9. Photographic bokeh

Official Problem Statement

Bokeh is an effect observed in photographs that show objects at different distances from the camera and have both sharp and out-of-focus areas. If a slit is placed in front of the lens of the camera, the image may have unusual anisotropic distortions in out-of-focus areas, while in-focus areas remain undistorted and sharp. Investigate how the properties and sharpness of resulting images depend on relevant parameters.

Geometric Optics — Circle of Confusion

When a point source lies outside the camera’s focal plane, the cone of light from the lens forms a Circle of Confusion (CoC) on the sensor instead of a point:

$$c = frac{f^2}{N,d_o}left|frac{d_o – d_text{focus}}{d_text{focus}}right|$$

Thin-lens law to convert object distance to image distance:

$$frac{1}{d_o} + frac{1}{d_i} = frac{1}{f}$$

Depth of Field

Depth of Field (DoF) is the range of object distances for which CoC stays below an acceptable threshold c_max (typically c_max ≈ d_sensor/1500 for digital cameras):

$$text{DoF} approx frac{2,N,c_text{max},d_o^2}{f^2}$$

Key relationship: smaller N (wider aperture) → shallower DoF → stronger bokeh.

Fourier Optics — Bokeh Shape

In Fraunhofer (far-field) diffraction, the intensity distribution at the focal plane is the squared magnitude of the 2D Fourier transform of the aperture function A(u,v). This is the Point Spread Function (PSF):

$$text{PSF}(x,y) = left|mathcal{F}{A(u,v)}right|^2$$

For a circular aperture of diameter D:

$$text{PSF}(r) = left[frac{2J_1(pi D r/lambda f)}{pi D r/lambda f}right]^2$$

where J₁ is the first-order Bessel function. First zero at r = 1.22λf/D (Rayleigh criterion).

Diffraction and Optimal Aperture

At very small apertures, diffraction dominates over geometric blur. The optimal f-number balancing geometric aberrations and diffraction is:

$$N_text{opt} approx sqrt{frac{f cdot c_text{max}}{1.22,lambda}}$$

For f = 50mm, c_max = 0.03mm, λ = 0.55μm: N_opt ≈ f/8.

Cat’s-Eye Bokeh

Off-axis point sources produce elliptical “cat’s-eye” bokeh due to mechanical vignetting: the lens barrel clips the effective aperture as seen from the edge of the sensor, making it elliptical rather than circular. This effect is desirable in cinematic lenses.

Experimental Setup

Constructing aperture filters

• Cut aperture shapes (circle, hexagon, heart, star) from black card stock.
• Attach to the lens front or hold in a filter holder.
• Photograph a background of point light sources (LED arrays, city lights at night).
• Keep focused object distance fixed; vary the background defocus distance.

Quantitative measurements

• Measure CoC diameter at various defocus distances; compare with formula.
• Measure DoF vs. f-number and verify the N·d_o²/f² scaling.
• Extract PSF by photographing a single LED from far away; compare FFT of bokeh image with aperture shape.

Core of the Problem — The Slit and Anisotropic Distortion

The problem statement specifically asks about a slit placed in front of the lens that causes “unusual anisotropic distortions.” This anisotropy follows directly from the 2D Fourier transform of the aperture function:

PSF width in each direction is inversely proportional to the aperture dimension in that direction:

$$Delta x_text{PSF} = frac{lambda f}{w}, qquad Delta y_text{PSF} = frac{lambda f}{H}$$

Since $H gg w$, we have $Delta x gg Delta y$ — the PSF is stretched horizontally (perpendicular to the slit). Out-of-focus points appear as elongated horizontal streaks on the sensor instead of circles.

Slit Aspect Ratio and Anisotropy

$$text{Bokeh stretch ratio} = frac{Delta x_text{PSF}}{Delta y_text{PSF}} = frac{H}{w}$$

A slit with $H/w = 10$ produces bokeh stretched 10× in the transverse direction. This linear relationship between slit geometry and bokeh distortion is directly testable.

Interactive Python Simulation — Aperture Shapes and PSF Patterns

This simulation computes the 2D Fourier transform of four aperture shapes and shows the corresponding bokeh (PSF) patterns:

References

1. Goodman, J.W. (2005). Introduction to Fourier Optics, 3rd ed. Roberts & Company.
2. Hecht, E. (2016). Optics, 5th ed. Pearson.
3. Born, M. & Wolf, E. (2013). Principles of Optics, 7th ed. Cambridge University Press.
4. Nasse, H.H. (2010). Depth of Field and Bokeh. Carl Zeiss Camera Lens Division Technical Note.
5. IYPT 2027 Problem 9. iypt.org

Technical Glossary