A fork grating is a disturbed one-dimensional amplitude grating, in which one or several bifurcations form the ‘fork’ structure. The magnitude l of the TC is the number of bifurcations or, equivalently, the difference in the number of grating lines on the upper and lower half. Figure 1(a) shows a fork grating with a single bifurcation. This pattern can be seen as the interference of a tilted plane wave, Fig. 1(b), with a helical wave, Fig. 1(c), shown as a vortex phase distribution. This combination could also be regarded as a carrier wave, Fig. 1(b), with an information-bearing wave, Fig. 1(c), resulting in the modulated signal, Fig. 1(a). In this context, Fig. 1 shows a transmission protocol, namely the encoding of a TC as a message. Therefore, a decoding process in the form of demodulation would allow the receiver to read the TC of the fork. In the following, this OAM transmission protocol in the near-field is described.
Let the grating plane be described by the coordinate system r = (x0, y0) so that the axis y0 is parallel to the grating lines far from the singularity. Further, the polar coordinates ϕ = arctan(x0/y0) and r = (x02 + y02)1/2 are introduced. Doing so, the amplitude transmittance function of a fork grating T(r) is decomposed in periodic functions of argument 2πx0/pc + lϕ, which relates to the TC l embedded in the helical wave and the grating period pc which is the inverse of the spatial carrier frequency used to modulate the vortex signal.
The amplitude transmittance function of the fork grating is represented by the Fourier series
$$T({r})=\sum\limits_{{n= - \infty }}^{\infty } {{T_n}} \exp \left[ {in\left( {\frac{{2\pi }}{{{p_c}}}{x_0}+l\phi } \right)} \right].$$
1
An incident plane wave propagating along the z-axis, perpendicular to the grating, possesses the complex amplitude distribution u0(r) at the grating plane z0 = 0. Immediately after the grating, the complex amplitude distribution ψ acquires the form:
$$\psi \left( {x,y,z=0} \right)={u_0}(x,y,z=0) \times T({x_0},{y_0})=\sum\limits_{{n= - \infty }}^{\infty } {{T_n}} \,{u_0}({\mathbf{r}})\exp \left[ {in\left( {\frac{{2\pi }}{{{p_c}}}x+l\phi } \right)} \right].$$
2
Eq. 2 implies that the grating period pc generates a linear phase in x-direction for the incoming beam. The second phase term lϕ leads to a continuous change of the horizontal period. The evolution of the wave function in z-direction involves an additional phase which we express using direction cosines,
$$\begin{gathered} \psi (x,y,z)=\int_{{ - \infty }}^{\infty } {\int_{{ - \infty }}^{\infty } A } \left( {\frac{\alpha }{\lambda },\frac{\beta }{\lambda },0} \right)\exp \left( {i\frac{{2\pi }}{\lambda }\sqrt {1 - {\alpha ^2} - {\beta ^2}} z} \right) \\ \times \exp \left[ {i2\pi \left( {\frac{\alpha }{\lambda }x+\frac{\beta }{\lambda }y} \right)} \right]d\frac{\alpha }{\lambda }d\frac{\beta }{\lambda }, \\ \end{gathered}$$
3
where
$$A\left( {\frac{\alpha }{\lambda },\frac{\beta }{\lambda },0} \right)=\int_{{ - \infty }}^{\infty } {\int_{{ - \infty }}^{\infty } {\psi (x,y,0)\exp } } \left[ { - i2\pi \left( {\frac{\alpha }{\lambda }x+\frac{\beta }{\lambda }y} \right)} \right]dxdy$$
4
are the Fourier components of the field distribution right after the grating. Eq. (4) describes a two-dimensional Fourier transform when considering kx = αk, ky = βk respectively. The direction cosines α, β, γ that describe the propagation vector k are interrelated through:
$$\gamma =\sqrt {1 - {\alpha ^2} - {\beta ^2}} .$$
5
Note that for α2 + β2 > 1, γ becomes complex and describes the rapidly attenuated evanescent waves.
