Enhancement of weak spin-Hall shift using higher-order helical-wavefront beams

Chandravati Prajapati; Nirmal K. Viswanathan

doi:10.1364/OSAC.1.000872

1. Introduction

Optical beam shift is a fundamental effect of a light beam as a result of spin-orbit interaction of light, which arises when a transversely-finite light beam interacts at a dielectric interface separating two media. The light beam after reflection, transmission or scattering shows shift of centroid of light beam with respect to the geometrical optics predictions due to mutual interaction between spin angular momentum (SAM) and orbital angular momentum (OAM). The beam shift perpendicular to the plane of incidence is called spin-Hall (SH) shift and shift in the plane of incidence is called Goos-Hänchen (GH) shift [1, 2]. The reason for these shifts lie in the fact that a light beam is comprised of large number of plane wave components having slightly different propagation vectors (k⃗) and different plane of incidence. After interaction with the interface, the plane wave components having different angle of incidence follow different Fresnel’s and Snell’s law that concoct angular gradient in the Fresnel’s coefficients and trigger GH shift. On the other hand, different plane of incidence of the plane wave components initiate change in geometric phase that trigger SH shift. These shifts are further classified into angular shift- the shift in the momentum space (angular deflection of the beam) and spatial shift- the shift in the coordinate space (position displacement of the beam). The angular shift is dependent on the beam size and hence increases with the propagation distance of beam, while the spatial shift is independent of beam size. Unified theory for these shifts are presented in [1–4] and Quantum mechanical treatment was used by Töppel et. al [5]. These effects are measured by [6, 7] using a position sensitive detector and CCD camera. They are measured using many techniques like interferometry, polarimatry and weak measurements [8–11].

Quantum mechanical treatment of GH and IF beam shifts presented in [5] is advantageous when understanding and employing the well known and very promising weak measurement technique to measure amplified beam shift. Weak measurement was introduced by Aharonov et. al [12] in 1988 which is an implication of quantum mechanical concept that, performing measurement of an observable without altering its value significantly. This can be achieved by making the coupling between the measuring device and measurement system weak. The perturbed value called “weak value” is exceptionally small, below the noise level that does not change the system after the measurement and allows us to use for further measurement in a specified way to amplify the measured value. The beam shifts which are very small in magnitude are best suited example of such system and can be amplified and measured using weak measurement method. This is achieved by pre-selecting the light beam in a well defined eigen state (polarization state) and, then allowing the system to interact weakly through reflection/transmission from a dielectric interface. Further, the system is post-selected in an eigen state which is nearly orthogonal to the pre-selected state, to amplify the beam shift by a large amount. First optical use of weak measurement was demonstrated by [13] for amplification and detection of small optical birefringence and was later used to measure small optical GH and SH shifts [10, 11], ultrasensitive beam deflection [14], longitudinal phase shift [15], small angular rotations [16] and circular birefringence [17].

Laguerre-Gaussian (LG) or vortex light beams have phase singularity at the centre of beam and have spiraling Poynting vector associated with them around the singular point. They carry an additional degree of freedom given by the orbital angular momentum (OAM) [18] which is associated with the charge of vortex beam. These beams have been demonstrated to have potential in fundamental and applied research topics including optical tweezers, optical trapping [19], optical communication [20] and quantum information science [21]. With innumerable application of LG beams their importance in spin-orbit interaction and contribution to beam shifts are imperative to study and understand. Aiello [22–25] addressed this problem to determine the role of functional shape of the beam on beam shifts for the reflected beam. They reported that the polarization-dependent angular shifts of the reflected beam coupled with the complex vortex structure of the beam and produces a spatial shift in the orthogonal direction i.e, angular GH shift and angular SH shift produces an additional spatial SH shift and spatial GH shift, respectively. Direct and accurate experimental measurement of these shifts for LG beams using a quadrant detector or position sensitive detector is a challenging task as they are prone to large intensity deformations as a result of relatively large beam shifts and leads to large error. Weak measurement on the other hand amplifies these shifts and measure them accurately as it enables extraction of beam shifts from the displaced intensity weighted centroid, which has remained unexplored till date for LG beams. The GH and SH shifts for LG beam of charge l = 1 was measured experimentally by [23, 26] using a quadrant detector and CCD camera. Loffler et al [27], experimentally observed OAM sidebands due to reflection and proved that these sidebands are evidence of the GH and SH shifts, but did not quantify them. We have used the weak measurement method to measure the SH shift for higher order LG beams of charge l = 1 to l = 3 including the Gaussian beam. We have used spiral phase plate (SPP) to generate helical wavefront LG beams. Though the SPP does not produce pure LG beam, the small magnitude of radial modes filtered out in the calculation and henceforth simulation do not appear to have any significant effect on the overall SH behaviour and matches well with the experimental results, and so the output beam is approximated as the LG beam. Theoretical analysis of the weak measurement for LG beams by studying the intensity distribution at cross position and away from cross position and subsequently the amplified SH shift is calculated. We then used weak measurement method to measure the amplified SH shift for TM and TE polarized higher-order LG beams. We observed that the SH shift increases with the order of LG beam due to the coupling between shifts in orthogonal direction and effect of OAM of the beam.

