Axial dependence of optical weak measurements in the critical region

The Goos–Hänchen [GH] shift [1–3] surely is one of the most intriguing research subjects to appear in literature in recent decades [4–12]. This shift, which is probably one of the clearest manifestations of the evanescent nature of light, represents an additional contribution to the geometrical optical path predicted by the Snell law [13, 14]. This quantum effect is still the subject of careful and broad investigation and continues to stimulate new discussion [15–22]. Of particular interest to the study presented in this paper is the GH shift frequency crossover [23]. For incidence angles ${{\theta }_{0}}$ far from the critical angle, ${{\theta }_{c}}$, it is well known that the GH shift is proportional to the wavelength, λ, of the optical beam [2, 4, 23, 24]. For incidence at critical angle the GH shift is amplified by a factor $\sqrt{{{{\rm w}}_{0}}/\lambda }$, where ${{{\rm w}}_{0}}$ is the beam waist. This amplification has been recently obtained analytically by using the stationary phase method [25, 26] and then confirmed by numerical calculations [23]. This frequency crossover will play a fundamental role in deriving the expected experimental curves for optical weak measurements in the critical (angle) region.

In a recent interesting experimental paper [27], by using the optical analog [28–30] of the electron spin weak measurement [31], the behavior of the GH shift curve has been reproduced in the region in which the incidence angles are far enough from the critical angle to permit some important approximations.

It is important to observe that in the optical analog of the electron spin weak measurement, polarized light plays the role of the spin $\frac{1}{2}$ particles and the laser beam replaces the coherent electron beam. Because the displacement produced by the optical system is a lateral shift rather than an angular deflection, as happens in the presence of the Stern–Gerlach magnet [31], we must consider spatial distributions instead of momentum distributions. Even though the physics is far from the same, with sufficient attention, an optical version of the electron spin weak measurement experiment can be constructed [28].

A unified linear algebra approach to dielectric reflection that recently appeared in the literature [29, 30] bases the analogy between weak values and optical beam shifts of polarized waves on the expectation value of the Artman operator. Such an operator is shown to be Hermitian for total internal reflection and non-Hermitian in the critical region [29]. For the mathematical details, we refer the reader to reference [30]. In our approach, we discuss the optical analog of the electron spin weak measurement, by analyzing, as done theoretically in reference [28] and experimentally in reference [27], the distance between the peaks of the outgoing optical beam. We recall that in the critical region, due to the breaking of symmetry [22], the peak and mean values do not necessarily coincide. In view of these comments, our discussion can be seen as a complementary work to the one which appears in reference [29, 30].

To make this introduction and the objective of our analysis clearer to the reader, we recall that in the optical analog of the quantum weak measurement [27, 28], the parameters which characterize the behavior of the distance between peaks in the experimental curves are

$$\begin{eqnarray}&&\epsilon ={{\epsilon }_{0}}+\Delta \epsilon ={\rm cos} (\alpha -\beta )/{\rm cos} (\alpha +\beta ),\end{eqnarray} \tag{ 1 }$$

where α and $\beta =\alpha +\frac{\pi }{2}\;+\;{{\gamma }_{0}}\;+\;\Delta \gamma$ are the polarization angles of the first and second polarizer (see figure 1), and

$$\begin{eqnarray}&&\Delta {{y}_{{{\rm GH}}}}=y_{{{\rm GH}}}^{^{[p]}}-y_{{{\rm GH}}}^{^{[s]}},\end{eqnarray} \tag{ 2 }$$

where $y_{{{\rm GH}}}^{^{[s,p]}}$ are, respectively, the GH shifts for s and p polarization. For small rotation angles, i.e., $\Delta \gamma \ll 1$, and for an incoming beam with an equal mixture of polarizations, i.e., $\alpha =\pi /4$, we have

$$\begin{eqnarray*}&&{{\epsilon }_{0}}+\Delta \epsilon ={\rm tan} ({{\gamma }_{0}}+\Delta \gamma )\approx {\rm tan} {{\gamma }_{0}}+\frac{\Delta \gamma }{{{{\rm cos} }^{2}}{{\gamma }_{0}}}.\end{eqnarray*}$$

For incidence angles far from the critical angle, the condition

$$\begin{eqnarray}&&\Delta \epsilon \gg \Delta {{y}_{{{\rm GH}}}}/{\rm w}(z)\approx \lambda /{\rm w}(z),\end{eqnarray} \tag{ 3 }$$

where ${\rm w}(z)={{{\rm w}}_{0}}\sqrt{1+{{\left( \lambda z/\pi \;{\rm w}_{0}^{2} \right)}^{^{2}}}}$, is easily satisfied and, as we shall see in detail later, the distance between the peaks of the outgoing beams for two opposite rotations in the second polarizer, i.e., ${{\beta }_{{\pm }}}=\frac{3}{4}\;\pi +{{\gamma }_{0}}\pm \mid \Delta \gamma \mid$, is given by

$$\begin{eqnarray}&&\Delta {{Y}_{{\rm max} }}\approx \;\;\Delta {{y}_{{{\rm GH}}}}/\mid \Delta \epsilon \mid .\end{eqnarray} \tag{ 4 }$$

Consequently, for polarizer rotations which satisfy the constraint in (3), the experimental curve of $\Delta {{Y}_{{{\rm max} }}}$ reproduces the GH shift curve amplified by the factor $1/\mid \Delta \epsilon \mid$. The GH behavior and its amplification (far from the critical region) were recently confirmed in the experimental investigation presented in reference [27].

