Efficient generation of propagation-invariant spatially stationary partially coherent fields

We propose and demonstrate a method for generating propagation-invariant spatially stationary fields in a controllable manner. Our method relies on producing incoherent mixtures of plane waves using planar primary sources that are spatially completely uncorrelated. The strengths of the individual plane waves in the mixture determine the exact functional form of the generated coherence function. We use light-emitting diodes as the primary incoherent sources and experimentally demonstrate the effectiveness of our method by generating several spatially stationary fields, including a type that we refer to as the regionwise spatially stationary field. We also experimentally demonstrate the propagation invariance of these fields, which is an extremely interesting and useful property of such fields. Our work should have important implications for applications that exploit the spatial coherence properties either in a transverse plane or in a propagation-invariant manner, such as correlation holography, wide-field optical coherence tomography, and imaging through turbulence.

Figure 1

Schematic illustration of how a propagation-invariant spatially stationary field can be generated using a spatially completely uncorrelated primary source.

Figure 2

(a) Schematic diagram of the experimental setup. A planar, spatially incoherent primary source is placed at the back focal plane of lens $L_1$. The cross-spectral density of the field produced by the source is measured using the spatial light modulator. The propagation length $z$ is the distance between the lens $L_1$ and the SLM, and the CCD camera is placed at the focal plane of lens $L_2$. (b) Representative experimental interference pattern produced by the double-slit simulated on the SLM and the associated one-dimensional plot.

Figure 3

(a) The CCD camera image of the LED. (b) Plot of intensity $I(\mathbf{\rho },z)$ as a function of $\mathbf{\rho }$ at $z=147$ cm. (c) Plots of $\left|W\right(\mathbf{\Delta }\rho ,z\left)\right|$ as a function of $\mathbf{\Delta }\rho$ at $z=147$ cm for various values of the offset parameter $\delta$. (d) Plot of $\left|W\right(\mathbf{\Delta }\rho ,z\left)\right|$ as a function of $\mathbf{\Delta }\rho$ for various values of $z$. In (c) and (d) the black dashed curves represent the theoretical prediction based on Eq. (5).

Figure 4

(a) The CCD camera image of the LED. (b) Plot of intensity $I(\mathbf{\rho },z)$ as a function of $\mathbf{\rho }$ at $z=65$ cm. (c) Plot of $\left|W\right(\mathbf{\Delta }\rho ,z\left)\right|$ as a function of $\mathbf{\Delta }\rho$ at $z=65$ cm for various values of the offset parameter $\delta$. (d) Plot of $\left|W\right(\mathbf{\Delta }\rho ,z\left)\right|$ as a function of $\mathbf{\Delta }\rho$ for various values of $z$. In (c) and (d) the black dashed curves represent the theoretical prediction based on Eq. (5).

Figure 5

(a) Diagram illustrating how the aperture size $D$ of the lens and the spatial width $s$ of the primary source fixes $z_{\max }$. Also shown are plots of the transverse coherence length as a function of $z$ for (b) $D=5.6$ mm and (c) $D=6.5$ mm.

Figure 6

(a) Diagram illustrating the generation of regionwise spatially stationary fields. (b) Plot of intensity $I(\mathbf{\rho },z)$ as a function of $\mathbf{\rho }$ at $z=65$ cm. (c) Plot of $\left|W\right(\mathbf{\Delta }\rho ,z\left)\right|$ as a function of $\mathbf{\Delta }\rho$ at $z=65$ cm for various values of the offset parameter $\delta$.