7.3 Optical resonators: stability and mode structure

Optical resonators are the heart of lasers, confining and amplifying light using mirrors and a gain medium. They come in various configurations, each with unique properties that affect stability and performance.

Understanding resonator stability is crucial for laser design. ABCD matrices help analyze light propagation, while the stability criterion determines if light remains confined. Transverse modes and resonance frequencies further shape the output beam's characteristics.

Optical Resonator Configuration and Stability

Components of optical resonators

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• Optical resonators confine and amplify light using two or more mirrors (flat or curved)
  • Common configurations: Fabry-Perot cavities and ring resonators
• Gain medium amplifies light through stimulated emission placed between the mirrors
  • Examples: solid-state crystals (Ti:Sapphire), semiconductors (GaAs), gases (He-Ne)
• Light circulates between the mirrors, passing through the gain medium multiple times
• Resonator geometry and mirror properties determine stability and mode structure

Stability analysis with ABCD matrices

• ABCD matrix formalism analyzes light propagation through optical systems
  • Each optical element represented by a 2x2 matrix
  • Matrices multiplied to determine overall system matrix
• Stability criterion for optical resonators: $0 \leq (A+D)^2/4 \leq 1$
  • A and D are elements of the system matrix
  • If satisfied, resonator is stable, and light remains confined within the cavity
• Stability depends on mirror curvature, spacing, and alignment
  • Planar mirror resonators are marginally stable
  • Confocal resonators (mirrors with equal radii of curvature separated by their focal length) are stable

Resonator Mode Structure and Properties

Transverse modes in stable resonators

• Transverse modes describe spatial distribution of light intensity perpendicular to propagation direction
• Hermite-Gaussian (HG) modes: orthogonal solutions to paraxial wave equation in Cartesian coordinates
  • Characterized by indices $n$ and $m$, representing nodes in $x$ and $y$ directions
  • Intensity profile described by Hermite polynomials and a Gaussian function
• Laguerre-Gaussian (LG) modes: orthogonal solutions in cylindrical coordinates
  • Characterized by radial index $p$ and azimuthal index $l$
  • Intensity profile described by Laguerre polynomials and a Gaussian function
  • Non-zero $l$ LG modes have a phase singularity at the center, resulting in a donut-shaped intensity profile
• Higher-order modes have larger transverse extent and more complex intensity patterns than the fundamental mode (HG00 or LG00)

Resonance frequencies and spectral range

• Resonance frequencies: frequencies at which light constructively interferes after one round trip
  • Determined by condition: $2L = m\lambda$, where $L$ is resonator length, $m$ is an integer, and $\lambda$ is wavelength
  • Resonance frequencies given by $f_m = mc/(2L)$, where $c$ is speed of light
• Free spectral range (FSR): frequency spacing between adjacent resonance frequencies
  • FSR = $c/(2L)$
  • Inversely proportional to resonator length
• Resonators with larger FSR support fewer longitudinal modes within a given gain bandwidth

Effects of perturbations on resonators

• Mirror misalignment reduces coupling efficiency and increases losses
  • Angular misalignment tilts wavefront, causing beam to walk off mirrors
  • Lateral misalignment shifts beam position and reduces overlap with gain medium
• Apertures within resonator limit transverse extent of modes
  • Smaller apertures increase diffraction losses and suppress higher-order modes
  • Carefully designed apertures used for mode selection and improving beam quality
• Other perturbations (thermal lensing in gain medium, mirror vibrations) affect stability and mode structure
  • Thermal lensing: non-uniform heating of gain medium leads to spatially varying refractive index
  • Vibrations cause fluctuations in resonator length and alignment, leading to frequency and intensity noise in output beam
• Proper design, alignment, and stabilization techniques optimize resonator performance in applications (lasers, optical filters)