Sensing optical cavity mismatch with a mode-converter and quadrant photodiode

We present a new technique for sensing optical cavity mode mismatch and alignment by using a cylindrical lens mode converting telescope, radio-frequency quadrant photodiodes, and a heterodyne detection scheme. The telescope allows the conversion of the Laguerre-Gauss bullseye mode (${\mathrm{LG}}_{01}$) into the 45° rotated Hermite-Gauss (“pringle”) mode (${\mathrm{HG}}_{11}$), which can be easily measured with quadrant photodiodes. We show that we can convert to the HG basis optically, measure mode mismatched and alignment signals using widely produced radio-frequency quadrant photodiodes, and obtain a feedback error signal with heterodyne detection.

Figure 1

The concept of sensing mode mismatch using a mode converter and heterodyne detection on a quadrant photodiode. A ${\mathrm{LG}}_{01}$ mode is converted to a 45°-rotated ${\mathrm{HG}}_{11}$ mode with a $\frac{\pi }{2}$ mode converter. The mode converter consists of two cylindrical lenses spaced by $f\sqrt{2}$ where $f$ is focal length. The incoming beam waist size $w_0$ is centered between the two cylindrical lenses and related to the cylindrical focal length via $f(w_0)=\frac{\pi w_0^2}{\lambda }/(1+\frac{1}{\sqrt{2}})$ [7].

Figure 2

Beam decomposition of the $|{\mathrm{LG}}_{01}⟩$ and $|{\mathrm{HG}}_{11}^{45°\mathrm{rot}}⟩$ mode in the HG basis. Shown are the (real) field amplitudes of the two relevant HG modes (left) and the resulting modes (right). The intensity profile for each field is plotted in the bottom right corner for each field image.

Figure 3

A cylindrical lens mode converter is shown. The beam shape is plotted with respect to the cylindrical lens focusing and non focusing axis. As the beam passes through the mode converter a factor of 90° or $\frac{\pi }{2}$ is accumulated in the focusing axis while the nonfocusing axis experiences normal Gouy phase accumulation.

Figure 4

Combining the photodiode segments can yield pitch, yaw, sum, and beam waist size or position. Left: quadrant photodiode (QPD). Right: bullseye phododiode (BPD).

Figure 5

A locked pre-mode-cleaner produces a beam that enters a 25 MHz EOM then propagates to a four segment thermal lens actuator. The thermal lens allows for pitch, yaw and beam size control. The cavity is aligned and well mode-matched to the beam. The cavity reflected beam is split into three paths. The first path leads to a single segment Pound-Drever-Hall (PDH) locking photodiode. The second and third path lead to a Gouy phase telescope that also shapes the beam for the $\frac{\pi }{2}$ mode converter. The quadrant photodiode (RFQPD) is in the path with the mode converter while the bullseye photodiode (RFBPD) is not. The Gouy phase at both bullseye photodiode and quadrant photodiode are similar. After demodulation, the signals are combined in the data acquisition system. Pitch, yaw, sum and mode-matching error signals are extracted. The cavity reflected power is attenuated by a factor of 0.30 before reaching the quadrant photodiode and 0.12 before reaching the bullseye photodiode.

Figure 6

The telescope design from the mode cleaner to the two mirror cavity is shown. Beam size and Gouy phase accumulation are plotted against propagation distance. Lenses and the cavity waist location are represented by vertical lines further described in Table 1.

Figure 7

Similar to Fig. 6 the as built telescope layout is shown. Here the beam propagates from the mode cleaner to the two-mirror cavity input coupler and reflects instead of transmits. The reflected beam then propagates to beam focusing lenses and a cylindrical lens mode converter. The reflected beam terminates at either a QPD or BPD wavefront sensor at similar Gouy phases. The BPD is located on a separate beam with the same profile as the blue y-axis plot, see Fig. 5. The vertical (Y) and horizontal (X) beam axis are shown. The only difference between the beams is seen at the $\frac{\pi }{2}$ mode converter. We show that one axis is focused while the other remains unchanged which adds a 90° Gouy phase difference between the axes.

