Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam

We introduce one kind of partially coherent beam with a nonconventional correlation function named the Hermite-Gaussian correlated Schell-model (HGCSM) beam. It is found that a HGCSM beam exhibits self-splitting properties on propagation in free space, i.e., the initial single beam spot evolves into two or four beam spots in the far field depending on the initial beam orders, which are closely related to the beam width and coherence widths of the HGCSM beam, and a focused HGCSM beam exhibits splitting and combining properties near the focal plane. Furthermore, we report experimental generation of a HGCSM beam and demonstrate splitting and combining properties of a focused HGCSM beam in experiment. The phenomenon of correlation-induced self-splitting will be useful for attacking multiple targets, trapping multiple particles, and guiding atoms.

Figure 1

Density plot of the square of the modulus of the DOC of the HGCSM beam for different values of the beam orders $m$ and $n$ with ${\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$.

Figure 2

Propagation factor $M_x^2$ of a HGCSM beam versus the beam order $m$ for different values of the initial coherence width ${\delta }_{0x}$.

Figure 3

Normalized intensity distributions of a HGCSM beam with $m=0,n=1,{\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ (a) in the ${\rho }_y-z$ plane on propagation, and (b) in the ${\rho }_x-{\rho }_y$ plane at several propagation distances in free space.

Figure 4

Normalized intensity distributions of a HGCSM beam with $m=1,n=0,{\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ (a) in the ${\rho }_x-z$ plane on propagation, and (b) in the ${\rho }_x-{\rho }_y$ plane at several propagation distances in free space.

Figure 5

Normalized intensity distributions of a HGCSM beam with $m=1,n=1,{\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ (a) in the $\rho -z$ plane with ${\rho }_x={\rho }_y$ on propagation, and (b) in the ${\rho }_x-{\rho }_y$ plane at several propagation distances in free space.

Figure 6

Normalized intensity distributions of a HGCSM beam with $m=2,n=2,{\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ (a) in the $\rho -z$ plane with ${\rho }_x={\rho }_y$ on propagation, and (b) in the ${\rho }_x-{\rho }_y$ plane at several propagation distances in free space.

Figure 7

Normalized intensity distributions of a HGCSM beam with $m=1,n=1$, and ${\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ in the $\rho -z$ plane with ${\rho }_x={\rho }_y$ on propagation in free space for different values of ${\sigma }_0$.

Figure 8

Normalized intensity distributions of a HGCSM beam with $m=1,n=1,{\sigma }_0=1\mathrm{mm}$ in the $\rho -z$ plane with ${\rho }_x={\rho }_y$ on propagation in free space for different values of ${\delta }_{0x}$ and ${\delta }_{0y}$.

Figure 9

Normalized intensity distributions of a focused HGCSM beam with $m=0,n=1$, and ${\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ (a) in the ${\rho }_x-{\rho }_y-z$ plane on propagation, and (b) in the ${\rho }_x-{\rho }_y$ plane at several propagation distances in free space.

Figure 10

Normalized intensity distributions of a focused HGCSM beam with $m=1,n=1$, and ${\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ (a) in the ${\rho }_x-{\rho }_y-z$ plane on propagation, and (b) in the ${\rho }_x-{\rho }_y$ plane at several propagation distances.

Figure 11

Normalized intensity distribution of a focused HGCSM beam with $m=1,n=1,{\delta }_{0x}={\delta }_{0y}=0.2\mathrm{mm}$ in the $\rho -z$ plane with ${\rho }_x={\rho }_y$ on propagation in free space for different values of the focal length $f$ of the thin lens.

Figure 12

Experimental setup for generating a HGCSM beam, measuring the square of the modulus of its degree of coherence and its focused intensity. RM, reflecting mirror; BE, beam expander; SLM, spatial light modulator; CA, circular aperture; RGGD, rotating ground disk; L, ${\mathrm{L}}_1,{\mathrm{L}}_2,{\mathrm{L}}_3$, thin lenses; GAF, Gaussian amplitude filter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; ${\mathrm{PC}}_1,{\mathrm{PC}}_2$, personal computers.

Figure 13

Phase grating for the beam whose intensity is given by Eq. (6) with different values of the beam orders $m$ and $n$. (a) $m=1,n=0$; (b) $m=0,n=1$; (c) $m=1,n=1$; (d) $m=2,n=2$.

Figure 14

Experimental results of (a) the intensity distribution and (b) the corresponding cross line (dotted curve) of the generated HGCSM beam just behind the GAF. The solid curve denotes the theoretical fit of the experimental results with ${\sigma }_0=1\mathrm{mm}$.

Figure 15

Experimental results of the square of the modulus of the DOC of the generated HGCSM beam for different values of the beam order $m$ and $n$ and the corresponding cross line (dotted curve). The solid line denotes the theoretical fit of the experimental results with ${\delta }_0=0.2\mathrm{mm}$.

Figure 16

Experimental results of the intensity distribution of a focused HGCSM beam for different values of $m$ and $n$ with ${\sigma }_0=1\mathrm{mm}$ and ${\delta }_0=0.2\mathrm{mm}$ at several propagation distances.