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As observed at the begining of this introduction, the frequency crossover in the critical region [23] leads to

$$\begin{eqnarray}&&\Delta y_{{{\rm GH}}}^{^{[{\rm cri}]}}\;\propto \;\sqrt{\lambda \;{\rm w}(z)}\;\gg \;\lambda .\end{eqnarray} \tag{ 5 }$$

This critical GH shift behavior stimulates investigation into what happens for incidence angles near the critical angle, where, due to the amplification $\sqrt{{\rm w}(z)/\lambda }$, the condition in (3) may no longer be valid. This should modify the shape of the experimental curves, and a new formula should be introduced to estimate the GH shift by measuring the distance between peaks, $\Delta {{Y}_{{\rm max} }}$. In view of a possible experimental implementation of optical weak measurements for incidence near the critical angle, we analyze the expected experimental curves for mixed polarized laser Gaussian beams with $\lambda =633\;{\rm nm}$ and ${{{\rm w}}_{0}}=200,\;300,{\rm and}\;500\;\mu {\rm m}$ propagating through BK7 and fused silica dielectric blocks.

This paper is organized as follows. In section 2, we give the transmission coefficient for the beam propagating through the dielectric structure of figure 2 and calculate the axial dependence of the GH shift for BK7 (figure 3) and fused silica (figure 4) blocks. In section 3, we introduce the idea of weak measurement in optics and analyze the effect that the polarizer and analyzer have on the s and p polarized waves of the outgoing beam. For critical incidence, new parameters have to be introduced (figure 5). The analysis of the distance between the main peaks of the outgoing beams for two opposite rotation angles of the second polarizer is presented in section 4. There a new analytical relation between the GH shift and the distance between peaks is also introduced. In this section, we also present the expected experimental curves for incoming Gaussian beams with different beam waists (${{{\rm w}}_{0}}=200,\;300,{\rm and}500\;\mu {\rm m}$) propagating through BK7 (figure 6) and fused silica (figure 7) dielectric blocks. The axial dependence of optical weak measurements is clear in the plots and is one of the important results of our analysis. The final section gives our conclusions and outlook.

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To obtain the mathematical expression for the transmitted beam, $E_{{{\rm out}}}^{^{[s,p]}}$, propagating in the y–z plane through the dielectric block (see figures 1 and 2), let us first introduce the Gaussian wave number distribution which determines the shape of the incoming beam, $E_{{{\rm in}}}^{^{[s,p]}}$:

$$\begin{eqnarray}&&g(\theta -{{\theta }_{0}})=\frac{k{{{\rm w}}_{0}}}{2\sqrt{\pi }}\;{\rm exp} \left[ -{{(k{{{\rm w}}_{0}})}^{^{2}}}{{(\theta -{{\theta }_{0}})}^{^{2}}}/4 \right].\end{eqnarray} \tag{ 6 }$$

In the electric amplitude expressions the superscript notation distinguishes between s and p polarized light. By using the paraxial approximation ($k\;{{{\rm w}}_{0}}\gtrsim 10$), the incoming electric field, which moves from the source laser to the left side of the dielectric block, can be represented by [13, 14]:

$$\begin{eqnarray}\begin{array}{rcl} E_{{{\rm in}}}^{^{[s,p]}}(y,z) & = & {{E}_{0}}\;{{e}^{ik\,z}}\int _{-\pi /2}^{+\pi /2}\;{\rm d}\theta \;g(\theta -{{\theta }_{0}}) \\ {} & {} & \times {\rm exp} \left[ i\left( \theta -{{\theta }_{0}} \right)k\;y-i\frac{{{(\theta -{{\theta }_{0}})}^{^{2}}}}{2}\;k\;z \right] \\ {} & = & \frac{{{E}_{0}}\;{{e}^{ik\,z}}}{\sqrt{1+2\;i\frac{z}{k\;{\rm w}_{0}^{2}}}}{\rm exp} \left[ -\frac{{{y}^{2}}}{{\rm w}_{0}^{2}+2\;i\;\frac{z}{k}} \right]. \\ \end{array}\end{eqnarray} \tag{ 7 }$$

For Gaussian lasers with a small beam waist with respect to the dimensions of the dielectric block, we can use the step technique of quantum mechanics [32–36] and give the Fresnel coefficients in terms of angles θ, ψ, and φ (see figure 2).

$$\begin{eqnarray*}&&{\rm sin} \theta =n\;{\rm sin} \psi \quad {\rm and}\quad \varphi =\psi +\frac{\pi }{4}.\end{eqnarray*}$$

