OBSERVATIONS OF THE GOOS-HÄNCHEN AND IMBERT-FEDOROV SHIFTS VIA WEAK MEASUREMENT. A Thesis. Presented to the. Faculty of. San Diego State University

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1 OBSERVATIONS OF THE GOOS-HÄNCHEN AND IMBERT-FEDOROV SHIFTS VIA WEAK MEASUREMENT A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Physics by Oxana Kazarina Fall 2014

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3 iii Copyright c 2014 by Oxana Kazarina

4 iv What is still in reflection that needs to be studied? Question of perplexed former professor to Nathaniel Hermosa

5 v ABSTRACT OF THE THESIS Observations of the Goos-Hänchen and Imbert-Fedorov Shifts via Weak Measurement by Oxana Kazarina Master of Science in Physics San Diego State University, 2014 The Goos-Hänchen (longitudinal) and Imbert-Fedorov (transverse) shifts show that the center of the reflected beam is shifted and doesn t follow the laws of geometrical optics. The beam has to have finite width and be totally internally reflected at an interface of two media having different indices of refraction. Because these shifts are very small (on the wavelength scale) the weak measurements, known from the quantum mechanics, allow to amplify and measure such small phenomena. In this paper the Goos-Hänchen and Imbert-Fedorov spatial shifts were measured experimentally via weak measurement technique which was observed for a linearly polarized beam at total internal reflection.

6 vi TABLE OF CONTENTS PAGE ABSTRACT... LIST OF FIGURES... ACKNOWLEDGMENTS... v viii xi CHAPTER 1 INTRODUCTION LIGHT REFLECTION... 4 Snell s Law and Fresnel Equations... 5 Critical Angle... 5 The Brewster Angle... 6 Total Internal Reflection... 7 Phase Changing on Reflection... 7 The Evanescent Wave in Total Reflection GOOS-HÄNCHEN AND IMBERT-FEDOROV SHIFTS Goos-Hänchen Shift Imbert-Fedorov Shift WEAK MEASUREMENT What is Weak Measurement? Example Involving Spin- 1 2 Particles Optical Analog of Weak Measurement GOOS-HÄNCHEN AND IMBERT-FEDOROV SHIFTS FROM A QUANTUM-MECHANICAL PERSPECTIVE EXPERIMENTAL WEAK MEASUREMENT OBSERVATION OF THE GOOS-HÄNCHEN SHIFT EXPERIMENTAL WEAK MEASUREMENT OBSERVATION OF THE IMBERT-FEDOROV SHIFT CONCLUSION REFERENCES... 42

7 vii APPENDICES A SPIN HALL EFFECT OF LIGHT VIA WEAK MEASUREMENT B PARAXIAL WAVE EQUATION... 48

8 viii LIST OF FIGURES PAGE Figure 1. Diagram for incident, reflected and transmitted rays at an XY-plane interface when the electric field is perpendicular to the plane of incidence. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993) Figure 2. Geometry of rays reflected by a plane interface. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993) Figure 3. Specular reflection of light at a dielectric surface. (a) S-polarization. (b) P-polarization. (c) Polarization at Brewster s angle. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993) Figure 4. Phase shift of electric field for internally reflected rays, with n=n 1 /n 2 =1/1.5. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993) Figure 5. Diagram showing the direction of the Poynting vector for the case of TIR. Source: G. R. Fowles, Introduction to Modern Optics (Courier Dover Publications, Mineola, NY, 1975) Figure 6. A method for illustrating the penetration of light into the rare medium. Source: G. R. Fowles, Introduction to Modern Optics (Courier Dover Publications, Mineola, NY, 1975) Figure 7. Principle of the ray path. I-incident beam. R 1 -reflected beam shows the behavior similar to internally reflected beam; R 2 -reflected beam shows the behavior expected by Goos and Hänchen. D-Goos-Hänchen shift. Source: R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964) Figure 8. Illustration of the Goos-Hänchen shift for a finite plane wave. W is the limited incident plane wave. Φ 1 and Φ 2 are the two energy fluxes. Source: R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964) Figure 9. GH and IF spatial shifts at total internal reflection. Source: F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004) Figure 10. Pre-selection and post-selection. Source: O. Hosten and P. Kwiat, Science 319, 787 (2008) Figure 11. Stern-Gerlach magnet layout for the AAV experiment. Source: I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, 2112 (1989)

9 Figure 12. Graph of the function ϕ(p; ɛ,, λ) (see Equation 32) as a function of P p/λ with ɛ=0.2 and λ =0.01. Source: I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, 2112 (1989) Figure 13. An optical analog of the AAV experiment. A broad, coherent beam passes through a polarizer (P) and analyzer (A). Between them is a birefringent crystal (C) which produces a small lateral displacement between x (o ray) and z (e ray) polarizations (see inset). Source: I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, 2112 (1989) Figure 14. Experimental setup for the observation of the Goos-Hänchen shift via weak measurement. L1 and L2 - lenses for the beam collimation, L3 and L4 - lenses for the beam focusing, P1 and P2 - Glan-Thompson polarizers, QWP - quarter-wave plate, HWP - half-wave plate, Win Cam D - camera. Source: G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 38, 1232 (2013) Figure 15. Beam profile of the light transmitted through the whole system. (a) Gaussian beam profile; (b) Beam profile for the crossed polarizers with ɛ = 0 rad Figure 16. Experimental data (squares) with the error bars are compared with the theoretical predictions (solid line) for GH shift in TIR Figure 17. Experimental setup for the observation of the Imbert-Fedorov shift via weak measurement. L1 and L2 - lenses of the beam expander, L3 and L4 - lenses for the beam focusing and collimating, respectively, P1 and P2 - Glan-Thompson polarizers, QWP - quarter-wave plate, HWP - half-wave plate, Win Cam D - camera. Sourse: G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 39, 2266 (2014) Figure 18. Beam profile of the light transmitted through the whole system. (a) The beam profile is still Gaussian; (b) Beam profile after the analyzer transmission is minimized Figure 19. Spatial IF beam shift for a 45 linearly polarized incident Gaussian beam. Dots with the error bars represent the experimental results and the solid line - the theoretical prediction. (The y axis on the right is the amplified data with an amplification factor of 500) Figure 20. SHEL. A linearly polarized and incident on an air-glass boundary beam of light divides very slightly into two components - right and left circular polarizations, or spin parallel and anti-parallel to the propagation direction, right after the refraction at the interface. Views from different angles. Source: 2Physics, (2008), com/2008/02/observation-of-spin-hall-effect-of.html ix

10 Figure 21. The SHEL demonstration at an air-glass interface. (A) Parallel ( + ) and anti-parallel ( ) spin components of a wave packet incident at angle θ I experience a split by a fraction of the wavelength upon refraction by an angle θ T. (B) To satisfy transversality different plane-wave components obtain different polarization rotations upon refraction. The input polarization is in the x I direction for all constituent plane waves. Arrows show the polarization vectors connected with each plane wave before and after refraction. The insets explain the vectors orientation. (C) Theoretical shifts of the spin components for horizontally and vertically polarized incident photons with wavelength λ = 633 nm. Source: O. Hosten and P. Kwiat, Science 319, 787 (2008) x

11 xi ACKNOWLEDGMENTS First, I would like to thank Dr. Anderson for the amazing education and experience in optics gained through this research. I would like to thank Dr. Kusnetsova and Prof. Salamon for taking the time to be on my thesis committee and review my work. In addition, thanks to Prof. Luis Araujo from Brazil, who works in close collaboration with Dr. Anderson, for assistance of building the system for the experiment, guidance through the measurements, and interaction via s during research analysis. Also, thanks to the amazing faculty and staff at SDSU that have shown dedication to the education of students. Thanks to my classmates for the amicableness and tenderness. I appreciate my dear parents, family, and friends for your support, inspiration, and believing in me. I also would like to thank SDSU for the generous financial support and opportunities provided during my two years studying here.

