Propagation-dependent beam profile distortion associated with the Goos-Hanchen shift

Yuhang Wan; Zheng Zheng; Jinsong Zhu

doi:10.1364/OE.17.021313

1. Introduction

The nonspecular reflection phenomena of a bounded light beam from a flat surface, where the actual reflected beam deviates from the path predicted by the ray optics with lateral/focal/angular shifts and/or beam waist modifications, have long been studied [1–4]. Among them, the lateral shift, named Goos-Hanchen (GH) shift, is the most widely studied one. Artmann first theoretically identified the cause of the GH effect as the different phases experienced by different spatial frequency components of the beam using the stationary-phase approach [5]. While the GH shift is negligible under common conditions, various configurations using absorbing media [6], left-handed metamaterial [7, 8] and surface plasmon resonance (SPR) devices [9] could hugely enhance the effect. Many experimental demonstrations [9, 10] and applications [11–13] of the enhanced GH effect have ensued.

Traditionally, the beam profile affected by the angular-dependent reflectivity is analyzed at the reflection plane in the theoretical studies, and the GH shift is quantitatively evaluated as the change in the peak position [14] or the centroid [15] of the profile. In contrast, in the experiments, this displacement is measured by position sensitive detectors placed in the far field, whose output is proportional to the weighted average position of the beam. More recently, the role of beam propagation in the GH phenomenon has attracted more attention. Theoretical studies reveal that the GH centroid shift could vary significantly at different distances from the reflection plane, and the propagation-dependent GH shift variation is derived based on the second-order truncated functional expansion of the reflectivity [15].

With the emerging of novel micro-photonic devices leveraging the GH effect [16], understanding the impact of the GH effect on the propagating wavefront, instead of just its averaged position, becomes increasingly important. Also, in the applications with potentially large GH effect, such as SPR sensing systems [17], accurately measuring and interpreting the beam shape reflected from a surface becomes essential.

In this paper, we further explore the role of propagation on the profile of the reflected focused Gaussian beam associated with the GH effect using the diffraction theory. Taking an SPR configuration as an example, our results show that strong and fast-varying distortions not yet fully elucidated in previous studies can occur in the near-field region mostly caused by the nonlinear phase shift introduced by the reflection, while in far field, the beam distribution is mainly modulated by the intensity reflectance as people would commonly observe.

2. Theoretical model

The schematic diagram of a focused beam incident upon a reflecting interface whose reflectivity can give rise to significant GH shift is shown in Fig. 1 . Three different coordinate systems are employed in our model: the incident coordinate (x_i,z_i) for the input Gaussian beam focused at the z_i=0 plane (Bi plane), the interface coordinate (x_o,z_o) in which the z_o=0 plane (Bo plane) is the physical reflecting surface, and the reflected coordinate (x_r,z_r) for the reflected beam [18]. Two-dimensional coordinates are assumed for simplicity.

Fig. 1 Schematic diagram of the configuration under study

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The optical field distribution along the Bi plane with the beam-waist radius w₀ is given by:

E_{B i} (x_{i}, 0) = \exp [- {(x_{i} / w_{0})}^{2}] / \sqrt{π} w_{0}

From Eq. (1), following an approach similar to that in [19], the field distribution along the Br plane (z_r=0) can be found via coordinate transformations and Fourier transforms as:

E_{B r} (x_{r}, 0) = \frac{1}{2 π \cos θ} \int_{- \infty}^{+ \infty} r (k_{x}) \exp [- {(\frac{k_{x} - k \sin θ}{2 \cos θ} w_{0})}^{2}] \exp (i k_{x} x_{r} \cos θ - i k_{z} x_{r} \sin θ) d k_{x}

where θ is the incident angle, k is the wavenumber in the incident medium, r(k_x) is the angular-dependent, complex reflectivity, k_x and k_z are the wavenumbers in the x_o and z_o directions respectively, and k_x²+k_z²=k².