The transmission coefficient which characterizes the outgoing beam is obtained by the transmission through the left/right sides and the reflection between the up/down sides of the dielectric block. After simple algebraic manipulations (for more details see reference [37]), we find

$$\begin{eqnarray}\begin{array}{rcl} {{T}^{^{[s]}}}(\theta ) & = & \frac{4\;n{\rm cos} \psi {\rm cos} \theta }{{{\left( \;{\rm cos} \theta +n{\rm cos} \psi \right)}^{^{2}}}} \\ {} & {} & \times {{\left( \;\frac{n{\rm cos} \varphi -\sqrt{1-{{n}^{2}}{{{\rm sin} }^{2}}\varphi }}{n{\rm cos} \varphi +\sqrt{1-{{n}^{2}}{{{\rm sin} }^{2}}\varphi }} \right)}^{^{2}}}{\rm exp} [i\;{{\phi }_{{{\rm Snell}}}}] \\ \end{array}\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} {{T}^{^{[p]}}}(\theta ) & = & \frac{4\;n{\rm cos} \psi {\rm cos} \theta }{{{\left( \;n{\rm cos} \theta +{\rm cos} \psi \right)}^{^{2}}}} \\ {} & {} & \times {{\left( \;\frac{{\rm cos} \varphi -n\sqrt{1-{{n}^{2}}{{{\rm sin} }^{2}}\varphi }}{{\rm cos} \varphi +n\sqrt{1-{{n}^{2}}{{{\rm sin} }^{2}}\varphi }} \right)}^{^{2}}}{\rm exp} [\;i\;{{\phi }_{{{\rm Snell}}}}], \\ \end{array}\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray*}&&{{\phi }_{{{\rm Snell}}}}=k\left[ \sqrt{2}n{\rm cos} \varphi \overline{AB}+(n{\rm cos} \psi -{\rm cos} \theta )\frac{\overline{AD}}{\sqrt{2}} \right].\end{eqnarray*}$$

For $n{\rm sin} \varphi \lt 1$, the outgoing beam,

$$\begin{eqnarray}\begin{array}{rcl} E_{{{\rm out}}}^{^{[s,p]}}(y,z) & = & {{E}_{0}}\;{{e}^{ik\,z}}\int _{-\pi /2}^{+\pi /2}\ \ {\rm d}\theta \;{{T}^{^{[s,p]}}}(\theta )g(\theta -{{\theta }_{0}}) \\ {} & {} & \times {\rm exp} \left[ \;i\left( \theta -{{\theta }_{0}} \right)k\;y-\;i\ \frac{{{(\theta -{{\theta }_{0}})}^{^{2}}}}{2}k\;z \right], \\ \end{array}\end{eqnarray} \tag{ 10 }$$

is centered at

$$\begin{eqnarray}\begin{array}{rcl} {{y}_{{{\rm Snell}}}} & = & -{{\left[ \;\frac{\partial {{\phi }_{{{\rm Snell}}}}}{k\;\partial \theta } \right]}_{{0}}} \\ {} & = & {\rm cos} {{\theta }_{0}}\left[ ({\rm tan} {{\psi }_{0}}+1)\overline{AB} \right. \\ {} & {} & +\left. ({\rm tan} {{\psi }_{0}}-{\rm tan} {{\theta }_{0}})\frac{\overline{AD}}{\sqrt{2}}\; \right]. \\ \end{array}\end{eqnarray} \tag{ 11 }$$

This represents the well-known geometrical shift predicted by the Snell law in ray optics.

For $n{\rm sin} \varphi \gt 1$ an additional phase comes from the internal reflection coefficients in (8) and (9):

$$\begin{eqnarray}\begin{array}{rcl} \left\{ \phi _{{{\rm GH}}}^{^{[s]}},\phi _{{{\rm GH}}}^{^{[p]}} \right\} & = & -4\left\{ {\rm arctan} \left[ \frac{\sqrt{{{n}^{2}}{{{\rm sin} }^{2}}\varphi -1}}{n{\rm cos} \varphi } \right] \right., \\ {} & {} & \left. {\rm arctan} \left[ \frac{n\sqrt{{{n}^{2}}{{{\rm sin} }^{2}}\varphi -1}}{{\rm cos} \varphi } \right] \right\} \\ \end{array}\end{eqnarray} \tag{ 12 }$$

and a new shift (the well known GH shift) must be considered. The numerical data for the propagation through BK7 (n = 1.515) and fused silica (n = 1.457) are plotted in figures 3 and 4. The data clearly show the amplification for incidence in the critical region, and they are in excellent agreement with the analytical prediction for incidence far from the critical angle,

