Questions: Paraxial Ray Approximation in Geometrical Optics

Question 1

A photographer shoots wide-open (large aperture), and the center of the image is sharp but the edges are blurry. Which phenomenon best explains this?

• A. Diffraction, because the aperture is too small to resolve edge detail
• B. Spherical aberration, because marginal rays hitting the outer lens fall outside the paraxial regime and focus at a different distance
• C. Chromatic aberration, because different wavelengths bend differently at the lens edge
• D. Vignetting, because the lens blocks light at steep angles

Answer: (B)

Wide aperture means more light passes through the outer portions of the lens, where rays strike at steeper angles. These marginal rays are non-paraxial: the approximation sin θ ≈ θ breaks down, so they refract more strongly than paraxial rays and focus at a shorter distance. The result is that the image plane satisfying the paraxial thin-lens equation is sharp for center rays but blurry for edge rays — spherical aberration. This is exactly why professional lenses use aspherical elements: to satisfy the exact focusing condition for marginal rays without relying on the paraxial assumption.

Question 2

Why does the paraxial approximation make lens and mirror optics analytically tractable in a way that full trigonometric ray tracing does not?

• A. It eliminates reflections at lens surfaces, reducing the number of equations needed
• B. It replaces sin θ ≈ θ, making Snell's law linear and ensuring all rays from one point converge to one image point
• C. It assumes rays travel parallel to the optical axis, so only one angle needs to be tracked
• D. It ignores diffraction effects, simplifying the wave optics to pure geometry

Answer: (B)

The key is linearity. Full Snell's law is n₁ sin θ₁ = n₂ sin θ₂ — a transcendental equation. Under the paraxial approximation sin θ ≈ θ, it becomes n₁θ₁ = n₂θ₂, which is linear. Linear optics is powerful: superposition holds, rays from different points don't interfere with each other, and the entire system behavior is captured by a small number of parameters (focal length, object distance). Crucially, all paraxial rays from a single object point converge to a single image point — the condition for a well-formed image. The thin-lens equation 1/f = 1/d_o + 1/d_i holds exactly in this regime. Without linearity, you'd need to ray-trace every individual ray numerically.

Question 3

A spherical lens will focus all paraxial rays from a single object point to a single image point, but marginal (non-paraxial) rays from the same point will focus at a slightly different distance.

• T. True
• F. False

Answer: True

This is the definition and consequence of spherical aberration. Paraxial rays — those traveling close to the optical axis at small angles — satisfy the approximation sin θ ≈ θ, producing the clean convergence guaranteed by the linear paraxial theory. Marginal rays hit the lens at steeper angles where the approximation fails; they bend more strongly and focus closer to the lens. The result is a distribution of focal points along the axis rather than a single point — the 'circle of least confusion' in the image plane. Aspherical lenses fix this by deliberately departing from a spherical shape so that the exact refraction condition is satisfied for rays at all heights.

Question 4

The paraxial approximation holds as long as the wavelength of light is much smaller than the lens aperture — making it a wave optics condition rather than a geometric one.

• T. True
• F. False

Answer: False

The paraxial approximation is purely geometric and depends on the angle of rays to the optical axis, not on wavelength. It states that sin θ ≈ θ (equivalently tan θ ≈ θ), which holds when θ is small in radians — roughly θ < 0.2 rad for errors under 1%. The wavelength-vs-aperture condition you might be thinking of is the criterion for diffraction vs. geometric optics: geometric optics holds when λ is much smaller than aperture features. These are entirely separate conditions. A system can be in the geometric optics regime (small wavelength) but violate the paraxial approximation if marginal rays hit the lens at steep angles.

Question 5

Explain why the paraxial approximation produces a simple, linear relationship between object distance, image distance, and focal length — and describe what physically breaks down when the approximation fails.

Model answer: The approximation replaces sin θ with θ, linearizing Snell's law. This ensures that all paraxial rays from one object point converge to exactly one image point, and that the convergence depends linearly on object distance — yielding the thin-lens equation 1/f = 1/d_o + 1/d_i. When the approximation fails (marginal rays at steep angles), the exact sin θ refraction causes different rays to focus at different distances, destroying the single focal point and producing spherical aberration.

Linearity is the operative mathematical concept. Linear systems obey superposition, can be described by matrices (the ray-transfer matrix formalism builds on this directly), and guarantee that each object point maps to a unique image point. Once non-linear terms in the angle become significant — scaling as θ³ and higher — the one-to-one mapping from object point to image point breaks. Each annular zone of the lens focuses at a slightly different distance, smearing what should be a point into a circle. Aspherical surfaces and multi-element designs correct for these higher-order terms.