As Eq. (2) cannot be integrated directly, further simplification is needed for later analytical analysis. Under the paraxial approximation, k_z can be expanded at k_x=ksinθ as: k_z=kcosθ -(k_x-ksinθ)tanθ. Our studies show that for the cases examined in this paper the above 1st-order Taylor series expansion is accurate enough. Then Eq. (2) can be rewritten as:

E_{B r} (x_{r}, 0) = \frac{1}{2 π \cos θ} \int_{- \infty}^{+ \infty} r (k_{x}) \exp [- {(\frac{k_{x} - k \sin θ}{2 \cos θ} w_{0})}^{2}] \exp [i x_{r} (\frac{k_{x} - k \sin θ}{\cos θ})] d k_{x}

As the field distribution on the Br plane is obtained, the distribution at the observation plane Cr after propagating a distance of Z can be calculated by applying the diffraction theory:

E_{C r} (x, Z) = \frac{Z}{i λ} \int_{- \infty}^{+ \infty} E_{B r} (x_{r}, 0) \frac{\exp (i k r_{x x_{r}})}{r_{x x_{r}}^{2}} d x_{r}

where

r_{x x_{r}}

is the distance from a diffraction point x_r in the Br plane to an observation point x in the Cr plane. For simplicity, we use E(x) in place of E_Cr(x,Z) in the following derivations.

Near-field approximation: Under the Fresnel approximation, the observed field distribution E(x), which we use in place of E_Cr(x,Z) in the following derivations, is written as:

E (x) = \frac{1}{i λ Z} e^{i k Z} \cdot e^{i \frac{k}{2 Z} x^{2}} \int_{- \infty}^{+ \infty} E_{B r} (x_{r}, 0) \exp (i \frac{k}{2 Z} x_{r}^{2}) \exp (- i \frac{k}{Z} x x_{r}) d x_{r}

By denoting the normalized deviation of the incident k_x vector σ=(k_x-ksinθ)/(kcosθ) and substituting Eq. (3) into Eq. (5), E(x) can be written as:

E (x) = \frac{1}{i λ Z} e^{i k Z} \cdot e^{i \frac{k}{2 Z} x^{2}} \frac{k}{2 π} \int_{- \infty}^{+ \infty} r (σ) \exp [- {(\frac{k σ w_{0}}{2})}^{2}] d σ \int_{- \infty}^{+ \infty} \exp (i k x_{r} σ) \exp (i \frac{k}{2 Z} x_{r}^{2}) \exp (- i \frac{k}{Z} x x_{r}) d x_{r}

As the second integral on the right-hand side can be integrated over x_r and leaves only k_x term

\sqrt{2 π Z / k} \cdot e^{i π / 4} \cdot \exp [- i k {(Z σ - x)}^{2} / (2 Z)]

, by denoting

C_{0} = k^{2} / \sqrt{2 π k Z} \cdot e^{- i π / 4} \cdot e^{i k Z}

, we get:

E (x) = \frac{C_{0}}{2 π} \int_{- \infty}^{+ \infty} r (σ) \exp [- {(\frac{k σ w_{0}}{2})}^{2}] \exp (- i \frac{k Z}{2} σ^{2}) \exp (i k x σ) d σ

Far-field approximation: Under the Fraunhofer approximation, the exp[ikx_r²/(2Z)] term in Eq. (5) is negligible, and the integral term is similar to the Fourier transform of E_Br(x_r,0). From Eq. (3), we get:

E_{B r} (u, 0) = k \cdot F T^{- 1} {r (σ) \exp [- {(k σ w_{0} / 2)}^{2}]}

where u=kx_r. That is the inversed Fourier transform from the spatial angular variable σ to u. Substituting Eq. (8) into Eq. (5), E(x) can be written as the Fourier transform of E_Br(u,0) from the variable u to the spatial angular variable x/Z:

E (x) = \frac{1}{i k λ Z} e^{i k Z} \cdot e^{i \frac{k}{2 Z} x^{2}} \int_{- \infty}^{+ \infty} E_{B r} (u, 0) \exp (- i \frac{x}{Z} u) d u

E (x / Z) \propto F T [E_{B r} (u, 0)] \propto F T (F T^{- 1} {r (σ) \exp [- {(k σ w_{0} / 2)}^{2}]}) \propto r (σ) \exp [- {(k σ w_{0} / 2)}^{2}]

As σ=x/Z is approximated in the far field, the field distribution along Cr plane is written as:

E (x) = C_{1} \cdot r (x / Z) \exp [- {(x / w_{z})}^{2}

where

C_{1} = - i / λ Z \cdot e^{i k Z} \cdot e^{i k x^{2} / (2 Z)}

, and the waist radius at Z is w_z=2Z/kw ₀ in the far-field area.