Figure 8

Power mode overlap $P=|\frac{2i\sqrt{z_{R1}{z}_{R2}}}{q_1-q_2^{*}}{|}^2$ between the optical cavity and the input beam is shown as the contour lines in percentage. The thermal lens actuation path is seen in red. As the thermal lens actuator changes the input beam into the cavity, the power mode overlap decreases. Mode mismatching can reach well below 10%.

Figure 9

Measured mode-mismatch response is shown as 9% mode mismatch is manually induced using the calibrated thermal lens actuator. The thermal lens actuator is equally heated radially thus ensuring only mode mismatch was induced. Counts from our digital data acquisition system are converted into mode mismatch $\epsilon$ as stated in 3c. From first principles our calibration also yields a $|\epsilon {|}^2=9\%$ mode-mismatch at the maximum value on this plot. Finally we also observed a 9% drop in cavity transmitted power.

Figure 10

Large signal gain for mode sensing outside the linear regime. Plotted is the imaginary part of the matrix element in Eq. (c9), as a function of waist size (top) and waist location (bottom). For each plot the diode was placed in the optimal sensing Gouy phase. The solid traces blue, red and green are for a QPD placed after a mode-converter, a BPD with inner segment radius $r^{\prime }=w\sqrt{0.5\ln 2}$ (no bias), and a BPD with inner segment radius $r=w\sqrt{0.5}$, in that order. All solid traces are calculated taking into account modes up or order $n$, $m=20$.The blue dash-dotted trace is the linear approximation from Eqs. (c5) and (c7). Finally, the cavity is kept on resonance during the sweep—this affects the large signal behavior of all traces, as well as the small signal gain (slope) of the green trace in the lower plot (BPD largest gain). The small signal gains of the blue (QPD) and red (BPD no bias) are independent of any length offset.

Figure 11

A 1 watt laser produces a beam at 1064 nanometer wave length. The beam passes through an electro-optic modulator (EOM) resonant at 9 MHz. The beam then passes through a beam splitter then into a hemispherical resonant optical cavity. The beam reflected from the cavity is then directed back to the beam splitter where now the reflected beam is directed to two paths. The first path contains two radio-frequency bullseye photodiodes (RFBPD) of varying radii. finesse automatically changes the bullseye photodetector size to match the beam incident on it. Second, the beam passes through a beam shaping telescope then to a mode converter before finally arriving at two radio-frequency quadrant photodiodes (RFQPD). Each style of photodiode has one photodiode that measures the beam at 0° Gouy phase and a second photodiode that measures the beam at 45° Gouy phase. This Gouy phase separation is ideal for measuring both beam waist size and beam waist location.

Figure 12

matlab was used to model the $\frac{\pi }{2}$ mode converter using a Fourier optics representation of lenses. The cylindrical lenses both had a focal length of $f=0.1\mathrm{m}$ and were separated by $f\sqrt{2}$. The input beam waist was located half way between the cylindrical lenses and had a size of $w_0=(f\lambda (1+1/\sqrt{2})/\pi {)}^{1/2}$. We propagate the $|{\mathrm{HG}}_{02}⟩$, $|{\mathrm{HG}}_{20}⟩$, and $|{\mathrm{LG}}_{01}⟩$ modes through the mode converting telescope. HG modes oriented parallel or perpendicular to the lens focusing axis will experience no structural change in intensity profile (bottom right of each field image). HG modes parallel to the lens focusing axis will get a sign flip in field. The $|{\mathrm{LG}}_{01}⟩$ mode converts into a 45° rotated $|{\mathrm{HG}}_{11}⟩$ mode. We can also see that alignment HG modes will be unaffected while mode mismatch $|{\mathrm{LG}}_{01}⟩$ modes will be perfectly converted into the HG basis.

Figure 13

Mode mismatch error signals generated by the finesse with matlab simulation. See Appendix pp4 for more details.