Assuming a plane wave with constant amplitude across the grating, u0(r) = 1, the Fourier components of the field distribution are dominated by the grating, see Eq. (1), and are
$$\begin{gathered} A\left( {\frac{\alpha }{\lambda },\frac{\beta }{\lambda },0} \right)=\int_{{ - \infty }}^{\infty } {\int_{{ - \infty }}^{\infty } {\sum\limits_{{n= - \infty }}^{\infty } {{T_n}} \exp \left[ {in\left( {\frac{{2\pi }}{{{p_c}}}x+l\phi } \right) - i2\pi \left( {\frac{\alpha }{\lambda }x+\frac{\beta }{\lambda }y} \right)} \right]} } \,dxdy \\ \approx \sum\limits_{{n= - \infty }}^{\infty } {{T_n}} \delta \left[ {\left( {\frac{n}{{{p_c}}} - \frac{\alpha }{\lambda }} \right)} \right]\delta \left[ {\frac{\beta }{\lambda }} \right]. \\ \end{gathered}$$
6
The approximation is valid as long as the vortex phase is much smaller than the integration range in x. In other words, the TC must be much smaller than the number of illuminated lines. Inserting into Eq. (3) yields
$$\begin{gathered} \psi (x,y,z) \approx \int_{{ - \infty }}^{\infty } {\int_{{ - \infty }}^{\infty } {\sum\limits_{{n= - \infty }}^{\infty } {{T_n}} \delta \left[ {\left( {\frac{n}{{{p_c}}} - \frac{\alpha }{\lambda }} \right)} \right]\delta \left[ {\frac{\beta }{\lambda }} \right]\exp \left( {i\frac{{2\pi }}{\lambda }\sqrt {1 - {\alpha ^2} - {\beta ^2}} z} \right)} } \\ \quad \quad \times \exp \left[ {i2\pi \left( {\frac{\alpha }{\lambda }x+\frac{\beta }{\lambda }y} \right)} \right]d\frac{\alpha }{\lambda }d\frac{\beta }{\lambda } \\ =\sum\limits_{{n= - \infty }}^{\infty } {{T_n}} \exp \left( {i\frac{{2\pi }}{\lambda }\sqrt {1 - \frac{{{n^2}{\lambda ^2}}}{{{p_c}^{2}}}} z} \right) \times \exp \left[ {i2\pi \left( {\frac{n}{{{p_c}}}x} \right)} \right]. \\ \end{gathered}$$
7
This field distribution is identical to the original diffraction plane, provided that
$$\exp \left( {i\frac{{2\pi }}{\lambda }\sqrt {1 - {n^2}\frac{{{\lambda ^2}}}{{{p_c}^{2}}}} z} \right)= \pm 1$$
8
up to a constant phase factor, which, in Fresnel approximation yields the well-known Talbot distance [32] where self-imaging is to be expected
$${L_T}=\frac{{p_{c}^{2}}}{{{n^2}\lambda }}.$$
9
The Talbot effect repeats the field distribution, and hence the phase, of the DOE in multiples of the fundamental Talbot length at z = q LT, where \(\in \mathbb{N}\) .
In order to assess the self-imaging, the diffracted beam was measured and simulated in a stack of planes behind the grating. We simulated the near-field diffraction using the angular spectral method [33]. Self-imaging in real space is reflected by self-imaging in Fourier space, in that the Fourier transform at the Talbot plane matches the Fourier transform of the grating, given by Eq. (2). We have shown earlier that the self-imaging can be identified by applying the normalized Pearson correlation coefficient (PCC) to the stack of Fourier spectra [34]:
$$R=\frac{{\sum\nolimits_{0}^{m} {\sum\nolimits_{0}^{n} {\left( {{A_{mn}} - \bar {A}} \right)\left( {{B_{mn}} - \bar {B}} \right)} } }}{{\sqrt {\left( {\sum\nolimits_{0}^{m} {\sum\nolimits_{0}^{n} {{{\left( {{A_{mn}} - \bar {A}} \right)}^2}} } } \right)\left( {{{\sum\nolimits_{0}^{m} {\sum\nolimits_{0}^{n} {\left( {{B_{mn}} - \bar {B}} \right)} } }^2}} \right)} }}.$$
10
A mn and Bmn are the values at row m and column n in the discrete power spectra A and B, respectively. \(\stackrel{-}{A}\) and \(\stackrel{-}{B}\) are the average values of A and B. The summation is limited to the most significant frequency range comprising the fundamental frequency and some higher harmonics. The PCC is calculated for each pair from the full stack of Fourier spectra. It is well known that the far-field diffraction pattern corresponds to the Fourier transform of the diffraction element, which in turn also corresponds to the Fourier transform of the intensity distribution at the Talbot planes. Similarly, the Fourier spectrum of a binary fork grating represents the field distribution of the far-field. The spatial carrier frequency corresponds to the first diffraction order in the far-field, which allows not only to identify the ring shape but also to retrieve the phase vortex in the near-field.
For this purpose, we apply the Fourier Transform method of phase retrieval [35], which is performed by band-pass filtering the region around the carrier frequency, followed by the inverse Fourier transform. The wrapped phase of the image is then found, and the phase vortex is retrieved after subtracting the spatial carrier.