A Second-Order Optical Butterworth Fabry-Pérot Filter

TL;DR

This work tackles the challenge of achieving narrow, flat-top optical passbands by realizing a second-order Butterworth filter in a compact single-cavity platform. The method relies on coupling the two orthogonal polarization modes via intra-cavity birefringence to form two dressed resonance modes with a flat-top transmission on resonance and rapid roll-off off-resonance, described by the transfer function $T(g)=8 g^2 \kappa^2 / (4 g^4 + 4 g^2 (\kappa^2 - 8\delta^2) + (\kappa^2 + 8\delta^2)^2)$, with optimal coupling at $g=\sqrt{2}\kappa/2$. Experimentally, a $F=45$ second-order filter is demonstrated with a passband width of $2.68(1)$ GHz, a stopband suppression up to $43$ dB at FSR/2, and an insertion loss of $2.2(1)$ dB for an FSR of ~125 GHz, confirming the Butterworth-like flat-top response and its robustness to coupling control. The approach shows promise for narrower filters and can be extended to higher-order filters and various applications such as laser stabilization, LIDAR sensitivity, and Raman spectroscopy.

Abstract

Filters with flat-top pass-bands are a key enabling technology for signal processing. From communication to sensing, the ability to choose a pass \emph{band}, rather than a single pass \emph{frequency}, while still efficiently suppressing backgrounds at other frequencies, is a critical capability for ensuring both detection sensitivity and power efficiency. Efficient transmission of a single frequency can be achieved by a single-pole resonator -- which in optics is a Fabry-Pérot cavity offering linewidths from kHz to GHz and beyond. Coupling multiple resonators allows for the construction of flat-top multi-pole filters. These, although straightforward from RF to THz where resonators are macroscopic and tunable, are more difficult to control in the optical band and typically realized with dielectric stacks, whose passband widths exceed 100 GHz. Here, we bridge the gap to narrower bandwidth flat-top filters by proposing and implementing a second-order Butterworth-type optical filter in a single two-mirror Fabry-Pérot cavity, by coupling the two polarization modes. We demonstrate a pass-band width of 2.68(1)~GHz, a maximum stopband suppression of 43~dB, and a passband insertion loss of 2.2(1)~dB, with out-of-band power suppression falling as the fourth power of detuning. This approach is viable down to much narrower filters, and has the potential to improve high-frequency phase noise performance of lasers, enhance the sensitivity of LIDARs, and provide higher quality narrowband filtering, for example, for Raman spectroscopy.

Figures (5)

• Figure 1: Theoretical second-order coupled cavity's transfer function $T$ (red solid line) as a function of detuning, compared with the bare cavity Lorentzian transmission (gray dashed line). The two transfer function spectra share the same full width at half maximum (FWHM) and peak transmission, while the second-order filter exhibits a flatter top and faster roll-off at its wings.
• Figure 2: Experimental setup of the second-order filter. (a) Polarization basis coupled cavity scheme for the second-order optical filter. The polarization of the input light (red) is aligned such that it transmits through the polarizing beam splitter (PBS$_1$), a Faraday Rotator (FR), and PBS$_2$. The two PBSs and the FR are part of Thorlab's IO-3-780-HP optical isolator. The polarization is then matched to the cavity polarization mode by a $\lambda/2$ wave plate (WP). It is then incident on a curved, partially reflective mirror M$_1$. The coupling of the polarization modes is achieved through an intra-cavity birefringent optic that acts like a $\epsilon \lambda$ WP---two implementations of such an optic are discussed in the main text. The cavity length is stabilized by the piezoelectric stack (PZT) on the high-reflectivity end mirror (M$_2$). The cavity is single-ended, and all light goes back through M$_1$. The output (purple, shown as a bigger beam for convenience) carries a polarization superposition of the input light and its orthogonal component. The orthogonal polarization (blue) is reflected by PBS$_2$ and is the 'output' light of the filter. The input polarization component (red) gets reflected by PBS$_1$ and is the 'rejected' light of the filter. (b) CNC-machined monolithic mount of the cavity structure to suppress acoustic noise.
• Figure 3: Controlling the birefringence splitting. (a) The simulated filtered spectrum as a function of the induced birefringent coupling strength $g$. (b) Experimentally measured spectra of the second-order filter. The green arrow indicates the optimal filter performance in the simulated and measured spectra. Coupling strength $g$, obtained from fitting the transmission to \ref{['eq:BirefringenceTransmission']}, is shown for each spectrum. The desired flat-transmission performance is achieved at the optimal coupling value of $g=\sqrt{2}\kappa/2 \approx 0.71 \kappa$ where $\kappa$ is the linewidth of the coupled cavity system.
• Figure 4: Filtered transmission spectrum of the second-order filter under the optimal coupling with an FSR of 125 GHz, a FWHM of $2.68(1)~\mathrm{GHz}$, and an insertion loss of $2.2(1)~\mathrm{dB}$. Transmission is normalized to input power. Inset left: the log-linear scale plot exhibits a $\sim40~\mathrm{dB}$ suppression of background at $FSR/2$ away from center. Inset right: the log-log scale plot showing the fast fall-off (black points) along with the theoretically predicted fall-off (red dashed line) of a second-order filter.
• Figure 5: Higher-order filter schemes (beams are drawn as a guide for the eye). (a) A general schematic for an $N$th-order filter with $N$ coupled cavities. (b) A schematic for an even $2N$th-order filter with N cavities. Each cavity has 2 polarization modes. Input light (red) is aligned to one of these polarization modes. These modes are coupled (violet) via a single intra-cavity birefringent optic to give coupled $2N$ cavity modes.