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OPTICS LETTERS / Vol. 39, No. 15 / August 1, 2014
Polarimetric measurement method to calculate optical beam shifts Chandravati Prajapati,* Shankar Pidishety, and Nirmal K. Viswanathan School of Physics, University of Hyderabad, Hyderabad 500046, India *Corresponding author: [email protected] Received April 16, 2014; revised June 5, 2014; accepted June 16, 2014; posted June 20, 2014 (Doc. ID 210278); published July 22, 2014 Stokes polarimetry measurements are carried out to calculate the spatial and angular Goos–Hänchen and Imbert–Fedorov shifts of a Gaussian beam reflected at glass–air interface, by measuring the phase difference between the TE and TM components and the amplitude of reflection. Variation of the beam shifts as a function of input beam polarization is also measured. The results obtained here are in good agreement with the theoretical predictions and the results obtained using a position sensitive detector. The polarimetric measurement method is accurate, independent of the intensity distribution of the beam, and opens up a new method to study the beam shift problem. © 2014 Optical Society of America OCIS codes: (240.3695) Linear and nonlinear light scattering from surfaces; (260.5430) Polarization; (260.2130) Ellipsometry and polarimetry. http://dx.doi.org/10.1364/OL.39.004388
Optical beam shifts are fundamentally important effects with a significant impact on the emerging research of subwavelength optics, near-field optical microscopy, nanophotonics, nanoplasmonics and near-field optical sensors [1–6]. The wavelength-order spatial and microradian angular deviation of bounded beams from the laws of geometric optics of reflection and transmission at a planar interface is further categorized into Goos– Hänchen (GH) shift, in the plane of incidence, and Imbert–Fedorov (IF) shift, perpendicular to the plane of incidence [1,7]. These spatial and angular beam shifts depend strongly on the polarization state of the beam and the nature of the dielectric interface and are due to the geometrical corrections to Fresnel’s and Snell’s plane wave equations for practical bounded beams. Goos and Hänchen first measured the shift for total internal reflection (TIR) beams [8], which were theoretically explained by Artmann [9] using stationary phase approximation. The IF shift for TIR beams was first predicted by Fedorov [10] and theoretically solved and measured by Imbert using the energy conservation formalism [11]. Although at the critical angle the Artmann formula gives infinite GH shift, the experimentally measured shifts are always finite for focused beams as explained in Refs. [12,13]. The early reports on spatial GH and IF beam shifts under TIR conditions were later extended to include angular GH and IF shifts under partial reflection conditions and for beams with orbital angular momentum [1,2,14–20]. The position sensitive detector (PSD) is extensively used as the device of choice to measure the beam shifts by tracking the intensity-weighted centroid of the reflected beam for incident beams in orthogonal polarization basis sets TE and TM, 45°, or RC and LC, using a polarization modulation device [21]. The PSD measurements however depend strongly on the beam intensity distribution leading to error in the measurements. An interferometric method was recently proposed and used [22,23] to measure the spatial GH and IF shifts of Gaussian beams upon reflection from aluminium mirror and glass plates. The method is quite accurate but 0146-9592/14/154388-04$15.00/0
requires lot of computation to extract the phase, especially when the shift is large compared to the wavelength of light. More recently, a weak measurement approach was also taken to observe GH and IF shifts under the TIR condition [24,25]. In this Letter, we propose and demonstrate the polarimetric technique to experimentally measure the spatial and angular GH and IF shifts of total and partially reflected beams by measuring the phase difference between TE and TM components and the amplitude of reflection. For the GH shift, the spatial part and the angular part depend, respectively, on the derivative of the phase and the amplitude of reflection with respect to the angle of incidence. For the IF shift, the spatial part and the angular part depend, respectively, on the phase difference and the amplitude difference between the reflected beams’ TE and TM components. Being very small in magnitude, the beam shift measurements present challenges which were overcome by interferometric and weak measurement methods [22–25]. The alternate polarimetric measurement method presented here is a simple, versatile and beam-shape and intensity-independent technique in contrast to the PSD measurement method and is better suited for measuring the weak values of polarization components and generalized 3D beam shifts [26,27]. Suitable theoretical formalism is used to describe completely all of the beam shifts under different reflection conditions (total, partial, Brewster and metallic) and for any classical beam [2,19,20]. The reflection of polarized light at the dielectric interface changes its amplitude and introduces phase with respect to the incident beam. The reflected light beams pass through a QWP (which gives a retardation of ϕ between the x and y components) and a polarizer P2 oriented at angle θ (which together forms the Stokes polarimetry measurement setup). The intensity of the reflected beam is recorded for six different ϕ; θ orientations from which the Stokes parameters are calculated [28]. From the Stokes parameters, the phase difference between the TE and TM components is calculated as © 2014 Optical Society of America
