Young’s double-slit experiment with a partially coherent vortex beam

Tianchi Chen; Tianchi Chen; Xingyuan Lu; Xingyuan Lu; Jun Zeng; Zhuoyi Wang; Hao Zhang; Chengliang Zhao; Chengliang Zhao; Bernhard J. Hoenders; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.410812

1. Introduction

Even after the passage of two centuries, Young's double-slit experiment is still the most classic method of verifying the coherence properties of light [1]. It has been performed successfully both in space and time domain [2], with light, electrons [3], surface plasmon polariton [4], atoms and molecules [5,6]. With the deepening and diversification of research, more and more variables are added to the study of Young's double-slit interference, e.g. polarization [7,8] and orbital angular momentum (OAM) [9]. Reciprocally, the interference patterns are also very helpful for the quantitative characterization of these features. It is well known that the measurement of OAM states of vortex beams is of great importance in the applications employing OAM-carrying vortex beams [10–13]. In the Young’s double-slit experiment based on the vortex beam, the carried topological charge (TC) will manifest itself in the twist and displacement of the interference fringes [9].

Partially coherent vortex beam (PCVB), which is proposed by Gori and coworkers [14,15], has displayed some unique properties, such as robust coherence singularities [16] and more freedom in radiation force control [17]. However, as the degree of coherence decreases, the interference phenomenon will become much weaker [18,19] due to the modes overlap [20]. Consequently, the methods based on interference intensities for measuring the TC of coherent vortex beams, such as multi-pinholes [21], annular gratings [22] and angular slits [23,24], might become less powerful and eventually even fail. Instead of analyzing the intensity, it makes more sense studying the distribution of cross-spectral density (CSD) in the partially coherent case [25,26]. For low coherent PCVB [14], we can see the dark rings in the CSD amplitude on the focal plane [26], and the number of these dark rings reveal the magnitude of the TC. The only drawback is that the sign feature of the TC, which also contains a significant amount of information, is lost. Chen et. al. obtained the complete information of the TC, including magnitude and sign, by measuring the CSD distribution after a couple of cylindrical lenses [27]. However, the topological information displayed by the amplitude is not obvious enough.

In this paper, we perform the Young’s double-slit interference experiment with a PCVB. In addition to the interference fringes intensities, we care more about the amplitude and phase distribution of the CSD on the focal plane. Due to the interference by the double-slit, each ring dislocation in the CSD amplitude becomes a pair of singularities, which creates the opportunity for the simultaneous measurement of magnitude and sign of the TC. The direction of phase winding around singularities tells the sign of the TC of the input PCVB and their number is twice the magnitude of the TC.

2. Theory

As we know, a partially coherent standard Laguerre-Gaussian (LG) beam is a typical PCVB. At the source plane z = 0, the CSD of a partially coherent standard LG_pl beam can be expressed as follows [28,29]

(1)$$\begin{array}{ll} W({{{\textbf r}_1},{{\textbf r}_2}} )&= {\left( {\frac{{2{r_1}{r_2}}}{{\omega_0^2}}} \right)^l}L_p^l\left( {\frac{{2r_1^2}}{{\omega_0^2}}} \right)L_p^l\left( {\frac{{2r_2^2}}{{\omega_0^2}}} \right)\exp \left( { - \frac{{r_1^2 + r_2^2}}{{\omega_0^2}}} \right)\\ &\times \exp [{\textrm{i} \cdot l \cdot ({{\theta_1} - {\theta_2}} )} ]\exp \left[ {\frac{{{{({{{\textbf r}_1} - {{\textbf r}_2}} )}^2}}}{{\sigma_g^2}}} \right] \end{array}$$

where r is the vector coordinate in the source plane, r and θ are the radial and azimuthal coordinates in the source plane. p is the radial index of Laguerre polynomial modes, which represents the number of radial nodes in the intensity distribution, and the quantity l denotes TC, ${\sigma _g}$ is the transverse coherence width, “*” is the complex conjugate, ${\omega _0}$ is the beam width of the fundamental Gaussian mode. For the convenience of explanation, the following discussion is carried out in Cartesian coordinate system.