2. Weak measurement theory for Laguerre-Gaussian beams

Consider a Laguerre-Gaussian (LG) beam pre-selected in a linear polarization state Ψ_i, given as $Ψ_{i} = E (r, ϕ) \times {[\begin{array}{l} cos α & sin α \end{array}]}^{T}$ , where α is the angle made by polarization axis with x axis, T is transpose of matrix and E (r, ϕ) describes the functional shape of LG beam. E(r, ϕ) for higher-order LG beam of azimuthal index l and radial index p, taken to be zero for simplicity is given as

E (r, ϕ) = \sqrt{\frac{2}{π | l |}} \frac{1}{ω (z)} L_{| l |} (\frac{2 r^{2}}{ω {(z)}^{2}}) {(\frac{\sqrt{2} r}{ω (z)})}^{| l |} exp (\frac{- r^{2}}{ω {(z)}^{2}}) exp (i l ϕ),

where,

r = \sqrt{x^{2} + y^{2}}

,

ϕ = arctan (\frac{y}{x})

,

ω (z) = k_{0} ω_{0}^{2} \sqrt{1 + \frac{z^{2}}{L^{2}}}

is the beam size at distance z, ω₀ is the beam waist and

L = \frac{k_{0} ω_{0}^{2}}{2}

is Rayleigh range.

The LG beam is then allowed to undergo weak interaction and then post-selected in a polarization state Ψ_f, where the polarization axis is chosen at an angle β = 90° + α ± Δ, which is nearly orthogonal to the initial pre-selected polarization state for Δ << 1 radians. In the cross position i.e, Δ = 0, the eigen state of weak value are separated out even if they are very small in magnitude. In the present case of weak interaction, we assume that a small magnitude of transverse SH shift arise in the beam cross section as a result of weak interaction of light beam with the dielectric interface and wherein the shift in the longitudinal direction has been ignored. The SH shift is the eigen value of circular polarization state which is different for right and left circular polarization. The SH shift for right and left circular state of polarization (SoP) of the beam have same magnitude but opposite in sign. The two circular SoP of beam are displaced opposite to each other and eventually get separated out in the cross position.

Assuming that the shape of beam does not change and the transverse SH splitting between RCP and LCP components in the beam is denoted by 2〈Y〉 and the phase difference that arise between the two orthogonal components due to SH effect is given as δ, the state for shifted LG beam as a result of SH shift in terms of RCP and LCP components is written as,

Ψ_{w} = \sqrt{\frac{2}{π | l |}} \frac{1}{ω (z)} {(\frac{\sqrt{2} \sqrt{(x^{2} + {(y - 〈 2 Y 〉)}^{2})}}{ω (z)})}^{| l |} exp (i l ϕ) [\begin{array}{l} (\frac{1 + i}{2}) exp (\frac{- x^{2} - {(y + 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) cos α \\ e^{i δ} (\frac{1 - i}{2}) exp (\frac{- x^{2} - {(y - 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) sin α \end{array}] .

The LG beam is then post selected by projecting at an angle β. The state of post-selected beam Ψ_f is written as

Ψ_{f} = ([\begin{matrix} cos β \\ sin β \end{matrix}] [\begin{matrix} cos β & sin β \end{matrix}]) Ψ_{w} = {[\begin{matrix} cos β & sin β \end{matrix}] Ψ_{w}} [\begin{matrix} cos β \\ sin β \end{matrix}],

\begin{array}{l} Ψ_{f} = & \sqrt{\frac{2}{π | l |}} \frac{1}{ω (z)} {(\frac{\sqrt{2} \sqrt{(x^{2} + {(y - 〈 2 Y 〉)}^{2})}}{ω (z)})}^{| l |} exp (i l ϕ) \times \\ {(\frac{1 + i}{2}) exp (\frac{- x^{2} + {(y + 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) cos α cos β + e^{i δ} (\frac{1 - i}{2}) exp (\frac{- x^{2} - {(y - 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) sin α sin β} [\begin{array}{l} cos β \\ sin β \end{array}] . \end{array}