$$\begin{eqnarray}\begin{array}{rcl} \left\{ y_{{{\rm GH}}}^{^{[s]}},y_{{{\rm GH}}}^{^{[p]}} \right\} & = & -{{\left[ \left\{ \frac{\partial \phi _{{{\rm GH}}}^{^{[s]}}}{k\partial \theta },\frac{\partial \phi _{{{\rm GH}}}^{^{\,[p]}}}{k\;\partial \theta } \right\} \right]}_{{0}}} \\ {} & = & \frac{4{\rm cos} {{\theta }_{0}}{\rm sin} {{\varphi }_{0}}}{k{\rm cos} {{\psi }_{0}}\sqrt{{{n}^{2}}{{{\rm sin} }^{2}}{{\varphi }_{0}}-1}} \\ {} & {} & \times \left\{ 1,\frac{1}{{{n}^{^{2}}}{{{\rm sin} }^{2}}{{\varphi }_{0}}-{{{\rm cos} }^{2}}{{\varphi }_{0}}} \right\}, \\ \end{array}\end{eqnarray} \tag{ 13 }$$

where the ${{{\rm w}}_{0}}$ dependence disappears [2, 4, 23, 24]. The plots of the GH shift for BK7 and fused silica clearly show an axial dependence. This axial dependence, which has been recently investigated and recognized as a possible source of angular deviations in the optical path predicted by the Snell law [37], must be seen in the optical weak measurement curves as well. Understanding how this axial dependence modifies the optical weak measurement curves in the critical region is one of the main objectives of our study.

On the basis of the results presented in the preceding section, we can approximate the s and p polarized outgoing beams as follows:

$$\begin{eqnarray*}\begin{array}{rcl} E_{{{\rm out}}}^{^{[s,p]}}(y,z) & \approx & \frac{{{E}_{0}}{{e}^{ik\,z}}}{\sqrt{1+2\;i\frac{z}{k\;{\rm w}_{0}^{2}}}}\mid T_{0}^{^{[s,p]}}\mid \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{ll} {} & {} & \times {\rm exp} \left[ -\frac{{{\left( y-{{y}_{{{\rm Snell}}}}-y_{{{\rm GH}}}^{^{[s,p]}} \right)}^{2}}}{{\rm w}_{0}^{2}+2\;i\;\frac{z}{k\;}} \right. \\ {} & {} & \left. +i\left( {{\phi }_{{{\rm Snell},0}}}+\phi _{{{\rm GH},0}}^{^{[s,p]}} \right) \right], \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where for $y_{{{\rm GH}}}^{^{[s,p]}}$, which represents the only entry for which we do not have a full analytical expression, we must to use the numerical data plotted in figures 3 and 4.

The intensity of the outgoing beam coming out from the dielectric block and moving toward the analyzer (${{z}_{{{\rm out}}}}\lt z\lt {{z}_{{{\rm A}}}}$ in figure 1) is given by

$$\begin{eqnarray}&&\begin{array}{l} {{I}_{{{\rm out}}}}(y,{{z}_{{{\rm out}}}}\lt z\lt {{z}_{{{\rm A}}}}) \\ \quad =\mid {\rm sin} \;\alpha \;E_{{{\rm out}}}^{^{[s]}}(y,z)+{\rm cos} \;\alpha E_{{{\rm out}}}^{^{[p]}}(y,z){{\mid }^{^{2}}} \\ \quad \propto \mid \tau {\rm tan} \alpha {\rm exp} \left[ -{{\left( \frac{y-{{y}_{{{\rm Snell}}}}-y_{{{\rm GH}}}^{^{[s]}}}{{\rm w}(z)} \right)}^{^{2}}}+\;i\;\Delta {{\phi }_{{{\rm GH}}}}\; \right] \\ \qquad +{\rm exp} \left[ -{{\left( \frac{y-{{y}_{{{\rm Snell}}}}-y_{{{\rm GH}}}^{^{[p]}}}{{\rm w}(z)} \right)}^{^{2}}}\; \right]{{\mid }^{^{2}}}, \\ \end{array}\end{eqnarray} \tag{ 15 }$$

where $\tau =\mid \;T_{0}^{^{[s]}}/\;\;T_{0}^{^{[p]}}\mid$ and $\Delta {{\phi }_{{{\rm GH}}}}=\phi _{{{\rm GH},0}}^{^{[s]}}-\phi _{{{\rm GH},0}}^{^{[p]}}$. The ${{\theta }_{0}}$ dependence of τ and $\Delta {{\phi }_{{{\rm GH}}}}$ is plotted in figures 5(a) (BK7) and (b) (fused silica). After removing the phase difference between the s and p polarized light by the analyzer located at $z={{z}_{{{\rm A}}}}$ and combining s and p polarization by the second polarizer located at $z={{z}_{{\beta }}}$, the outgoing beam intensity becomes

$$\begin{eqnarray}&&\begin{array}{l} {{I}_{{{\rm out}}}}(Y,z\gt {{z}_{{\beta }}}) \\ \quad \propto \mid \tau \;{\rm tan} \alpha {\rm tan} \beta \;{\rm exp} \left[ -{{\left( \frac{Y+\;\frac{\Delta {{y}_{{{\rm GH}}}}}{2}}{{\rm w}(z)} \right)}^{^{2}}} \right] \\ \quad +\;{\rm exp} \left[ -{{\left( \frac{Y-\;\frac{\Delta {{y}_{{{\rm GH}}}}}{2}}{{\rm w}(z)} \right)}^{^{2}}}\; \right]{{\mid }^{2}}, \\ \end{array}\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray*}&&Y=y-{{y}_{{{\rm Snell}}}}-\frac{y_{{{\rm GH}}}^{^{[s]}}+y_{{{\rm GH}}}^{^{[p]}}}{2}\quad {\rm and}\quad \Delta {{y}_{{{\rm GH}}}}=y_{{{\rm GH}}}^{^{[p]}}-y_{{{\rm GH}}}^{^{[s]}}.\end{eqnarray*}$$