Now, we define the plane before the double-slit as the source plane. We set ‘b’ as the slit width, ‘a’ as the distance from the slits to the center [as is shown in the illustration of Fig. 4(a)]. According to the Ref. [30], we can express any axisymmetric field in a simple, unified, analytical way [31]:

(2)$$T(\textrm{r}) = \sum\limits_{m = 1}^M {{A_m}} \exp \left\{ {\left[ { - \frac{{{B_m}}}{{{{(b/2)}^2}}}{{(x - a - \frac{b}{2})}^2}} \right] + \exp \left[ { - \frac{{{B_m}}}{{{{(b/2)}^2}}}{{(x + a + \frac{b}{2})}^2}} \right]} \right\},$$

where ${A_m}$ and ${B_m}$ are the expansion and Gaussian coefficients, respectively, which can be obtained by minimizing $\int\limits_0^\infty {{{[{T({\textbf r},{A_m},{B_m}) - {T_{\textrm{obj}}}({\textbf r})} ]}^2}} d{\textbf r}\textrm{ }$, i.e. the difference between the analytical expression and the real object function. A table of the coefficients ${A_m}$ and ${B_m}$ can be found in Refs. [30,32]. With the increase of M, the accuracy of the simulation will improve. Here we take M as 10, which is enough to assure sufficient accuracy [31–33].

When a partially coherent standard LG_pl beam illuminates a double-slit, the CSD of the transmitted beam just behind the double-slit plane can be expressed as [30]

(3)$${W_2}({{\bf r}_1},{{\bf r}_2}) = {W_1}({{\bf r}_1},{{\bf r}_2})T({{\bf r}_1}){T^\ast }({{\bf r}_2}).$$

where ${W_1}({{\bf r}_1},{{\bf r}_2})$ denotes the CSD just before the two slits. The CSD function of a partially coherent beam passing through an ABCD optical system can be dealt with the generalized Collins integral equation:

(4)$$\begin{array}{ll} W({\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2}) &= {\left( {\frac{1}{{\lambda |B |}}} \right)^2}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{W_2}(} } } } {{\bf r}_1},{{\bf r}_2})\\ &\textrm{ } \times \exp \left[ { - \frac{{ik}}{{2{B^\ast }}}({A^\ast }x_1^2 - 2{x_1}{\rho_{x1}} + {D^\ast }\rho_{x1}^2) - \frac{{ik}}{{2{B^\ast }}}({A^\ast }y_1^2 - 2{y_1}{\rho_{y1}} + {D^\ast }\rho_{y1}^2)} \right]\\ &\textrm{ } \times \exp \left[ {\frac{{ik}}{{2B}}(Ax_2^2 - 2{x_2}{\rho_{x2}} + D\rho_{x2}^2) + \frac{{ik}}{{2B}}(Ay_2^2 - 2{y_2}\rho_{y2}^2 + D\rho_{y2}^2)} \right]d{x_1}d{x_2}d{y_1}d{y_2}, \end{array}$$

where $\boldsymbol{\mathrm{\rho}}$ denotes the vector coordinates in the observation plane, ${\boldsymbol{\mathrm{\rho}} }\textrm{ = }({{\rho_x},{\rho_y}} )$ and $\textrm{r = }({x,y} )$. Substituting from Eqs. (1)–(3) into Eq. (4), we can get the final expression of the CSD after free space transmission by choosing $A = 1,B = z,C = 0,D = 1$, or onto the focal plane with $A = 0,B = f,C ={-} 1/f,D = 1$, where f is the focal length. The main derivation and the final version of the analytical expression of $W({\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2})$ has been given in Supplement 1.

3. Simulation

Based on Eqs. (S9)-(S11) in Supplement 1, we studied the intensities of the interference fringes of PCVB passing through a double-slit and transmitting in the free space, under different degree of spatial coherence:${\sigma _g} = 0.3{\omega _0}$, ${\sigma _g} = 0.5{\omega _0}$ and ${\sigma _g} = 10{\omega _0}$(nearly completely coherent). Here, p=0, ${\omega _0} = 1\textrm{mm}$, a=0.0725 mm, b=0.18 mm and the free space transmission distance z=800 mm. From Fig. 1, we can only qualitatively conclude that, for the high coherent case, the interference fringes will twist to different directions for different signs of the TC and the displacements depend on the magnitude of the TC. The results are the same as in Ref. [9]. As the degree of coherence decreases, the interference patterns of different modes overlap, and both the twist and displacement become not that obvious.

Fig. 1. Interference fringes of a partially coherent vortex beams passing through a double-slit under different topological charges and spatial coherence.

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Compared with the analysis of the intensity, the investigation of the CSD distribution brings more information about the PCVB. By choosing a reference point ${{{\boldsymbol{\mathrm{\rho}}} }_\textrm{2}} = ({0,0} )$, we study the phase and amplitude distribution of the CSD function after the PCVB transmitting through a double-slit and propagating to the focal plane. Here, the focal length is f=800 mm and the other parameters keep the same as Fig. 1.