August 1, 2014 / Vol. 39, No. 15 / OPTICS LETTERS
ϕs − ϕp arctan
S3 − η: S2
(1)
Here η δx − δy is the phase difference between the TE and TM components of the incident beam. The value of η is different for different polarizations. For TE and TM beams η 0; for 45° beams, η 0 or π; and for LCP and RCP beams, η π∕2. The amplitude of reflection is calculated as s S0 S1 jr s j 2E 20x
s S0 − S1 : and jr p j 2E 20y
(2)
For practical beams such as the one used here, we use the beam shift formulas of Aiello and Woerdman [17], which can be used to calculate both spatial and angular shifts under any reflection condition. The paraxial approximation of the Gaussian beam in angular spectrum representation and a Taylor series expansion of the reflection coefficients around the angle of incidence is carried out to study the deviation of the beam from the geometrical ray path. The deviations are considered up to first-order from which the GH and IF shifts are calculated as the shift of the centroid of the reflected beam. The formulas for dimensional GH and IF shifts are ∂ ln r p ∂ ln r s 2 2 2 2 λ jr s j as I ∂θi jr p j ap I ∂θi GH 2π jr s j2 a2s jr p j2 a2p ∂ ln r p ∂ ln r s 2 2 2 2 jr jr j a R j a R s p s p ∂θi ∂θi Z 2 r2 jr s j2 a2s jr p j2 a2p k L
(3)
and ap as cot θi λ IF − jr s j2 jr p j2 sin η 2π jr s j2 a2s jr p j2 a2p 2jr s jjr p j sinη ϕs − ϕp 2 − jr j2 cot θ jr a a j s p s p i Z cos η; − 2 r2 2 2 2 2 k L jr s j as jr p j ap
(4)
where as E 0x and ap E 0y are the amplitude of TE and TM components of the incident beam; θi is the angle of incidence; Z r kzr , where zr is the propagation distance; L kω20 ∕2 is the Rayleigh range, ω0 being the beam waist; and k 2πn∕λ0 is the wave vector with λ0 as the wavelength of light in vacuum and n the refractive index of the dielectric medium. The first term in Eqs. (3) and (4) corresponds to spatial shift (in meters) and the second term gives the angular shift (in radians). Z r ∕kL is the beam propagation enhancement factor. Under the condition Z r ∕kL ≪ 1, for a well-collimated beam, formulas (3) and (4) have dominant first terms and give the spatial GH and IF shifts, but for focused beams, the second-term contribution is significant giving a large angular shift. In our case, the beam is not wellcollimated and the beam size is nearly 500 μm, giving
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Z r ∕kL ≈ 0.5, so a significant amount of angular shift can be observed in addition to the spatial shift. The small angular shifts can be further increased significantly by focusing the beam [18]. For TIR cases, at θi > θC , jr s j jr p j 1, and the second term in the equation for GH and IF shifts becomes zero. For an incident angle less than θC , the light beam undergoes through partial reflection and jr s j ≠ 1, jr p j ≠ 1. There is no variation in phase; therefore, the first term in Eqs. (3) and (4) becomes zero. At a particular angle θB , where jr p j 0, called the Brewster angle, the beam is largely distorted and the angular GH shift is found to be very large. A schematic of the experimental setup is shown in Figure 1. The polarized incident beam is reflected from the hypotenuse of a 45°–90°–45° BK7 glass prism (P) mounted on a precision rotation stage and is analyzed using a QWP and P2 combination [29]. A He–Ne laser at 632.8 nm is used and the incident beam of polarization TE, TM, linear 45°, right circular (RC) and left circular (LC) are produced with the help of polarizer P1 and HWP/QWP and six intensity measurements are performed over the range of angles of incidence from 28° to 50°, which includes total internal, partial and Brewster reflections. From the six intensity measurements, the Stokes parameters are calculated from which the phase difference between TE and TM components of the beam and the amplitude of reflection coefficients are calculated using Eqs. (1) and (2). The beam shifts are calculated using Eqs. (3) and (4). Figure 2 is the angular GH shift plot for the TM polarized beam obtained from theory (blue line) and measured from the experiment (red symbol). As we move toward θB , the negative shift increases and becomes very large just below θB . At θB , it jumps to a very large positive value due to the cross-polarization effect [18,30], which then decreases with the incidence angle. Though the theory predicts a shift of 10 μrad around the Brewster angle, we measured a shift of 8 μrad and the reason for this small deviation is due to errors associated with the measurement. It is important to note in Fig. 2 that the angular GH shift for the TM beam starts to increase as we approach the critical angle of 41.4°. Figure 3 is the plot of angular GH shift obtained from (a) the experiment and (c) the theory for TE (magenta), 45° (black and red) and RC/LC (blue and green)
Fig. 1. Schematic of the experimental setup. WP, half/quarter wave plate; QWP, quarter-wave-plate; P1 and P2, Glan– Thompson polarizer; D, detector; and P, BK7 prism.