From the above approximations, we can see from Eq. (11) that, in the far field, the intensity profile of the reflected beam is directly modulated by the angular-dependent reflectance as people normally observe in the experiments. While in near field, however, the profile of the reflected beam is determined by both the intensity and the phase term of the reflectivity as shown in Eq. (7), where the phase term of the integral varies fast at different Z, and can cause strong and fast-varying distortions.

3. Results and discussions

To further investigate the phenomenon, we apply the above theoretical model to an SPR configuration, where SPR with a sharp spectral response results in strongly enhanced GH effect [9]. The medium that the beam propagates in is assumed to be the BK7 glass (ε₁ = 2.295), and a 35nm thick layer of silver (ε₂ = −18+0.5i) is considered to be at the glass and air (ε₃ = 1) interface, which generates the SPR effect for the TM-polarized input beam. The optical wavelength is set at 633 nm and the incident angle is set around 42.99^◦ (the SPR angle). The reflectivity from the glass/silver/air interface is calculated using the Fresnel equations [20]. Figure 2(a) shows the calculated reflectivity (amplitude and phase) around the SPR angle. For a focused beam that covers a range of different k_x vectors, the r(σ) (i.e. r(k_x)) is calculated through the relationship θ=sin⁻¹(k_x/k). r(σ) can also be written as R(σ)exp(iϕ(σ)), where R(σ) is the amplitude response and ϕ(σ) is the phase response.

Fig. 2 (a) Reflectivity under different incident angles; (b)The field distribution in the Br plane with different focused beams at the incident angle of 42.99° (soild line: calculated by Eq. (3), dashed line: calculated by Eq. (12))

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First, we consider the beam profile observed at the Br plane, as most theoretical work studying the GH shift assumed. For a quasi-collimated or loosely focused beam (w ₀ = 200 μm, beam divergence angle = 0.12°, or a larger beam), as shown in Fig. 2(a), the corresponding range of σ is small. When ϕ(σ) ’s expanded into ϕ₀+ϕ₁σ+ϕ_n(σ), with ϕ₁ the first derivative of ϕ and ϕ_n(σ) the higher order term, the ϕ_n(σ) term can be omitted. Thus, Eq. (3) becomes:

E_{B r} (x_{r}, 0) = \frac{e^{i ϕ_{0}} k}{2 π} \int_{- \infty}^{+ \infty} R (σ) \exp [- {(\frac{k σ w_{0}}{2})}^{2}] \exp [i k σ (x_{r} + \frac{ϕ_{1}}{k})] d σ

It is the Fourier transform of the Gaussian function modulated by R(σ)with a shift of ϕ₁/k, just as the classical theoretical model of the Goos-Hanchen shift predicts [5].

However, for a relatively tightly focused beam (such as w₀ = 20 μm, beam divergence angle = 1.15°), the higher order terms of ϕ(σ) over a much larger angular spectrum range become non-negligible. We found that their effect on the spatial distribution of the beam is significant. Figure 2(b) shows the field amplitude distribution of the reflected beam under different input beam sizes. For a quasi-collimated or loosely focused beam, the beam profile calculated using Eq. (3) agrees very well with the result calculated using Eq. (12) by omitting the higher order phase terms. While when the focus is relatively tight, the beam profile not only significantly deviates from the input beam shape, but also shows strong difference with the result calculated using Eq. (12), illustrating the effect of the nonlinear phase on the beam shape. Correspondingly, the centroid position of the reflected beam in the Br plane, as well as the GH shift, diverges from the traditional theoretical prediction. Figure 3 shows that the traditional Artmann’s formula [5] accurately predicts the GH shift under different incident angles when the beam size is relatively large but fails to match the results under a tightly focused beam, similar to what had been reported in [14].

