Mode Matching
The single lens, and its position, that couples one Gaussian beam waist into another, plus the best coupling of a lens you already have.
Matching lens focal length f
36.22 mm
The one positive lens that couples 100 %
Lens distance from input waist s₁
43.72 mm
Lens distance to target waist s₂
156.3 mm
Shortest usable focal length f₀ = πw₁w₂/λ
20.27 mm
Magnification w₂/w₁
4.000
Rayleigh ranges z_R1 / z_R2
5.067 / 81.07 mm
Your lens: best distance from input waist
49.28 mm
Your lens: coupling η
97.67 %
Best over positions between the waists
f₀ = πw₁w₂/λ; for f ≥ f₀: s₁ = f ± (w₁/w₂)√(f² − f₀²), s₂ = f ± (w₂/w₁)√(f² − f₀²), s₁ + s₂ = d.
Coupling of beams q₁, q₂ at one plane (q = z + izR): η = 4 Im q₁ Im q₂ / |q₁ − q₂*|², which includes both the size and the wavefront curvature mismatch.
Best coupling versus lens focal length
- Best η at each focal length
- Matching lens (η = 100 %)
- Your lens
Beam radius along the axis
- Matching lens
- Your lens at its best position
- Target beam
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Paraxial TEM₀₀ beams and a thin lens, closed-form solution. Coupling η is the exact overlap of the two Gaussian beams, including wavefront curvature; aberrations, tilt and lateral offset are ignored.
- Kogelnik H., Li T. (1966). Laser beams and resonators. Appl. Opt. 5, 1550–1567. doi:10.1364/ao.5.001550
- Self S. A. (1983). Focusing of spherical Gaussian beams. Appl. Opt. 22, 658. doi:10.1364/ao.22.000658
- Joyce W. B., DeLoach B. C. (1984). Alignment of Gaussian beams. Appl. Opt. 23, 4187. doi:10.1364/ao.23.004187