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Fig. 2. Plot of angular GH shift for the TM-polarized beam obtained from theory (blue line) and from experimental measurement (red symbol).
polarized beams. The angular GH shift for the TE polarized beam is positive and increases with the angle of incidence as θi approaches θC . For 45° polarized beams, the shifts are nearly the same and negative but near θC they become positive and increase; for RC and LC polarized beams, the behavior is the same as that of 45° polarized beams and beams small in magnitude. The angular GH shift for all the polarized beams is large near the critical angle. Theory results overlap for 45° and RC/LC polarized beams. There is good agreement between the theory and experiment. For the theoretical simulation of beam shifts, we used as 1, ap 0 for the TE polarized beam, as 0, ap 1 for the TM polarized p p beam, and as 1∕ 2, ap 1∕ 2 for the 45° and the RC-/LC-polarized beam. The value of ω0 is taken from the experiment and the value of zr is measured to be 18.4 cm. Theory suggests that there is no angular IF shift for RC and LC polarized beams, as η π∕2. As we calculate the angular IF shift by measuring the phase introduced upon reflection, both TE and TM beams acquire beam shifts that could be because of the splitting of TE and TM polarized beams into RC and LC components (Spin
Fig. 3. (a) Experimental measurement and (c) theory simulation of the angular GH shift for TE (magenta); 45° (black, red); RC/LC (blue, green) polarized beams. (b) Experimental measurement and (d) theory simulation of angular IF shift for TE and TM (magenta, orange) and 45° (black, red) polarized beams.
Hall Effect of Light) [26]. We plotted the angular IF shift for TE, TM, and 45° polarized beams obtained from (b) the experiment and (d) the theory. The angular IF shift for 45° beams (black and red curves) are nearly the same magnitude but opposite in sign. The magnitude of shift for the TM (orange) beam is less compared to that for the TE (magenta) beam because of the small reflectivity of the TM beam especially near θB although both shifts increase with the angle of incidence. A good agreement between theory and experiment is found. Figure 4 is the plot of the spatial GH shift obtained from (a) the experiment and (c) the theory and plot of spatial IF shifts obtained from (b) the experiment and (d) the theory for TE and TM (magenta, orange), 45° (black, red), and RC and LC (blue, green) polarized beams. The spatial GH shift has a peak shift at θC 41.4° for all polarizations with varying magnitude. The shift is nearly the same for 45° and for RC and LC polarized beams. The experimental spatial GH shift is around 1.2 μm for the TE beam and 1.7 μm for the TM beam, while theory has 1.2 μm for the TE beam and 1.8 μm for the TM beam. The spatial IF shift for the 45° polarized beam is zero below θC which then increases and becomes almost constant for larger angles. The spatial IF shift for 45° polarized beams are the same in magnitude but opposite in sign and the same is true for RC and LC polarized beams. The spatial IF shift for RC and LC polarized beams has a peak at θC which then decreases with the angle of incidence. The IF shift for the TE beam is negative and is positive for the TM polarized beam, both with a peak at the critical angle. A good agreement between theory and experiment is found. Apart from the measurement of spatial and angular beam shifts using the polarimetric method, where all of the measured behavior matches well with the theoretical predictions and with other reports [18,21] we also studied the variation of spatial GH and IF shifts with changing-input beam polarization. The input beampolarization orientation is changed from TE to TM passing through all linear states and through all elliptical/ circular states. Figure 5 is a plot of the variation of
Fig. 4. Experimental measurement of (a) spatial GH shift and (b) spatial IF shift for TE and TM (magenta, orange), 45° (black, red), and RC/LC (blue, green) polarized beams. Theory simulation of (c) spatial GH shift and (d) spatial IF shift.
