Generation and Propagation of a Hermite-Gaussian Correlated Schell-Model LG_0l Beam

Abstract

A partially coherent beam under the combined action of a Hermite-Gaussian correlated function and vortex phase, named the HGCSMLG_0l beam has been explored both theoretically and experimentally. The statistical properties, such as the intensity and distribution of the degree of coherence (DOC) on propagation are analyzed in detail, based on the deduced equations. We find that the intensity is determined dominantly by the non-conventional correlated function when the coherence length is comparatively small and by vortex phase when the coherence length is large. The modulus of the DOC is not vulnerable to coherence width, rather, it is affected by both non-conventional correlated function and vortex phase. Our results are verified well by the experiment results.

1. Introduction

The light beam with a spiral structure of the phase (i.e., vortex phase), named the vortex beam has continuously been under the spotlight in optical research area, due to its unique properties, e.g., orbital angular momentum (OAM), and phase singularity [1,2,3]. The OAM of a vortex beam can be used for particle trapping, detection of spinning object, free-space optical communications, and quantum information processing [4,5,6,7]. At the beginning, research attention was concentrated on the fully coherent vortex beams [8,9,10,11]. The quantitative measurement methods of the topological charge (TC) or the OAM state have been developed [12,13,14,15,16,17]. Until recently, a partially coherent (PC) beam, obtained by reducing the spatial coherence width, has demonstrated its superiority to a complete coherent counterpart in various applications [18,19,20,21,22,23,24,25,26]. In other words, controlling the spatial coherence width provides an effective way of manipulating beam properties. It is naturally to extend the coherent vortex beams to partially coherent vortex light (PCV). The PCV light, whose correlation function satisfies the Gaussian distribution, is a conventional PCV beam (e.g., Gaussian Schell-model (GSM) vortex light). Compared with PC light without vortex phase, and its complete coherent counterpart, the GSM vortex beam has advantages in many aspects. It can be shaped into prescribed beam profiles by modulating the coherence width of the incident beam to capture particles no matter which index of refraction is larger or smaller than that of the ambient [26,27]. PCV light has advantages of attenuating the beam-width spreading, beam wander, and scintillation induced by atmospheric turbulence [28,29,30]. PCV light also displays self-reconstruction abilities in the distribution of the DOC on propagation, which can be used for information encryption and decryption [31].

Now, more and more attention has been paid to PC light with non-Gaussian distribution correlation function, which is named the special correlated (or nonconventional) PC light. A variety of these type of beams have been constructed and generated experimentally [20,32,33,34,35,36,37,38,39], ever since the conditions which guarantee the real existence of cross-spectral density function or matrix, were discussed in [40,41]. These non-Gaussian distribution correlation functions bring about the new changes and characteristics of the beam on propagation, which can be applied to reduce scintillation in turbulence and realize optical super resolution [42,43,44]. In this article, we will discuss the combined action of special correlation structure and the vortex phase on the beam’s propagation properties, both theoretically and experimentally. An LG_0l beam with a Hermite-Gaussian correlated function named HGCSMLG_0l beam is introduced and explored in detail. We find that the correlation function and TC together determine the focused spot and the DOC distribution on the propagation displays’ rotating properties, which are induced by the vortex phase. By measuring the distribution of DOC, the handedness and the TC of the vortex phase can be obtained. The above theory results are verified well by the experiment results.

2. Propagation Equation of a HGCSMLG_0l Beam

For a PC LG_0l beam, the cross-spectral density (CSD) function can be described as follows [45,46],

$$W\left(r_1,r_2\right)={\left(\frac{2r_1{r}_2}{4{\sigma }_0^2}\right)}^{\left|l\right|}\exp \left(-\frac{r_1^2+r_2^2}{4{\sigma }_0^2}\right)\gamma \left(r_2-r_1\right)\exp \left[-il\left({\phi }_1-{\phi }_2\right)\right],$$