12 1 CHAPTER 1 INTRODUCTION The history of light reflection began in the antique period. Euclid already had a mathematical description of the law of reflection in 300 BC. 1 In the 18th century, Isaac Newton suspected that light can be reflected before hitting the surface. Although not contradicting Euclid, it seems that Newton saw that reflection was not as simple as what Euclid thought. 2 By the 20th century, researchers took a second look at reflection mostly because a real physical beam had a finite width. In 1947 two German scientists Goos and Hänchen reported the first quantitative measurement of a longitudinal shift of the centroid of the beam. 3 A year later, Artmann gave a mathematical description of the Goos-Hänchen shift. This was very close to Newton s representation of the shift. However, Goos and Hänchen saw that the reflection with a dielectric happens at the back of the reflection surface (internal reflection). 2 A transverse shift of the beam was also theoretically observed by Fedorov in and experimentally verified by Imbert in This phenomenon is called the Imbert-Fedorov effect. In the last few years, there were also observations of angular deviations in both longitudinal and transverse directions, which show real deflection from the law of reflection. 6, 7 At the present time, the Goos-Hänchen and Imbert-Fedorov shifts are attracting fast growing attention due to the development of nano-optics applying light evolution at subwavelength scales. 8 Now, it is well known that there are four basic deviations from the geometrical optics picture that can happen when light is reflected: lateral and angular Goos-Hänchen shifts and lateral and angular Imbert-Fedorov shifts. All these shifts are on the order of fractions of a wavelength for the spatial shifts and micro- to milliradians for the angular shifts. 2

13 Light reflection and refraction at a plane dielectric interface are presented in almost all optical systems and are the most basic optical processes. The interaction of a plane wave with an interface is described by Snell s law and the Fresnel equations. This shows the geometrical-optics picture of light evolution. However, for a real optical beam which has a finite width, the situation becomes more complicated. At the wavelength scale the reflected and transmitted beams somewhat do not obey the rules of geometrical-optics evolution. 8 Based on commonly accepted terminology, the shifts of Goos-Hänchen and Imbert-Fedorov in this paper are referred to as GH and IF, respectively. Among the various experimental techniques for the observation of optical beam shifts, the use of weak measurements have been very successful. Weak measurements were introduced in quantum mechanics. This approach was used for the first time by Hosten and Kwiat (see Appendix A) to observe the spin-hall effect of light (SHEL). 6 This phenomenon is a spin-dependent displacement for photons transmitted through an interface and orthogonal to the plane of incidence. 9 Recently the GH and IF shifts were observed via a weak 9, 10 measurement approach. This research is mainly concentrated on the longitudinal spatial Goos-Hänchen and transverse spatial Imbert-Fedorov shifts. Two experiments are performed using the weak measurement technique. The results of these experiments show that not only the GH shift can be seen for linearly polarized light beam at total internal reflection but also IF shift can be observable too, contradicting the common belief of existence for the circular or elliptical polarizations. First, the theory of light reflection is used in this paper to introduce Snell s law and the Fresnel equations along with the important features of the total internally reflected light. The critical angle represents an angle of incidence at which the total internal reflection occurs. The Brewster angle represents an angle of incidence at which light having a particular polarization can be transmitted through a transparent dielectric surface, with no reflection. The expression for the relative phase difference is given in the phase changing on reflection section. The special case of total internal reflection is presented in the evanescent wave section which helps to understand the occurrence of GH and IF shifts. Next, the theory of GH and IF effects are introduced in detail. 2

14 3 Because the GH and IF shifts were experimentally observed using weak measurement technique, the weak measurement itself is described including example involving spin- 1 2 particles as well as an optical version of the effect. Then, the classical optical effects GH and IF shifts are studied using a quantum-mechanical formalism. Finally, the two experiments of this research are going to be presented. The shifts were observed via weak measurement for the linearly polarized light in total internal reflection. The amplification factors of both effect were obtained correctly and experimental results show excellent agreement with theoretical predictions in both experiments.

15 4 CHAPTER 2 LIGHT REFLECTION Figure 1 11 shows the light beams diagram: Figure 1. Diagram for incident, reflected and transmitted rays at an XY-plane interface when the electric field is perpendicular to the plane of incidence. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993). It is well known that the incident light consists of plane harmonic waves: 11 E i = E 0 e i(k i r w i t). Then the reflected and transmitted waves can be expressed: E r = E 0 e i(kr r wrt),

16 5 E t = E 0 e i(kt r wtt). SNELL S LAW AND FRESNEL EQUATIONS The interaction of a plane wave with an interface is described by well-known Snell s law and the Fresnel equations. 8 Snell s law is: n 1 sin θ i = n 2 sin θ t. (1) By introducing a relative refractive index n = n 2 n 1 the Equation 1 can be rewritten as: n = sin θ i sin θ t. (2) The Fresnel reflection and transmission coefficients for the s-polarized beam (TE polarization) are: r s = E r = cos θ n2 sin 2 θ E i cos θ + n 2 sin 2 θ, (3) t s = 2 cos θ cos θ + n 2 sin 2 θ. (4) The Fresnel reflection and transmission coefficients for the p-polarized beam (TM polarization) are: r p = E t = n2 cos θ n2 sin 2 θ E i n 2 cos θ + n 2 sin 2 θ, (5) t p = E t E i = 2n cos θ n 2 cos θ + n 2 sin 2 θ. (6) CRITICAL ANGLE In order to discuss the reflection of light, it is necessary to distinguish between two physically different situations: 11 external reflection: n 1 < n 2 or n = n 2 n 1 > 1, internal reflection: n 1 > n 2 or n = n 2 n 1 < 1.

17 6 In external reflection, the incident beam comes from the side with the smaller index of refraction. In internal reflection, the incident beam is in the medium with larger index of refraction. For the internal reflection (n < 1) there will be values of θ, that is: 12 θ c = arcsin n. (7) That angle is called the critical angle. The critical angle of incidence is reached when the angle of refraction reaches 90 (see Figure 2 11 ). Figure 2. Geometry of rays reflected by a plane interface. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993). THE BREWSTER ANGLE Another interesting feature of the incident light is that: if after reflection the light is linearly polarized, then the angle of incidence is called the polarizing angle or Brewster angle θ p : Figure 3 11 shows the polarization at Brewster s angle. θ p = arctan n. (8)

18 7 Figure 3. Specular reflection of light at a dielectric surface. (a) S-polarization. (b) P-polarization. (c) Polarization at Brewster s angle. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993). Polarizing angles are not the same for the external reflection when n 2 > n 1 and internal reflection when n 2 < n 1. For the BK7 prism case (n 1 =1.51) when light travels from air to glass (external reflection) θ p = For reflection from glass to air (internal reflection) θ p = TOTAL INTERNAL REFLECTION The phenomenon when the internal angle of incidence is equal to or greater then the critical angle (θ i > θ c or θ i > arcsin n) is called total internal reflection or TIR (Figure 2). It is the case when all the incoming energy is reflected back into the incident medium. As incident angle becomes larger, the reflected beam gets stronger while the transmitted beam gets weaker, until θ t vanishes and θ r carries off all the energy at θ r = θ c. 1 PHASE CHANGING ON REFLECTION In the case of external reflection, a π-phase shift of E occurs at any of incidence for the s-polarization and for θ > θ p for the p-polarization. When reflection is internal a π-phase shift occurs for the p-polarization for θ < θ p. However, when θ > θ c = arcsin n, in case of total internal reflection the phase changes are: ( ) sin 2 θ n δ p = 2 arctan 2, (9) n 2 cos θ ( ) sin 2 θ n δ s = 2 arctan 2. (10) cos θ

19 8 The expression for the relative phase difference is: δ = δ p δ s ( cos θ ) sin 2 θ n = 2 arctan 2 sin 2. θ (11) Graphs of δ, δ s and δ p are shown on the Figure Figure 4. Phase shift of electric field for internally reflected rays, with n=n 1 /n 2 =1/1.5. Source: F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics (Prentice-Hall International, Englewood Cliffs, NJ, 1993).