August 1, 2014 / Vol. 39, No. 15 / OPTICS LETTERS
Fig. 5. Experimental measurement of (a) spatial GH shift and (b) spatial IF shift with varying input beam polarization from TE to TM passing through linear states (red) and through elliptical/ circular states (black).
maximum spatial GH and IF shifts with polarization orientation at θC 41.4°. The GH shift is minimum for the TE beam and increases as we move toward the TM beam. The IF shift is minimal for TE and TM beams and increases for linear 45° and circular polarization and is maximal for 45° and circular polarized beams, as suggested from the theory [4]. In conclusion, we proposed and demonstrated a polarimetric method to measure all of the beam shifts: spatial and angular in the plane of incidence (GH shift) and perpendicular to the plane of incidence (IF shift). We also verified that the 45° polarized beam behaves in the same manner as the RC/LC polarized beam, which proves the fact that the phase changes in the same way between these orthogonal states. The results obtained using this method are in good agreement with the theory and the results obtained from PSD measurements. More detailed experimental measurements including variations in the beam shift due to beam size, angular dependence of reflectivity and measurement errors of the Stokes vectors are currently underway and will be reported elsewhere. The polarimetric measurement method is simple, versatile and independent of the intensity distribution of the beam. In addition, the polarimetric method can provide a significantly new insight into measuring the spatial and angular variation of beam shifts leading naturally up to weak measurements and the measurement of generalized beam shifts. The authors acknowledge the Department of Science and Technology in India for financial support for the project.
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References 1. K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013). 2. A. Aiello, New J. Phys. 14, 013058 (2012). 3. S. A. Taya, E. J. El-Farram, and T. M. El-Agez, Int. J. Electon. Commun. (AEÜ) 66, 204 (2012). 4. F. I. Baida, D. V. Labeke, and J.-M. Vigoureux, J. Opt. Soc. Am. A 17, 858 (2000). 5. H. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007). 6. X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006). 7. K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006). 8. F. Goos and H. Hanchen, Ann. Phys. 436, 333 (1947). 9. K. Artmann, Ann. Phys. 2, 87 (1948). 10. F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955). 11. C. Imbert, Phys. Rev. D 5, 787 (1972). 12. B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971). 13. H. M. Lai, F. C. Cheng, and W. K. Tang, J. Opt. Soc. Am. A 3, 550 (1986). 14. J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973). 15. Y. M. Antar and W. M. Boerner, Can. J. Phys. 52, 962 (1974). 16. C. C. Chan and C. Tamir, Opt. Lett. 10, 378 (1985). 17. A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437 (2008). 18. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009). 19. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010). 20. C. Prajapati and D. Ranganathan, J. Opt. Soc. Am. A 29, 1377 (2012). 21. F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004). 22. C. Prajapati, D. Ranganathan, and J. Joseph, J. Opt. Soc. Am. A 30, 741 (2013). 23. C. Prajapati, D. Ranganathan, and J. Joseph, Opt. Lett. 38, 2459 (2013). 24. G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 38, 1232 (2013). 25. G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 39, 2266 (2014). 26. J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012). 27. W. Nasalski, Opt. Commun. 197, 217 (2001). 28. D. H. Goldstein, Polarized Light, (CRC Press, 2011). 29. S. Pidishety, V. Kumar, and N. K. Viswanathan, Opt. Lett. 37, 4233 (2012). 30. A. Aiello, M. Merano, and J. P. Woerdman, Opt. Lett. 34, 1207 (2009).
