# Generation and Propagation of a Hermite-Gaussian Correlated Schell-Model LG\({}_{0l}\) Beam
## 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 \(\lambda\) = 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 HGCSMLG\({}_{0l}\) beam. Figure 8: The modulus of the focal plane DOC (\(|\mu(\rho,0;z)|\)) of an HGCSMLG\({}_{0l}\) beam for different \(m\), \(n\) and the TC. For the first row the solid curve is the cross line \(x=0\). For the second and third row the solid curve is the cross line \(y=0\). 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 (\(|u(\rho,0;z)|\)) 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 HGCSMLG0l 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. Figure 10: Experimental result of the intensity of a focused HGCSMLG0l beam at different distances for different beam orders \(m\), \(n\) with \(l=2\). Figure 9: A scheme of optical layout for generating and measuring HGCSMLG0l. Laser, 532 nm; BE, beam expander; SLM1, SLM2, spatial light modulator; CA, circular aperture; L1, L2, L3, lenses; RGGD, rotating ground glass disk; GAF, Gaussian amplitude filter; BS, beam splitter; BPA, beam profile analyzer. Figure 11: The modulus of the DOC (\(|\mu(\mathbf{\rho},0;z)|\)) of a focused HGCSMLG\({}_{0l}\) beam at \(z=0.68f\) with \(m=1\), \(n=1\).