20 9 Summarized results for the case of internal reflection are: , θ < θ p 0 δ p =, θ p < θ < θ ( ) c (12) sin 2 θ n 2 arctan 2, θ > θ n 2 c cos θ 0, θ < θ ( ) c δ s = sin 2 θ n 2 arctan 2 (13), θ > θ c cos θ THE EVANESCENT WAVE IN TOTAL REFLECTION Even though the incident energy is totally reflected when the angle of incidence is equal to or greater than the critical angle, there is still an electromagnetic wave field in the region beyond the boundary. This field is called the evanescent wave and can be expressed as: where α = k t sin 2 θ n 2 E t = E 0t e iωt e ixkt sin θ/n e αz (14) 1 is the real and positive number. The last function of the Equation 14 is e αz. It represents a rapid exponential decrease in the amplitude of the wave as it propagates into the rare medium along the z-direction and becomes negligible at a distance of only a few wavelengths. When the wave penetrates into the rare medium by an amount z = 1 α = the amplitude is decreased by a factor 1/e. λ sin 2π 2 θ 1 n 2 In fact it may seem that the principle of conservation of energy is violated by the wave existence in the rare medium. Instead, a study of the direction of the energy flow by means of the Poynting vector diagram as shown on Figure 5, 12 proves that the energy circulates around and goes back into the dense medium as required by energy considerations. Thus, the energy of this evanescent wave returns to its original medium unless a second medium is introduced 11, 12 into its region of penetration. (15)

21 10 Figure 5. Diagram showing the direction of the Poynting vector for the case of TIR. Source: G. R. Fowles, Introduction to Modern Optics (Courier Dover Publications, Mineola, NY, 1975). The fact that the wave actually penetrates into the rare medium can be demonstrated experimentally. One way to do this is by placing the long faces of two degree prisms close together, but not in actual contact, as can be seen from Figure 6.12 Figure 6. A method for illustrating the penetration of light into the rare medium. Source: G. R. Fowles, Introduction to Modern Optics (Courier Dover Publications, Mineola, NY, 1975).

22 11 Light from a source S is found to be partially transmitted. The amount of transmission depends on the separation of the prism faces. 12 Therefore, when the disturbing influence in the rare medium doesn t exist, the evanescent wave must return to the denser medium, due to total reflection of the light energy. However, in the case of a very narrow beam of light, the reflected beam is physically shifted from the geometrically predicted beam by a very small amount. This phenomenon was studied by Goos and Hänchen in 1947 and is called the Goos-Hänchen shift. 12

23 as total. 3 In Figure 7, 15 an incident ray of light I hit the boundary surface at an angle within the 12 CHAPTER 3 GOOS-HÄNCHEN AND IMBERT-FEDOROV SHIFTS GOOS-HÄNCHEN SHIFT An incident beam upon an interface, separating two different dielectric media at an angle of incidence greater than the critical angle, enters some distance inside the medium of lower refractive index. And then it reenters the medium of higher refractive index at a point on the interface laterally shifted from the point of entry. This separation distance is called the Goos-Hänchen lateral shift. After being laterally displaced from initial position, the beam proceeds into the medium of higher refractive index with an angle of reflection equal to the angle of incidence. 13 This phenomena was interpreted by Goos and Hänchen in 1947 as the tunneling of the light beam through the air. 14 The actual Goos and Hänchen experiment demonstrates what happens at total reflection if the incident wave enters into the medium of lower index and reemerges into the medium of higher index. 15 The theory of total reflection states that such reflection doesn t appear suddenly at the boundary surface between the optically denser and the less dense medium. Instead, the energy penetrates a short distance into the less dense medium before the total reflection takes place. After the boundary, it shows a rapid, exponential decrease, representing a transversely attenuated wave - evanescent wave. The most remarkable feature of this process is that the light energy flows from the denser into the less dense medium at one point, and comes back with its entire strength to the denser medium at another point. Thus, the reflection is justified total reflection range. If total reflection were to occur right at the boundary at point 1, the totally reflected light ray would follow the path R 1.

24 13 Figure 7. Principle of the ray path. I-incident beam. R 1 -reflected beam shows the behavior similar to internally reflected beam; R 2 -reflected beam shows the behavior expected by Goos and Hänchen. D-Goos-Hänchen shift. Source: R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964). However, if the ray first enters the less dense medium, it will leave it again nearly at the point 2 and travel along R 2. The described phenomenon will occur only with a narrow beam with a certain small angle of aperture. In that case, the initially imagined ray R 1 will be subject to a displacement D and causes the actual ray R 2 to appear. The displacement D is very small with order of magnitude: length of a light wave. 3 According to, 15 the cause of the shift in TIR is energy inflow into the medium of lower index on one side of the beam and outflow into the medium of higher index on the other side of the beam. The overall phenomenon of some energy transfer from one side of the beam to the other results in a translation. As it can be seen from Figure 8 15 the time-average flux of energy Φ 1 for the plane wave across a strip of width D is the Goos-Hänchen shift. The D shift must be equal to the time-average flux of energy Φ 2 in the whole medium of lower index, parallel to the plane of separation. This statement is evident when the right side of the reflected beam is taken into account. If there were no shift, the reflected beam would follow the path Ax. Because of the shift, the actual path is A x. Therefore, the energy of the beam, between Ax and A x can only be the energy of the surface wave coming back into the medium of higher index. This provides a straightforward evaluation of the shift. 15

25 14 Figure 8. Illustration of the Goos-Hänchen shift for a finite plane wave. W is the limited incident plane wave. Φ 1 and Φ 2 are the two energy fluxes. Source: R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964). Therefore, mathematical representations of the shifts for p and s polarized beams near the critical angle of total reflection are: D p = sin(θ) πn 2 D s = sin(θ) π [ ] λ, (16) sin 2 (θ) n 2 [ ] λ. (17) sin 2 (θ) n 2 When the incident angle θ is greater then, but close to, the critical angle in TIR and sin(θ) = n = n 2 n 1, then Equations 16 and 17 become D p = 1 n 2 1 πn 1 [ D s 1 = πn 1 [ n 2 n 2 ] λ, (18) sin 2 (θ) n 2 ] λ. (19) sin 2 (θ) n 2 IMBERT-FEDOROV SHIFT From the previous chapters it is already known that an optical beam subjected to TIR on a dielectric plane between media of high and low indices of refraction is slightly shifted from the position of the incident beam. The effect is called the Goos-Hänchen (GH) shift and

26 15 corresponds to a longitudinal displacement in the incident plane, and occurs for the linearly polarized beam. For the different polarization states, for instance circular or elliptical, there are two simultaneous displacements: one shift (longitudinal) in the incident plane that could still be considered as being due to the GH effect and one (transverse) in the perpendicular direction, also known as the Imbert-Fedorov (IF) effect. 16 Figure 9 16 shows GH and IF spatial shifts at total internal reflection. Figure 9. GH and IF spatial shifts at total internal reflection. Source: F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, 1863 (2004). Imbert stated in his work 5 that the new lateral shift of the reflected beam should be considered as the energy flux but in the transverse direction, similar to GH energy flux in longitudinal direction. Futhermore, the size of the transverse shift is much smaller then the size of the GH shift. However, in contrast with the commonly assumed polarization dependency of the shift, the IF effect is also observable for a linearly polarized beam at TIR. 9, 17 Additionally, it is shown in 17 that the dependency of the IF shift on the polarization states is in good agreement with the experimental data and the shift doesn t vanish for the linearly polarized light.

