Ray Optics: Paraxial Tricks & ABCD Matrices (IPhO Optics)

ABCD (ray transfer) matrices are a speed tool. They let you collapse a pile of lenses, gaps, and mirrors into one matrix, then read off whether the system images, where the image plane is, and what the magnification is.

1. Definitions (Must Know)

• Paraxial approximation: angles to the axis are small, so sin θ ≈ θ and tan θ ≈ θ (with θ in radians).
• Ray vector (one common convention): r = (y; θ) where y is height above the axis and θ is the ray angle.
• ABCD matrix: r₂ = Mr₁ with M = (A, B; C, D).
• Free-space propagation by distance L: P(L) = (1, L; 0, 1).
• Thin lens of focal length f: L(f) = (1, 0; -1/f, 1).
• Imaging condition (from plane 1 to plane 2): B = 0 for the total matrix between those planes.
• Transverse magnification (when B = 0): m = A.
• Effective focal length (same medium, this convention): f_eff = -1/C for the system matrix.
• Afocal system: C = 0 (parallel rays in give parallel rays out).

2. Key Ideas (What Earns Marks)

• Pick a consistent sign convention and stick to it (distances along the axis, ray angles, and focal lengths).
• Multiply matrices in the physical order the ray experiences them (the rightmost matrix acts first).
• For imaging problems, build the matrix from object plane to candidate image plane, then set B = 0.
• If you only need the effective focal length, compute the system matrix once and read f_eff from C.
• Always sanity-check with a limiting case: remove a lens, set a spacing to zero, or take an object very far away.

3. Detailed Explanations

3.1 Where the matrices come from

For free propagation over distance L, a ray at angle θ drifts by Lθ:

which is exactly

A thin lens changes the angle but not the height at the lens plane:

so

3.2 Composition rule

If a ray passes through element 1 and then element 2, the total transformation is

So you multiply matrices left-to-right in the order of propagation, with the rightmost acting first.

3.3 Imaging and the meaning of B

The output height is

For a point in the object plane, different rays correspond to different θᵢₙ. For those rays to reunite at a single image point in the output plane, yₒᵤₜ must not depend on θᵢₙ, so imaging requires B = 0.

When B = 0, the transverse magnification is simply

3.4 Effective focal length and afocality

For a thin lens, C = -1/f. For a general system (same medium), you can define an effective focal length by matching that structure:

If C = 0, the system is afocal: parallel input rays remain parallel at the output.

4. Common Mistakes

• Multiplying matrices in the wrong order.
• Using degrees inside small-angle formulas (ABCD is paraxial; keep θ in radians).
• Mixing sign conventions for object distance, image distance, and focal length mid-problem.
• Thinking C = 0 is the imaging condition (it is the afocal condition in this convention).
• Forgetting that ABCD matrices are paraxial: they do not capture large-angle aberrations.

5. Exam Tips

• For a single lens imaging problem, ABCD is a clean way to re-derive 1/s + 1/s' = 1/f without memorizing.
• If the system looks messy, compute the combined matrix once, then use one simple condition ( B = 0 or C = 0).
• Always report whether the image is inverted by checking the sign of m.

6. Worked Examples

7. Mind Stretchers

8. Practice