1. Introduction

Light reflection on a boundary between two dielectric media is of fundamental importance in optics, and Fresnel reflection formulae were established in the 19th century for plane waves. Reflected light beams, however, deviate from the classical ray tracing, equivalently, geometric optics and specular reflection. This nonspecular reflection has attracted considerable attention, and various effects such as lateral shift and angular deflection have been investigated exclusively for Gaussian beams [1–4].

The most well-known among the nonspecular phenomena is the Goos-Hänchen (GH) shift in the total reflection region [5]. The magnitude of the shift is usually as small as the optical wavelength, but increases enormously up to the order of $\sqrt{w/k}$ when the incidence angle is close to the critical angle [1]. Here, w is the transverse beam size, and k is the wave number in the dielectric medium on the incident side. Note that the nonspecular effects generally contain deformations of the intensity distributions (IDs) inside the reflected beam even though the GH shift is approximately a simple translational displacement of the specular reflection. Thus, there is no general definition for the magnitude of the nonspecular lateral shift.

The transverse shift in the total reflection region was first predicted by Federov [6], and experimentally verified by Imbert [7]. It is due to the polarization states of the light beam and has been referred to as either the transverse GH shift or the Imbert-Federov (IF) shift. The direction of the shift depends on the sign of the change in the polarization ellipticity at the reflection. Unlike the GH shift, the magnitude of the IF shift is always the order of the optical wavelength and vanishes for the s- and p-polarized beams and for the critical and grazing incidences. Recently, Onoda et al. [8] theoretically derived the IF shift from the conservation law of total angular momentum consisting of spin and extrinsic orbital angular momenta [9] between the incident and reflected photons.

In 1992, Allen et al. pointed out that the Laguerre-Gaussian (LG) beams with a non-zero azimuthal index possess an orbital angular momentum [10]. Thereafter, the LG beams have been extensively studied in various fields such as singular optics, manipulation of atoms and macroscopic particles, and quantum information [11]. The nonspecular reflection of the general light beams with the orbital angular momentum was theoretically discussed by Fedoseyev, who predicted that the center of the intensity gravity (CIG) of the reflected beam would give rise to the transverse shift in the partial reflection region [12]. The electromagnetic energy inside such a reflected beam is redistributed so that the intensity increases on one side of the incident plane and decreases on the other while the beam frame remains unmoved. The direction of the transverse deformation depends on the sign of the orbital angular momentum of the incident beam. The transverse shift of the CIG of the reflected LG beam is as large as the order of l$\sqrt{w/k}$ near critical incidence, where l is the azimuthal index of the LG beam. The p-polarized LG beam also shows the large transverse deformation near the Brewster angle, which was verified quite recently; the measured shift of the CIG was a few times larger than the optical wavelength [13].

In this paper, we first derive a general expression of the field distribution of reflected Hermite-Gaussian (HG) and LG beams using angular spectrum analysis [14] together with a paraxial approximation. We then describe the observation of huge transverse deformations in the reflected p- and s-polarized LG beams with |l| ≤ 3, and compare the observed IDs with our calculations.

2. Theoretical Background

2.1 Coordinate system

We consider the reflection of a monochromatic paraxial beam at a plane interface of two semi-infinite transparent dielectric substances. The refractive indices are n _i and n _t, and the relative refractive index is defined as n _rel = n _t/n _i. Figure 1 depicts the incident and reflected beams as well as their axis coordinates. The z-axis for the incident beam is chosen on the optical axis of the incident beam with the incident angle θ ₀, and the beam waist is located at z = 0 if the entire system is filled with the substance of the refractive index n _i. The Z-axis and the position of Z = 0 for the reflected beam coincide with the optical axis and the waist of the specular reflection beam, respectively. The x- and X-axes are in the incidence plane, and the y- and Y-axes are taken so that both the coordinate systems are right-handed.

 

Fig. 1. Geometry and coordinate systems of incident and reflected beams.

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2.2 Incident HG beam

To derive the ID of the reflected LG beam, we first consider an incident HG beam because the symmetry of the system simplifies the calculations based on the HG mode. The complex electric field amplitude of the parallel (p, x-direction) or perpendicular (s, y-direction) component is given as

where j denotes either p or s, $E_{\mathrm{i}}^j$ is the constant amplitude, $v_{n\mathit{,}m}^{\text{HG}}$ (x,y,z) is the HG-mode solution of the paraxial wave equation, and n and m are the mode indices for the x- and y-axes, respectively [15]. A time dependence of e^-iωt is implied and suppressed for simplicity. At the beam waist, z = 0, the HG-mode function is explicitly written as

with

and

Here, w ₀ is the beam radius at the beam waist, $N_n^{\text{HG}}$ is the normalization constant, and H_n is the n-th Hermite polynomial. The amplitude of the angular spectral components is given by Fourier transformation [16] as