27 16 CHAPTER 4 WEAK MEASUREMENT There are many analogies that exist between phenomena in quantum theory and in classical wave optics. Some of them include the transverse optical polarization and spin- 1 as 2 2-state systems, and the Schrödinger equation and paraxial equation of light. The main goal here is to describe and explore the strong analogy between the spatial shift of light beams experienced on reflection from a planar interface and the idea of weak quantum measurement. 18 WHAT IS WEAK MEASUREMENT? In quantum mechanics, weak measurement is some sort of interaction or disturbance, where the observed system or particle is very weakly coupled to the measuring device. So, wave function does not collapse but continues on unchanged. A pointer initially points at zero before the measurement would point at the weak value after the measurement. Therefore, the measuring device pointer is shifted by the weak value after the measurement. Moreover, the system is not disturbed by the measurement. This formalism is within the boundaries of the theory and does not contradict with any fundamental concept, particularly - Heisenberg s uncertainty principle. In addition, in weak measurement theory, the system and the measurement device together are quantum systems. Weak measurement has two steps: in the first step the quantum measurement device is weakly coupled to the quantum system; in the second step the measurement device is measured strongly. The collapsed state of the measurement device represents the outcome of the weak measurement process. 19 In 1988 the idea of weak measurements and weak values was first developed by Yakir Aharonov, David Albert and Lev Vaidman. It is very useful for gaining information about preand post-selected systems described by the two-state vector formalism. The two-state vector is represented by:

28 17 Φ Ψ where the state Φ develops backwards from the future (future-to-past) and the state Ψ develops forward from the past (past-to-future). If φ i and φ f are the pre- and post-selected quantum mechanical states, respectively, then the weak value of the observable  is defined as A w = φ f  φ i φ f φ i. (20) This quantity is very important because it represents the outcome of the operation  φ i, post-selected into a new state φ f rather than being compared with the initial state φ i. The weak value is only defined when φ f is not orthogonal to φ i, otherwise the denominator would be equal to zero. 20 In addition, the weak value may become a big number - superweak, especially when φ i and φ f are almost orthogonal. Thus, the denominator of Equation 20 becomes vanishingly small. 18 EXAMPLE INVOLVING SPIN- 1 2 PARTICLES In the paper 21 Aharonov, Albert and Vaidman (AAV) illustrated the following experiment of the standard Stern-Gerlach type which measured a spin component of a spin- 1 2 particle. A beam of spin- 1 2 particles moves in the y direction with defined speed. The spins of the particles point in the xz plane at an angle α to the x axis. The spatial wave function of the particles has a Gaussian shape with width in the z direction. Therefore, the beam is diverging with a momentum spread p z = 1/(2 ). The prepared beam goes through a Stern-Gerlach device which does the measurement of the z component of the spin. This creates a relation between the spin operator ˆσ z and the z coordinate through the coupling Hamiltonian H = λg(t)ẑˆσ z (21) where λ is proportional to the particle s magnetic moment and the localized function g(t) is normalized to 1. The quantum system s state Ψ corresponds to the spin state of the particle, whereas the state of the measuring device Φ corresponds to the spatial wave function.

29 18 It should be considered in a weak measurement of λσ z that the beam splitting δp z, induced by the Stern-Gerlach magnet, is small compared to momentum spread p z = 1/(2 ) - the overall p z spread of the beam. In that case, the σ z =+1 and -1 components of the beam continue to overlap strongly and are not purely separated as they would be in a strong measurement (Figure 10 6 ). Figure 10. Pre-selection and post-selection. Source: O. Hosten and P. Kwiat, Science 319, 787 (2008). To post-select the spin state, the beam has to be passed through a second magnet with a strong Stern-Gerlach field aligned in the x direction. As a result, the beam is split into two well-separated beams and the σ x =+1 beam is selected and displayed on a distant screen (Figure ). Figure 11. Stern-Gerlach magnet layout for the AAV experiment. Source: I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, 2112 (1989). The initial spin state is the +1 eigenstate of (cos α)σ x + (sin α)σ z : Ψ in = 1 cos α 2 + sin α 2 2 cos α 2 sin α 2, (22) and the final state is the +1 eigenstate of ˆσ x : Ψ f = (23)

30 19 Therefore, Ψ f Ψ in = cos α 2, Ψ f ˆσ z Ψ in = sin α 2, (24) so that the weak value of the spin component σ z is A ω = Ψ f λ ˆσ z Ψ i Ψ f Ψ i The initial spatial wave function is expressed as: = (λ σ z ) ω = λ tan α 2. (25) ) φ in (q) q Φ in = exp ( z2 f(x, y). (26) 4 2 The precise x and y dependences are unimportant and should be ignored from now on. The final (p-representation) wave function is φ f (p) p Φ f ( cos α ) ( exp [ 2 p z λ tan α ) ] 2. (27) 2 2 The observed distribution is Gaussian and centered on tan α/2. However, the restriction which is [ λ 1 min tan α 2, cot α ]. (28) 2 implies that, for a given, one cannot approach too close to α = π. Otherwise the measured value, tan α/2, can be much greater than unity. To understand this better, the special example is shown where α = π 2ɛ with ɛ 1. Then the final wave function becomes: and is valid when φ f (p) ɛ exp ( [ 2 p z λ ) ] 2 ɛ (29) The corresponding exact result is: λ ɛ 1. (30) φ f (p) = ϕ(p z ; ɛ,, λ) (31)

31 20 where ϕ is defined by ϕ(p; ɛ,, λ) 1 (1 + ɛ) exp[ 2( 2 (p λ) 2 ] (1 ɛ) exp[ 2 (p + λ) 2 ] ). (32) Equation 32 shows that the wave function is a superposition of two Gaussians, centered on p = ±λ, corresponding to the σ z = ±1 eigenstates. The near cancellation of the two terms leaves a small difference function ϕ(p), which turns out to be approximately Gaussian and peaked at p = λ/ɛ. This fact is illustrated in Figure 12, 22 which shows ϕ(p) for ɛ=0.2 and λ =0.01. Figure 12. Graph of the function ϕ(p; ɛ,, λ) (see Equation 32) as a function of P p/λ with ɛ=0.2 and λ =0.01. Source: I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, 2112 (1989). If the initial and final spins are selected to be at angles α and β (previously, β was taken to be zero) to the x axis, respectively, then the weak value of the operator λˆσ z would be A ω = λ/ɛ, where ɛ = ɛ (1/2) (α, β) cos [ 1(α β)] 2 sin [. (33) 1(α + β)] 2 The final wave function would be [ ] ( 1 φ f (p) = cos (α β) exp [ 2 p z λ ) ] 2 2 ɛ (34)

32 21 and is valid if The exact corresponding result can be [ λ min ɛ, 1 ]. (35) ɛ [ ] 1 φ f (p) = sin (α + β) ϕ(p z ; ɛ,, λ) (36) 2 in terms of the function introduced in Equation 32. The above formulas are valid for any value of ɛ. However, the interesting effects occur when ɛ is small. OPTICAL ANALOG OF WEAK MEASUREMENT In an optical analog of the previous experiment, the beam of spin- 1 2 particles is replaced by a Gaussian-mode laser beam and the pre-selection and post-selection Stern-Gerlach magnets are replaced by optical polarizers. 23 The setup is shown in Figure Figure 13. An optical analog of the AAV experiment. A broad, coherent beam passes through a polarizer (P) and analyzer (A). Between them is a birefringent crystal (C) which produces a small lateral displacement between x (o ray) and z (e ray) polarizations (see inset). Source: I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, 2112 (1989).