Observing that

$$\begin{eqnarray*}&&{\rm tan} \alpha {\rm tan} \beta =\frac{\epsilon -1}{\epsilon +1}\approx \frac{{{\epsilon }_{0}}-1}{{{\epsilon }_{0}}+1}+\frac{2}{{{({{\epsilon }_{0}}+1)}^{^{2}}}}\;\Delta \epsilon ,\end{eqnarray*}$$

the choice of an appropriate rotation ${{\gamma }_{0}}$ makes it possible to fix the parameter ${{\epsilon }_{0}}(={\rm tan} {{\gamma }_{0}})$ to

$$\begin{eqnarray}&&{{\epsilon }_{0}}=\frac{\tau -1}{1+\tau }\quad \Rightarrow \quad \frac{{{\epsilon }_{0}}-1}{{{\epsilon }_{0}}+1}=-\frac{1}{\tau }.\end{eqnarray} \tag{ 17 }$$

The angular dependence of ${{\gamma }_{0}}$ is plotted in figures 5(b) (BK7) and (d) (fused silica) for incidence angles in the critical region. This choice enables rewriting of the outgoing intensity in terms of the parameters τ and $\Delta \epsilon$ as follows:

$$\begin{eqnarray}&&\begin{array}{l} {{I}_{{{\rm out}}}}(Y,\Delta \epsilon ) \\ \quad \propto \mid \left[ \;\frac{{{(1+\tau )}^{^{2}}}}{2\;\tau }\;\Delta \epsilon -1 \right]\;{\rm exp} \left[ -{{\left( \frac{Y+\;\frac{\Delta {{y}_{{{\rm GH}}}}}{2}}{{\rm w}(z)} \right)}^{^{2}}} \right] \\ \quad +{\rm exp} \left[ -{{\left( \frac{Y-\;\frac{\Delta {{y}_{{{\rm GH}}}}}{2}}{{\rm w}(z)} \right)}^{^{2}}}\; \right]{{\mid }^{^{2}}} \\ \quad \approx {{\left\{ 2\left[ \frac{{{(1+\tau )}^{^{2}}}}{4\;\tau }\;\Delta \epsilon +\frac{\Delta {{y}_{{{\rm GH}}}}}{{{{\rm w}}^{2}}(z)}Y \right]{\rm exp} \left[ -\frac{{{Y}^{^{2}}}}{{{{\rm w}}^{2}}(z)} \right] \right\}}^{^{2}}}. \\ \end{array}\end{eqnarray} \tag{ 18 }$$

Finally, noting that, in the critical region, ${{(1+\tau )}^{^{2}}}\approx 4\;\tau$ (see figures 5(b)–(d), we can get a further simplification of the outgoing beam intensity:

$$\begin{eqnarray}&&{{I}_{{{\rm out}}}}(Y,\Delta \epsilon )\;\propto {{\left[ \Delta \epsilon +\frac{\Delta {{y}_{{{\rm GH}}}}}{{{{\rm w}}^{2}}(z)}Y \right]}^{^{2}}}{\rm exp} \left[ -\frac{2\;{{Y}^{^{2}}}}{{{{\rm w}}^{2}}(z)} \right].\end{eqnarray} \tag{ 19 }$$

The starting point in optical weak measurement experiments is to set the angles of the first and second polarizers to

$$\begin{eqnarray*}&&\{{{\alpha }_{0}},{{\beta }_{0}}\}=\{\frac{\pi }{4},\frac{3\;\pi }{4}+{{\gamma }_{0}}\}.\end{eqnarray*}$$

For this choice ($\Delta \epsilon =0$) we find that the outgoing intensity,

$$\begin{eqnarray}&&{{I}_{{{\rm out}}}}(Y,0)\propto {{Y}^{^{2}}}{\rm exp} \left[ -\frac{2\;{{Y}^{^{2}}}}{{{{\rm w}}^{2}}(z)} \right],\end{eqnarray} \tag{ 20 }$$

is a symmetric function with two peaks centered at $Y_{{{\rm max} }}^{^{\pm }}=\pm {\rm w}(z)/\sqrt{2}$ and a minimum centered at ${{Y}_{{{\rm min} }}}=0$. By changing the angle of the second polarizer from ${{\beta }_{0}}$ to ${{\beta }_{0}}+\Delta \gamma ,$ we break the symmetry. In terms of $\Delta \epsilon =\Delta \gamma /{{{\rm cos} }^{2}}{{\gamma }_{0}}$, we find