As shown in Fig. 2, we calculate the amplitude distributions of the CSD function of the PCVB with TCs of +1, +2, −2, and +3 under various spatial coherence for ${\sigma _g} = 0.3{\omega _0}$, ${\sigma _g} = 0.5{\omega _0}$ and ${\sigma _g} = 0.8{\omega _0}$. Compared with the results shown in [26], dark rings no longer exist, but are replaced by some dark slits. It shows that the TC is half the number of dark slits. To observe more directly, we draw the crossline of the amplitude distributions along the vertical direction. A minimal value corresponds to a dark slit. We can immediately see that when l=+1, +2, −2, there are 2, 4, and 4 singularities, respectively. However, for larger TC or larger spatial coherence, the number of dark slit is a bit ambiguous, as is shown in the last column and last row of Fig. 2. When the spatial coherence is relative higher, only two clear dark slits can be seen for |l|=2.

Fig. 2. Theoretical simulations of the amplitude distribution of the CSD function and the crossline along vertical direction for a partially coherent vortex beams passing through a double-slit with different topological charges and spatial coherence.

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To reveal the sign information of the TC, we further calculated the phase distribution of the CSD in Fig. 3. Instead of a phase jump, there is a phase singularity located in the center of each dark slit. These singularities with zero CSD amplitude and unknown phase are the so-called coherence singularities [16]. Interestingly, each phase winding has the same direction. In detail, rotating counter-clockwise from −π to π (from blue to red) corresponds to a positive sign of the TC, whereas rotating clockwise (from blue to red) represents a negative sign. Similarly, the number of coherence singularities is 2|l|. In Fig. 3, we have marked the coherence singularities with rotating arrows. When l=+1, +2, −2 and +3, there are 2, 4, 4 and 6 singularities, respectively.

Fig. 3. Theoretical simulations of the phase distribution of the CSD function for a partially coherent vortex beams passing through a double-slit with different topological charges and spatial coherence.

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From Fig. 1 to Fig. 3, we can see that the amplitude distributions in Fig. 2 display more and also quantitative information of TC than the free space interference patterns in Fig. 1, but we can only get the magnitude for relative smaller TC in relative smaller spatial coherent cases. Compared with the amplitude distribution, coherence singularities distributions have more visual version for the exact determination of TC value. More prominently, both the magnitude and sign of TC can be read out from an individual phase pattern. We can clearly identify positive and negative TC from the winding direction of phase around each singularity. Compared with amplitude analysis, in the case of TC=3, and, the determination of TC is not ambiguous any more. Both in Fig. 2 and Fig. 3, we give a comparison of difference spatial coherence. Especially in Fig. 3, it shows that even when the degree of spatial coherence decreases, the relation ${\sigma _g} = 0.8{\omega _0}$ ship between the number and phase winding direction of coherent singularities and exact values of TC is still valid and clear.

4. Experimental measurement

Figure 4 shows the experimental setup for generating a partially coherent LG_0l beam and measuring its amplitude and phase distribution after passing through a double-slit and focused on the observation plane (SLM₂). First, a laser emits a coherent laser beam with a wavelength of $\lambda = 532\textrm{nm}$. The beam is expanded by the beam expander (BE) and focused by the lens L₁ onto a rotating ground glass disk (RGGD). Then, after the transmitted beam has passed through the lens, we obtain a partially coherent beam with tunable spatial coherence [34]. The generated partially coherent beam is reflected by the mirror (M) and then shines on the first spatial light modulator SLM₁. Here, SLM₁ is used to load the phase grating designed by the method of computer-generated holograms, and then generate partially coherent LG_0l beam. The LG_0l beam passes through the double-slit and is focused by the lens L₃ onto the SLM₂. SLM₂ is used to introduce an adjustable phase-perturbation $\varphi$ in the center. It is worth noting that the focused PCVB should also be aligned with the center SLM₂. The output beam from SLM2 is reflected by the beam splitter BS, and goes to the charge coupled detector (CCD) located on the Fourier plane. In order to make the following derivation clearer, the coordinate symbols of double-slit, SLM₂ and CCD plane have been marked correspondingly in Fig. 4.

Fig. 4. Experimental setup of generating a partially coherent LG_0l beam and measuring its amplitude and phase distribution. BE, beam expander; L₁, L₂, L₃ and L₄, thin lenses, with focal length 150 mm, 200 mm, 800 mm and 400 mm, respectively; RGGD, rotating ground glass disk; BS, beam splitter; M, mirror; SLM₁, SLM₂, spatial light modulator; CCD, charge coupled detector. Illustration (a) shows the geometric meaning of the double-slit parameters. r, ${{\boldsymbol{\mathrm{\rho}}} }$ and k marked in brackets after double-slit, SLM2 and CCD are the coordinate symbols used in the article.