33 The initial and final polarizations selected by a polarizer and analyzer are at angles α and β to the x axis, respectively. The weak measuring device is placed between them. It is a slab of weakly birefringent crystal, which represents a small lateral displacement between the ordinary and extraordinary rays. These rays are arranged to correspond to the x and z polarizations, respectively. To be able to calculate the lateral displacement for a given crystal, it has to be small compared to the width of the beam. Displacement produced by the birefringent material is lateral, and not an angular deflection as in the case of a Stern-Gerlach magnet. Therefore, the z coordinate has to be considered instead of p z distribution; and δ 1/(2 ) will exchange roles, relative to the previous discussion. The initial beam has a Gaussian profile with a large spatial width, 1/(2δ). When the beam is passed through the polarizer, its wave function becomes q Φ in Ψ in = exp( δ 2 z 2 ) cos α sin α 22 (37) where Ψ is a two-vector and not a spinor. After passing through the birefringent material the o ray (x-polarization) and e ray (z-polarization) are shifted by different amounts a 1 and a 2, so the emerging beam will be cos α exp[ δ2 (z a 1 ) 2 ] sin α exp[ δ 2 (z a 2 ) 2 ]. (38) The analyzer projects out the β component of polarization. Thus, the produced spatial wave function looks like: q Φ f = cos β cos α exp[ δ 2 (z a 1 ) 2 ] + sin β sin α exp[ δ(z a 2 ) 2 ]. (39) This can be rewritten in terms of the ϕ function introduced in Equation 32: The parameters are given by φ f (q) q Φ in = cos(α + β)ϕ(z a; ɛ, δ, λ). (40) and a = 1 2 (a 1 + a 2 ), λ = 1 2 (a 1 a 2 ) (41)

34 ɛ = ɛ (1) (α, β) 23 cos(α β) cos(α + β). (42) Here ɛ is a whole angle. Because of this, the interesting region in which ɛ is very small now corresponds to nearly orthogonal polarizers. The beam shift can be seen by changing ɛ. 22

35 24 CHAPTER 5 GOOS-HÄNCHEN AND IMBERT-FEDOROV SHIFTS FROM A QUANTUM-MECHANICAL PERSPECTIVE According to, 24 a close connection between the beam shifts and the weak measurements exists. In the experiments 6, 9, 10 weak measurements are used to enhance and therefore, make beam shift effects observable. Knowing the correspondence between the paraxial wave equation and the Schrödinger equation (see Appendix B), the electric field of a paraxial beam can be described as the wave function of a nonrelativistic quantum-mechanical spin- 1 particle. The two unit vectors that 2 span the transverse plane perpendicular to the beam propagation axis z are e 1 = (1, 0) and e 2 = (0, 1). The light beam polarization can be written as a two-component spinor: A a 1 e 1 + a 2 e 2 = {a 1, a 2 } where a a 2 2 = 1. The three 2 2 matrices that represent the Fresnel reflection (F), the longitudinal Goos-Hänchen (GH) shift and the transverse Imbert-Fedorov (IF) shift, respectively, are: F r p(θ) 0, (43) 0 r s (θ) i ln r p 0 GH θ 0 i ln r, (44) s θ ( 0 i 1 + r ) p cot θ IF r s ( i 1 + r ) s. (45) cot θ 0 r p r p (θ) and r s (θ) are the Fresnel reflection coefficients evaluated at the incident angle θ (note the correspondence p with p-polarization and s with s-polarization).

36 25 In the weak measurement experiment when a polarizer is placed in front of the detector and oriented along the direction B b 1 e 1 + b 2 e 2 = {b 1, b 2 } with b b 2 2 = 1 the polarization of the beam will be projected along this direction. Beam shifts are defined by the displacement of the centroid of the beam distribution after reflection with respect to the reflected beam centroid. Thus, the expectation value (weak value) of the shift in the reference frame attached to the reflected beam has to be calculated. The post-selected enhanced GH and IF shifts are: GH B GH A B A = i b r p 1 a θ 1 + b 2 rs a θ 2, (46) b 1r p a 1 + b 2r s a 2 IF B IF A B A b = i(r p + r s ) cot θ 1a 2 b 2a 1. (47) b 1r p a 1 b 2r s a 2 become: If a 1 =0 (s input polarization) or a 2 =0 (p input polarization) then Equations 46 and 47 GH p = i r p r p θ, GH s = i r s r s θ, (48) ( IF p = i b r ) ( s cot θ, IF b s = i b r ) p cot θ. (49) 1 r p b 2 r s From Equation 48 it follows that the ordinary GH shift cannot be enhanced if one has either s (a 1 = 0) or p (a 2 = 0) input polarization. However, the IF shift can be enhanced for the same kind of input states.

37 26 CHAPTER 6 EXPERIMENTAL WEAK MEASUREMENT OBSERVATION OF THE GOOS-HÄNCHEN SHIFT The measurement technique of this experiment is an optical analog of the quantum weak measurement concept introduced in Chapter 5. The GH shift of linearly polarized beam is going to be experimentally observed in TIR and compared with the theoretical predictions. The prism, as the weak measuring device, introduces a small lateral displacement D p for the p polarization (D s for the s polarization) of a Gaussian beam in TIR. The Cartesian coordinate system is considered with the direction of the beam propagation along the z axis with the vertical direction along the y axis and x - perpendicular to both z and y. The initial state (pre-selection) is selected by the polarizer P1 and the final state (post-selection) by the analyzer - quarter-wave plate (QWP), half-wave plate (HWP) and the polarizer P2. The weak measurement system works as follows: the polarizer is set to α = 45 and the analyzer to β = α ɛ. With ɛ 1 the initial and final states are nearly orthogonal. Then the emerging beam is laterally shifted by the ± 1 2 GH cot(ɛ), where GH = D p D s. The separation GH cot(ɛ) is measured in between the two beams with the two polarizations settings α = 45 and β = α + 90 ± ɛ, respectively. Therefore, the small beam displacement GH due to the GH effect is increased by a factor of cot(ɛ). The light is produced from a 633 nm He-Ne laser. The beam expander is built with lenses L1 and L2 to generate a collimated Gaussian to decrease the number of problems with multiple reflections in the prism. A lens L3 focuses the light from a collimated beam to the size of beam waist ω 0 =45µm. After a polarizer P1 is set at α = 45, the beam experiences TIR in a BK7 prism with n=1.51 at 633 nm. When the beam is emerged from the analyzer the shift is measured on the camera Win Cam D.

38 27 The experiment runs as follows: first, the P2 is set to β = α + 90 ; second, the QWP and the HWP are set in order to minimize the light transmitted to the camera; third, the P2 is set to β = α ɛ and measurements are taken, and right after P2 is set to β = α + 90 ɛ, measurement are taken again. Thus, the beam displacement GH cot(ɛ) can be measured with the camera. The experimental setup is shown on the Figure Figure 14. Experimental setup for the observation of the Goos-Hänchen shift via weak measurement. L1 and L2 - lenses for the beam collimation, L3 and L4 - lenses for the beam focusing, P1 and P2 - Glan-Thompson polarizers, QWP - quarter-wave plate, HWP - half-wave plate, Win Cam D - camera. Source: G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 38, 1232 (2013). The angle ɛ is selected to be rad (0.5 ) as experiment 10 suggested as an appropriate compromise in between transmitted power P t =10 3 P (where P is the maximum power that can be transmitted through polarizer P2) and lateral displacement amplification cot(ɛ) = 115. Also, according to, 25 strong enhancements can be expected close to angles at

39 28 which no light is transmitted for a fixed initial and final polarizations. Therefore, in order to see the shift, the strong enhancement is a necessity. As we can see, the left side of the Figure 15 corresponds to a measurement of a weak value, the beam profile of the transmitted beam is still Gaussian. Figure 15. Beam profile of the light transmitted through the whole system. (a) Gaussian beam profile; (b) Beam profile for the crossed polarizers with ɛ = 0 rad. However, the right side corresponds to crossed polarizer or orthogonal initial and final states. Therefore, the transmitted beam is not anymore Gaussian but has two peaks which are separated by a distance approximately 2ω, where ω is the beam waist expected for the input Gaussian beam in the camera plane. The experimental data is shown in Figure 16 and corresponds to the GH shift GH derived from measurements. The horizontal axis represents the angle of incidence and the vertical axis - the GH shift of a p-polarized beam with respect to a s-polarized beam. The experimental data (dots) are the measurements divided by the amplification factor cot(ɛ) = 115. The solid line shown on the graph is the theoretical predictions for a Gaussian beam (D p D s ), where D p and D s are derived from the Equations 16 and 17, respectively. From the experimental data, there were no access to the individual GH shift for p polarization (D p ) or s polarization (D s ). What was measured is the amplified difference between the GH shift for p and s polarization; that is, (D p D s ) cot(ɛ). The data gathered during the experiment is in micrometer scale and the amplification factor is cot(ɛ)=115. In order to get GH = (D p D s ) in micrometers, the experimental data has to be divided by the amplification factor 155.