(1)

where $r_1$, $r_2$ are two arbitrary points in the source plane, ${\sigma }_0$, $\phi$ and l are the beam width, azimuth angle, and the topological charge, respectively, $\gamma \left(r_2-r_1\right)$ denotes the correlation function. In order to explore the effects of a special correlation function on a PCV, the conventional Gaussian distribution function will be replaced by a Hermite-Gaussian distribution function, which can be expressed by:

$$\begin{array}{l}\gamma \left(r_2-r_1\right)=\frac{H_{2m}\left[\left(x_2-x_1\right)/\sqrt{2}{\delta }_0\right]}{H_{2m}\left[0\right]}\exp \left[-\frac{{\left(x_2-x_1\right)}^2}{2{\delta }_0^2}\right]\\ \times \frac{H_{2n}\left[\left(y_2-y_1\right)/\sqrt{2}{\delta }_0\right]}{H_{2n}\left[0\right]}\exp \left[-\frac{{\left(y_2-y_1\right)}^2}{2{\delta }_0^2}\right],\end{array}$$

(2)

where ${\delta }_0$ is the spatial coherence width, $H_{2m}$, and $H_{2n}$ are the Hermite polynomial with beam order 2m, and 2n, respectively. This new constructed beam is called the HGCSMLG_0l beam.

According to the generalized Huygens-Fresnel principle [26,37], a HGCSMLG_0l beam passing through an ABCD optical system can be dealt with the generalized Collins integral equation

$$\begin{array}{l}W\left({\mathit{\rho }}_1,{\mathit{\rho }}_2;z\right)=\frac{1}{{\left(\lambda B\right)}^2}{\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-\infty }^{\infty }W\left(r_1,r_2\right)}}\exp \{-\frac{ik}{2B}[A\left(r_1^2-r_2^2\right)\\ -2r_1\cdot {\mathit{\rho }}_1+2r_2\cdot {\mathit{\rho }}_2+D\left({\rho }_1^2-{\rho }_2^2\right)]\}{d}^2{r}_1{d}^2{r}_2,\end{array}$$

(3)

where ${\mathit{\rho }}_1$ and ${\mathit{\rho }}_2$ are two arbitrary points in the output plane, $k$ and A, B, D are the wave-number and the transfer matrix elements of an optical system.

By substituting Equations (1) and (2) into Equation (3) and the integrating them, the CSD of the outgoing HGCSMLG_0l beam can be acquired with the intermediate steps listed in Appendix A.

$$\begin{array}{l}W\left({\mathit{\rho }}_1,{\mathit{\rho }}_2;z\right)=\mathsf{\Pi }{\displaystyle \sum _{c_1=0}^{\left|l\right|}{\displaystyle \sum _{c_2=0}^{\left|l\right|}{\displaystyle \sum _{k_1=0}^m{\displaystyle \sum _{k_2=0}^n{\displaystyle \sum _{p_1}^{2k_1}{\displaystyle \sum _{p_2}^{2k_2}{\displaystyle \sum _{d_1=0}^{b_4}{\displaystyle \sum _{a_1=0}^{\left[d_1/2\right]}{\displaystyle \sum _{d_2=0}^{b_2}{\displaystyle \sum _{a_2=0}^{\left[d_2/2\right]}{\left(\frac{1}{2{\delta }_0^2}\right)}^{k_1+k_2}{\left(\frac{i\sqrt{2}}{\sqrt{{\alpha }_1^{\ast }}{\delta }_0^2}\right)}^{d_1+d_2-2a_1-2a_2}}}}}}}}}}}\\ \times \frac{{\left(-1\right)}^{b_6}{\left(i\right)}^{b_5}{\pi }^2{d}_1!d_2!\left|l\right|!\left|l\right|!}{a_1!a_2!c_1!c_2!k_1!k_2!}\frac{2^{-3\left(b_2+b_4\right)/2-b_x-b_y}{\left({\alpha }_1^{\ast }\right)}^{-(b_2+b_4+2)/2}{\alpha }_2{}^{-(b_x+by+2)/2}}{\left(d_1-2a_1\right)!\left(d_2-2a_2\right)!\left(\left|l\right|-c1\right)!\left(\left|l\right|-c2\right)!}\\ \times \left(\begin{array}{c}n-1/2\\ n-k_2\end{array}\right)\left(\begin{array}{c}m-1/2\\ m-k_1\end{array}\right)\left(\begin{array}{c}2k_2\\ p_2\end{array}\right)\left(\begin{array}{c}2k_1\\ p_1\end{array}\right)\left(\begin{array}{c}{b}_4\\ d_1\end{array}\right)\left(\begin{array}{c}{b}_2\\ d_2\end{array}\right)H\left({\rho }_{x1},{\rho }_{x2},{\rho }_{y1},{\rho }_{y2}\right),\end{array}$$