$$\begin{eqnarray}&&{{Y}_{{{\rm min} }}}(\Delta \epsilon )=-\frac{\Delta \epsilon }{\Delta {{y}_{{{\rm GH}}}}}\;{{{\rm w}}^{2}}(z)\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&\begin{array}{l} Y_{{{\rm max} }}^{^{\pm }}(\Delta \epsilon ) \\ \quad =\frac{-\Delta \epsilon \;\pm \sqrt{{{(\Delta \epsilon )}^{^{2}}}+2[\Delta y_{{{\rm GH}}}^{^{2}}/\;\;{{{\rm w}}^{2}}(z)]}}{2\Delta {{y}_{{{\rm GH}}}}}\;{{{\rm w}}^{2}}(z). \\ \end{array}\end{eqnarray} \tag{ 22 }$$

It is clear that for positive $\Delta \epsilon$ (counterclockwise rotation $\Delta \gamma$ around ${{\gamma }_{0}}$), $Y_{{{\rm max} }}^{+}(\mid \Delta \epsilon \mid )$ represents the position of the main peak of the outgoing beam. For negative $\Delta \epsilon$, the main peak is instead centered at $Y_{{{\rm max} }}^{-}(-\mid \Delta \epsilon \mid )$. By using equation (22), the distance between these peaks is given by

$$\begin{eqnarray}\begin{array}{rcl} \Delta {{Y}_{{{\rm max} }}} & = & Y_{{{\rm max} }}^{^{+}}(\mid \Delta \epsilon \mid )-Y_{{{\rm max} }}^{^{-}}(-\mid \Delta \epsilon \mid ) \\ {} & = & \frac{-\mid \Delta \epsilon \mid +\sqrt{\mid \Delta \epsilon {{\mid }^{^{2}}}+2[\;\Delta y_{{{\rm GH}}}^{^{2}}/{{{\rm w}}^{2}}(z)]}}{\Delta {{y}_{{{\rm GH}}}}}\;{{{\rm w}}^{2}}(z). \\ \end{array}\end{eqnarray} \tag{ 23 }$$

In the region $0\leqslant \mid \Delta \epsilon \mid \leqslant \Delta {{y}_{{{\rm GH}}}}/{\rm w}(z)$, we find

$$\begin{eqnarray}&&\sqrt{2}\;{\rm w}(z)\;\leqslant \;\;\Delta {{Y}_{{{\rm max} }}}\;\leqslant \;(\sqrt{3}-1)\;{\rm w}(z)\;.\end{eqnarray} \tag{ 24 }$$

This clearly shows that by increasing the value of $\mid \Delta \epsilon \mid$, we reduce the distance between the peaks. For $\mid \Delta \epsilon \mid \gg \Delta {{y}_{{{\rm GH}}}}/{\rm w}(z),$

$$\begin{eqnarray}&&\Delta {{Y}_{{{\rm max} }}}\;\approx \Delta {{y}_{{{\rm GH}}}}/\mid \Delta \epsilon \mid .\end{eqnarray} \tag{ 25 }$$

For incidence angles far from the critical region, because the GH shift is proportional to the wavelength of the laser beam, the condition

$$\begin{eqnarray*}&&\frac{\Delta {{y}_{{{\rm GH}}}}}{{\rm w}(z)}\;\approx \;\frac{\lambda }{{\rm w}(z)}\ll \;\mid \Delta \epsilon \mid \end{eqnarray*}$$

can be easily satisfied. Thus, far from the critical region, the experimental curves of $\Delta {{Y}_{{{\rm max} }}}$ reproduce the GH curves amplified by the factor $1/\mid \Delta \epsilon \mid$.

In the critical region, the frequency crossover and the axial dependence, shown in figures 3 (BK7) and 4 (fused silica), work against the validity of the constraint $\mid \Delta \epsilon \mid \gg \Delta {{y}_{{{\rm GH}}}}/{\rm w}(z)$. This means that in this region, the experimental curves of $\Delta {{Y}_{{{\rm max} }}}$ do not necessarily reproduce the GH curves. The expected experimental curves of the distance between peaks are plotted for different values of $\mid \Delta \epsilon \mid$ in figures 6 (bk7) 7 (fused silica). The plots confirm that, at critical incidence, the amplification does not reproduce the $1/\mid \Delta \epsilon \mid$ proportionality. The axial dependence is removed by increasing the beam waist ${{{\rm w}}_{0}}$.