40 29 Figure 16. Experimental data (squares) with the error bars are compared with the theoretical predictions (solid line) for GH shift in TIR. Then the measured data must be compared with the theoretical prediction. The Equation 17 has to be subtracted from Equation 16 which will give (D p D s ). The result is in nanometers, so it has to be divided by 1000 to convert it to micrometers. After these corrections, the theoretical shift still had to be multiplied by 4.5 to get it to fit with the experimental data. This value of 4.5 could be a geometrical amplification factor coming from the fact that the beam was expanding. The ratio z/z 0 is equal to 5, where z is the focal length f=50mm of lens L4, and z 0 =10mm is Rayleigh range. Therefore, it can be assumed that the factor of 4.5 is actually the z/z 0 ratio. The difference between these two values could be due to the fact that the beam was not focused exactly at the point where it was expected. The absence of the experimental data around 45 of incidence is because of inevitable problems with multiple reflections in the prims. According to the paper, 10 the p and s components of the beam emerge in phase from the weak measuring device. In TIR the p and s components have different phase shifts δ p

41 and δ s that rely on the angle of incidence. The wave function of the beam after passing the polarizer P1 is (as the Equation 37 where δ = 1/ω and z = x): ] cos(α) exp [ (x)2 ω 2 ] sin(α) exp [ (x)2 = ω 2 ] 1 exp [ (x)2 2 ω 2 ] 1 exp [ (x)2 2 ω 2 30, (50) the p and the s beam components (see Equation 38 where a 1 = D p and a 2 = D s ) emerging from TIR are cos(α)e i δ 2 exp [ (x D ] p) 2 ω 2 sin(α)e i δ 2 exp [ (x D ] s) 2 ω 2 = [ 1 e i δ 2 exp (x D p) 2 2 ω 2 [ 1 e i δ 2 exp (x D s) 2 2 ω 2 ] ] (51) where δ = δ p δ s is the phase difference between p and s (Equation 11), D p and D s (Equations 16 and 17) are the spatial GH shifts due to TIR. The multiplication by a complex exponential affects neither the relative phase nor polarization. 10 The QWP and the HWP are used to compensate for the relative phase.in this particular case of representing this compensation the Jones matrix is diagonal. With a non-diagonal matrix the two polarization components would be mixed and, therefore, the beam shifts would be mixed as well: e i δ e i δ 2. (52) The beam coming out from the retarder has the form (as Equation 38): cos(α) exp [ (x D ] p) 2 ω 2 sin(α) exp [ (x D ] s) 2 = ω 2 1 exp [ (x D ] p) 2 2 ω 2 1 exp [ (x D ] s) 2 2 ω 2. (53) After light emerged from the polarizer P2 we have the spatial wave function as the same as Equation 39: f(x) = cos(β) cos(α) exp [ (x D ] p) 2 + sin(β) sin(α) exp [ (x D ] s) 2. (54) ω 2 ω 2

42 31 It also can be written as Equation 40: f(x) = cos(α + β)φ(x a; ɛ, ω 0, GH ) (55) where a = 1 2 (D p + D s ), tan(ɛ) ɛ, and GH = (D p D s ). The φ is given by φ(x) = 1 [ (1 + ɛ) exp [ (x 1 2 GH) 2 ] 2 ω 2 [ (1 ɛ) exp (x GH) 2 ]] ). (56) ω 2 It is obvious to see from the last equation that the wave function is a superposition of two Gaussians, centered at x = ± 1 2 GH. Also, for 1 2 GH/ω ɛ 1 the two Gaussians interfere destructively and produce a single Gaussian. The centroid of this single Gaussian is shifted by the weak value 1 2 GH/ɛ 1 2 GH cot(ɛ). 10 The results of the GH experiment are good. By choosing a fixed value for ɛ = ± rad a precise amplification of cot(ɛ) = 115 was obtained for the GH curve and experimental data was able to fit with the graph of theoretical predictions.

43 32 CHAPTER 7 EXPERIMENTAL WEAK MEASUREMENT OBSERVATION OF THE IMBERT-FEDOROV SHIFT In the IF shift experiment the weak value of the linear polarized light beam was measured. First, the polarization of a Gaussian beam that undergoes TIR was pre-selected. Then the IF shift acted as the weak measuring effect, and the final polarization state was post-selected by the analyzer. Obviously, the displacement of the beam centroid on a position of sensitive detector was amplified. Also, the experimental results were compared with the theoretical predictions. The experimental setup is shown on the Figure Figure 17. Experimental setup for the observation of the Imbert-Fedorov shift via weak measurement. L1 and L2 - lenses of the beam expander, L3 and L4 - lenses for the beam focusing and collimating, respectively, P1 and P2 - Glan-Thompson polarizers, QWP - quarter-wave plate, HWP - half-wave plate, Win Cam D - camera. Sourse: G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 39, 2266 (2014). The IF effect can be observed when the incident Gaussian beam is totally reflected by a BK7 prism. The light from a He-Ne 633 nm laser is focused to a spot with

44 33 the size ω 0 = 45µm by a lens L3. Then the beam is p polarized using a Glan-Thompson polarizer P1. After reflection an analyzer - quarter-wave plate (QWP), half-wave plate (HWP) and the polarizer P2 post-selects the desired polarization states. A lens L4 (focal length f=250 mm) is used for the emerging beam collimation from the analyzer. The output signal is read by a Win Cam D. The polarizers P1 and P2 are crossed. Hence, P2 transmits s polarized light only. Then the QWP and the HWP are set in order to minimize the light transmission through P2. In that case the transmitted beam has a double peak intensity profile (Figure 18). Figure 18. Beam profile of the light transmitted through the whole system. (a) The beam profile is still Gaussian; (b) Beam profile after the analyzer transmission is minimized. The two peaks have separation of 2ω where ω is the beam waist for the input Gaussian beam on the Win Cam D. According to the paper 9 the IF effect was measured with an experimental scheme derived from quantum mechanics, even though this effect is completely classical. matrix: 24 In the TIR case in a fully quantum-mechanical form the IF shift is given by the 2 2 ( ) 0 i 1 + r 1 r IF = 2 cot(θ) ( ) i 1 + r 2 r 1 cot(θ) 0 ( ) 0 i 1 + r 1 r = cot(θ) ( ) 2. i 1 + r 2 r 1 0 (57)

45 34 From 9 the IF shift is given as: IF = ( 1 + cos(δ) ) cot(θ)σ 2 sin(δ) cot(θ)σ 1 (58) where δ = δ p δ s is the phase difference between p and s polarized beams in TIR, θ is the incident angle and σ 1, σ 2 are the Pauli matrices: σ 1 = 0 1 σ 2 = 0 i. (59) 1 0 i 0 As a result, IF = ( 1 + cos(δ) ) cot(θ) 0 i sin(δ) cot(θ) 0 1 i i ( 1 + cos(δ) + i sin(δ) ) = cot(θ) i ( 1 + cos(δ) i sin(δ) ). 0 (60) From Equations 57 and 60 it can be seen that the operator IF is hermitian. The [ ] [ ] component σ 1 is diagonal ([ ]) in the linear polarization basis 1/ 2 1/ and 1/ 2 2 1/ ; the 2 ] ] component σ 2 is diagonal ([ ]) in the circular polarization basis and. It [ 1/ 2 i/ 2 [ 1/ 2 i/ 2 should be noted that the p polarization is p = (1, 0) and the s polarization is s = (0, 1). If pre-selected polarization of the incident beam is p polarized and post-selected final polarization is ψ = (ɛ, 1), ɛ is a small angle, then the weak value of the IF matrix is: ψ IF p ψ p = ( ) 0 i ( 1 + cos(δ) + i sin(δ) ) ɛ 1 cot(θ) i ( 1 + cos(δ) i sin(δ) ) ( ) ɛ where