(4)

with

$$\mathsf{\Pi }=\frac{1}{{\left(2{\sigma }_0^2\right)}^{\left|l\right|}{\lambda }^2{B}^2}\frac{m!\sqrt{\pi }}{\mathsf{\Gamma }\left(m+1/2\right)}\frac{n!\sqrt{\pi }}{\mathsf{\Gamma }\left(n+1/2\right)}\exp \left(-\frac{ikD}{2B}\left({\rho }_1^2-{\rho }_2^2\right)\right),$$

(5)

$$\begin{array}{l}H\left({\rho }_{x1},{\rho }_{x2},{\rho }_{y1},{\rho }_{y2}\right)=H_{b_y}\left(i\frac{{\beta }_y}{2\sqrt{{\alpha }_2}}\right)H_{b_x}\left(\frac{i{\beta }_x}{2\sqrt{{\alpha }_2}}\right)H_{b_2-d_2}\left(\frac{k{\rho }_{x2}}{\sqrt{2{\alpha }_1^{\ast }}B}\right)\\ \times H_{b_4-d_1}\left(\frac{k{\rho }_{y2}}{\sqrt{2{\alpha }_1^{\ast }}B}\right)\exp \left(\frac{{\beta }_x^2+{\beta }_y^2}{4{\alpha }_2}\right)\exp \left(-\frac{k^2\left({\rho }_{x2}^2+{\rho }_{y2}^2\right)}{4{\alpha }_1^{\ast }{B}^2}\right),\end{array}$$

(6)

$$\begin{array}{l}{b}_1=\left|l\right|+p_1-c_1;b_2=\left|l\right|+2k_1-p_1-c_2,b_3=c_1+p_2;\\ b_4=c_2+2k_2-p_2;b_5=c_1+c_2-b_2-b_4-b_x-b_y;\\ b_6=a_1+a_2+k_1+k_2+p_1+p_2+c_{\gamma },b_x=b_1+d_2-2a_2;\\ b_y=b_3+d_1-2a_1;c_{\gamma }=\{\begin{array}{c}{c}_{1}l\ge 0\\ c_{2}l<0\end{array};\end{array}$$

(7)

$$\begin{array}{l}{\alpha }_1=\frac{1}{4{\sigma }_0^2}+\frac{1}{2{\delta }_0^2}+\frac{ikA}{2B};{\alpha }_2={\alpha }_1-\frac{1}{4{\alpha }_1^{\ast }{\delta }_0^4};{\beta }_y=\frac{ik{\rho }_{y1}}{B}-\frac{ik{\rho }_{y2}}{2{\alpha }_1^{\ast }B{\delta }_0^2};\\ {\beta }_x=\frac{ik{\rho }_{x1}}{B}-\frac{ik{\rho }_{x2}}{2{\alpha }_1^{\ast }B{\delta }_0^2}.\end{array}$$

(8)

The intensity and the DOC of the HGCSMLG_0l beam in the exiting plane are expressed as;

$$I\left(\mathsf{\rho },z\right)=W\left(\mathsf{\rho },\mathsf{\rho };z\right),$$

(9)

$$\mu \left({\mathsf{\rho }}_1,{\mathsf{\rho }}_2;\mathrm{z}\right)=\frac{W\left({\mathsf{\rho }}_1,{\mathsf{\rho }}_2;z\right)}{\sqrt{W\left({\mathsf{\rho }}_1,{\mathsf{\rho }}_1;z\right)W\left({\mathsf{\rho }}_2,{\mathsf{\rho }}_2;z\right)}}.$$

(10)

Submitting Equations (4)–(8) into Equations (9)–(10), one can analyze the distributions of the intensity and the DOC of the propagating HGCSMLG_0l beam conveniently.