For experimental use, it is convenient to express the GH shift, $\Delta {{y}_{{{\rm GH}}}}$, in terms of the experimental quantity $\Delta {{Y}_{{{\rm max} }}}$. From equation (23), we obtain

$$\begin{eqnarray}&&\Delta {{y}_{{{\rm GH}}}}=\frac{2\mid \Delta \epsilon \mid \;{{{\rm w}}^{2}}(z)}{2\;{{{\rm w}}^{2}}(z)-\Delta Y_{{{\rm max} }}^{^{2}}}\;\Delta {{Y}_{{{\rm max} }}}.\end{eqnarray} \tag{ 26 }$$

The error in the GH shift is given by

$$\begin{eqnarray*}&&\begin{array}{l} \frac{\sigma (\Delta {{y}_{{{\rm GH}}}})}{\Delta {{y}_{{{\rm GH}}}}} \\ \quad =\sqrt{\begin{array}{ccccccccccccccc} {{\left[ \frac{\sigma (\mid \Delta \epsilon \mid )}{\mid \Delta \epsilon \mid } \right]}^{^{2}}} \\ \quad +{{\left[ \frac{2\;{{{\rm w}}^{2}}(z)+\Delta Y_{{{\rm max} }}^{^{2}}}{2\;{{{\rm w}}^{2}}(z)-\Delta Y_{{{\rm max} }}^{^{2}}}\;\frac{\sigma (\Delta {{Y}_{{{\rm max} }}})}{\Delta {{Y}_{{{\rm max} }}}} \right]}^{^{2}}} \\ \quad +{{\left\{ \frac{2\;\Delta Y_{{{\rm max} }}^{^{2}}}{2\;{{{\rm w}}^{2}}(z)-\Delta Y_{{{\rm max} }}^{^{2}}}\;\frac{\sigma [{\rm w}(z)]}{{\rm w}(z)} \right\}}^{^{2}}} \\ \end{array}}. \\ \end{array}\end{eqnarray*}$$

Recalling that for $\Delta \epsilon =0$, the distance between the peaks gives direct information about the beam waist, $\Delta {{Y}_{{{\rm max} }}}=\sqrt{2}\;{\rm w}(z)$, we can use

$$\begin{eqnarray*}&&\sigma (\Delta {{Y}_{{{\rm max} }}})=\sigma [{\rm w}(z)]\end{eqnarray*}$$

in the preceding error formula and obtain

$$\begin{eqnarray*}&&\begin{array}{l} \frac{\sigma (\Delta {{y}_{{{\rm GH}}}})}{\Delta {{y}_{{{\rm GH}}}}} \\ \quad =\sqrt{\begin{array}{ccccccccccccccc} {{\left[ \frac{\sigma (\mid \Delta \epsilon \mid )}{\mid \Delta \epsilon \mid } \right]}^{^{2}}}+{{\left[ \frac{2{{{\rm w}}^{2}}(z)+\Delta Y_{{{\rm max} }}^{^{2}}}{2{{{\rm w}}^{2}}(z)-\Delta Y_{{{\rm max} }}^{^{2}}} \right]}^{^{2}}} \\ \quad \times \left\{ 1+{{\left[ \frac{2\Delta Y_{{{\rm max} }}^{^{3}}/{\rm w}(z)}{2{{{\rm w}}^{2}}(z)+\Delta Y_{{{\rm max} }}^{^{2}}} \right]}^{^{2}}} \right\}{{\left[ \frac{\sigma (\Delta {{Y}_{{{\rm max} }}})}{\Delta {{Y}_{{{\rm max} }}}} \right]}^{^{2}}} \\ \end{array}}. \\ \end{array}\end{eqnarray*}$$

In the region ${\rm w}(z)\leqslant \Delta {{Y}_{{{\rm max} }}}\leqslant \sqrt{2}\;{\rm w}(z)$, we find

$$\begin{eqnarray*}&&13\leqslant {{\left[ \frac{2{{{\rm w}}^{2}}(z)+\Delta Y_{{{\rm max} }}^{^{2}}}{2\;{{{\rm w}}^{2}}(z)-\Delta Y_{{{\rm max} }}^{^{2}}} \right]}^{^{2}}}\left\{ 1+{{\left[ \frac{2\;\Delta Y_{{{\rm max} }}^{^{3}}/\;{\rm w}(z)}{2\;{{{\rm w}}^{2}}(z)+\Delta Y_{{{\rm max} }}^{^{2}}} \right]}^{^{2}}} \right\}\leqslant \infty .\end{eqnarray*}$$

To avoid great standard deviations, we must work in the region $\Delta {{Y}_{{{\rm max} }}}\leqslant {\rm w}(z)$, where

$$\begin{eqnarray*}&&\begin{array}{l} \sqrt{{{\left[ \frac{\sigma (\mid \Delta \epsilon \mid )}{\mid \Delta \epsilon \mid } \right]}^{^{2}}}+{{\left[ \frac{\sigma (\Delta {{Y}_{{{\rm max} }}})}{\Delta {{Y}_{{{\rm max} }}}} \right]}^{^{2}}}} \\ \quad \leqslant \frac{\sigma (\Delta {{y}_{{{\rm GH}}}})}{\Delta {{y}_{{{\rm GH}}}}} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}&&\begin{array}{l} \quad \leqslant \sqrt{{{\left[ \;\frac{\sigma (\mid \Delta \epsilon \mid )}{\mid \Delta \epsilon \mid } \right]}^{^{2}}}+13{{\left[ \frac{\sigma (\Delta {{Y}_{{{\rm max} }}})}{\Delta {{Y}_{{{\rm max} }}}} \right]}^{^{2}}}}. \\ \end{array}\end{eqnarray} \tag{ 27 }$$

