1. Introduction

In recent three decades, singular optics has grown as an interesting and highly applied field which has been well extended to other fields of physics such as acoustic, electron and neutron physics, etc. A variety of different methods have been introduced for both generation and detection of optical vortices [1]. In some of methods used for the generation of vortex beams, a Gaussian laser mode passes through some optical elements such as spiral phase plates [2,3], forked gratings [4–9], spiral zone plates [10–12], and Q-plates [13,14] in which the vortices, based on the effect of the element on the impinging beam, form at different propagation distances. A considerable number of methods have been also proposed to characterize optical vortices specifically to determine their TCs, including the diffraction from different structured apertures [15–19] and gratings [2025], interferometry [26], and moiré deflectometry [27].

On the other hand, generation of optical vortex arrays (vortex lattices) has recently found interesting applications in the manipulation of microoptomechanical pumps, microlithography, orbital angular momentum (OAM) multiplexing, and quantum processing [28]. One of the methods used for the generation of optical vortex arrays is based on the near-field diffraction of a vortex beam from optical gratings [29–33]. Moreover, a detailed study on the near- and far-field diffraction from 2D orthogonal multiplicatively separable and nonseparable periodic structures has been presented in Refs. [34,35]. It is shown that the spatial spectrum or far-field diffraction pattern of these structures is a 2D orthogonal lattice of impulses.

In this work we introduce a family of pure phase 2D orthogonal multiplicatively separable periodic structures having sinusoidal or binary phase profiles including a phase singularity. Indeed, the proposed structure is multiplication of two pure phase 1D fork-shaped gratings (FSGs) defined along $x$ and $y$ directions, having TCs $l_x$ and $l_y$ and phase variation amplitudes $\gamma _x$ and $\gamma _y$, respectively. Diffraction of a Gaussian beam by these structures leads to generation of a 2D array of vortex beams with different TCs and power sharings. We show that the power share of any diffraction order equals the absolute square of the corresponding Fourier coefficient which is a two variable function of $\gamma _x$ and $\gamma _y$. Therefore, sharing of the incident power among different diffraction orders can be managed by changing the values of $\gamma _x$ and $\gamma _y$. It is also shown that the TC of $(m,n)$th diffraction order is $l_{m,n}=-\left ( ml_x+nl_y \right )$. Then, the value of $l$ can be manipulated by adjusting the values of $l_x$ and $l_y$.

It is worth noting that in a very close work [36] generation of a 2D orthogonal array of optical vortices using a pure phase 2D FSG having a modified blazed profile has been proposed. This modified blazed profile is defined by an anti-blazing function that is obtained by a numerical method, Gerchberg-Saxton algorithm [37]. Using different anti-blazing functions leads to different power distributions in the diffraction orders. In another work the produced array of optical vortices have been used for free space information transfer by the same research group [38]. It is also worth mentioning that the optical vortex arrays in both Refs. [36,38] were generated under plane wave illumination of the modified blazed based pure phase 2D FSGs. While in the current work we present a full mathematical insight to the power sharing and TC management by explicit solving the Fresnel integral and using the 2D Fourier analyses of the transmittance. This kind of facing to the issue enables us to continuously calculate the diffraction behaviour in terms of $\gamma _x$, $\gamma _y$, $l_x$, and $l_y$. The main advantage of our approach is that we can determine the mentioned parameters of the grating to obtain a desired power sharing optimally. For instance, by choosing a binary profile for the grating and $\gamma _x+\gamma _y=\pi$ all of the second (even) orders completely vanish.

