Image enhancement in a miniature self-imaging degenerate optical cavity

Image-enhancing optical cavities offer an intriguing platform for nonlinear optical processing of two-dimensional signals, with potential applications in optical information processing and optically implemented artificial neural networks. Here, we analyze the performance of a self-imaging degenerate cavity via numerical simulations and show the existence of a cavity-size-dependent minimum spread in the transverse-mode resonances. This nondegeneracy, in turn, leads to an inherent trade-off between the resonant enhancement and the fidelity of the intracavity image as the cavity finesse changes.

Figure 1

Cavity geometry and mode spectrum. (a) Schematic diagram of the cavity. The cavity consists of a flat mirror, a thin lens with focal length $f$, and a curved mirror with radius of curvature $R$, separated by $d_1$ and $d_2$. The optical elements have a transverse extent defined by the aperture diameter $a$. (b) Mode spectrum of a cavity with $f=R=d_1/2=1$ mm, $a=300\mu \mathrm{m}$, and $d_2=d_1+\mathrm{\Delta }{d}_2$. Mode degeneracy for this cavity is predicted to occur for $\mathrm{\Delta }{d}_2=0$. The operating wavelength $\lambda$ is $1\mu \mathrm{m}$. The amplitude reflection coefficients of the mirrors are set to 0.96, while the lens exhibits unity transmission; the inset shows a zoomed-in plot of the first set of peaks. The profile of the fundamental mode (first peak) has been overlaid in (a), where the color represents the field intensity. (c) The transverse profiles of the first four peaks are well approximated by the Laguerre-Gauss (LG) modes.

Figure 2

Mode spectrum vs cavity size. Plot of the cavity-mode spectrum as a function of the change in $d_2=d_1+\mathrm{\Delta }{d}_2-N\lambda /2$, with $\mathrm{\Delta }{d}_2$ on the $y$ axis and axial mode shift number $N$ on the $x$ axis. As $N$ increases from zero, moving the cavity to a different axial mode, the transverse modes first become increasingly closely packed before reaching the minimum spread, after which they begin to spread out again. The color bar represents the square root of the field intensity. Note that for large cavities, the degeneracy condition is reached at $N=0$, as shown in Fig. 7 in Appendix pp2.

Figure 3

Minimum mode spread vs cavity size. Plot of the minimum spread as a function of $d_1$, which parametrizes the cavity length as well as $f$, $R$, and $d_2$. The aperture diameter of the cavity is set to $\sqrt{50}w$, where $w$ is the largest beam width of the fundamental mode inside the cavity. The spread is smaller for larger $d_1$, indicating that the cavity becomes more degenerate as its dimensions grow. The curve is only a guide to the eyes.

Figure 4

Injecting an image into the cavity. (a) Plot of the intracavity field intensity enhancement (blue, left arrow) and RMSE (red, right arrow) vs $\mathrm{\Delta }{d}_2$. The enhancement is the highest where the cavity modes occur, i.e., $\mathrm{\Delta }{d}_2\approx 0$. (b) Enhancement (blue, left arrow) and RMSE (red, right arrow) vs cavity size, parametrized by $d_1$. As can be predicted from the result of Fig. 3, as the cavity size decreases, the image becomes both less intense and less similar to the incident image. The error bars are smaller than the points. The curves are only guides to the eyes. (c) Intensity profiles of the incident image, along with the intracavity image for $d_1=0.2,0.3,0.4,1.0$, and 2.0 mm.

Figure 5

Intracavity image vs cavity finesse. (a) Plot of the enhancement (blue, left arrow) and RMSE (red, right arrow) as a function of the cavity finesse. For high mirror reflectivity and high finesse, the modes become more enhanced, but the disparity among the individual modes' enhancement grows. This, in turn, increases the RMSE and distorts the image. On the other hand, for low mirror reflectivity and low finesse, the image is less enhanced but appears more similar to the incident image. The error bars are smaller than the points. The curves are only guides to the eyes. (b) Transverse profiles of the intracavity field intensity for $F\approx 15,31$, and 160.

Figure 6

Intracavity power buildup. Plot of the intracavity power (sum over all intensity values in the simulation grid at the plane of the flat mirror) vs the number of cavity round trips. The power increases rapidly and saturates after a sufficient number of round trips, determined by the cavity finesse, or, in terms of the simulation variables, the amplitude reflection coefficient $r$ of the two cavity mirrors.

Figure 7

Paraxial degenerate cavity. (a) Mode spectrum of a degenerate cavity under the paraxial approximation. $f=R=d_1/2=1$ mm, $d_2=d_1+\mathrm{\Delta }{d}_2$, $\lambda =1\mu \mathrm{m}$, and $r=0.96$, the same parameters as in Fig. 1. Unlike the individual peaks that spread out in Fig. 1, here, the modes are highly degenerate, appearing as single peaks at every half wavelength. (b) Zoomed-in spectrum for $\mathrm{\Delta }{d}_2=0$. As predicted by paraxial cavity theory, the modes are completely degenerate. The linewidth of the degenerate modes exhibits excellent agreement with the theoretical value given by $\lambda /2F$, where the finesse $F$ is a function of only $r$. (c) Mode spectrum vs cavity size. The cavity parameters are the same as those used for Fig. 2. Whereas the length of the nonparaxial cavity in the main text must be offset by an $N\ne 0$ number of half wavelengths to achieve optimal degeneracy, the paraxial cavity is completely degenerate for $N=0$, as predicted by the paraxial cavity theory. The color bar represents the field intensity.

Figure 8

Mode spectrum vs aperture size. Plot of the cavity-mode spectrum as a function of the detuning $\mathrm{\Delta }{d}_2$ ($y$ axis) and the aperture diameter ($x$ axis). As the aperture diameter increases, more peaks are observed in the spectrum, indicating the onset of progressively higher order modes. At the same time, the modes move closer to one another, even when the size of the aperture starts to become more than an order-of-magnitude bigger than the beam width. The color bar represents the square root of the field intensity.

Figure 9

Intracavity image quality vs incident image size. (a) Plot of the enhancement (blue, left arrow) and the RMSE (red, right arrow) of the intracavity image as a function of the size of the incident image, as its lateral size is changed from 0.1 to 0.9 of the aperture diameter. The error bars are smaller than the points. The curves are only guides to the eyes. (b) Transverse profiles of the incident image and the intracavity field intensity for different incident image sizes. All three profiles exhibit similar enhancement, but the profile for the fractional size of 0.4 exhibits the highest fidelity. The color bars indicate the intensity.

Figure 10

Cavity-mode spectrum with nonlinear media. (a) Mode spectrum of a cavity ($f=R=d_1/2=1$ mm, $a=300\mu \mathrm{m}$, $d_2=d_1+\mathrm{\Delta }{d}_2,\lambda =1\mu \mathrm{m}$, $r=0.993$) containing a saturable absorber. As $\alpha$ increases, the peaks become attenuated, but their locations remain the same. (b) Mode spectrum of a cavity [same geometry as (a)] containing a Kerr medium. As $n_2$ increases, the peaks become attenuated and shift as well as changing shapes.