The intensity of the output beam is calculated using ${| Ψ_{f} |}^{2} = Ψ_{f}^{*} Ψ_{f}$ and given as

\begin{array}{l} I_{Ψ_{f}} & = & A [exp (\frac{- 2 x^{2} - 2 {(y - 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) {cos}^{2} α {cos}^{2} β] \\ + & A [exp (\frac{- 2 x^{2} - 2 {(y + 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) {sin}^{2} α {sin}^{2} β] \\ + & A [(2 sin δ) exp (\frac{- 2 x^{2} - {(y + 〈 Y 〉)}^{2} - {(y - 〈 Y 〉)}^{2}}{ω {(z)}^{2}}) cos α cos β sin α sin β], \end{array}

where A is given as

A = \frac{2}{π | l |} \frac{1}{ω {(z)}^{2}} {(\frac{2 (x^{2} + {(y - 〈 2 Y 〉)}^{2})}{ω {(z)}^{2}})}^{| l |} .

The first and second terms in Eq. (5) corresponds to intensity distribution of displaced RCP and LCP components of LG beam and third term corresponds to the cross intensity distribution of RCP and LCP components in the beam. For $β = α + \frac{π}{2}$ , at the cross position, the two circular polarized components of LG beam destructively interfere and give Hermite-Gaussian like intensity distribution with two shifted peaks separated by $\sqrt{2} ω_{0}$ . For Δ ≠ 0 and Δ << 1, a distorted LG beam shifted by $\frac{1}{2} (2 〈 Y 〉 cot Δ) = \frac{〈 Y 〉}{Δ}$ is observed. Figure 1 shows the intensity profile of 45° linearly polarized Laguerre-Gaussian beam of order l = 1, first row, l = 2, second row, and l = 3, third row, for cross-position (Δ = 0) and slightly away from cross position, Δ = ±1° (1.745 × 10⁻² radians). For simulation, value of SH shift 〈Y〉 is taken to be 0.5 μm, beam waist is 30 μm and wavelength is 632.8 nm. From the Fig. 1, it can be seen that there is amplification of $\frac{〈 Y 〉}{Δ} = 28 μ m$ is observed. The difference of centroid of RCP beam for Δ = 1° to that of LCP beam for Δ = −1° gives the amplified SH splitting. It can also observed that SH shift increases with increasing order of LG beam.

Fig. 1 Intensity profile of beam after incorporating SH shift of 0.5 μm using weak measurement for linear 45° polarized beams plotted for higher-order LG beams at Δ = 0 (first column), Δ = 1° (second column) and Δ = −1° (third column). First row is for l = 1, second row for l = 2 and third row for l = 3 mode LG beams.

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3. Weak measurement treatment of the spin-Hall effect for Laguerre-Gaussian beams

The weak measurement scheme is applied to the higher-order LG beams to calculate the amplified spin-Hall shift using Jones polarization formalism. Consider a LG beam of azimuthal order l pre-selected in a state Ψ₁ using a polarizer P₁ by orienting at an angle α with the x axis. The LG beam is then allowed to reflect from a BK7 glass prism and then post-selected using a polarizer P₂ in a state Ψ₂ by projecting at an angle β = 90° + α ± Δ with x axis. Since Δ is very small (Δ << 1) the post-selected state is projected nearly orthogonal to the pre-selected state and thereby amplifying the SH shift. For the case of LG beams, the complex shifts in x-direction (GH) and y-direction (SH) couple with the spatial profile of beam in orthogonal directions and therefore we need to consider both the beam shifts in the theoretical calculations.

For the case of linear polarized LG beam at α polarizer orientation, the state |Ψ₁ > is written as

| Ψ_{1} > = [\begin{matrix} cos α \\ sin α \end{matrix}] = [\begin{matrix} a_{1} \\ a_{2} \end{matrix}],

where a₁ = cos α and a₂ = sin α, and

a_{1}^{2} + a_{2}^{2} = 1

. The state of LG beam is written as E(r, ϕ) |Ψ₁ >, E (r, ϕ) is functional shape of the beam given in Eq. (1).