3. Theoretical Analysis the Propagation of an HGCSMLG_0l Beam

Applying the formulae obtained in the last section, the statistical properties of a propagating HGCSMLG_0l beam will be analyzed. It is supposed that the beam propagates in free space after passing through a focal lens locating at $z=0$ with $f=400\mathrm{mm}$. The related parameters are defined as $\lambda =632.8\mathrm{nm},{\sigma }_0=1\mathrm{mm},{\delta }_0=1\mathrm{mm}.$

In Figure 1, Figure 2 and Figure 3, we calculated the normalized intensity of a focused HGCSMLG_0l beam at different distances with $m=0,n=1$ and the increasing values of the TC. One finds from the first row of Figure 1, Figure 2 and Figure 3, that the HGCSMLG_0l beam reduces to the HGCSM beam for $l=0$. As expected, a propagating HGCSM beam will split from the original one spot to four spots gradually [37]. From the second row and the third row of Figure 1, Figure 2 and Figure 3, the HGCSMLG_0l beam exhibits interesting variance of the beam profile under the combined action of unconventional correlation function and the vortex phase. As the second row of Figure 1, Figure 2 and Figure 3 shows, the intensity distribution of the HGCSMLG0l beam evolves increasingly from a donut shape into a two-beamlets array distribution (see Figure 1d2 and Figure 2d2) or four-beamlets array distribution (see Figure 3d2) in the focal plane, while a PC LG_0l beam remains in a circular symmetry while propagating in free space [45,46]. It is clear that the new special correlation function plays a dominant role in the transmission rule of the beam. As the third row of Figure 1, Figure 2 and Figure 3 shows, the light spot of the focal plane does not separate into two (or four) beamlets with the increasing TC. It can be explained that the vortex phase plays an anti-splitting role.

Figure 4 shows the normalized focal plane intensity of an HGCSMLG_0l beam, with $l=2$ for different beam orders m, n and the spatial coherence width ${\delta }_0$. One can see that the beam orders m, n and the spatial coherence width ${\delta }_0$ determine the splitting array form of focal plane. Once the spatial coherence width ${\delta }_0$ increases to a certain value, the far-field beamlets array distribution would disappear and degenerate into a semi-dark hollow beam profile. We can draw a conclusion that the beam shape in the focal plane depends on the correlation function, as the initial coherence width is not large, and as the initial coherence width becomes large the focused beam profile is mainly determined by the vortex phase. As a result, coherence width should be suitably chosen to obtain the splitting array form of beam profile on propagation.

Figure 5, Figure 6 and Figure 7 displays the modulus of the DOC ($\left|\mu \left(\rho ,0;z\right)\right|$) of a focused HGCSMLG_0l beam at different distances with different m, n and the increasing values of the TC. The first rows of Figure 5, Figure 6 and Figure 7 show that the variation of the modulus of the DOC of a focused HGCSM beam, for different beam order, with increasing distance z. The distribution of the DOC of the HGCSM beam displays an array distribution near the source plane ($z=1\mathrm{mm}$) which is the same as the source plane [19] and remains array distribution as the beam propagates. It is not until it reaches the focal plane that the DOC distribution evolves into one beam spot. The second and the third rows of Figure 5, Figure 6 and Figure 7 show the propagation properties of the DOC of a focused HGCSMLG_0l beam at different distances for different beam order m, n and increasing values of the TC. One can find the vortex phase brings about obvious changes of the DOC distribution even in the plane ($z=1\mathrm{mm}$) near the source plane. The DOC distribution exhibits a “rotating effect” as the beam propagates along the z axis. We can decide the handedness of the TC by the rotation direction of the distribution of the DOC. As the rotation direction of windmills (see Figure 7b2–c2) is anti-clockwise, the value of TC is positive ($l>0$). While on the contrary, the value of TC is negative ($l<0$) (see Figure 7b3–c3).

Figure 8 shows the modulus of the focal plane DOC of an HGCSMLG_0l beam for different m, n and the TC. The dark rings’ number of the focal plane DOC increases as the value of TC increases. For $m=0$, $n=1$ (see Figure 8a1–d1) and $m=1$, $n=0$ the dark rings’ number along the y direction and the x direction equals to the TC number l separately. For $m=1$, $n=1$ (see Figure 8a3–d3), the dark rings’ number along the x or y direction is one less than the TC number l if $l>2$. Thus, the TC number can be determined by the number of the dark rings. It can be concluded that we can distinguish the handedness and value of the TC from the DOC distribution, in spite of form changes of the DOC distribution, caused by the new correlation function.