2. Pure phase 1D FSGs with sinusoidal and binary profiles

By considering $(x,y)$ as Cartesian coordinates and $(r,\theta )$ as the corresponding polar coordinates, generalized transmission function of a 1D fork-shaped structure can be written as [39,40]:

For a pure phase periodic structure the absolute square of the corresponding Fourier coefficient, ${ \left | {{t_m}\left ( {{\gamma _x}} \right )} \right |}^2$, equals the power share of the $m$ the diffraction order from the power of the incident/transmitted light beam, see Appendix A. Using this fact the power shares of the diffraction orders of a pure phase 1D FSG with a sinusoidal profile are illustrated in terms of $\gamma _x$ in the first row of Fig. 2. Considering ${ \left | {{t_m}\left ( {{\gamma _x}} \right )} \right |}^2=J_m^2\left ( {{\gamma _x}} \right )$ we can find some values of $\gamma _x$ for which power sharing gets interesting properties, see Fig. 3 of Ref. [42]. For instance, considering $\gamma _x=2.4048$, as the first root of the function $J_0(\gamma _x)$, zero (central) diffraction order vanishes as is shown in the second row of Fig. 2. Considering $\gamma _x=1.4347$, as the first intersection of the functions $J_0(\gamma _x)$ and $J_1(\gamma _x)$, the power shares of the zero and first diffraction orders are equal as is seen in the third row of Fig. 2. Furthermore, by setting $\gamma _x=2.6298$, as the first intersection of the functions $J_1(\gamma _x)$ and $J_2(\gamma _x)$, the power shares of the first and second diffraction orders are equal as is shown in the fourth row of Fig. 2. By setting $\gamma _x=1.8412$ the function $J_1(\gamma _x)$ reaches to its maximum, then the power share of the first diffraction order is maximized, see the fifth row of Fig. 2. Moreover, the power sharing among diffracted beams is calculated in the first row of Visualization 1 for a pure phase 1D FSG with a sinusoidal profile by changing $\gamma _x$.

 

Fig. 1. Phase profiles of $T_x(x,y)$ with $l_x=\,2$ (first column), $T_y(x,y)$ with $l_y\,=\,3$ (second column), $T(x,y)$ with $l_x=\,2 , l_y\,=\,3$ (third column), and $T(x,y)$ with $l_x=l_y=0$ (fourth column) having $\gamma _x=\gamma _y=\frac {\gamma }{2}$ and $\Lambda _x=\Lambda _y=\,0.1$ mm. The first and second rows are for the sinusoidal and binary transmittances, respectively.

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Fig. 2. First row: power shares of the zero to second diffraction orders of a pure phase 1D FSG with a sinusoidal profile in terms of $\gamma _x$. Second to fifth rows: power shares of the zero to third diffraction orders for different values of $\gamma _x$. See the first row of Visualization 1.

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Fig. 3. First row: power shares of the zero, first, and third diffraction orders of a pure phase 1D FSG with a binary profile in terms of $\gamma _x$. Second to third rows: power shares of the zero to third diffraction orders for different values of $\gamma _x$. See the second row of Visualization 1.

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By using Eq. (4), the power shares of the diffraction orders for a pure phase 1D FSG with a binary profile are calculated in terms of $\gamma _x$ and the results are depicted in the first row of Fig. 3. As is expected we only have the odd diffraction orders, except the zero order. Furthermore, we have $\left | {{t_m}\left ( {\pi - {\gamma _x}} \right )} \right | = \left | {{t_m}\left ( {{\gamma _x}} \right )} \right |$ that leads to symmetry of the plots around $\gamma _x=\frac {\pi }{2}$. For $\gamma _x=\frac {\pi }{2}$ the zero (central) diffraction order vanishes and the first diffraction order is maximum as is shown in the second row of Fig. 3. By setting $\left | t_0\left ( {{\gamma _x}} \right )\right |=\left | t_{\pm 1}\left ( {{\gamma _x}} \right )\right |$ in Eq. (4) we get ${\gamma _x} = {\tan ^{ - 1}}(\pi /2) \approx 1.004$ and ${\gamma _x} =\pi - {\tan ^{ - 1}}(\pi /2) \approx 2.138$, the intersections of blue solid and dashed red curves in the first row of Fig. 3. For these two values of $\gamma _x$ the power shares of the zero (central) and first diffraction orders are equal, see third row of Fig. 3. Moreover, for a pure phase 1D FSG with a binary profile, the power sharing among diffracted beams by changing $\gamma _x$ is illustrated in the second row of Visualization 1.