From the aforementioned condition on $\Delta {{Y}_{{{\rm max} }}}$, by using equation (26) we obtain

$$\begin{eqnarray}&&\mid \Delta \epsilon \mid \geqslant ,\frac{\Delta {{y}_{{{\rm GH}}}}}{2{\rm w}(z)}.\end{eqnarray} \tag{ 28 }$$

The choice of $\mid \Delta {{\epsilon }_{{{\rm min} }}}\mid =\Delta {{y}_{{{\rm GH}}}}/2\;{\rm w}(z)$ in the second polarizer thus is an additional experimental constraint to avoid great standard deviations.

The possibility of using the weak measurement of the electron spin component [31] in optics [28, 29] has recently stimulated an experiment [27] based on the interference between different polarizations, in which the GH shift curves are reproduced in the region of the validity of the standard analytic formula (13). Nevertheless, the analytical shift (13) diverges when the incidence angle approaches the critical angle. In a recent paper [23], this divergence was removed and an analytic formula, valid for $2z\ll k{\rm w}_{0}^{2}$, was proposed for the GH shift at the critical angle:

$$\begin{eqnarray}&&\begin{array}{l} {{\left\{ y_{{{\rm GH}}}^{^{[s]}},y_{{{\rm GH}}}^{^{[p]}} \right\}}_{{{\rm cri}}}} \\ \quad \approx \frac{\sqrt{k\;{\rm w}(z)}}{k} \\ \quad \times \sqrt{2\;\sqrt{2\;\pi }\frac{\sqrt{2-{{n}^{2}}+2\;\sqrt{{{n}^{2}}-1}}}{{{n}^{^{2}}}-1+\sqrt{{{n}^{2}}-1}}}\left\{ 1,{{n}^{2}} \right\}. \\ \end{array}\end{eqnarray} \tag{ 29 }$$

This closed formula, which is in excellent agreement with the numerical data plotted in figures 3(c) and 4(c), clearly shows the crossover frequency at the critical angle. The amplification $\sqrt{k\;{\rm w}(z)}$ at the critical angle suggested studying with more care the behavior of the peaks in the beam in optical weak measurements for incidence within the critical region. Indeed, in such a region, due to this amplification, the condition $\mid \Delta \epsilon \mid \gg \Delta {{y}_{{{\rm GH}}}}/{\rm w}(z)$ and the consequent proportionality among the experimental curves of the distance between peaks and the GH curves are no longer valid; see, for example, figures 6(c) and 7(c).

In our study, we have also found an axial dependence in optical weal measurements. This axial dependence can affect the experimental curves and, for small second polarizer rotations and small values of the beam waist, produces a practically flat region; see the plots in figures 6(a) and 7(a) for $\mid \Delta \epsilon \mid =0.01$. To minimize the axial dependence, we have to work with a laser beam with ${{{\rm w}}_{0}}\geqslant 500\;\mu {\rm m}$. It is important to observe here that, also for ${{{\rm w}}_{0}}=500\;\mu {\rm m}$, the curve amplification $1/\mid \Delta \epsilon \mid$ can be reproduced far from the critical region only.

In view of a possible experimental analysis of the study presented in this article, we have also estimated in which region we reach the better standard deviation for $\Delta {{y}_{{{\rm GH}}}}$ in the critical region. By using the second polarizer angle constraint, equation (28), and the analytical formula for the GH shift at the critical angle, equation (29), we find

$$\begin{eqnarray}&&\mid \Delta \epsilon \mid \geqslant \sqrt{\sqrt{\frac{\pi }{2}}\frac{\sqrt{2-{{n}^{2}}+2\sqrt{{{n}^{2}}-1}}}{{{n}^{^{2}}}-1+\sqrt{{{n}^{2}}-1}}}\frac{{{n}^{2}}-1}{\sqrt{k\;{\rm w}(z)}}.\end{eqnarray} \tag{ 30 }$$

This implies that for laser beams with ${{{\rm w}}_{0}}=500\;\mu {\rm m}$ and a camera at $z\leqslant 50\;{\rm cm}$, $\mid \Delta {{\epsilon }_{{{\rm min} }}}\mid \approx 0.015$ for both BK7 and fused silica dielectric blocks.

We conclude this work by observing that our analysis does not take into account cumulative dissipations and imperfections in the dielectric prism (such as the misalignment of its surfaces) and beam reshaping caused by interference. A phenomenological way to include misalignment effects is given in reference [37]. An interesting discussion of the origin of negative and positive lateral shifts in a dielectric slab is investigated in reference [38] from the viewpoint of the interference between multiple light beams.

The authors thank the Capes (M P A) and CNPq (S D L and G G M) for financial support. One of the authors (S D L) is greatly indebted to S A Carvalho for interesting comments and stimulating discussions. Finally, the authors thank the referees for their useful observations and suggestions.