Each partial wave has the wave vector $k_x\hat{\mathit{x}}+k_y\hat{\mathit{y}}+\sqrt{k^2-k_x^2-k_y^2}\hat{\mathit{z}}$, where x̂, ŷ, and ẑ are the unit vectors of the axes. It is incident to the dielectric interface with the incidence angle θ scattered around the principal incidence angle θ ₀. The incident paraxial beam is mainly composed of the partial waves having small angular deviations, k_x /k and k_y /k, within the paraxial parameter 1/kw ₀. The parallel angular deviation, k_x /k, introduces the first order change in the incidence angle θ as $\theta ={\theta }_0-\frac{k_x}{k}$, whereas the perpendicular deviation, k_y /k, is not took into account here because it contributes to the second-order corrections at most. In general, the x- (y-) polarized paraxial beam has the z- and/or y- (x-) field component, and the magnitude is the order of 1/kw ₀ smaller than that of the x- (y-) component [17]. In the paper, however, we consider only the x- (y-) field component of the partial waves.

2.3 Reflection of LG beam

Each partial wave reflects on the interface according to the Fresnel reflection formulae. In the partial reflection region (sinθ < n _rel), the complex amplitude reflection coefficient is given as

where $E_{\mathrm{r}}^j$ is the constant electric field amplitude of the reflected beam, m_p = $n_{\text{rel}}^2$, and m_s = 1. In the total reflection region (sinθ > n _rel), it is represented as

The reflected partial wave has the wave vector $k_X\hat{\mathit{X}}+k_Y\hat{\mathit{Y}}+\sqrt{k^2-k_X^2-k_Y^2}\hat{\mathit{Z}},$, and each component is given as (k_X , k_Y , k_Z )= (-k_x ,k_y , k_z ) Therefore, the spatial field distribution of the reflected beam at the beam waist, Z = 0, is obtained as

through inverse Fourier transformation up to the first-order of the angular deviation.

The electric field amplitude of the incident LG beam is given as

where p and l are the radial and azimuthal indices characterizing the mode, and the LG-mode function at the beam waist, z = 0, is represented by

with the normalization constant

where $L_p^{\left|l\right|}$ is the associated Laguerre polynomial. The LG_p,l beam has p annular nodes, an optical vortex with charge l on the optical axis, and an orbital angular momentum of lħ directing toward the z axis per photon. When the coordinates for the HG and LG modes are connected through x = r cosϕ and y = rsinϕ, the LG mode can be expanded in terms of the HG modes as

and the real expansion coefficients are given as

where N is the mode order given by n + m for the HG_n,m mode and 2p + |l| for the LG_p,l mode. Using Eqs (8), (12), and (13) and the superposition principle, we can generally calculate the field distribution of the reflected LG beam for any incidence angle. The position of the dielectric interface does not affect the field distribution of the reflected beam. Here, no propagation effects are considered because we observed the IDs of the reflected beam only at the beam waist. In the calculation, we numerically integrate Eq. (8) over the range of $-\frac{6}{k{w}_0}<\frac{k_X}{k}<\frac{6}{k{w}_0},$, assuming n _rel = 1/1.515 for the interface between the glass of BK7 and the air.

2.4 Quasi-LG beam generated by a computer-generated hologram

To generate LG beams, we used three off-axis fork-like holograms [18]. When the hologram with an |l| -branched fork is illuminated by a Gaussian beam, the plus/minus first-order diffraction beam has the doughnut intensity profile characteristic to the LG_p=0,±|l| beam far down from the hologram.

We take the cylindrical coordinate for the incident LG beam in Fig. 1. The z-axis is the optical axis of the first-order diffraction beam having the beam waist at z = 0 and the Rayleigh length of z_r . The azimuthal angle is referred to the x-axis, which corresponds to the direction of the fork of the hologram in the present configuration. When the hologram is positioned at z = z _h < 0, |z _h| >> z_r , the field amplitude near the beam waist, z≈0, is expressed in terms of the LG beams [19] as

with

where E ₀ is the constant field amplitude, the parameter w ₀ in $v_{p\mathit{,}l}^{\text{LG}}$ is identical to that of the incident Gaussian beam, and a_p,l is normalized such that $\sum _{p=0}^{\infty }{\mid a_{p,l}\mid }^2=1.$. Even though the LG_p=0,l beam has the largest contribution, the series contains infinite number of the LG_p>0,l beam components. Their amplitude converges very slowly as p, and they contribute more as |l|. For example, the values of |a _p=0,l|² are 0.785, 0.500, and 0.295 for l = ±1, ±2, and ±3, respectively. Therefore, the resultant beam is hereafter referred as to the quasi-LG_p=0,l beam. In the calculation, we truncate the series up to p = 5, 20, and 30 for l = ±1, ±2, and ±3, respectively, which gives the values of ${\displaystyle \sum _{p=0}^{\infty }}{\mid a_{p,l}\mid }^2$ up to be 0.959, 0.955, and 0.932. Here we do not consider any details of the transmission function of the hologram because they affect only the diffraction efficiency.