46 35 ( ) 0 i ( 1 + cos(δ) + i sin(δ) ) ψ IF p = ɛ 1 cot(θ) i ( 1 + cos(δ) i sin(δ) ) = cot(θ) ( i ( 1 + cos(δ) i sin(δ) ) iɛ ( 1 + cos(δ) + i sin(δ) )) 1 0 = cot(θ) ( i ( 1 + cos(δ) i sin(δ) )) = cot(θ) ( sin(δ) + i(1 + cos(δ) ) and ψ p = ( ) ɛ = ɛ. As a result, ψ IF p ψ p = cot(θ) ( ) sin(δ) + i(1 + cos(δ). (61) ɛ The real part of the weak value comes from σ 1 and it introduces the IF shift of a (1/ 2, 1/ 2) linearly polarized beam. The imaginary part comes from σ 2 and it introduces the IF shift of a (1/ 2, i/ 2) circularly polarized beam. If pre-selected polarization of incident beam is p polarized, and post-selected final polarization is φ = (ɛ, i) then the weak value of the IF matrix is: φ IF p φ p = ( ) 0 i ( 1 + cos(δ) + i sin(δ) ) ɛ i cot(θ) i ( 1 + cos(δ) i sin(δ) ) ( ) ɛ i 1 0 where

47 36 ( ) 0 i ( 1 + cos(δ) + i sin(δ) ) φ IF p = ɛ i cot(θ) i ( 1 + cos(δ) i sin(δ) ) = cot(θ) (1 + cos(δ) i sin(δ) iɛ ( 1 + cos(δ) + i sin(δ) )) 1 0 = cot(θ) ( 1 + cos(δ) i sin(δ) ) and φ p = ( ) ɛ i 1 0 = ɛ. As the result, φ IF p φ p = cot(θ) ( ) 1 + cos(δ) i sin(δ). (62) ɛ Here the real part of the weak value comes from σ 2 and the imaginary part - from σ 1. The experimental set up permits to measure the two contribution σ 2 and σ 1 of the IF shift and observe the imaginary part of these weak values. As explained in 7, 9 the imaginary part increases as the beam propagates. These dimensionless weak values are rewritten as beam shifts as: y ψ = λ ( cot θ sin δ + z ) (1 + cos δ), (63) 2π ɛ z 0 y φ = λ ( cot θ 1 + cos δ z ) sin δ 2π ɛ z 0 (64) where y ψ and y φ are the centroids position of the post-selected beams in the direction orthogonal to the incident plane. The real part of the complex weak value is comparable to λ/(2π) where λ is the light wavelength. Because the imaginary part increases with the propagation distance from the waist of the beam, 6, 7 it scales with λ/(2π) (z/z 0 ). Here z 0 = π ω2 0 is the Rayleigh range, λ with ω 0 as the beam waist of focused light beam at the back of the prism, and z is the focal

48 37 length of the lens L4. In practice if z/z 0 is sufficiently big then the imaginary part contribution of the weak value is much more increased than the real one. In this particular experimental set up the propagation distance z is equal to the focal length f=250mm, and Rayleigh range is z 0 =10mm so z = f = 25 z 0. With such a z/z 0 ratio, the length of beam propagation distance is long enough. Therefore, the terms that don t contain the z/z 0 can be ignored in Equations 63 and 64 because they are small. Therefore, y ψ = λ cot θ 2π ɛ = λ 2π cot θ ɛ z λ cot θ z cos(δ) z 0 2π ɛ z 0 z (1 + cos(δ)), (65) z 0 y φ = λ cot θ z sin(δ). (66) 2π ɛ z 0 Under these circumstances, Equation 65 represents the IF shift of a circularly polarized beam and Equation 66 represents the IF shift of a linearly polarized beam. In the measurement of the spatial IF shift for the 45 linear polarized Gaussian experiment the HWP was rotated of an angle (+ɛ/2) and then of ( ɛ/2). The relative position of the beam centroid was measured in one case with respect to the other. According to these settings, the Jones matrices for the QWP (Q) and for the HWP (H) are: Q = i 0, 0 1 H = R( (±ɛ/2)) H 0 R(±ɛ/2) (67) = 1 (±ɛ/2) ±ɛ/2 ±ɛ/ (±ɛ/2) 1 = 1 ɛ2 /4 ±ɛ ±ɛ 1 + ɛ 2 /4 = 1 ±ɛ. ±ɛ 1

49 38 The approximations 1 ɛ 2 /4 1 and 1 + ɛ 2 /4 1 were made due to ɛ 2 /4 1 (ɛ = rad and ɛ 2 = rad 2 ). After the measurement the two post-selected final states are: ( ) s HQ = ±ɛ i 0 ±ɛ ( ) = 0 1 i ±ɛ ±iɛ 1 = (±iɛ, 1) = (±ɛ, i). The experimental data is shown in the next Figure 19 and corresponds to the IF shift IF derived from the measurements. (68) Figure 19. Spatial IF beam shift for a 45 linearly polarized incident Gaussian beam. Dots with the error bars represent the experimental results and the solid line - the theoretical prediction. (The y axis on the right is the amplified data with an amplification factor of 500).

50 The horizontal axis represents the angle of incidence, the primary vertical axis - the IF shift for 45 linearly polarized beam, and the secondary axis - the amplified data. The solid line shown on the graph is the theoretical predictions for the spatial shift of a 45 linearly polarized incident Gaussian beam, derived from the Equation 66. What was measured during the experiment is the amplified shift for 45 linearly polarized beam. In the measurements ɛ was set to rad. Then, the total amplification factor is 1 ɛ z = = 500. z 0 The data gathered during the experiment is in micrometer scale and the amplification factor is 500. In order to get IF in micrometers, the experimental data has to be divided by the amplification factor 500. Then the measured data must be compared with the theoretical prediction. The results of Equation 66 has to be divided by 1000 to convert it to micrometers. After these corrections, the theoretical shift still had to be multiplied by 1.76 to get it to fit with the experimental data. This value of 1.76 could be a geometrical amplification factor coming from the fact that the beam was not absolutely collimated after lens L4 and was expanding. The results of the IF experiment are good. By choosing a fixed value for ɛ = ±0.049 rad a precise amplification of 1/ɛ z/z 0 =500 was obtained of the IF curve and experimental data were able to fit with the graph of theoretical predictions. Unfortunately, the data near 45 of incidence couldn t be recorded. Most likely, it could be because of unavoidable problems with multiple reflections in the prism. Also, when light was transmitted through both polarizers and the prism, a pattern of interference fringes was seen. This could produce the actual noise in the data. This particular result of IF effect is significant because it shows that with weak measurement approach the linearly polarized dependent part of the IF shift can also be observed. 39

51 40 CHAPTER 8 CONCLUSION The realization of the weak value measurement was demonstrated during the experiments. It was confirmed that this approach is very valuable for observing tiny displacements in TIR and is allowed for a neat amplification of optical effects. While measuring spatial beam shifts of tens or hundreds of nanometers is in general a challenging task, measuring amplified light beam shifts of tens or hundreds of microns is quite easy. The paper began with the theory of light reflection where Snell s law, the Fresnel equations along with TIR, and special cases of TIR were introduced. The theory of GH and IF effects were covered, and the concept of weak measurement showed how these effects can be enhanced for the measurements in the laboratory. Quantum-mechanical notation provided a useful tool to derive spatial GH and IF shifts with finite beam width at TIR. During the research, the phenomena of the GH and IF shifts have been presented using the optical analog of quantum weak measurement. This principle was certainly valid for quantum states. For instance, the initial and final states were the polarization states of the laser beam. The eigenvalues of the weak measurement were the displacements of the beam. The weak value was only defined when the initial state was not orthogonal to the final state. Therefore, the shift couldn t be detected when the polarizer and the analyzer were orthogonal to each other. Analysis of the data from both experiments showed the agreement with Imbert predictions that the IF is much smaller than the GH. It also can be seen from the figures with the experimental data. The amplified factor of IF shift is much bigger than the amplified factor of GH shift. The double peak intensity profile was seen for the minimum analyzer transmission in both experiments. In addition, the separation between the two peaks in both experiments was equal to 2ω where ω was the beam waist of the centroid at the camera plane.