The LG beam undergoes external reflection and the amplitude and phase of beam are expected to modify under the influence of GH and SH shifts. The amplitude of reflected beam can be written as

E^{r} (r, ϕ) = [\begin{matrix} r_{p} (1 + \frac{k_{X}}{k} 𝒳_{p}) & r_{p} \frac{k_{y}}{k} 𝒴_{p} \\ - r_{s} \frac{k_{y}}{k} 𝒴_{s} & r_{s} (1 + \frac{k_{X}}{k} 𝒳_{s}) \end{matrix}] E (r, ϕ),

where, r_p,s are Fresnel’s coefficients for reflection which is dependent on angle of incidence, θ. 𝒳_p,s and 𝒴_p,s corresponding to GH and SH beam shifts respectively are given as

\begin{array}{l} 𝒳_{p, s} = - i \frac{\partial ln r_{p, s}}{\partial θ}, \\ 𝒴_{s, p} = \pm i (1 + \frac{r_{p, s}}{r_{s, p}}) cot θ . \end{array}

The reflected LG beam after weak interaction, is post-selected in a state |Ψ₂ > using a polarizer oriented at an angle α + 90° ± Δ. The state |Ψ₂ > is written as

| Ψ_{2} > = [\begin{matrix} a_{2} \pm Δ a_{1} \\ - a_{1} \pm Δ a_{2} \end{matrix}] .

The state of reflected beam is written as E^r (r, ϕ) |Ψ₂ >.

The amplified SH shift can be found by calculating the expectation value of y– position of reflected beam between the pre-selected and post-selected states of LG beams,

y_{S H}^{w} = \frac{〈 Ψ_{2} | E^{r *} (r, ϕ) (y) E (r, ϕ) | Ψ_{1} 〉}{E^{r *} (r, ϕ) (y) E (r, ϕ) 〈 Ψ_{2} | Ψ_{1} 〉} .

The real and imaginary parts of Eq. (10) correspond to spatial and angular SH shifts. This can be further simplified for TM and TE polarized beams for which |Ψ₁ > is given as

{[\begin{array}{l} 1 & 0 \end{array}]}^{T}

and

{[\begin{array}{l} 0 & 1 \end{array}]}^{T}

, respectively. The value of |Ψ₂ > is given as

{[\begin{array}{l} \pm Δ & - 1 \end{array}]}^{T}

and

{[\begin{array}{l} 1 & \pm Δ \end{array}]}^{T}

. On solving Eq. (10), the weak value of SH shift for TM and TE polarized beams, respectively, is found to be,

y_{SH}^{w} = \frac{{\pm Δ r_{p, s} (1 + \frac{k_{X}}{k} 𝒳_{p, s}) + r_{s, p} \frac{k_{y}}{y} 𝒴_{s, p}} E^{*} (r, ϕ) (y) E (r, ϕ)}{\pm Δ} .

This equation provides two-fold information-first, amplification of the angular SH shift by a factor $\frac{1}{Δ}$ due to weak measurement and second, coupling of GH and IF shifts in orthogonal direction. For the case of TM and TE polarized beams under the weak measurement, there is no amplification of angular and spatial GH shift. On substituting the value of 𝒴_p,s from Eq. (8) we get imaginary term with amplification factor, which shows that there is amplification of only angular SH shift [28,29]. Due to complex spatial profile of LG beams there is coupling between the spatial profile of beam and complex shifts in orthogonal direction, leading to modification of GH and SH shifts. The imaginary part of the shift is coupled with OAM number of beam and results in real shift in the orthogonal direction in addition to the actual spatial and angular shifts. Thus, the angular GH shift couples with OAM of LG beam and results in an additional spatial SH shift [2,24,25].

The final amplified SH shift for TM and TE polarized beams has three terms given as,

{SH}_{w}^{p, s} = \frac{1}{k_{0}} ℜ [i (1 + \frac{r_{s, p}}{r_{p, s}}) cot θ] - l \frac{1}{k_{0}} ℑ (\frac{\partial r_{s, p}}{\partial θ}) + \frac{Z}{Δ L} \frac{(1 + | l |)}{k_{0}} ℑ [i (1 + \frac{r_{s, p}}{r_{p, s}}) cot θ] .