Similarly $T_y(x,y)$ can be defined by replacing $x$ by $y$ in the Eqs. (1)–(4). The second column of Fig. 1 shows the phase distributions of $T_y(x,y)$ with $l_y\,=\,3$ and $\Lambda _x=\,0.1$ mm having sinusoidal (first row) and binary (second row) profiles.

3. Pure phase 2D FSGs: power sharing among diffraction orders

Let us now define a 2D pure phase FSG with the following transmittance:

Substituting $T_x(x,y)$ and $T_y(x,y)$ from Eq. (2) in Eq. (5) we get

For the sinusoidal profile we have ${t_{m,n}(\gamma _x, \gamma _y) }= {i^{m + n}}{J_m}(\gamma _x ){J_n}(\gamma _y )$ which is generally a two-variable function. Nevertheless, by considering $\gamma =\gamma _x+\gamma _y$ as a constant we get $t_{m,n}(\gamma _x) = i^{m + n}J_m(\gamma _x )J_n(\gamma -\gamma _x )$ which is a one-variable function. Using this identity, the power contributions of the zero to second diffraction orders of a sinusoidal pure phase 2D FSG are plotted in Figs. 4 and 5 in terms of $\gamma _x$ for the $\gamma =\pi$ (first column) and $\gamma =\pi / 2$ (second column). In the first column of Fig. 4, for the zero (central) diffraction order we have ${t_{0,0}}({\gamma _x}) = {J_0}({\gamma _x}){J_0}\left ( {\pi - {\gamma _x}} \right )$ then $\gamma _0 \approx 2.4048$ (the first root of the function $J_0(\gamma _x)$), and $\pi -\gamma _0 \approx 0.7368$ can be considered as the roots of ${t_{0,0}}({\gamma _x})$. It is also seen that setting $\gamma _x=\gamma _0$ the values of $\left | {{t_{0, \pm 1}}({\gamma _x})} \right |$ go to zero and the values of $\left | {{t_{ \pm 1,0}}({\gamma _x})} \right |$ reach to the corresponding maxima. Similarly setting $\gamma _x=\pi -\gamma _0$ the values of $\left | {{t_{0, \pm 1}}({\gamma _x})} \right |$ reach to their maxima and the values of $\left | {{t_{ \pm 1,0}}({\gamma _x})} \right |$ go to zero. In Visualization 2 the power sharing among diffracted beams is illustrated for $\gamma =\pi$ by changing $\gamma _x$. Looking at the second column of Fig. 4, we see that the zero (central) diffraction order is dominant and has the most power share comparing the higher orders. Moreover, the power sharing among diffracted beams for $\gamma =\pi /2$ by changing $\gamma _x$ is illustrated in Visualization 3. Figure 5 shows the power contributions of the second diffraction orders. As is seen, for $\gamma =\pi / 2$ (second column) all the second order power shares are less than $7\%$.

 

Fig. 4. The power share of the zero and first diffraction orders of a pure phase 2D FSG with a sinusoidal profile having $\gamma =\pi$ (first column) and $\gamma = \pi / 2$ (second column) in terms of $\gamma _x$. See Visualization 2 and Visualization 3.

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Fig. 5. The power share of the second diffraction orders in terms of $\gamma _x$ for a a pure phase 2D FSG with a sinusoidal profile having $\gamma =\pi$ (first column) and $\gamma =\pi / 2$ (second column).

