Cauchy-Riemann beams

Leveraging operator techniques, we address the paraxial wave equation governing a field formed by the multiplication of a Gaussian function and an entire function; notably, the latter adheres to the Laplace equation, ${\nabla }_{\perp }^{2}f(x+iy)=0$, a direct consequence of satisfying the Cauchy-Riemann equations. Our theoretical and experimental exploration brings to light the intrinsic rotation of this field during propagation, elucidated by the incorporation of the quantum (Bohm) potential. This straightforward result holds promise, enabling the analytical deduction of the Fraunhofer or Fresnel diffraction pattern. Essentially, it simplifies the extraction of the Fresnel or Fourier transform from a function satisfying the Cauchy-Riemann equations.

Figure 1

Intensity distribution of the field with $f(x+iy)=cos\left[\frac{2\pi }{T}(x+iy)\right]$ at $z=0.0\text{m}$, (a) theoretical, and (b) experimental; in (c) and (d) we present the same situation, but with the propagation distance $z=0.5\text{m}$. (e) and (f) depict the theoretical and experimental results, respectively, when $f(x+iy)=J_1\left[\frac{2\pi }{T}(x+iy)\right]$ at $z=0.0\text{m}$; and at $z=0.5\text{m}$ we have (g) and (h). The parameters are $g=1.25\times {10}^7{\text{m}}^{-2}$, $T=0.0008\text{m}$, and $\lambda =633\text{nm}$, all within a viewing window in millimeters.

Figure 2

The experimental setup employed for synthesizing a desired optical field involves a linearly polarized He-Ne laser ($\lambda =633\text{nm}$) directed onto a spatial light modulator (SLM). The SLM exhibits the required synthetic phase hologram. By utilizing a $4f$-optical system and a binary spatial filter (SF), we can selectively isolate one of the diffracted orders from the synthetic phase hologram. The chosen order carries the desired complex field spatial distribution, allowing for a precise synthesis of the desired optical field after an optical Fourier transform operation.

Figure 3

The left column represents the intensity of the field for $f(x+iy)=exp\left[\eta {(x+iy)}^2\right]$, while the right column shows the experimental results. The parameters used in the experimental setup are as follows: $T=8\times {10}^{-4}\text{m}$, $g=\frac{12}{T^2}(1+i)$, and $\eta =\frac{2}{3}g$. (a) and (b) depict the intensity distribution at $z=0.0\text{m}$, and (c) and (d) display the intensity at $z=0.45\text{m}$ using Eq. (6).

Figure 4

The Bohm potential $V_{\text{B}}(x,y)$ for the Cauchy-Riemann field given in (18) at distances $z$ from 0.0 to $1.5\times {10}^{-7}$ in increments of $0.5\times {10}^{-7}$: (a) $z=0.0$, (b) $z=0.5\times {10}^{-7}$, (c) $z=1.0\times {10}^{-7}$, and (d) $z=1.5\times {10}^{-7}$. The independent variables, $x$ and $y$, vary from $-2.0\times {10}^{-4}$ to $2.0\times {10}^{-4}$, and the range of the Bohm potential is scaled in each of the figures. The values of the parameters are $g=1.25\times {10}^7$ and $\eta =1.0\times {10}^7$. The colors, being scaled, are solely to emphasize the “rotation” of the Bohm's potential.