First term in Eq. (12) is spatial SH shift due to Gaussian envelope and independent of OAM number (l) of beam and it is remained unamplified. The second term is spatial SH shift that contributes from coupling between angular GH shift and OAM and linearly proportional to (l) and remains unamplified. The third term is angular SH shift which is linearly dependent on OAM number as (1 + |l|) and is amplified due to weak measurement. The amplification factor as a result of weak measurement is given as $\frac{1}{Δ} \frac{Z}{L}$ .

4. Experimental measurements and discussion of results

A Gaussian light beam of wavelength 632.8 nm and beam size approximately 470 μm from a vertically polarized He-Ne laser passes through a Spiral Phase Plate (SPP) from RPC Photonics, USA (Model no. VPP-m633) which transforms the Gaussian beam to a higher-order Laguerre-Gaussian (LG) beam. The LG beam of definite order then passes through a Half-wave plate (HWP) and a Polarizer P₁ which serve as the pre-selection device of weak measurement. The pre-selected LG beam is focused using a biconvex lens of focal length 50 mm to a beam size of 30 μm onto the BK7 right angle glass prism (n = 1.51) surface which is mounted on a rotation stage. The experimental set up is shown in Fig. 2. The beam undergoes external reflection from the glass surface where the weak interaction occurs. The reflected light beam is then collimated using a lens of focal length 75 mm. The beam is then post-selected in an orthogonal polarization state with respect to P₁ using a polarizer P₂. The intensity images are recorded using a CCD camera (Spiricon, USA) of pixel size 4.4 μm connected to a computer. When the polarizer P₂ is positioned orthogonal with respect to P₁, the intensity distribution shows two lobe Hermite-Gaussian type in the beam cross section and oriented vertically, which are in fact two separated orthogonal circular polarized RCP and LCP components in the beam cross section separated due to SH shift. With P₂ rotated away from the crossed position on one side, the intensity of one of the polarized components is separated. The intensity profile of TM (first and second row) and TE (third and fourth row) polarized LG beam of order l = 1 and l = 2, at 46° is shown in Fig. 3. First column is intensity profile of LG beams just after SPP for the order l = 0 to l = 3. Second column is intensity profile at crossed position and third and fourth columns are intensity profile for Δ = ±1.36 × 10⁻² radians away from crossed polarization.

Fig. 2 Schematic of experimental Set-up. SPP (Spiral Phase Plate) is used to generate higher-order Laguerre-Gaussian beams; P₁ and P₂ are Glan Thompson Polarizers used to pre-select and post-select the states respectively; L₁ and L₂ are lenses of focal length 5 and 7.5 cm respectively; HWP, a Half-Wave Plate; CCD, a camera. For external reflection, we used BK7 glass prism.

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Fig. 3 First column is intensity profile of input LG beam of order l = 0 to l = 3. Second, third and fourth column is intensity profile of LG beam of order l = 1 (first and second row), and l = 2, (third and fourth row) for TM and TE polarized beams at 46° for crossed position (first column) and ±Δ = 1.36 × 10⁻² radians away from crossed polarization (third and fourth column).

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We have used TM and TE polarized LG beams of order l = 1 to l = 3 for pre-selection by rotating the polarizer P₁ to an angle α = 0° and α = 90° respectively. For post-selection, the polarizer P₂ is rotated to an angle β = 90° ± Δ and β = 0° ± Δ respectively, where Δ = 0 is the crossed position measurement and Δ = 1.36 × 10⁻² radians for away from crossed position measurements. The SH Shift is calculated by finding the difference of intensity weighted centroid of images recorded at +Δ and −Δ orientations. Measurements are performed by varying the angle of incidence θ from 26° to 70°. The SH shift is extracted from the recorded images and plotted as a function of angle of incidence for LG beams of order l = 0 to l = 3 shown in Fig. 4 for TM and TE polarized beams. It is observed that the SH shift increases with increasing order of the LG beam. For TM polarized beam [Fig. 4(a)], the SH shift increases with angle of incidence till the Brewster angle (56.5°), and then jumps from high positive value to high negative value. The SH shift further decreases in the negative direction with increasing θ, returning back to its starting value. This behavior is the signature of phase jump from π to 0° at the Brewster angle of incidence. The dashed line in the graph are results obtained from the theoretical Eq. (12) and symbols are results from the weak measurement. Black color dashed line and symbol is plotted for Gaussian beam (l = 0), red for LG, l = 1, blue for LG, l = 2 and green for LG, l = 3 mode beams. The theoretical value of SH shift increases very rapidly near the Brewster angle of incidence while in experiment, it increases slowly due to the cross-polarization effect. The SH shift for TE polarized beams Fig. 4(b) increases with angle of incidence up to 46° and then decreases. Black color dashed line and symbol is plotted for Gaussian beam (l = 0), red for LG, l = 1, blue for LG, l = 2 and green, for LG, l = 3 mode beams.

