Problems

Problem 2.1 Principle of Fermat and Snell’s Law.

Consider a layer of thickness $d$ and refractive index $n_2$ which is sandwiched between two half spaces with refractive indices $n_1$ and $n_3$ as shown in Figure 1. A ray from point $P=(x_P,y_P)$ with $y_P>d$ passes through point $Q=(x_Q,y_Q)$ with $y_Q<0$.

(a) Write a formula for the OPL of the ray from $P$ to $Q$ as shown in Figure 1.

(b) Find the equations satisfied by $x_A$ and $x_B$ at a stationary OPL. Hint: set the partial derivatives of the OPL with respect to $x_A$ and $x_B$ equal to zero.

(c) Express the equations derived under b) in terms of $\sin \theta_1$, $\sin \theta_2$ and $\sin \theta_3$ and derive that Snell’s Law holds for the angles $\theta_1$ and $\theta_3$:

Note that the relationship between $\theta_3$ and $\theta_1$ is independent of the refractive index $n_2$ and the thickness $d$ of the middle layer.

(d) Does Snell’s Law Eq. (1) hold irrespective of the number of layers and their thicknesses in between the two half spaces? Explain your answer.

(e) Derive Eq. (1) by using the boundary conditions for the tangential components of the electromagnetic field, using Maxwell’s boundary conditions for electromagnetic fields.

Problem 2.2 Perfect focusing by an ellipsoid and a hyperboloid.

Suppose that two media have indices $n_1>n_2$ and the source $S$ is far to the left in medium $n_2$. We seek an interface that focuses the parallel incident rays at $F$ in medium $n_1$, as in Figure 2. A plane $DD'$ perpendicular to the incident beam is a constant-phase reference plane.

(a) If point $A$ is on the interface sought, derive that

where the constant is the same for all points $A$ on the interface.

(b) Show that by moving the plane $DD'$ parallel to itself we can achieve that for the new plane $DD'$

where $e=n_2/n_1<1$. This is the focus–directrix definition of an ellipse in cross section, or an ellipsoid of revolution.

(c) Suppose next that $n_2>n_1$. By the same phase-matching argument, show that the interface is a hyperboloid with $F$ as a focus.

(d) Use the previous results to describe a lens with refractive index $n_2>n_1$ and having hyperboloid surfaces which perfectly image two given points S and P in the ambient medium with refractive index $n_1$.

Problem 2.3 Perfect focusing by a parabolic mirror

Next we consider perfect focusing of parallel rays in air ($n=1$) by a mirror. Let there be a parallel bundle of rays in air ($n=1$) and suppose we want to focus all rays at point $F$.

(a) We draw a plane $\Sigma_1$ perpendicular to the rays as shown in Figure 4. The rays that hit $\Sigma_1$ have traversed the same optical path length. Draw a second surface $\Sigma_2$ parallel to $\Sigma_1$. Consider rays hitting the mirror at $A_1$ and $A_2$. Derive that

(b) Derive that

(c) Show that Eq. (4) is satisfied for points $A$ for which $|AF|=|AD|$, and conclude that the mirror is a paraboloid with $F$ as focus and $\Sigma_2$ as directrix.

Problem 2.4 Imaging of a virtual object.

A virtual object is at a distance $d_0$ behind a converging lens. The incoming rays would converge there if the lens were absent, as their dashed continuations in Figure 5 show. Construct the refracted rays for a lens of focal length $f$.

(a) Construct the image when $f=2$ cm, $d_0=4$ cm and the height of the object is 1 cm.

(b) Is the image real or virtual? Inverted or upright?

(c) Calculate the location of the image using the lens formula and compare it with your drawing. What is the magnification?

Problem 2.5 Suppose we have two thin lenses ${\cal L}_1$ and ${\cal L}_2$. Lens ${\cal L}_1$ is convergent (i.e. positive) with focal distance $f_1>0$ and lens ${\cal L}_2$ is divergent (i.e. negative) with focal distance $f_2<0$. The distance between the lenses is $d$. Let there be an object at distance $2f_1$ in front of lens ${\cal L}_1$, as shown in Figure 6.

(a) Derive the condition on the distance $d$ between the lenses such that the final image is real.

(b) Let $f_1=3~\text{cm}$ and $f_2=-2~\text{cm}$. What should be the distance $d$ such that the final image is real and the magnification is 2?

Problem 2.6 System matrix for imaging by a spherical surface.

Consider a spherical surface with radius of curvature $R$, with a medium of refractive index $n_1$ ($n_2$) to the left (right) of the surface).