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For the binary profile, considering Eq. (4), when $m$ and/or $n$ gets non-zero even values ${t_{m,n}(\gamma _x, \gamma _y) }=0$. It means the $(m,n)$ the diffraction order vanishes if $m$ and/or $n$ is a non-zero even number. Here again, considering $\gamma =\gamma _x+\gamma _y$ as a constant, we have ${t_{m,n}(\gamma _x) }= {t_m}(\gamma _x ){t_n}(\gamma -\gamma _x )$ where ${t_m}(\gamma _x )$ and ${t_n}(\gamma -\gamma _x )$ are obtained from Eq. (4). For the special case $\gamma =\pi$, using Eq. (4), we get

Therefore, in this case we have $\left | {{t_{0,0}}({\gamma _x})} \right | = {\cos ^2}({\gamma _x})$ which goes to zero for ${\gamma _x}=\pi /2$ and reaches to its maximum for ${\gamma _x}=0,\, \pi$, see the first column, first row of Fig. 6. In this case we also have $\left | {{t_{ \pm 1, \pm 1}}({\gamma _x})} \right | = 4{\sin ^2}({\gamma _x})/{\pi ^2}$ which go to zero for ${\gamma _x}=0,\, \pi$ and reach to their maxima for ${\gamma _x}=\pi /2$. Moreover, we get $\left | {{t_{0, \pm 1}}({\gamma _x})} \right | = \left | {{t_{ \pm 1,0}}({\gamma _x})} \right | = \sin (2{\gamma _x})/\pi$ which goes to zero for ${\gamma _x}=0,\, \pi /2, \, \pi$ and reaches to its maximum for ${\gamma _x}=\pi /4, \, 3\pi /4$. For ${\gamma _x} = {\tan ^{ - 1}}(\pi /2)$ and ${\gamma _x} = \pi - {\tan ^{ - 1}}(\pi /2)$ at which $\left |t_0(\gamma _x)\right |= \left |t_{\pm 1}(\gamma _x)\right |$ we get $\left | {{t_{ \pm 1, \pm 1}}({\gamma _x})} \right | = \left | {{t_{0, \pm 1}}({\gamma _x})} \right |= \left | {{t_{\pm 1,0}}({\gamma _x})} \right |= \left | {{t_{0, 0}}({\gamma _x})} \right |$ by considering Eq. (8). It means that for these values of ${\gamma _x}$ zero and all the first diffraction orders have the same power shares. In Visualization 4 the power sharing among diffracted beams is illustrated for $\gamma =\pi$ by changing $\gamma _x$. Moreover, By comparing second columns of Figs. 4 and 6 we see that for $\gamma =\pi /2$, the power shares of the first diffraction orders for the binary profile are considerably more than sinusoidal profile. The power sharing among diffracted beams by changing $\gamma _x$ is illustrated for $\gamma =\pi /2$ in Visualization 5. It should be noted that power contributions of the second diffraction orders vanish for binary profile and therefore we do not illustrated them.

 

Fig. 6. The power share of the zero and first diffraction orders of a pure phase 2D FSG with a binary profile having $\gamma =\pi$ (first column) and $\gamma =\pi / 2$ (second column) in terms of $\gamma _x$. See also Visualization 4 and Visualization 5.

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As a different approach we consider the total phase amplitude $\gamma$ as a variable parameter in which it is equally shared between the corresponding phase amplitudes along $x$ and $y$ directions, namely $\gamma _x=\gamma _y=\frac {\gamma }{2}$. Then the he Fourier coefficient ${t_{m,n}(\gamma _x, \gamma _y) }$ gets the following form:

Using this identity, the different diffraction orders’ power shares are calculated and the results are depicted in the first column of Fig. 7. For $\gamma _x=\gamma _y=\gamma /2=1.4347$ at which $\left |t_0(\gamma /2)\right |= \left |t_{\pm 1}(\gamma /2)\right |$ we get $\left | {{t_{ \pm 1, \pm 1}}({\gamma })} \right | = \left | {{t_{0, \pm 1}}({\gamma })} \right |= \left | {{t_{\pm 1,0}}({\gamma })} \right |= \left | {{t_{0, 0}}({\gamma })} \right |$ by considering Eq. (9). Consequently, for $\gamma =2\times 1.4347$ all zero and the first diffraction orders have the same power shares. Furthermore, for $\gamma _x=\gamma _y=\gamma /2=1.0825$ the values of $\left | {{t_{0, \pm 1}}({\gamma })} \right |= \left | {{t_{\pm 1,0}}({\gamma })} \right |$ reach to their maximum. In Visualization 6 the power sharing among diffracted beams is animated for a sinusoidal profile and $\gamma _x=\gamma _y=\gamma /2$ by changing $\gamma$.

 

Fig. 7. First column: the power share of the zero (first row), first (second row), and second (third row) diffraction orders in terms of $\gamma$ for a 2D FSG with a sinusoidal profile. Second column: the power share of the zero (first row), first (second row), and third (third row) diffraction orders in terms of $\gamma$ for a 2D FSG with a binary profile. See Visualization 6 and Visualization 7.

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For the binary profile, by using Eqs. (4) and (9), the diffraction orders’ power shares are calculated and the results are depicted in the second column of Fig. 7. Comparing Eqs. (8) and (9) we see that $\left |{t_{m,n}}(\gamma )\right |$ and $\left |{t_{m,n}}(\gamma _x )\right |$ have the same functionality except a coefficient $\frac {1}{2}$ in the arguments of Eq. (9). Therefore power share plots in the second column (first and second rows) of Fig. 7 is the half of the plots of the first column of Fig. 6 which are horizontally stretched twice. Looking at the third row of Fig. 7, it is seen that the power shares of the second diffraction orders of the sinusoidal profile are about the power shares of the third diffraction orders of the binary profile. The power sharing among diffracted beams is animated for a binary profile and $\gamma _x=\gamma _y=\gamma /2$ by changing $\gamma$ in Visualization 7.

4. Diffraction of a Gaussian beam from pure phase 2D FSGs

Now suppose that the defined structures by Eq. (6) are illuminated by a Gaussian beam, fixing the waist position in the aperture plane $z = 0$, we can write:

It should be mentioned that $\left ( {\frac {{m\lambda z}}{\Lambda _x },\frac {{n\lambda z}}{\Lambda _y }} \right )$ show the coordinates of $(m,n)$th diffraction order of the grating at the observation plane, see Eq. (45) of Ref. [34]. Therefore Eq. (13) can be considered as a transformation of coordinates to the center of $(m,n)$th diffraction order. Now let us define $\rho _{m,n}$ and $\varphi _{m,n}$ so that ${X_m} = {\rho _{m,n}}\cos \left ( {{\varphi _{m,n}}} \right )$ and ${Y_n} = {\rho _{m,n}}\sin \left ( {{\varphi _{m,n}}} \right )$, then we have $\rho _{m,n}=\sqrt {X_m^2+Y_n^2}$ and ${\varphi _{m,n}} = {\tan ^{ - 1}}\left ( {\frac {{{Y_n}}}{{{X_m}}}} \right )$. In the polar coordinates, Eq. (14) gets the following form:

Considering $\int\limits_0^{2\pi } {{e^{i(l - s)\theta }}d\theta = 2\pi {\delta _{l,s}}}$ in which $\delta _{l,s}$ indicates the Kronecker delta, Eq. (19) leads to

By using the following reference integral [44]:

Equation (22) is the main result and specifies the complex amplitude of diffraction of a Gaussian beam from a 2D FSG at any arbitrary propagation distance $z$. Equation (22) implies that $(m,n)$th diffraction order is a vortex beam which its TC equals $l_{m,n}=-\left ( ml_x+nl_y \right )$ and its power share equals ${\left | {{t_{m,n}}(\gamma _x, \gamma _y)} \right |^2}$.