3. Experimental

Figure 2 illustrates the experimental setup. The light source is a linearly polarized 633-nm single-frequency He-Ne laser (Spectra Physics 117A). The emitted Gaussian beam passes through two polarizer plates in order to adjust the polarization and intensity (not shown in Fig. 2) and an off-axis fork-like hologram [18], which is computer-generated and reduced in size by photographing onto a 35-mm negative film. The pair of first-order diffraction beams are the quasi-LG_p=0,±|l| beams, and each of them has an orbital angular momentum with the opposite sign. They diverge with a diffraction angle of 6 milliradians and are led into a convex lens with a focal length of 25 cm. A portion of the beams is divided by a beam splitter and always monitored with a charge-coupled device (CCD) plate for adjusting the position of the hologram so that the optical vortex is located on the optical axis. The quasi- LG_p=0,±|l| beams are incident to a right angle prism of BK7, and the bright zero-order-diffracted beam is blocked by a beam stopper. The incidence angle to the basal plane of the prism is varied around the critical angle θ_c of 41.30° by rotating the prism holder. The reflected beams are observed in the plane of the beam waist using another CCD plate without any lens system. The observation plane is apart from the lens by 32 cm, and the distance between the hologram and the lens is a few centimeters. The former is significantly larger than the Rayleigh length of the converging beam of 11.7 cm, while the latter is sufficiently smaller than that of the emitting laser beam. Therefore, Equations (14,15) are valid in the present configuration.

 

Fig. 2. Experimental setup. CCD: charge coupled device

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4. Results

Figure 3 shows the observed and calculated IDs of the reflected p-polarized quasi- LG_p=0,l±1 beams. The five columns correspond to incidence angles from θ_c - 0.132° to θ_c + 0.132° in steps of 0.066°.The first and second rows are the observed and calculated IDs for the l = 1 beam, and the third and fourth rows are those for the l = -1 beam. Each frame represents a region of 600×600 μm², and the X- and Y-axes direct rightward and upward, respectively. The calculated IDs for the ±|l| beams are mirror images of each other through the X-axis. The incidence angle in the observation was not directly measured but determined to be equal to the critical angle when the observed ID was similar to that calculated for the critical incidence. We are able to determine the critical angle with an accuracy of 0.02° because the total intensity and the ID of the reflected beam are sensitive to the incident angle near the critical angle. The prism was then rotated so that the beam image on the CCD plate moved a distance corresponding to a 0.10° rotation of the prism. This induces a 0.066° change in incident angle to the base surface of the prism near the critical incidence. The value of kw ₀ is determined to be (1.50 ± 0.05) × 10³ by fitting the calculated intensity profile on the Y-axis to the observed one. The large transverse deformation is observed when the angular spectrum lies in the range of Δθ ≈ (kw ₀)^-1 ≈ 0.03° from the critical angle. The observed and calculated IDs agree well, and some differences are attributed to the imperfection of the hologram. The deformation direction that depends on the sign of the incident angular momentum is consistent with both the prediction by Fedoseyev [12] and the IF shift of the circularly polarized beam. The observed IDs were reversed with respect to the X-axis when the hologram turned so that the fork was directed in the opposite direction. The quasi-LG beam enters the prism at the incident angle of about 5.6°, then reflects on the basal plane, and eventually leaves it for the air at the incident angle of about 3.7°. These nonspecular transmissions also induce the transverse deformation. However, we do not explicitly consider them because the transverse shift of the CIG is three orders of magnitude smaller than that induced by the reflection on the basal plane.

 

Fig. 3. Observed and calculated IDs of reflected p-polarized quasi- LG_p=0,l=′1 beams. The X-and Y-axes direct rightward and upward, respectively.