OPTICS LETTERS / Vol. 39, No. 15 / August 1, 2014
Polarimetric measurement method to calculate optical beam shifts Chandravati Prajapati,* Shankar Pidishety, and Nirmal K. Viswanathan School of Physics, University of Hyderabad, Hyderabad 500046, India *Corresponding author: [email protected] Received April 16, 2014; revised June 5, 2014; accepted June 16, 2014; posted June 20, 2014 (Doc. ID 210278); published July 22, 2014 Stokes polarimetry measurements are carried out to calculate the spatial and angular Goos–Hänchen and Imbert–Fedorov shifts of a Gaussian beam reflected at glass–air interface, by measuring the phase difference between the TE and TM components and the amplitude of reflection. Variation of the beam shifts as a function of input beam polarization is also measured. The results obtained here are in good agreement with the theoretical predictions and the results obtained using a position sensitive detector. The polarimetric measurement method is accurate, independent of the intensity distribution of the beam, and opens up a new method to study the beam shift problem. © 2014 Optical Society of America OCIS codes: (240.3695) Linear and nonlinear light scattering from surfaces; (260.5430) Polarization; (260.2130) Ellipsometry and polarimetry. http://dx.doi.org/10.1364/OL.39.004388
Optical beam shifts are fundamentally important effects with a significant impact on the emerging research of subwavelength optics, near-field optical microscopy, nanophotonics, nanoplasmonics and near-field optical sensors [1–6]. The wavelength-order spatial and microradian angular deviation of bounded beams from the laws of geometric optics of reflection and transmission at a planar interface is further categorized into Goos– Hänchen (GH) shift, in the plane of incidence, and Imbert–Fedorov (IF) shift, perpendicular to the plane of incidence [1,7]. These spatial and angular beam shifts depend strongly on the polarization state of the beam and the nature of the dielectric interface and are due to the geometrical corrections to Fresnel’s and Snell’s plane wave equations for practical bounded beams. Goos and Hänchen first measured the shift for total internal reflection (TIR) beams [8], which were theoretically explained by Artmann [9] using stationary phase approximation. The IF shift for TIR beams was first predicted by Fedorov [10] and theoretically solved and measured by Imbert using the energy conservation formalism [11]. Although at the critical angle the Artmann formula gives infinite GH shift, the experimentally measured shifts are always finite for focused beams as explained in Refs. [12,13]. The early reports on spatial GH and IF beam shifts under TIR conditions were later extended to include angular GH and IF shifts under partial reflection conditions and for beams with orbital angular momentum [1,2,14–20]. The position sensitive detector (PSD) is extensively used as the device of choice to measure the beam shifts by tracking the intensity-weighted centroid of the reflected beam for incident beams in orthogonal polarization basis sets TE and TM, 45°, or RC and LC, using a polarization modulation device [21]. The PSD measurements however depend strongly on the beam intensity distribution leading to error in the measurements. An interferometric method was recently proposed and used [22,23] to measure the spatial GH and IF shifts of Gaussian beams upon reflection from aluminium mirror and glass plates. The method is quite accurate but 0146-9592/14/154388-04$15.00/0
requires lot of computation to extract the phase, especially when the shift is large compared to the wavelength of light. More recently, a weak measurement approach was also taken to observe GH and IF shifts under the TIR condition [24,25]. In this Letter, we propose and demonstrate the polarimetric technique to experimentally measure the spatial and angular GH and IF shifts of total and partially reflected beams by measuring the phase difference between TE and TM components and the amplitude of reflection. For the GH shift, the spatial part and the angular part depend, respectively, on the derivative of the phase and the amplitude of reflection with respect to the angle of incidence. For the IF shift, the spatial part and the angular part depend, respectively, on the phase difference and the amplitude difference between the reflected beams’ TE and TM components. Being very small in magnitude, the beam shift measurements present challenges which were overcome by interferometric and weak measurement methods [22–25]. The alternate polarimetric measurement method presented here is a simple, versatile and beam-shape and intensity-independent technique in contrast to the PSD measurement method and is better suited for measuring the weak values of polarization components and generalized 3D beam shifts [26,27]. Suitable theoretical formalism is used to describe completely all of the beam shifts under different reflection conditions (total, partial, Brewster and metallic) and for any classical beam [2,19,20]. The reflection of polarized light at the dielectric interface changes its amplitude and introduces phase with respect to the incident beam. The reflected light beams pass through a QWP (which gives a retardation of ϕ between the x and y components) and a polarizer P2 oriented at angle θ (which together forms the Stokes polarimetry measurement setup). The intensity of the reflected beam is recorded for six different ϕ; θ orientations from which the Stokes parameters are calculated [28]. From the Stokes parameters, the phase difference between the TE and TM components is calculated as © 2014 Optical Society of America
