim, and D. E. Kim, “Measurement of the degree of polarization of the spectra from laser produced recombining Al plasmas,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 017401 (2002). [CrossRef]
  11. J. C. Kieffer, J. P. Matte, H. Pepin, M. Chaker, Y. Beaudoin, T. W. Johnston, C. Y. Chien, S. Coe, G. Mourou, and J. Dubau, “Electron Distribution Anistropy in Laser-Produced Plasmas from X-Ray Line Polarization Measurements,” Phys. Rev. Lett. 68, 480–483 (1992). [CrossRef] [PubMed]
  12. W. J. Bock, and W. Urbanczyk, “Measurement of polarization mode dispersion and modal birefringence in highly birefringent fibers by means of electronically scanned shearing-type interferometry,” Appl. Opt. 32, 5841–5848 (1993). [CrossRef] [PubMed]
  13. J. B. Kortright, M. Rice, Z. Hussain, H. A. Padmore, A. Adamson, W. R. A. Huff, A. T. Young, E. J. Moler, S. A. Kellar, R. X. Ynzunza, F. J. Palomares, H. Daimon, E. D. Tober, and C. S. Fadley, “Polarization measurement and vertical aperture optimization for obtaining circularly polarized bend-magnet radiation,” Rev. Sci. Instrum. 67, 3363 (1996). [CrossRef]
  14. N. Chinone, and R. Ulrich, “Elasto-optic polarization measurement in optical fiber,” Opt. Lett. 6, 16–18 (1981). [CrossRef] [PubMed]
  15. M. Fridman, G. Machavariani, N. Davidson, and A. A. Friesem, “Fiber lasers generating radially and azimuthally polarized light,” Appl. Phys. Lett. 93, 191104 (2008). [CrossRef]
  16. 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. Phys. Lett. 77, 3322–3324 (2000). [CrossRef]
  17. K. S. Youngworth, and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]
  18. A. V. Nesterov, and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33, 1817–1822 (2000). [CrossRef]
  19. K. Venkatakrishnan, and B. Tan, “Interconnect microvia drilling with a radially polarized laser beam,” J. Micromech. Microeng. 16, 2603–2607 (2006). [CrossRef]
  20. M. O. Scully, “A simple laser linac,” Appl. Phys. B 51, 238–241 (1990). [CrossRef]
  21. E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46, 6640–6653 (1992). [CrossRef] [PubMed]
  22. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28, 807–809 (2003). [CrossRef] [PubMed]
  23. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Space-variant polarization scrambling for image encryption obtained with subwavelength gratings,” Opt. Commun. 261, 5–12 (2006). [CrossRef]
  24. E. Hecht, Optics, (Addison Wesley, San Francisco, California) 2002, Chap. 8.

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