For better demonstration of the results, let us consider some examples. As the first example, we consider the diffraction of a Gaussian beam with $\lambda =532$ nm and $w_0\,=\,2.5$ mm from four typical pure phase 2D FSGs having $l_x=l_y=1$ and $\Lambda _x=\Lambda _y=0.1$ mm with sinusoidal and binary profiles. The resulted diffraction patterns at distance $z=2$ m from the gratings are illustrated in Fig. 8 in which $\gamma _x=\gamma _y=\frac {\pi }{2}$ in the first column, and $\gamma _x=\gamma _y=\frac {\pi }{4}$ in the second column. Considering $l_x=l_y$ and according to $l_{m,n}=-\left ( ml_x+nl_y \right )$, the intensity patterns of $(m,n)$th diffraction orders over the second and fourth Cartesian quadrants ($n=-m$) are not doughnut-shaped and their TCs are zero. Considering the first row of Fig. 8, from the power sharing perspective, the power shares of the $(\pm 1, \pm 1 )$th orders are dominant (negligible) in the first (second ) column. This feature was predicted in the first row of Fig. 4, by comparing red dashed plots in the first and second columns. Furthermore, in the second row, the considerable power sharing difference between the first and second columns of Fig. 8 can be also explained by comparing the first and second columns of Fig. 6. The diffraction patterns of Fig. 8 are animated in Visualization 8, Visualization 9, Visualization 10 and Visualization 11 under propagation from $z=0$ to $z=2$ m.

 

Fig. 8. Diffraction patterns at distance $z=2$ m from four typical pure phase 2D FSGs having $l_x=l_y=1$, $\Lambda _x=\Lambda _y$ = 0.1 mm, $\gamma _x=\gamma _y=\frac {\pi }{2}$ (first column), $\gamma _x=\gamma _y=\frac {\pi }{4}$ (second column), having sinusoidal (first row), and binary (second row) profiles illuminated by a Gaussian beam with $\lambda$ = 532 nm and $w_0$ = 2.5 mm. See Visualization 8, Visualization 9, Visualization 10 and Visualization 11.

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To explicitly demonstrate the TCs of the diffraction orders, and verify the identity $l_{m,n}=-\left ( ml_x+nl_y \right )$ for TC of the $(m,n)$th diffraction order, we depict the corresponding phase profiles of the intensity patterns presented in the first row of Fig. 8 in Fig. 9 using Eq. (22). For the sake of simplicity in representation we set $\Lambda =500$ nm instead of $\Lambda =532$ nm so that the coordinates of the center of $(m,n)$th diffraction order at distance $z=2$ m is obtained $(10m,10n)$ in millimetres. For instant, the phase profile of the diffraction order $(1,1)$ around the point $(x=10\, \text {mm}, y=10\,\text {mm})$ includes two singularities with TC=-1 so that the total TC equals −2. As another example, looking carefully at the $(-2,-1)$th diffraction order, it includes three singular points around the point $(x=-20\, \text {mm}, y=-10\,\text {mm})$ with TC=+1 so that the total TC equals +3.

 

Fig. 9. The corresponding phase profiles of the diffraction patterns illustrated in the first row of Fig. 8. The central areas of diffraction orders are covered by the patterns.