August 1, 2014 / Vol. 39, No. 15 / OPTICS LETTERS
ϕs − ϕp arctan
S3 − η: S2
(1)
Here η δx − δy is the phase difference between the TE and TM components of the incident beam. The value of η is different for different polarizations. For TE and TM beams η 0; for 45° beams, η 0 or π; and for LCP and RCP beams, η π∕2. The amplitude of reflection is calculated as s S0 S1 jr s j 2E 20x
s S0 − S1 : and jr p j 2E 20y
(2)
For practical beams such as the one used here, we use the beam shift formulas of Aiello and Woerdman [17], which can be used to calculate both spatial and angular shifts under any reflection condition. The paraxial approximation of the Gaussian beam in angular spectrum representation and a Taylor series expansion of the reflection coefficients around the angle of incidence is carried out to study the deviation of the beam from the geometrical ray path. The deviations are considered up to first-order from which the GH and IF shifts are calculated as the shift of the centroid of the reflected beam. The formulas for dimensional GH and IF shifts are ∂ ln r p ∂ ln r s 2 2 2 2 λ jr s j as I ∂θi jr p j ap I ∂θi GH 2π jr s j2 a2s jr p j2 a2p ∂ ln r p ∂ ln r s 2 2 2 2 jr jr j a R j a R s p s p ∂θi ∂θi Z 2 r2 jr s j2 a2s jr p j2 a2p k L
(3)
and ap as cot θi λ IF − jr s j2 jr p j2 sin η 2π jr s j2 a2s jr p j2 a2p 2jr s jjr p j sinη ϕs − ϕp 2 − jr j2 cot θ jr a a j s p s p i Z cos η; − 2 r2 2 2 2 2 k L jr s j as jr p j ap
(4)
where as E 0x and ap E 0y are the amplitude of TE and TM components of the incident beam; θi is the angle of incidence; Z r kzr , where zr is the propagation distance; L kω20 ∕2 is the Rayleigh range, ω0 being the beam waist; and k 2πn∕λ0 is the wave vector with λ0 as the wavelength of light in vacuum and n the refractive index of the dielectric medium. The first term in Eqs. (3) and (4) corresponds to spatial shift (in meters) and the second term gives the angular shift (in radians). Z r ∕kL is the beam propagation enhancement factor. Under the condition Z r ∕kL ≪ 1, for a well-collimated beam, formulas (3) and (4) have dominant first terms and give the spatial GH and IF shifts, but for focused beams, the second-term contribution is significant giving a large angular shift. In our case, the beam is not wellcollimated and the beam size is nearly 500 μm, giving
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Z r ∕kL ≈ 0.5, so a significant amount of angular shift can be observed in addition to the spatial shift. The small angular shifts can be further increased significantly by focusing the beam [18]. For TIR cases, at θi > θC , jr s j jr p j 1, and the second term in the equation for GH and IF shifts becomes zero. For an incident angle less than θC , the light beam undergoes through partial reflection and jr s j ≠ 1, jr p j ≠ 1. There is no variation in phase; therefore, the first term in Eqs. (3) and (4) becomes zero. At a particular angle θB , where jr p j 0, called the Brewster angle, the beam is largely distorted and the angular GH shift is found to be very large. A schematic of the experimental setup is shown in Figure 1. The polarized incident beam is reflected from the hypotenuse of a 45°–90°–45° BK7 glass prism (P) mounted on a precision rotation stage and is analyzed using a QWP and P2 combination [29]. A He–Ne laser at 632.8 nm is used and the incident beam of polarization TE, TM, linear 45°, right circular (RC) and left circular (LC) are produced with the help of polarizer P1 and HWP/QWP and six intensity measurements are performed over the range of angles of incidence from 28° to 50°, which includes total internal, partial and Brewster reflections. From the six intensity measurements, the Stokes parameters are calculated from which the phase difference between TE and TM components of the beam and the amplitude of reflection coefficients are calculated using Eqs. (1) and (2). The beam shifts are calculated using Eqs. (3) and (4). Figure 2 is the angular GH shift plot for the TM polarized beam obtained from theory (blue line) and measured from the experiment (red symbol). As we move toward θB , the negative shift increases and becomes very large just below θB . At θB , it jumps to a very large positive value due to the cross-polarization effect [18,30], which then decreases with the incidence angle. Though the theory predicts a shift of 10 μrad around the Brewster angle, we measured a shift of 8 μrad and the reason for this small deviation is due to errors associated with the measurement. It is important to note in Fig. 2 that the angular GH shift for the TM beam starts to increase as we approach the critical angle of 41.4°. Figure 3 is the plot of angular GH shift obtained from (a) the experiment and (c) the theory for TE (magenta), 45° (black and red) and RC/LC (blue and green)
Fig. 1. Schematic of the experimental setup. WP, half/quarter wave plate; QWP, quarter-wave-plate; P1 and P2, Glan– Thompson polarizer; D, detector; and P, BK7 prism.
