Kvant Physics Problem 377
The system consists of a distant pointlike lamp emitting monochromatic light of wavelength $\lambda$ in the visible range, typically $\lambda \sim 5.
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Problem
If one looks while squinting at distant bright lamps, one usually sees vertical or slightly inclined columns of light extending downward and upward from the lamp. Explain this phenomenon. Devise and carry out experiments to test your explanation.
Hint. The surface of the cornea is never dry. It is always covered by a layer of tear fluid.
Setup and Assumptions
The system consists of a distant pointlike lamp emitting monochromatic light of wavelength $\lambda$ in the visible range, typically $\lambda \sim 5.5 \times 10^{-7},\text{m}$. The optical system includes the human eye, modeled by a circular aperture of diameter $D$ representing the pupil, and a thin tear film coating the corneal surface.
The observable quantity is the angular distribution of perceived intensity around the image of the lamp on the retina, specifically the appearance of vertically extended light streaks above and below the lamp.
The tear film is assumed to be a thin, weakly refracting layer whose thickness and surface profile contain small anisotropic irregularities. The pupil is assumed to change shape under squinting into a horizontally elongated slit of width $a \ll D$. Atmospheric effects are neglected, and the lamp is treated as a spatially incoherent point source at optical infinity.
Physical Principles
Light propagation through a finite aperture produces diffraction governed by the Huygens–Fresnel principle. For a single slit of width $a$, the angular intensity distribution is given by
$$I(\theta) = I_0 , \text{sinc}^2!\left(\frac{\pi a \sin\theta}{\lambda}\right),$$
where $\text{sinc}(x) = \frac{\sin x}{x}$.
Refraction through a weakly varying thin film follows Snell’s law in differential form, producing local angular deflections proportional to transverse gradients of optical path length.
Scattering from an ensemble of fine structures introduces angular redistribution of intensity, with characteristic directions determined by the geometry of those structures.
The observed retinal image corresponds to the angular intensity distribution integrated over the pupil function and modified by phase shifts introduced by the tear film.
Derivation
When the eye is not squinting, the pupil is approximately circular and diffraction effects are weak in any preferred direction. The angular resolution is then isotropic with characteristic scale $\Delta \theta \sim \lambda/D$.
During squinting, the eyelids constrain the pupil into a horizontal slit of effective height $a$ while maintaining a much larger horizontal extent. The aperture function becomes strongly anisotropic, and diffraction must be treated separately in the vertical and horizontal directions.
Let the slit be oriented horizontally, so that its narrow dimension is vertical with width $a$. The far-field diffraction pattern in the vertical angular coordinate $\theta_y$ follows
$$I(\theta_y) = I_0 , \text{sinc}^2!\left(\frac{\pi a \theta_y}{\lambda}\right),$$
where the small-angle approximation $\sin\theta_y \approx \theta_y$ has been used.
The first minima occur at
$$\frac{\pi a \theta_y}{\lambda} = \pi,$$
which gives the characteristic angular width
$$\theta_y \sim \frac{\lambda}{a}.$$
Since $a$ is reduced by squinting to values on the order of $a \sim 0.5,\text{mm} = 5 \times 10^{-4},\text{m}$, the angular spread becomes
$$\theta_y \sim \frac{5.5 \times 10^{-7},\text{m}}{5 \times 10^{-4},\text{m}} = 1.1 \times 10^{-3},\text{rad}.$$
At a typical viewing distance of a lamp $L \sim 10,\text{m}$, the apparent vertical smearing scale on the retina corresponds to a linear extent
$$h \sim L \theta_y = 10,\text{m} \cdot 1.1 \times 10^{-3},\text{rad} = 1.1 \times 10^{-2},\text{m} = 1.1,\text{cm}.$$
This produces a visible vertical column of light extending above and below the lamp image.
The tear film contributes an additional mechanism. Gravity-driven flow of the tear layer forms elongated microchannels and thickness gradients predominantly in the vertical direction on the corneal surface. Each channel acts as a weak cylindrical lens with its axis approximately horizontal, producing preferential vertical focusing and defocusing of rays. The combined effect of many such structures adds an extended vertical halo, reinforcing the diffraction-induced streak.
Result
The dominant angular extent of the vertical light column under squinting is
$$\theta_y \sim \frac{\lambda}{a}.$$
The corresponding linear extent at distance $L$ is
$$h \sim \frac{L\lambda}{a}.$$
Substituting $\lambda = 5.5 \times 10^{-7},\text{m}$, $a = 5 \times 10^{-4},\text{m}$, and $L = 10,\text{m}$ yields
$$h \sim \frac{10 \cdot 5.5 \times 10^{-7}}{5 \times 10^{-4}},\text{m} = 1.1 \times 10^{-2},\text{m}.$$
$$\boxed{h \approx 1.1,\text{cm}}$$
Sanity Checks
The expression $h \sim L\lambda/a$ has dimensions of length since $\lambda/a$ is dimensionless and multiplied by $L$, confirming dimensional consistency. Reducing the slit width $a$ increases $h$, matching the observation that stronger squinting enhances the vertical columns.
In the limit of no squinting, $a \to D$, the angular spread reduces to $\theta \sim \lambda/D$, restoring near-isotropic diffraction and eliminating pronounced vertical elongation. In the opposite limit of extreme squinting, $a \to 0$, the model predicts diverging angular spread, corresponding physically to loss of directional resolution.
An experimental test follows from varying $a$ by controlled squinting while observing a distant point source: the vertical length of the streak should scale inversely with the slit width. Closing one eyelid produces a stronger single horizontal slit and increases the vertical streak symmetry.
Applying artificial tears smooths the corneal surface, reducing refractive index gradients in the tear film and diminishing the cylindrical lens contribution. Drying the eye slightly or blinking changes the pattern dynamically, confirming the role of the tear layer.
Rotating the head while maintaining the same squint direction rotates the slit orientation, and the streaks rotate accordingly, confirming that the effect is tied to aperture geometry rather than external optical artifacts.