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As the second example, the diffraction of a Gaussian beam with $\lambda =532$ nm and $w_0=2.5$ mm from six pure phase 2D FSGs having $l_x=2, l_y=3$ and $\Lambda _x=\Lambda _y=0.1$ mm and different values of $\gamma _x$ and $\gamma _y$ with sinusoidal and binary profiles. The resulted diffraction patterns at a distance $z=2$ m from the gratings are depicted in Fig. 10 in which the first and second rows are allocated to the sinusoidal and binary profiles, respectively. The values $\gamma _x$ and $\gamma _y$ have been chosen based on the extrema and intersections of the plots of the first-order power shares in Figs. 4, 6, and 7. As is seen in the first column of Fig. 4, for a grating with a sinusoidal profile and $\gamma = \pi$, for $\gamma _x=\gamma _0 \approx 2.4048$ and $\gamma _x=\pi -\gamma _0 \approx 0.7368$ the power shares of $(\pm 1,0)$/$(0,\pm 1)$ and $(0,\pm 1)$/$(\pm 1,0)$ diffraction orders reach to maximum/zero, respectively, and the $(0,0)$ diffraction order vanishes for both values of $\gamma _x$. This feature is verified and demonstrated at the first row, the first and second columns, of Fig. 10. Moreover, as was mentioned in the explanations of the first column of Fig. 7, for $\gamma _x=\gamma _y=1.4347$ power is shared equally among zero and all the first diffraction orders. Thus, a $3 \times 3$ array of diffracted beams is generated with different TCs and having total power share of $81\%$ from the incident beam’s power, see the first row third column of Fig. 10. This feature might find application in free-space optical communications. For the binary profile, returning to the first column of Fig. 6, we see that for $\gamma _x=\gamma _y=\pi /2$, the $(0,0)$, $(0,\pm 1)$ and $(\pm 1,0)$ diffraction orders vanish. This result is reflected in the second row first column of Fig. 10. As an interesting result, we see that the total power share of four illustrated vortices is more than $65\%$ of the incident beam’s power. This feature can be used for generation of high power optical vortices. In the second row second column of Fig. 10 the values of $\gamma _x$ and $\gamma _y$ are chosen so that the power shares of $(0,\pm 1)$ and $(\pm 1,0)$ diffraction orders gets their maximum values. In the second row third column we choose the values of $\gamma _x$ and $\gamma _y$ so that all the zero and all the first diffraction orders have the same power shares. Therefore an isolated $3 \times 3$ array of diffracted beams is generated with total power share of $74\%$ from the incident beam’s power. As it was proved in the previous section, all of the second diffraction orders vanish for a pure phase FSG with a binary profile. In Visualization 12, Visualization 13, Visualization 14, Visualization 15, Visualization 16 and Visualization 17 the diffraction patterns of Fig. 10 are animated under propagation from $z=0$ to $z=2$ m.

 

Fig. 10. Diffraction patterns at distance z = 2 m from three different pure phase 2D FSGs with $l_x$ = 2, $l_y$ = 3, $\Lambda _x=\Lambda _y$ = 0.1 mm, and different values of $\gamma _x$ and $\gamma _y$ having sinusoidal (first row), and binary (second row) profiles illuminated by a Gaussian beam with $\lambda$ = 532 nm and $w_0$ = 2.5 mm. See Visualization 12, Visualization 13, Visualization 14, Visualization 15, Visualization 16 and Visualization 17.

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5. Experiments

We examine some of the presented theoretical results experimentally by impinging a Gaussian beam over some pure phase 2D FSGs generated by using a spatial light modulator (SLM). We used a conventional SLM extracted from a video projector (LCD projector KM3, model no. X50) for which the total phase amplitude of the prepared gratings is limited to $\gamma =\frac {\pi }{2}$. The second harmonic of an Nd:YAG diode-pumped laser beam having a Gaussian profile and collimated wavefront with a wavelength $\lambda =532$ nm impinges the SLM where the desired phase profile is imposed on it. At distance $z=300$ cm from the SLM, we recorded the diffraction pattern by a camera (Nikon D100). Six different 2D FSGs having sinusoidal and binary profiles with $l_x= l_y=5$, and different values of $\gamma _x$ and $\gamma _y$ are imposed on the SLM. The experimentally recorded and the corresponding calculated diffraction patterns are illustrated in Fig. 11. The same patterns for $l_x=3\; {\textrm {and}} \; l_y=-2$ are shown in Fig. 12. Because of $\gamma =\frac {\pi }{2}$ limitation of the used SLM we have $\gamma _x+\gamma _y=\frac {\pi }{2}$ for all of the illustrated cases.

