Kvant Physics Problem 193
Two one-dimensional paraxial optical systems are considered, both composed of thin lenses with identical focal length magnitude $f$ measured in meters.
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Problem
There are two systems of lenses with equal focal lengths. The optical axes of the lenses coincide (Fig. 1). The first system consists of converging lenses; in the second system, converging lenses alternate with diverging lenses. Find the trajectories of the rays in each system if the distance between the lenses is much smaller than the focal length.
Figure 1
Setup and Assumptions
Two one-dimensional paraxial optical systems are considered, both composed of thin lenses with identical focal length magnitude $f$ measured in meters. The optical axes of all lenses coincide, and ray propagation is described in the paraxial approximation by small angles $\theta$ with respect to the axis and transverse displacement $x$ from it.
In the first system all lenses are converging with focal length $f>0$, separated by a distance $d$ satisfying $d \ll f$. In the second system converging lenses with focal length $+f$ alternate with diverging lenses with focal length $-f$, again separated by the same distance $d \ll f$.
Each lens is treated as infinitely thin, so refraction occurs instantaneously at the lens plane. Free-space propagation between lenses is treated as uniform rectilinear motion. Only first-order terms in $d/f$ are retained, consistent with the paraxial and weak-element limits.
The unknown is the functional form of ray trajectories $x(z)$ along the optical axis coordinate $z$ for both systems.
Physical Principles
Paraxial ray propagation in free space obeys the linear relations
$$\frac{dx}{dz} = \theta,$$
where $\theta$ is the ray angle.
A thin lens of focal length $f$ produces an instantaneous change in slope given by
$$\Delta \theta = -\frac{x}{f},$$
with the sign convention that $f>0$ corresponds to a converging lens and $f<0$ to a diverging lens.
Over a small propagation step $d$, the change in angle per unit length can be written in the continuum limit as
$$\frac{d\theta}{dz} = \lim_{d \to 0} \frac{\Delta \theta}{d}.$$
Combining these relations yields a second-order differential equation for the ray trajectory,
$$\frac{d^2 x}{dz^2} = \frac{d\theta}{dz}.$$
Derivation
In the first system every interval of length $d$ contains one converging lens. Over such an interval the slope change is
$$\Delta \theta = -\frac{x}{f}.$$
Dividing by the period $d$ gives the effective rate of change of angle along the axis,
$$\frac{d\theta}{dz} = -\frac{x}{f d}.$$
Using $\frac{dx}{dz}=\theta$, differentiation with respect to $z$ yields
$$\frac{d^2 x}{dz^2} = -\frac{x}{f d}.$$
This equation has the form of a linear harmonic oscillator with angular spatial frequency squared $\omega^2 = \frac{1}{f d}$.
In the second system each period consists of a converging lens followed by a diverging lens, each separated by distance $d$ of the same order and both operating in the paraxial regime. The converging lens produces
$$\Delta \theta_1 = -\frac{x}{f},$$
while the diverging lens produces
$$\Delta \theta_2 = -\frac{x}{-f} = +\frac{x}{f}.$$
The total change in angle over one full pair is
$$\Delta \theta_{\text{pair}} = -\frac{x}{f} + \frac{x}{f} = 0.$$
The net angular change per unit length vanishes in the same continuum approximation,
$$\frac{d\theta}{dz} = 0,$$
so the trajectory equation becomes
$$\frac{d^2 x}{dz^2} = 0.$$
Result
For the first system the ray trajectory satisfies
$$\frac{d^2 x}{dz^2} + \frac{1}{f d} x = 0,$$
with general solution
$$x(z) = A \cos!\left(\frac{z}{\sqrt{f d}}\right) + B \sin!\left(\frac{z}{\sqrt{f d}}\right).$$
Here $f$ and $d$ are measured in meters, so $\sqrt{f d}$ has units of meters and the argument of the trigonometric functions is dimensionless.
For the second system the trajectory satisfies
$$\frac{d^2 x}{dz^2} = 0,$$
with general solution
$$x(z) = A z + B,$$
where $A$ is a dimensionless angle and $B$ has units of meters.
Sanity Checks
The equation for the first system contains the factor $\frac{1}{f d}$ with units $\mathrm{m^{-2}}$, consistent with the second derivative $\frac{d^2 x}{dz^2}$ which also has units $\mathrm{m^{-1}}$ per derivative of $z$, giving $\mathrm{m^{-2}}$ overall.
In the limit $d \to \infty$ with fixed $f$, the focusing strength $\frac{1}{f d}$ tends to zero, reducing the first system to straight-line propagation, consistent with isolated lenses having negligible cumulative effect.
In the second system cancellation of lens powers relies on equal magnitudes $|f|$; any mismatch would introduce a residual term proportional to $\frac{1}{f_1} + \frac{1}{f_2}$, which would restore weak focusing. The most sensitive step to sign error is the diverging lens relation $\Delta \theta = -x/f$, where $f<0$ must be inserted before performing algebraic cancellation.