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🔬Modern Optics

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7.2 ABCD matrix formalism for beam propagation

2 min readLast Updated on July 22, 2024

ABCD matrices simplify the analysis of light propagation through optical systems. These 2x2 matrices represent how each element affects a beam's position and angle, allowing us to model complex setups by multiplying matrices together.

This formalism is particularly useful for Gaussian beams, which are characterized by their waist size and location. By using ABCD matrices, we can easily calculate how a Gaussian beam changes as it passes through lenses, mirrors, and free space.

ABCD Matrix Formalism

Ray transfer matrices in beam propagation

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  • Ray transfer matrices (ABCD matrices) describe transformation of light ray or Gaussian beam propagating through optical system
    • 2x2 matrix represents each optical element
    • Matrices multiply to determine overall system effect on beam
  • ABCD matrices relate input and output beam parameters using equation:
    • (x2θ2)=(ABCD)(x1θ1)\begin{pmatrix} x_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} x_1 \\ \theta_1 \end{pmatrix}
      • x1x_1, θ1\theta_1 input beam position and angle
      • x2x_2, θ2\theta_2 output beam position and angle
      • AA, BB, CC, DD elements of ABCD matrix
  • ABCD matrix formalism particularly useful for analyzing Gaussian beam propagation
    • Gaussian beams characterized by beam waist size (w0w_0) and location (z0z_0)
    • Complex beam parameter (qq) describes Gaussian beam at any point along propagation

ABCD matrices of optical elements

  • Free space propagation matrix:
    • (1d01)\begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}
      • dd distance of propagation
  • Thin lens matrix:
    • (101f1)\begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix}
      • ff focal length of lens
  • Curved mirror matrix:
    • (102R1)\begin{pmatrix} 1 & 0 \\ -\frac{2}{R} & 1 \end{pmatrix}
      • RR radius of curvature (positive for concave, negative for convex)

Gaussian beam tracing with matrices

  • Trace Gaussian beam through optical system by multiplying ABCD matrices of each element in order encountered
    • Resulting matrix represents total beam transformation
  • Output complex beam parameter (q2q_2) related to input (q1q_1) by:
    • q2=Aq1+BCq1+Dq_2 = \frac{Aq_1 + B}{Cq_1 + D}
  • For resonators, determine ABCD matrix for one round trip
    • Resonator stability condition: 0(A+D)240 \leq (A+D)^2 \leq 4
    • If met, resonator supports stable Gaussian modes

Beam parameter calculations using matrices

  • Complex beam parameter (qq) related to beam size (ww) and radius of curvature (RR) by:
    • 1q=1Riλπw2\frac{1}{q} = \frac{1}{R} - i\frac{\lambda}{\pi w^2}
      • λ\lambda wavelength of light
  • Find beam parameters at output of optical system:
    1. Calculate input complex beam parameter (q1q_1)
    2. Determine ABCD matrix for entire optical system
    3. Use ABCD matrix to find output complex beam parameter (q2q_2)
    4. Extract beam size (w2w_2) and radius of curvature (R2R_2) from q2q_2


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.