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
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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 w_0 w 0 : Minimum beam radius located at z = 0 z = 0 z = 0
Beam radius w ( z ) w(z) w ( z ) : Radial distance where electric field amplitude falls to 1 / e 1/e 1/ e of maximum value at position z z z
w ( z ) = w 0 1 + ( z z R ) 2 w(z) = w_0\sqrt{1 + (\frac{z}{z_R})^2} w ( z ) = w 0 1 + ( z R z ) 2 , z R z_R z R is Rayleigh range
Radius of curvature R ( z ) R(z) R ( z ) : Wavefront radius at position z z z
R ( z ) = z [ 1 + ( z R z ) 2 ] R(z) = z[1 + (\frac{z_R}{z})^2] R ( z ) = z [ 1 + ( z z R ) 2 ]
Rayleigh range z R z_R z R : Distance from beam waist where beam radius increases by factor of 2 \sqrt{2} 2
z R = π w 0 2 λ z_R = \frac{\pi w_0^2}{\lambda} z R = λ π w 0 2 , λ \lambda λ is wavelength
He-Ne laser: λ = 632.8 \lambda = 632.8 λ = 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:
Beam radius w ( z ) w(z) w ( z ) increases with distance from beam waist
w ( z ) = w 0 1 + ( z z R ) 2 w(z) = w_0\sqrt{1 + (\frac{z}{z_R})^2} w ( z ) = w 0 1 + ( z R z ) 2
Radius of curvature R ( z ) R(z) R ( z ) changes from infinity at beam waist to minimum at Rayleigh range, then increases
R ( z ) = z [ 1 + ( z R z ) 2 ] R(z) = z[1 + (\frac{z_R}{z})^2] R ( z ) = z [ 1 + ( z z R ) 2 ]
Gouy phase ψ ( z ) \psi(z) ψ ( z ) : Additional phase shift of Gaussian beam compared to plane wave
ψ ( z ) = arctan ( z z R ) \psi(z) = \arctan(\frac{z}{z_R}) ψ ( z ) = arctan ( z R z )
Gouy phase shift is π / 2 \pi/2 π /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 = A q 1 + B C q 1 + D q_2 = \frac{Aq_1 + B}{Cq_1 + D} q 2 = C q 1 + D A q 1 + B , q 1 q_1 q 1 and q 2 q_2 q 2 are complex beam parameters at input and output
1 q = 1 R − i λ π w 2 \frac{1}{q} = \frac{1}{R} - i\frac{\lambda}{\pi w^2} q 1 = R 1 − i π w 2 λ
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
Thin lens focuses Gaussian beam to smaller beam waist
New beam waist size w 0 ′ w_0' w 0 ′ and location z 0 ′ z_0' z 0 ′ calculated using lens focal length f f f and input beam parameters:
w 0 ′ = w 0 1 + ( z 0 f ) 2 w_0' = \frac{w_0}{\sqrt{1 + (\frac{z_0}{f})^2}} w 0 ′ = 1 + ( f z 0 ) 2 w 0
z 0 ′ = f 2 f + z 0 [ 1 + ( f z 0 ) 2 ] z_0' = \frac{f^2}{f + z_0[1 + (\frac{f}{z_0})^2]} z 0 ′ = f + z 0 [ 1 + ( z 0 f ) 2 ] f 2
Collimation of Gaussian beams
Thin lens collimates diverging Gaussian beam producing larger beam waist and nearly flat wavefront
Lens focal length required for collimation equals radius of curvature of input beam at lens position
Collimated beam waist size w 0 ′ w_0' w 0 ′ :
w 0 ′ = w ( z ) f z w_0' = \frac{w(z)f}{z} w 0 ′ = z w ( z ) f , w ( z ) w(z) w ( z ) is beam radius at lens position, z z z is distance from input beam waist to lens
Used in telescopes and beam expanders (Galilean, Keplerian)