Decomposing Spatial Mode Superpositions with a Triangular Optical Cavity

We demonstrate that a triangular cavity can operate as an efficient discriminator for Hermite-Gaussian optical modes. The process is applied to distinct fields produced by a spatial light modulator, decomposing the generated paraxial field onto a basis of modes defined by the resonant optical cavity. We also present a proposal for the realization of a mode sorter based on a sequence of these cavities, without the need for optical isolation. We discuss its possible applications in sensitive measurements and distribution of information in optical systems. The high purity of the process should enable its use at the level of quantum information sharing.

Figure 1

An OAM holographic mask ($\ell =3$ for example) can be understood as the sum (mod $2\pi$) of a blazed grating plus an azimuthal phase profile.

Figure 2

Resonance peaks of a superposition of the ${\mathrm{HG}}_{00},{\mathrm{HG}}_{01},{\mathrm{HG}}_{10},{\mathrm{HG}}_{20}$, and ${\mathrm{HG}}_{30}$ modes (proportion $10:5:4:3:2$) as one microscopically scans the total length $2L$ of (a) a linear cavity and (b) a triangular cavity. The simulation considers the same total length ($2L=34.4$ cm) and the same mirror curvature for both cavities ($R_c=20$ cm). These numbers correspond to a regular spacing ${\epsilon }_{N+1}-{\epsilon }_N\simeq 0.38\lambda$ in a linear cavity and are chosen as realistic parameters matching the geometrical features of the triangular cavity used in the experiment.

Figure 3

(a) Experimental setup composed of a laser, a SLM controlled via Labview, an optical cavity whose length is swept by a piezoelectric transducer (PZT), and a beam splitter (BS) that splits the beam to a photodetector and a charge-coupled device (CCD) camera. (b) SLM Gaussian mask to control the output waist. (c) Mode matching scheme.

Figure 4

Misaligned zero-order Gaussian mode. (a) Scheme of the SLM mask. (b) Detected peak-intensity for each mode as a function of $\delta /w_0$, where $\delta$ is the horizontal mask misalignment relative to the beam axis. Dots are experimental data and solid lines are theoretical predictions. Blue represents the ${\mathrm{HG}}_{00}$ mode; orange represents the ${\mathrm{HG}}_{10}$ mode; green represents the ${\mathrm{HG}}_{20}$ mode; red represents the ${\mathrm{HG}}_{30}$ mode. (c) Decomposition of a displaced Gaussian function on the basis of centered Hermite-Gaussian functions. The Gaussian with a small displacement $\delta$ as compared to the waist $w_0$ is essentially the superposition of the centered Gaussian and the first-order HG function.

Figure 5

Mismatched Gaussian mode. (a) Scheme of the SLM mask. (b) Detected peak-intensity for each mode order as a function of $w/w_0$, where $w$ is the waist of the beam leaving the SLM and $w_0$ is the ideal SLM beam waist for cavity mode matching. Dots are experimental data and solid lines are theoretical predictions. Blue represents the ${\mathrm{HG}}_{00}$ mode; orange represents the second-order modes; green represents the fourth-order modes. (c) Decomposition of a Gaussian function of width $w$ on a basis of Hermite-Gaussian functions of width $w_0$. When the ratio $w/w_0$ is sufficiently close to 1, the mismatched Gaussian is essentially the superposition of the zero-order Gaussian and the second-order HG function.

Figure 6

(a) Misaligned mask for ${\mathrm{HG}}_{10}$. (b) Detected peak-intensity for each mode as a function of $\delta /w_0$, where $\delta$ is the horizontal mask misalignment relative to the beam axis. Dots are experimental data and solid lines are theoretical predictions. Blue represents the ${\mathrm{HG}}_{00}$ mode; orange represents the ${\mathrm{HG}}_{10}$ mode; green represents the ${\mathrm{HG}}_{20}$ mode; red represents the ${\mathrm{HG}}_{30}$ mode.

Figure 7

(a) Misaligned mask for ${\mathrm{LG}}_{10}$. (b) Detected peak intensity for each mode as a function of $\delta /w_0$, where $\delta$ is the horizontal mask misalignment relative to the beam axis. Dots are experimental data and solid lines are theoretical predictions. Blue represents the ${\mathrm{HG}}_{00}$ mode; orange represents the ${\mathrm{HG}}_{10}$ mode; green represents the ${\mathrm{HG}}_{01}$ mode; red represents the ${\mathrm{HG}}_{20}$ mode; purple represents the ${\mathrm{HG}}_{02}$ mode; brown represents the ${\mathrm{HG}}_{30}$ mode.

Figure 8

(a) Phase mask for ${\mathrm{HG}}_{20}$. (b) Detected peak intensity as a function of $d/w_0$, where $d$ is the width of the central region of the ${\mathrm{HG}}_{20}$ mask (with applied phase $\pi$). Dots are experimental data and solid lines are theoretical predictions. Blue represents the ${\mathrm{HG}}_{00}$ mode; orange represents the ${\mathrm{HG}}_{20}$ mode; green represents the ${\mathrm{HG}}_{40}$ mode; red represents the ${\mathrm{HG}}_{60}$ mode.

Figure 9

Experimental scheme proposed to realize a spatial mode sorter based on a sequence of triangular cavities. See details in the main text.