Fig. 3 Goos-Hanchen centroid shift in the Br plane under different incident angles for different input beam sizes. Solid line: calculated with the Artmann’s formula

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When the observation plane moves away from the Br plane, the beam position and shape also changes. Figure 4 shows the calculated beam centroid position at different distances from the Br plane under different incident angles. The propagation distance is normalized by the Fraunhofer distance Z_F, defined by Z_F= 180*w ₀ ²/λ. For a loosely focused beam, our simulated results based on Eq. (7) match well with the results obtained using the equations given in [15]. However, for the case of w ₀ =20μm, the magnitude and the trend of the GH shift change significantly as shown in Fig. 4(b). In contrast, the results based on previous studies are invariant to the beam size [15] and would remain the same as shown in Fig. 4 (a), which fails to predict the actual GH shift due to the omission of the higher-order terms.

Fig. 4 The centroid of the reflected beam at different distances under different incident angles. In (a), solid line: results based on our Eq. (7); dashed line: results based on Eq. (7) in [15].

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The difference in the propagation-dependent beam profile variations is even more profound. With the loosely focused beam (w ₀ = 200 μm), Fig. 5(a) shows that the shape of the beam hardly changes when it moves from the near field region into the far field region. To better illustrate the possible changes in the shape of the beam, the scale of the x axis is normalized by the beam size of a similar Gaussian beam at that distance in order to remove the effect of size expansion due to beam divergence. The more tightly focused beam experiences a much more significant change through the propagation, as shown in Fig. 5(b). Only after a certain distance (when approaching the far field area, Z > Z_F), the beam shape becomes stable and maintains that shape afterwards, and the steady beam profile matches what is described in Eq. (11) and is proportional to the angular-dependent reflectance. The strong distance-dependent distortion is caused by the interaction of the Z-dependent phase term exp(-ikZσ²/2) and the higher-order reflection phase terms of ϕ(σ).

Fig. 5 Propagation-dependent profile distortion of the focused reflected beam

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We note that there is one point in either plot of Fig. 4 where the lines cross each other, under which angle the GH shift is propagation-distance independent. Theory in [15] identifies this particular angle corresponding to the extreme position of the reflectance function R, e.g. the SPR angle in this case (42.99°). The result in Fig. 4 (a) is in line with that. Under that incident angle, when the beam is large, the reflected beam under the GH effect is just moving in parallel to that predicted by the geometric optics shifted by a fixed GH distance with little variation in its shape, as shown in Fig. 5 (a), just as many previous studies on GH effects envisioned. However, for the tightly focused beam, not only this angular position is shifted, to 43.06° in our case here, but the change in the beam shape is very large even when the centroid position remains unchanged (see Fig. 5(c)).

We also note that the phenomenon we discussed here is not limited to the SPR effect and could happen in other configurations of GH effect, where either the beam covers a large angular spectrum or the phase change is steep and, thus, highly nonlinear. For applications like high-resolution SPR array sensing, where the measurement of the reflected beam profile is used to re-construct the angular-dependent reflectivity, the distortions and their spatial variations might cause uncertainties in the measurement results if their presence is ignored, especially in a compact or integrated system.

4. Conclusions

By deriving the theoretical model governing the optical field distribution of a propagating reflected beam from a flat surface, we show that the beam profile can strongly deviate from the previously widely-studied shape at the reflecting plane in the near-field, before it converges to a profile decided by the intensity reflectance in the far-field. Our study shows that this fast-varying, propagation-dependent profile distortion cannot be accurately predicted by theories using only the lower-order terms of the reflectivity. Our results offer useful guidelines for future studies on devices and experiments with strong GH effects.

Acknowledgments

The work at Beihang University was supported by NSFC (60877054/60921001), 973 Program (2009CB930701), and the Innovation Foundation of BUAA for PhD Graduates.

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Propagation-dependent beam profile distortion associated with the Goos-Hanchen shift

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (12)

Optics Express