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Fig. 2. Plot of angular GH shift for the TM-polarized beam obtained from theory (blue line) and from experimental measurement (red symbol).
polarized beams. The angular GH shift for the TE polarized beam is positive and increases with the angle of incidence as θi approaches θC . For 45° polarized beams, the shifts are nearly the same and negative but near θC they become positive and increase; for RC and LC polarized beams, the behavior is the same as that of 45° polarized beams and beams small in magnitude. The angular GH shift for all the polarized beams is large near the critical angle. Theory results overlap for 45° and RC/LC polarized beams. There is good agreement between the theory and experiment. For the theoretical simulation of beam shifts, we used as 1, ap 0 for the TE polarized beam, as 0, ap 1 for the TM polarized p p beam, and as 1∕ 2, ap 1∕ 2 for the 45° and the RC-/LC-polarized beam. The value of ω0 is taken from the experiment and the value of zr is measured to be 18.4 cm. Theory suggests that there is no angular IF shift for RC and LC polarized beams, as η π∕2. As we calculate the angular IF shift by measuring the phase introduced upon reflection, both TE and TM beams acquire beam shifts that could be because of the splitting of TE and TM polarized beams into RC and LC components (Spin
Fig. 3. (a) Experimental measurement and (c) theory simulation of the angular GH shift for TE (magenta); 45° (black, red); RC/LC (blue, green) polarized beams. (b) Experimental measurement and (d) theory simulation of angular IF shift for TE and TM (magenta, orange) and 45° (black, red) polarized beams.
Hall Effect of Light) [26]. We plotted the angular IF shift for TE, TM, and 45° polarized beams obtained from (b) the experiment and (d) the theory. The angular IF shift for 45° beams (black and red curves) are nearly the same magnitude but opposite in sign. The magnitude of shift for the TM (orange) beam is less compared to that for the TE (magenta) beam because of the small reflectivity of the TM beam especially near θB although both shifts increase with the angle of incidence. A good agreement between theory and experiment is found. Figure 4 is the plot of the spatial GH shift obtained from (a) the experiment and (c) the theory and plot of spatial IF shifts obtained from (b) the experiment and (d) the theory for TE and TM (magenta, orange), 45° (black, red), and RC and LC (blue, green) polarized beams. The spatial GH shift has a peak shift at θC 41.4° for all polarizations with varying magnitude. The shift is nearly the same for 45° and for RC and LC polarized beams. The experimental spatial GH shift is around 1.2 μm for the TE beam and 1.7 μm for the TM beam, while theory has 1.2 μm for the TE beam and 1.8 μm for the TM beam. The spatial IF shift for the 45° polarized beam is zero below θC which then increases and becomes almost constant for larger angles. The spatial IF shift for 45° polarized beams are the same in magnitude but opposite in sign and the same is true for RC and LC polarized beams. The spatial IF shift for RC and LC polarized beams has a peak at θC which then decreases with the angle of incidence. The IF shift for the TE beam is negative and is positive for the TM polarized beam, both with a peak at the critical angle. A good agreement between theory and experiment is found. Apart from the measurement of spatial and angular beam shifts using the polarimetric method, where all of the measured behavior matches well with the theoretical predictions and with other reports [18,21] we also studied the variation of spatial GH and IF shifts with changing-input beam polarization. The input beampolarization orientation is changed from TE to TM passing through all linear states and through all elliptical/ circular states. Figure 5 is a plot of the variation of
Fig. 4. Experimental measurement of (a) spatial GH shift and (b) spatial IF shift for TE and TM (magenta, orange), 45° (black, red), and RC/LC (blue, green) polarized beams. Theory simulation of (c) spatial GH shift and (d) spatial IF shift.