 

Fig. 11. Experimentally recorded (first and third rows) and calculated (second and fourth rows) diffraction patterns of a Gaussian beam with $\lambda =532$ nm and $w_0=5$ mm from six different pure phase 2D FSGs with $l_x= l_y=5$, $\Lambda _x=\Lambda _y=0.1$ mm, and different values of $\gamma _x$ and $\gamma _y$ having sinusoidal (first and second rows), and binary (third and fourth rows) profiles at distance $z=300$ cm from the gratings.

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Fig. 12. The same patterns of Fig. 11 for $l_x$ = 3 and $l_y$ = −2.

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In the next stage, we directly measure the TCs of optical vortices generated over the diffraction orders of a pure phase 2D FSG to verify the identity $l_{m,n}=-\left ( ml_x+nl_y \right )$ experimentally. In this regard, we use the diffraction from a quadratic curved-line (parabolic-line) grating introduced in Ref. [24] as a simple and efficient tool for characterization of optical vortices. In this method, we let each of the generated vortex beams individually illuminates a quadratic curved-line grating (also known parabolic-line grating), see Fig. 13. The first diffraction order of the quadratic curved-line grating is only a set of elongated intensity spots along a straight line. The number of intensity spots over the first-order diffraction pattern and its orientation respectively determine the absolute value and sign of the TC of the impinging vortex beam. The absolute value of the TC is obtained as $\left | l \right | =N-1$ where $N$ denotes the number of elongated intensity spots. Alternatively, we can account the number of intensity nulls between elongated intensity spots as the absolute value of TC. If the orientation of the first-order diffraction pattern is placed along the first and third (second and fourth) Cartesian quadrants the sign of TC is negative (positive). Figure 14 illustrates the first-order diffraction patterns of the optical vortices generated in the second columns of Fig. 11 and Fig. 12 passing through a quadratic curved-line grating.

 

Fig. 13. Experimental setup for generating a 2D array of optical vortices by a 2D FSG imposed on an SLM and measuring of their TCs using a quadratic curved-line grating (also known parabolic-line grating, PLG). S.F. stands for spatial filter.

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Fig. 14. First-order diffraction patterns of the optical vortices generated in the second columns of Fig. 9 (first to third columns) and Fig. 10 (fourth to sixth columns) passing through a quadratic curved-line grating.

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6. Conclusion

A new class of pure phase 2D FGSs was introduced having topological defect numbers $l_x$ and $l_y$ and phase variation amplitudes $\gamma _x$ and $\gamma _y$ with sinusoidal and binary profiles. Diffraction of a Gaussian beam from this class of grating leads to generation of 2D arrays of optical vortices having different power and TC distributions. It was shown that the TC of $(m,n)$ diffraction order is $l_{m,n}=-\left ( ml_x+nl_y \right )$ and the power distribution among diffraction orders can be managed by adjusting the values of $\gamma _x$ and $\gamma _y$. For instance, setting $\gamma _x=\gamma _y=1.435$ for the sinusoidal profile and $\gamma _x=\gamma _y= {\tan ^{ - 1}}(\pi /2) \approx 1.004$ for the binary profile lead to generation of a $3 \times 3$ array of optical beams having equal powers and different TCs. This feature has application in OAM based free-space optical communications. Furthermore, by setting $\gamma _x=\gamma _y=\pi /2$ for the binary profile we obtain a $2 \times 2$ array of optical vortices with considerable power shares and different TCs. This feature might find application in the field of optical tweezers. Finally, some of the presented theoretical results experimentally verified by impinging a Gaussian beam over some pure phase 2D FSGs generated by using a spatial light modulator (SLM). Moreover, TCs of the generated optical vortices have been measured by using a parabolic-line grating to verify the identity $l_{m,n}=-\left ( ml_x+nl_y \right )$ experimentally.