Gaussian beams are the backbone of laser optics. They describe how laser light behaves as it travels, focusing on the beam's shape and intensity. Understanding these beams is crucial for working with lasers in various applications.

The notes cover the key aspects of Gaussian beams, including their electric field distribution, propagation characteristics, and how they interact with optical elements. This knowledge is essential for designing and optimizing laser systems in research and industry.

Gaussian Beam Fundamentals

Electric field of Gaussian beams

• Gaussian function describes transverse electric field distribution of Gaussian beams
  • Highest electric field amplitude at center decreases with radial distance
    • Models laser beams and light propagation in optical fibers (single-mode fibers)
  • Intensity distribution proportional to square of electric field amplitude
• Gaussian beam key parameters:
  • Beam waist $w_0$: Minimum beam radius located at $z = 0$
  • Beam radius $w(z)$: Radial distance where electric field amplitude falls to $1/e$ of maximum value at position $z$
    • $w(z) = w_0\sqrt{1 + (\frac{z}{z_R})^2}$, $z_R$ is Rayleigh range
  • Radius of curvature $R(z)$: Wavefront radius at position $z$
    • $R(z) = z[1 + (\frac{z_R}{z})^2]$
  • Rayleigh range $z_R$: Distance from beam waist where beam radius increases by factor of $\sqrt{2}$
    • $z_R = \frac{\pi w_0^2}{\lambda}$, $\lambda$ is wavelength
      • He-Ne laser: $\lambda = 632.8$ nm

Gaussian Beam Propagation

Evolution of beam parameters

• Gaussian beams propagate through free space with changing beam radius, radius of curvature, and phase
• Beam parameter evolution:
  1. Beam radius $w(z)$ increases with distance from beam waist
     • $w(z) = w_0\sqrt{1 + (\frac{z}{z_R})^2}$
  2. Radius of curvature $R(z)$ changes from infinity at beam waist to minimum at Rayleigh range, then increases
     • $R(z) = z[1 + (\frac{z_R}{z})^2]$
  3. Gouy phase $\psi(z)$: Additional phase shift of Gaussian beam compared to plane wave
     • $\psi(z) = \arctan(\frac{z}{z_R})$
     • Gouy phase shift is $\pi/2$ as beam propagates from $-\infty$ to $+\infty$
       • Important for mode matching in resonators (laser cavities)

ABCD matrix for optical systems

• ABCD matrix formalism analyzes Gaussian beam propagation through simple optical systems
  • 2x2 matrix represents each optical element relating input and output beam parameters
    • Optical elements: lenses, mirrors, free space propagation
  • Overall system matrix is product of individual element matrices in order encountered
• ABCD matrix transforms Gaussian beam parameters:
  • $q_2 = \frac{Aq_1 + B}{Cq_1 + D}$, $q_1$ and $q_2$ are complex beam parameters at input and output
    • $\frac{1}{q} = \frac{1}{R} - i\frac{\lambda}{\pi w^2}$
• Transformed complex beam parameter determines beam waist size and location after system propagation

Comparison of wave types

• Plane waves:
  • Infinite transverse extent and constant amplitude
  • Flat wavefronts perpendicular to propagation direction
  • No divergence or convergence upon propagation
    • Idealized and not physically realizable
• Spherical waves:
  • Amplitude decreases with distance from source
  • Spherical wavefronts centered at source
  • Diverge upon propagation
    • Produced by point sources (antennas)
• Gaussian beams:
  • Finite transverse extent with Gaussian amplitude distribution
  • Curved wavefronts approaching plane waves far from beam waist
  • Diverge upon propagation, slower than spherical waves
  • Maintain Gaussian profile during propagation
    • Realistic model for laser beams (HeNe, diode lasers)

Focusing of Gaussian beams

1. Thin lens focuses Gaussian beam to smaller beam waist
2. New beam waist size $w_0'$ and location $z_0'$ calculated using lens focal length $f$ and input beam parameters:
   • $w_0' = \frac{w_0}{\sqrt{1 + (\frac{z_0}{f})^2}}$
   • $z_0' = \frac{f^2}{f + z_0[1 + (\frac{f}{z_0})^2]}$

Collimation of Gaussian beams

1. Thin lens collimates diverging Gaussian beam producing larger beam waist and nearly flat wavefront
2. Lens focal length required for collimation equals radius of curvature of input beam at lens position
3. Collimated beam waist size $w_0'$:
   • $w_0' = \frac{w(z)f}{z}$, $w(z)$ is beam radius at lens position, $z$ is distance from input beam waist to lens
     • Used in telescopes and beam expanders (Galilean, Keplerian)