August 1, 2014 / Vol. 39, No. 15 / OPTICS LETTERS
Fig. 5. Experimental measurement of (a) spatial GH shift and (b) spatial IF shift with varying input beam polarization from TE to TM passing through linear states (red) and through elliptical/ circular states (black).
maximum spatial GH and IF shifts with polarization orientation at θC 41.4°. The GH shift is minimum for the TE beam and increases as we move toward the TM beam. The IF shift is minimal for TE and TM beams and increases for linear 45° and circular polarization and is maximal for 45° and circular polarized beams, as suggested from the theory [4]. In conclusion, we proposed and demonstrated a polarimetric method to measure all of the beam shifts: spatial and angular in the plane of incidence (GH shift) and perpendicular to the plane of incidence (IF shift). We also verified that the 45° polarized beam behaves in the same manner as the RC/LC polarized beam, which proves the fact that the phase changes in the same way between these orthogonal states. The results obtained using this method are in good agreement with the theory and the results obtained from PSD measurements. More detailed experimental measurements including variations in the beam shift due to beam size, angular dependence of reflectivity and measurement errors of the Stokes vectors are currently underway and will be reported elsewhere. The polarimetric measurement method is simple, versatile and independent of the intensity distribution of the beam. In addition, the polarimetric method can provide a significantly new insight into measuring the spatial and angular variation of beam shifts leading naturally up to weak measurements and the measurement of generalized beam shifts. The authors acknowledge the Department of Science and Technology in India for financial support for the project.
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References 1. K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013). 2. A. Aiello, New J. Phys. 14, 013058 (2012). 3. S. A. Taya, E. J. El-Farram, and T. M. El-Agez, Int. J. Electon. Commun. (AEÜ) 66, 204 (2012). 4. F. I. Baida, D. V. Labeke, and J.-M. Vigoureux, J. Opt. Soc. Am. A 17, 858 (2000). 5. H. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007). 6. X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 (2006). 7. K. Yu. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006). 8. F. Goos and H. Hanchen, Ann. Phys. 436, 333 (1947). 9. K. Artmann, Ann. Phys. 2, 87 (1948). 10. F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955). 11. C. Imbert, Phys. Rev. D 5, 787 (1972). 12. B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971). 13. H. M. Lai, F. C. Cheng, and W. K. Tang, J. Opt. Soc. Am. A 3, 550 (1986). 14. J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973). 15. Y. M. Antar and W. M. Boerner, Can. J. Phys. 52, 962 (1974). 16. C. C. Chan and C. Tamir, Opt. Lett. 10, 378 (1985). 17. A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437 (2008). 18. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337 (2009). 19. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, Phys. Rev. A 82, 023817 (2010). 20. C. Prajapati and D. Ranganathan, J. Opt. Soc. Am. A 29, 1377 (2012). 21. F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004). 22. C. Prajapati, D. Ranganathan, and J. Joseph, J. Opt. Soc. Am. A 30, 741 (2013). 23. C. Prajapati, D. Ranganathan, and J. Joseph, Opt. Lett. 38, 2459 (2013). 24. G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 38, 1232 (2013). 25. G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 39, 2266 (2014). 26. J. B. Götte and M. R. Dennis, New J. Phys. 14, 073016 (2012). 27. W. Nasalski, Opt. Commun. 197, 217 (2001). 28. D. H. Goldstein, Polarized Light, (CRC Press, 2011). 29. S. Pidishety, V. Kumar, and N. K. Viswanathan, Opt. Lett. 37, 4233 (2012). 30. A. Aiello, M. Merano, and J. P. Woerdman, Opt. Lett. 34, 1207 (2009).