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Here, we proposed an efficient method of measuring the CSD. First, without introducing the perturbation, the intensity captured by the CCD can be expressed as:

(5)$${I_0}({\textbf k} )= \int\!\!\!\int {W({{{{\boldsymbol{\mathrm{\rho}}} }_1},{{{\boldsymbol{\mathrm{\rho}}} }_2}} )\exp [{ - i \cdot 2\boldsymbol{\mathrm{\pi}}{\textbf k} \cdot ({{{{\boldsymbol{\mathrm{\rho}}} }_1} - {{{\boldsymbol{\mathrm{\rho}}} }_2}} )} ]\textrm{d}{{{\boldsymbol{\mathrm{\rho}}} }_1}\textrm{d}{{{\boldsymbol{\mathrm{\rho}}} }_2}} ,$$

where $W({{{{\boldsymbol{\mathrm{\rho}}} }_1},{{{\boldsymbol{\mathrm{\rho}}} }_2}} )$ is the CSD of the focused PVCB on SLM₂ plane. k is the vector coordinate in the CCD plane. When the perturbation is introduced at an on-axis point, that is, ${{{\boldsymbol{\mathrm{\rho}}} }_0}\textrm{ = }({0,0} )$, the intensity expression becomes:

(6)$$\begin{array}{ll} I({\textbf k} )&= {I_0}({\textbf k} )+ ({C - 1} ){({C - 1} )^\ast }W({{{{\boldsymbol{\mathrm{\rho}}} }_0},{{{\boldsymbol{\mathrm{\rho}}} }_0}} )+ \\ &+ ({C - 1} )\int {W({{{\boldsymbol{\mathrm{\rho}}} }_1},{{{\boldsymbol{\mathrm{\rho}}} }_0})\exp [ - i \cdot 2\boldsymbol{\mathrm{\pi}}{\textbf k}({{{\boldsymbol{\mathrm{\rho}}} }_1} - {{{\boldsymbol{\mathrm{\rho}}} }_0})]\textrm{d}{{{\boldsymbol{\mathrm{\rho}}} }_1}} \\ &+ {({C - 1} )^\ast }\int {W({{{\boldsymbol{\mathrm{\rho}}} }_0},{{{\boldsymbol{\mathrm{\rho}}} }_2})\exp [ - i \cdot 2\boldsymbol{\mathrm{\pi}}{\textbf k}({{{\boldsymbol{\mathrm{\rho}}} }_0} - {{{\boldsymbol{\mathrm{\rho}}} }_2})]\textrm{d}{{{\boldsymbol{\mathrm{\rho}}} }_2}} . \end{array}$$

with $C\textrm{ = }\exp ({\textrm{i}\varphi } )$, which is used for characterizing the phase perturbation, and the inverse Fourier transform of this intensity can be written as

(7)$$\begin{array}{ll} F{T^{ - 1}}[{I(\textbf{k})} ]({{\boldsymbol{\mathrm{\rho}}} }) &= F{T^{ - 1}}[{{I_0}(\textbf{k})} ]({{\boldsymbol{\mathrm{\rho}}} }) + ({C - 1} ){({C - 1} )^\ast }W({{{\boldsymbol{\mathrm{\rho}}} }_0},{{{\boldsymbol{\mathrm{\rho}}} }_0})\delta ({{\boldsymbol{\mathrm{\rho}}} })\\ &+ ({C - 1} )W({{{\boldsymbol{\mathrm{\rho}}} }_0} + {{\boldsymbol{\mathrm{\rho}}} },{{{\boldsymbol{\mathrm{\rho}}} }_0}) + {({C - 1} )^\ast }W({{{\boldsymbol{\mathrm{\rho}}} }_0},{{{\boldsymbol{\mathrm{\rho}}} }_0} - {{\boldsymbol{\mathrm{\rho}}} }). \end{array}$$

Here, $\delta ({{\boldsymbol{\mathrm{\rho}}} })$ is the Dirac function. By changing the phase assignments $\varphi$ for three times, three equations are obtained, and then the CSD function $W({{{\boldsymbol{\mathrm{\rho}}} },\textrm{0}} )$ can be calculated with the formula:

(8)$$W({{{\boldsymbol{\mathrm{\rho}}} },\textrm{0}} )= F{T^{ - 1}}\left\{ {\frac{{({C_ -^\ast{-} 1} )[{{I_ + } - {I_0}} ]- ({C_ +^\ast{-} 1} )[{{I_ - } - {I_0}} ]}}{{({{C_ + } - 1} )({C_ -^\ast{-} 1} )- ({C_ +^\ast{-} 1} )({{C_ - } - 1} )}}} \right\}.$$

$({{\textrm{I}_ + },{\textrm{I}_ - },{\textrm{I}_0}} )$ are three perturbed intensities captured by the CCD, corresponding to $({{C_ + },{C_ - },0} )$, where three different phase values ($\varphi = \textrm{2}\boldsymbol{\mathrm{\pi}}\textrm{/3}$, $\varphi ={-} \textrm{2}\boldsymbol{\mathrm{\pi}}\textrm{/3}$, and $\varphi = 0$) are assigned to the perturbation with the help of the SLM₂. The phase values should be chosen as far apart as possible within 2π. Then the intensity patterns can be maximally different, and therefore the calculation can be most noise robust.

5. Results and discussions

Figure 5 shows the free space interference intensities of a PCVB passing through the double-slit under various spatial coherence ${\sigma _g} = 0.3{\omega _0}$, ${\sigma _g} = 0.5{\omega _0}$ and ${\sigma _g}\textrm{ = }10{\omega _0}$ (nearly completely coherent). Here, ${\omega _0} = 1.5mm$, a=0.0725 mm, b=0.18 mm and the free space transmission distance z=800 mm. It shows that he larger the |l|, the more obvious the distortion. The TC with the opposite sign has the opposite twisting direction. Based on the above relationship, we can just qualitatively judge the topological charge from the interference intensity. As the degree of coherence decreases, the interference patterns become more and more blur.

Fig. 5. Experimental measurements of interference fringes of a partially coherent vortex beams passing through a double-slit under different topological charges and spatial coherence.

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As is discussed in Simulation section, determining the value of TC from phase distributions is not only better than the interference intensity, but also better than the amplitude distribution of CSD function. With the help of Eq. (8), we measured the CSD distributions on the focal plane (i.e., SLM₂ plane and f=800 mm) with different TCs for different spatial coherence, and showed corresponding phase distributions in Fig. 6. The experimental results agree well with the theoretical results in Fig. 3. When the phase around each coherence singularity is rotating counter-clockwise (clockwise) from –π to π, the sign of the TC is positive (negative), and the number of coherence singularities is equal to 2|l|. Compared with the interference patterns, the phase distribution can help obtain a more quantitative conclusion. From the Fig. 6, we can see that, not only the location but also the rotation of the phase singularity in the CSD is distinguishable, even when the spatial coherence decreases. In other words, the exact value of TC, including magnitude and sign, can be obtained by measuring the phase structures of CSD functions after Young’s double-slit interference.

Fig. 6. Experimental results of the phase distribution of the CSD function for a partially coherent vortex beams passing through a double-slit with different topological charges and spatial coherence.

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6. Conclusions

In this paper, we investigate the phase and amplitude of the CSD in Young’s double-slit interference experiment of a PCVB. We derive the analytical expressions of the CSD function of the PCVBs after passing through a double-slit which is subsequently focused on the observation plane. By choosing an on-axis reference point, we can observe the coherence singularities based on its phase and amplitude patterns. The results show that the magnitude of the TC is half of the number of coherence singularities. A phase that rotates counterclockwise from −π to π corresponds to a positive sign, whereas a phase that rotates clockwise from −π to π represents a negative sign. We also verified this rule in the experiment using the self-reference holography method, that is measuring the CSD of a PCVB after interference and focusing by introducing a phase perturbation point in the center. Our results provide a new way for simultaneously measuring the magnitude and sign of TC of a PCVB and will be useful for some applications where the exact measurement of the TC matters, like quantum information processing, optical storage, and communications [35,36].

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11525418, 11774250, 11974218, 91750201); Innovation Group of Jinan (2018GXRC010); Priority Academic Program Development of Jiangsu Higher Education Institutions; China Scholarship Council (201906920048); Local science and technology development project of the central government (YDZX20203700001766); Tang Scholar.

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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Young’s double-slit experiment with a partially coherent vortex beam

Abstract

1. Introduction

2. Theory

3. Simulation

4. Experimental measurement

5. Results and discussions

6. Conclusions

Funding

Disclosures

References

Supplementary Material (1)

Cited By

Figures (6)

Equations (8)

Optics Express