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Figure 4 presents the calculated and observed IDs and their intensity profiles along the X≈0 dotted white line of the reflected p-polarized quasi- LG_p=0,l=-1, LG_p=0,l=-2, and LG_p=0,l=-3 beams and the reflected s-polarized quasi- LG_p=0,l=-1 beam at the critical incidence. Each frame of the IDs represents a region of 1000×1000 μm², and the X- and Y-axes are set similarly as in Fig. 3. The observed and calculated intensity profiles are drawn in a red solid curve and a blue dotted curve, and normalized at the maximum. A green dotted curve indicates the calculated intensity profile of the pure LG_p=0,l beam with the value of kw ₀ assumed to be 1.75×10³. The observed and calculated IDs and the red and blue curves agree very well. However, the green curves for the reflected pure LG_p=0,l=2, and LG_p=0,l=-3 beams agree with the observed inner profile but not with the outer profile because the incident quasi-LG_p=0,l beams generated by the hologram contains considerable fractions of the higher p components for large |l|. The transverse deformation is more obvious for the larger |l| because the annular radius and the intensity inclination increase with |l|. The nonspecular effects are essentially due to the incidence-angle dependence of the reflection coefficient. Therefore, the transverse deformation is large for the high-order mode beam composed of wider angular spectral components and for the p-polarized beam whose reflection coefficient varies more steeply than the s-polarized beam.

 

Fig. 4. Observed and calculated IDs and their intensity profiles of reflected p-polarized quasi-LG_p=0,l=-1, LG_p=0,l=2, and LG_p=0,l=-3 beams and reflected s-polarized quasi LG_p=0,l=1 beam at critical incidence. Red solid curves are the observed intensity profiles, and blue dotted curves are the calculated ones. Green dotted curves are calculated for the pure LG_p=0,l beam with the different value of kw ₀.

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5. Discussion

Up to now, the nonspecular lateral shifts have been considered as subtle effects even at the critical incidence. Taking the present experimental condition as an example, the CIG of the reflected p-polarized quasi-LG_p,l=±1 beams shifts 10.3 μm towards the X-axis because of the GH shift and ±12.1 μm along the Y-axis due to the transverse deformation. Even though both shifts have a similar magnitude, the transverse deformation is far more prominent because the local intensity is considerably different between the positions transversely separated by the beam size. To evaluate this, we introduce a transverse/longitudinal intensity contrast

where I _1st and I _2nd are the first and second maximum intensities on the X = 0 / Y = 0 line, respectively. The intensity contrast vanishes for any incident LG and HG beams and represents what fraction of energy moves from the second maximum to the first maximum at the reflection. The transverse intensity contrasts for the reflected p- (s-) polarized quasi-LG beams with l=±1, ±2 and ±3 are calculated to be 0.149 (0.068), 0.208 (0.092), and 0.211 (0.092), respectively. The corresponding longitudinal intensity contrasts are as small as 0.0354 (0.0163), 0.0026 (0.00138), and 0.0296 (0.0136), which indicates that the GH shift is close to the simple translational displacement. Figure 5 shows the calculated transverse intensity contrasts versus the incidence angle near the critical angle. The black and red curves are for the quasi-LG_p=0,l±1 and LG_p=0,l±2 beams, and the thick and thin ones correspond to the p- and s-polarizations, respectively. Each curve takes the largest value at the incident angle of about θ_c - 0.03°. The curves of the quasi- LG_p=0,l±3 beams are not indicated because they are very close to those of the quasi-LG _p=0,l±2 beams.

 

Fig. 5. Calculated transverse intensity contrast. Black and red curves are for the quasi-LG_p=0,l±1 and LG_p=0,l±2 beams, and thick and thin ones correspond to the p- and s-polarizations, respectively.

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It is noted that the transverse shift can be observed only within the Rayleigh range z _R, which is 15.4 cm long in the present case, from the beam waist. At the plane of |Z| >> z _R, the field amplitude of the reflected beam at the transverse position (X, Y) is mainly determined by the particular angular spectral component having the wave vector of (k_X , k_Y )=k/Z(X,Y). There, the ID of the reflected beam is bright in the region of X/Z > θ_c -θ ₀ because the angular spectral components of k_X /k > θ_c - θ ₀ lie in the total reflection region. Therefore, any reflected paraxial beams seem to be longitudinally deformed. For the LG beams with non-zero orbital angular momentum, the ID rotates as the propagation in according to the Gouy phase, and the transverse deformation is observed at the beam waist. The calculation taking the beam propagation into account is described in a separate paper.

6. Conclusion

We demonstrate, for the first time, to the best of our knowledge, the huge transverse deformation of the reflected LG beams near the critical incidence. This will be practically applicable to diagnosis of the optical orbital angular momentum and orbital-angular-momentum-dependent manipulation of the light beams. They are closely related to quantum information technology using the orbital angular momentum of the photon as a bit.