[Paper Review] Variable Free Spectral Range Spherical Mirror Fabry-Perot Interferometer

Katherine Kerner, Simon Rochester+2 more|ArXiv.org|Jun 18, 2003

Advanced Measurement and Metrology Techniques3 citations

TL;DR

This paper presents a spherical Fabry-Perot interferometer with adjustable mirror spacing that achieves variable free spectral range (FSR) by tuning to higher-order degeneracy conditions, where every Nth transverse mode aligns with an axial mode. The method enables FSR reduction by a factor of N (up to N=15), producing narrow, resolvable fringes suitable for high-resolution laser frequency scanning without requiring multiple long interferometers.

ABSTRACT

A spherical Fabry-Perot interferometer with adjustable mirror spacing is used to produce interference fringes with frequency separation (c/2L)/N, N=2-15. The conditions for observation of these fringes are derived from the consideration of the eigenmodes of the cavity with high transverse indices.

Motivation & Objective

• To develop a compact, tunable Fabry-Perot interferometer with adjustable free spectral range (FSR) for high-resolution spectroscopy.
• To overcome the limitation of fixed FSR in conventional interferometers by exploiting higher-order mode degeneracies in spherical mirror cavities.
• To demonstrate experimentally that FSR can be reduced by a factor of N (up to 15) through precise mirror spacing control.
• To validate the theoretical model predicting mirror separations for N-fold degeneracy using ray-tracing and eigenmode analysis.
• To provide a practical, low-cost alternative to multiple long interferometers for applications requiring closely spaced frequency markers.

Proposed method

• The interferometer uses two spherical mirrors with radius of curvature R, separated by adjustable distance L, operating in a symmetric confocal-like configuration.
• The resonance condition is derived from the round-trip phase shift, including the Gouy phase shift, leading to the frequency formula ν(q,k) = (c/2L)[q + (k+1)/π arccos(1−L/R)].
• Degeneracy between axial modes and transverse modes (n,m) occurs when L/R = 1 − cos(lπ/N), with l and N mutually prime, l < N, enabling N-fold degeneracy.
• Mirror spacing is adjusted via a piezoelectric ceramic tube to tune the FSR, with the confocal condition (L=R) corresponding to N=2.
• Transmission fringes are observed during voltage-driven scanning of the mirror separation, and peak positions are used to verify resonance conditions.
• Theoretical predictions for L/R are compared with measured mirror spacings, with uncertainty ~7 microns.

Experimental results

Research questions

• RQ1 Can the free spectral range (FSR) of a spherical Fabry-Perot interferometer be reduced below the confocal value by tuning mirror spacing?
• RQ2 What specific mirror separations produce N-fold degeneracy between axial and transverse modes for N > 2?
• RQ3 How does the effective finesse and peak transmission scale with increasing N in the degenerate mode regime?
• RQ4 To what extent do experimental measurements of resonance conditions agree with theoretical predictions based on Gouy phase and cavity eigenmodes?
• RQ5 Can a single compact interferometer replace multiple long interferometers for high-resolution frequency scanning applications?

Key findings

• The interferometer successfully produced well-resolved transmission fringes with FSR reduced by a factor of N, up to N=15, by tuning mirror spacing to N-fold degeneracy conditions.
• Measured mirror spacings matched theoretical predictions from Eq. (3) within an experimental uncertainty of approximately 7 microns.
• For N=15, the FSR was reduced to c/(2L×15), enabling high-resolution frequency markers suitable for Doppler-free spectroscopy.
• Peak transmission decreased approximately as 2/N due to reduced degeneracy, but peak widths remained largely unchanged, preserving high mirror reflectivity-limited finesse.
• The effective finesse scaled as 2/N due to FSR reduction, but the overall FSR was reduced sufficiently to allow a single compact device to replace multiple longer interferometers.
• Experimental transmission patterns confirmed that N=15 fringes are resolvable and useful for scanning over narrow spectral features, such as individual peaks in the Rb D2 line absorption spectrum.