Fig. 4 Profile of S₃ component of Stokes parameter for LG beams of order (a) l = 0, (b) l = 1, (c) l = 2, and (d) l = 3 at Brewster angle 56.5°.

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The experimental results show that close to Brewster angle of incidence for TM poalrized beam, the SH shift does not increase as significantly as expected in the theory because of the cross-polarization effect [30 31], which is predominated around the Brewster angle. The TM and TE polarized Gaussian beams reflected at Brewster angle generate a two-lobe beam with both a dominant and a cross-polarized component and the beam splits into two lobe profile in the plane of incidence. This effect is dominant in the TM polarized beam due to which the intensity distribution of beam in various post-selection positions modify and thus the amplified SH shift due to weak measurement is also modified. For TE polarized beam this effect is very less and produces splitting in the vertical plane and therefore the shift remains unmodified matching well with theoretical simulations shown in Fig. 4(b) by dashed line. The same is valid for the case of TM polarized LG profile beams, as this effect is a polarization dependent effect. The splitting of TM and TE beams in right and left circular polarized components can be demonstrated by studying the S₃ components of Stokes parameter. We performed S₃ measurements of the Stokes parameters by projecting the weak circular polarized components of beam by inserting a quarter-Wave plate of retardation 90° before the polarizer P₂. The images are recorded by rotating the polarizer by ±4° away from this position and S₃ is calculated by subtracting the two images. Plot of Stokes S₃ parameter for various orders of TM polarized LG beam from l = 1 to l = 3 including Gaussian beam is shown in Fig. 5 at Brewster angle of 56.5°. Positive value of S₃ shows measure of left circular component and negative value of S₃ shows the measure of right circular component in the beam. Because of domination of cross-polarization effect at Brewster angle for TM polarized beam the S₃ profile is nearly in horizontal plane. For TE polarized beam, cross-polarization effect is very minimal and profile of S₃ remains in vertical plane even at the Brewster angle.

Fig. 5 Spin-Hall shift for (a) TM and (b) TE polarized beams plotted for various higher-order LG beams of order l = 0 (black), l = 1 (red), l = 2 (blue) and l = 3 (green) colors.

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5. Conclusion

We used quantum weak measurement method to measure the spin-Hall effect of light in higher-order Laguerre-Gaussian beams. First, intensity profile of higher-order LG beams at various post-selections have been studied theoretically and demonstrated experimentally. Then, weak measurement is performed by varying the angle of incidence from 26° to 70° and the SH shift is calculated from the intensity measurements. For the case of LG beams, there is coupling between GH and SH shifts because of complex spatial profile of beam which eventually modifid the resultant shift. The angular GH shift couples with the complex profile of LG beam and result in an additional spatial SH shift. Thus the SH shift has three terms, spatial SH shift due to Gaussian profile, spatial SH shift as a result of coupling of angular GH and OAM of the beam and the angular SH shift. Further, the Jones polarization treatment of weak measurement showed that there is amplification of only angular SH shift while other terms remain unamplified. The SH shift for TM and TE polarized LG beams of order l = 1 to l = 3 including Gaussian beam has been measured and calculated using weak measurement with amplification factor of $\frac{Z}{Δ L} = 1200$ . The Brewster cross-polarization effect around the Brewster angle of incidence for TM polarized LG beams modified the SH shift close to Brewster angle, thereby reducing the effective SH shift anticipated from theory. The overall SH shift increases with the order of LG beams. Stokes S₃ parameter is plotted to verify the splitting of TM and TE polarized LG beam in left and right circular polarized components.

Acknowledgments

Author acknowledges the financial support from the UGC, Dr. D S Kothari Postdoctoral Fellowship, and DST-SERB.

Disclosures

There are no conflicts of interest related to this article.

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Enhancement of weak spin-Hall shift using higher-order helical-wavefront beams

Abstract

1. Introduction

2. Weak measurement theory for Laguerre-Gaussian beams

3. Weak measurement treatment of the spin-Hall effect for Laguerre-Gaussian beams

4. Experimental measurements and discussion of results

5. Conclusion

Acknowledgments

Disclosures

References

Cited By

Figures (5)

Equations (13)

OSA Continuum