4. Experimental Verification

In this part, the generation of an HGCSMLG_0l beam is introduced and our theoretical predictions will be validated.

Figure 9 shows the optical layout for generating and measuring HGCSMLG_0l. For generating a HGCSMLG_0l beam, the coherent laser beam, with λ = 532 nm comes into beam expander (BE) and the emitted light is modulated by spatial light modulator (SLM1). A designed computer-generated hologram, for producing the unconventional correlation function, is loaded on SLM1 [37]. The diffracted light of first-order is focused onto a rotating ground glass disk with lens (L1, f = 150 mm). The scattered light is collected by a lens (L2, f = 200 mm) and is then filtered by a Gaussian amplitude plate. SLM2 is a phase spatial light modulator which is used to loading the designed computer-generated holograms for generating the vortex phase. After SLM2, we can obtain the HGCSMLG_0l beam. The TC and the beam orders m, n can be controlled by SLM2 and SLM1 separately. L3 is used to focus the generated HGCSMLG_0l beam. The beam profile analyzer is used to record the intensity information of the measured plane. The corresponding correlation function can be measured with the help of self-referencing holography, as mentioned in Ref. [47], in which the correlation function of the illumination source can be retrieved.

Figure 10 shows our experimental results in relation to the intensity of a focused HGCSMLG_0l beam at different distances for different beam orders m, n with $l=2$. As can be seen, the new unconventional correlation function has a decisive role in the intensity evolution properties of the HGCSMLG0l beam.

The intensity distribution morphs from a donut shape into an array form on propagation, which agrees well with Figure 3. Figure 11 shows the measured modulus of the DOC ($\left|\mu \left(\rho ,0;z\right)\right|$) of a focused HGCSMLG0l beam at $z=0.68f$, with $m=1,n=1$ compared with the theoretical results. We find that the distribution of the DOC displays a “rotating effect”. For $l>0$ and $l<0$, the direction of rotation is anti-clockwise and clockwise severally. So, the handedness of the vortex phase can be judged by measuring its DOC on propagation. Compared with the theoretical results, Figure 12 shows the measured modulus of the focal plane DOC of an HGCSMLG_0l beam with $l=2$ for different beam orders m, n. It can be found that the distribution of the DOC is dependent on the Hermite polynomial beam orders. For $m=0$ and $n=0$, the modulus of the DOC exhibits an elliptical distribution with the major axis along the y direction, and along x direction, respectively. For $m=1,n=1$, the distribution of the DOC has a rectangular distribution. Figure 13 shows the experimental modulus of the focal plane DOC of an HGCSMLG0l beam with $m=1,n=1$ and increasing values of TC. With the TC increasing, the dark rings’ number of the modulus of the focal plane DOC also increases as shown in Figure 8. Thus, we can determine the magnitude of the TC by measuring its DOC in the focal plane. In general, our experimental results agree well with the theoretical results.

5. Conclusions

Under interaction of nonconventional correlated function and vortex phase, the propagation rules of a partially coherent beam named HGCSMLG_0l beam are investigated theoretically and experimentally. Our results show that the behavior of the intensity of the beam on propagation is determined by the vortex phase, spatial coherence width and correlation function of the initial beam together. When the spatial coherence width of the initial beam is large, the vortex phase plays a dominant role, and the beam spot in the focal plane still displays a dark hollow beam profile, which can be used for trapping a particle with refractive index smaller than that of the ambient. When the spatial coherence width of the initial beam is small, the correlation function plays a dominant role. The beam spot in the focal plane displays two or four solid beam spots, which can be used for trapping multiple particles simultaneous with refractive indices larger than that of the ambient. For the effect of the coherence width on the modulus of the DOC is not as obvious as the intensity distribution, the modulus of the DOC is affected by both non-conventional correlated function and vortex phase. In spite of the form change of the modulus of the DOC caused by the non-conventional correlated function, the handedness and TC of vortex phase can be distinguished from the modulus of the DOC. The robust of the handedness, and TC of vortex phase, which is not vulnerable to coherence length and correlation function, may be useful in information transfer and recovery.