52 41 The results of both experiments are good. In both experiments by choosing a fixed value for ɛ, precise amplifications of the GH and IF curves were obtained. Experimental data was able to fit with the graph of theoretical predictions. Thus, the weak measurement provides an accurate amplification factor. The results of the second experiment proves that IF shift exists even for linearly polarized light. It could be possible to improve the results of the experiment by using a diode laser with wavelength of 800 nm and higher. This type of laser has a short coherence length to compare with HeNe laser. With HeNe laser we were getting lots of interference which could caused the appearance of fringes on the camera screen from the polarizers and the prism. This could produce the actual noise in the data that was seen on the experimental graphs, especially in the IF shift case. To sum up, the Goos-Hänchen and Imbert-Fedorov shifts are attracting more and more attention due to the development of nano-optics applying light evolution at subwavelength scales. Weak measurement is becoming more and more practical in quantum world. The results of the experiments described in this paper and other results 6, 9, 10, 23 show a great potential of weak measurement to observe tiny phenomena.

53 42 REFERENCES 1 E. Hecht, Optics (Addison-Wesley Longman, Inc., Chicago, IL, 2002). 2 N. Hermosa, BKR Blog, (2012), beam-shape-corrections-to-the-law-of-reflection/. 3 F. Goos, and H. Hanchen, Ann. Phys. 1, (1947). 4 F. I. Fedorov, J. Opt. 15, (2013). 5 C. Imbert, Phys. Rev. D 5, 787 (1972). 6 O. Hosten, and P. Kwiat, Science 319, (2008). 7 A. Aiello, and J. P. Woerdman, Opt. Lett. 33, (2008). 8 K. Y. Bliokh, and A. Aiello, J. Opt. 15, (2013). 9 G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 39, (2014). 10 G. Jayaswal, G. Mistura, and M. Merano, Opt. Lett. 38, (2013). 11 F. L. Pedrotti, and L. S. Pedrotti, Introduction to Optics ((Prentice-Hall International, Englewood Cliffs, NJ, 1993)). 12 G. R. Fowles, Introduction to Modern Optics (Courier Dover Publications, Mineola, NY, 1975). 13 S. R. Seshadri, J. Opt. Soc. Am. A 5, (1988). 14 W. Whewell, J. Opt. A: Pure Appl. Opt. 11, (2009). 15 R. H. Renard, J. Opt. Soc. Am. 54, (1964). 16 F. Pillon, H. Gilles, and S. Girard, Appl. Opt. 43, (2004). 17 C. Menzel, C. Rockstuhl, T. Paul, S. Fahr, and F. Lederer, Phys. Rev. A 77, (2008). 18 J. B. Götte, and M. R. Dennis, New J. Phys. 14, (2012). 19 B. Tamir, and E. Cohen, Quanta 2, 7 17 (2013). 20 J. B. Götte, and M. R. Dennis, New J. Phys. 14, (2012). 21 Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, (1988). 22 I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, Phys. Rev. D 40, (1989). 23 N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (1991). 24 F. Töppel, M. Ornigotti, and A. Aiello, New J. Phys. 15, (2013). 25 J. B. Götte, and M. R. Dennis, Opt. Lett. 38, (2013). 26 2Physics, (2008),

54 43 APPENDIX A SPIN HALL EFFECT OF LIGHT VIA WEAK MEASUREMENT

55 44 SPIN HALL EFFECT OF LIGHT VIA WEAK MEASUREMENT According to, 26 in February 2008 two physicists Onur Hosten and Paul G. Kwait from the University of Illinois at Urbana-Champaign demonstrated the optical analog of the Spin Hall effect in electronic systems - Spin Hall effect of light (SHEL). This effect is different from previously discussed GH and IF shifts. Hosten and Kwait have shown the generality of the effect for particles in a different nature. A new technique was used from quantum weak measurement to improve the tiny effect. The beam slightly splits into two beams, each containing different spin states. A beam of linearly polarized light is an equal combination of spin state of photons: parallel (s=+1 or right-circularly polarized) and anti-parallel (s=-1 or left circularly polarized) to the propagation direction. Due to refraction at an air-glass interface the beam changes its propagation. Hence, the beam center experiences a spin-dependent (or circular polarization-dependent) displacement perpendicular both to the initial propagation direction (it refracts or bends) and the change in the propagation direction (it shifts to the side, i.e. a lateral shift). Two different spin components (parallel and anti-parallel to the propagation direction) have opposite shifts. And when this beam changes its direction (after the refraction on an air-glass interface), it splits very slightly into two beams, each containing different spin states. Because of the rotational symmetry around the axis perpendicular to the interface (z-axis), the total angular momentum (spin and orbital) of light around this axis has to be conserved. Suppose that the light, before entering the air-glass interface, has the spin angular momentum. This spin angular momentum is either parallel or anti-parallel to the direction of the propagation and carries a certain component along the z-axis. After beam refracts at the interface, the spin still stays either parallel or anti-parallel to the new propagation direction. But now direction of the beam is changed, and the spin angular momentum component along the z-axis is changed, too. As was mentioned before, the angular momentum component consists of spin and orbital momentum components. Because the spin component was

56 45 changed after bending at the interface, the orbital component had to change too to keep the angular momentum conserved. Therefore, the spin Hall effect balances this change in the angular momentum component, and light laterally shifts itself from the z-axis. In the Hosten-Kwait experiment a laser beam of linearly polarized incident light on a glass prism at some angle was observed (Figure ). Figure 20. SHEL. A linearly polarized and incident on an air-glass boundary beam of light divides very slightly into two components - right and left circular polarizations, or spin parallel and anti-parallel to the propagation direction, right after the refraction at the interface. Views from different angles. Source: 2Physics, (2008), /02/observation-of-spin-hall-effect-of.html. After refraction, the parallel and anti-parallel spin components obtained opposite displacements out of the plane of incidence. However, because the separation between these two beams was very tiny on the nanometer scale and the beams widths on the millimeter scale, the two beams overlapped to a great extent. A new metrological method quantum weak measurement in pre- and post-selected systems was used to measure the minimal effect (Figure 21 6 ).

57 46 Figure 21. The SHEL demonstration at an air-glass interface. (A) Parallel ( + ) and anti-parallel ( ) spin components of a wave packet incident at angle θ I experience a split by a fraction of the wavelength upon refraction by an angle θ T. (B) To satisfy transversality different plane-wave components obtain different polarization rotations upon refraction. The input polarization is in the x I direction for all constituent plane waves. Arrows show the polarization vectors connected with each plane wave before and after refraction. The insets explain the vectors orientation. (C) Theoretical shifts of the spin components for horizontally and vertically polarized incident photons with wavelength λ = 633 nm. Source: O. Hosten and P. Kwiat, Science 319, 787 (2008). If the measurement were to be strong, the beams of different spin states would absolutely split from each other, and we would be able to tell the spin state by looking at the beam position. In the weak measurement the beams were still overlapping to a large degree and we could tell only very little about the spin state by looking at the beam position.