(a) Derive the ray matrix between a plane to the left of the surface at a distance $d_1$ from the vertex and a plane to the right of the surface at a distance $d_2$ from the vertex. (As always, the rays are assumed to propagate from left to right.)

(b) Derive the conditions such that the plane at distance $d_2$ is the image of the plane at distance $d_1$. Express the formula in coordinates $s_o=-d_1$, $s_i=d_2$ with respect to the vertex as origin and show that the result is identical to formula Eq. (26)

(c) Assume that $n_1=1$, $n_2=2$ and $R= 2$ cm. Show by construction using the paraxial version of Snell’s Law that when the object is virtual with $s_o=-4$ cm, the coordinate of the image point is given by $s_i= 8$ cm, in agreement with the formula derived in b).

(d) Construct again the image when $s_o=-4$ cm but now for the case that $R=-2$ cm and verify that the coordinate $s_i$ of the image point agrees with the formula derived in b).

Problem 2.7 The purpose of this problem is to derive the Lensmaker’s formula for a thin lens by applying the imaging formula of a spherical surface twice. The image $P$ of a point $S$ as shown in Figure 7 is computed in two steps. First the intermediate image $P'$ of $S$ by the spherical surface with vertex $V_1$ is computed and then this intermediate image is imaged by the spherical surface with vertex $V_2$.

(a) Use Eq. (26) to deduce that when $S$ is imaged by the first spherical surface as if the second spherical surface were absent, the $z$-coordinate $s_{i1}$ of the image $P'$ with respect to the origin at $V_1$ satisfies:

where $s_{o1}$ is the $z$-coordinate of $S$ with respect to the origin at $V_1$.

(b) Show that with respect to the origin at $V_2$ the $z$-coordinate of $P'$ is

(c) Show that the $z$-coordinate $s_{i2}$ of P with respect to the origin at $V_2$ satisfies

(d) Add the results of a) and c) to derive

(e) Derive the Lensmaker’s formula for the thin lens (see the Ray Matrix chapter) by taking the limit $d\rightarrow 0$ in d).

Problem 2.8 System matrix for focusing.

Consider a ray transfer matrix

between two planes.

(a) Suppose that any ray that is parallel to the optical axis in the first plane goes through a point on the optical axis in the second plane. This means that the second plane is the image focal plane of the system. What does this imply for the elements of the transfer matrix?

(b) Suppose that the first plane is the object focal plane, so that any ray emitted by the point on the optical axis in the first plane becomes collimated in the second plane. What does this imply for the elements of the transfer matrix?

(c) Consider two thin lenses with distance $d$ and focal distances $f_{1i}$ and $f_{2i}$. Derive the transfer matrix linking the plane immediately before the first lens with the plane immediately behind the second lens. You may assume that the lenses are in air with refractive index $n=1$.

(d) Use the condition that you found in a) to derive the image focal distance of a system consisting of two thin lenses with image focal distances $f_{1i}$ and $f_{2i}$ and distance $d$. Verify that the result agrees with the distance for the image focal plane Eq. (45). Hint: let $f_i$ be the distance of the image focal point of the two-lens system to the second lens. Write the transfer matrix between the plane immediately before the first lens and the plane through the image focal point.

(e) Add a third thin lens with image focal distance $f_{3i}$ in contact with the second lens. Derive the ray matrix of this system.

(f) Let $f_i$ be the distance from the image focal plane to the third lens. Use the condition derived in a) and the ray matrix derived in e) to derive $f_i$.

Problem 2.9 Matrix for two thin lenses.

(a) Consider two thin lenses in a medium of refractive index $n$. Their powers are $\mathcal{P}_1$ and $\mathcal{P}_2$, and the distance between them is $d$. Use the ray vector $(n\alpha,y)^{T}$, with angle $\alpha$ first and height $y$ second. Derive that the matrix between the planes immediately to the left of the first lens and the plane immediately to the right of the second lens is given by

(b) Define the combined power $\mathcal{P}_{\mathrm{eff}}=\mathcal{P}_1+\mathcal{P}_2-(d/n)\mathcal{P}_1\mathcal{P}_2$. Show that the image focal coordinate measured from the second lens and the object focal coordinate measured from the first lens are:

(c) With $\mathcal{P}_j=n/f_{ji}$, verify that these formulae agree with Eq. (46) and Eq. (45).

Problem 2.10 Entrance pupil of a system of two lenses. Consider a system of two lenses $L_1$ and $L_2$ with distance $d$. The left lens $L_1$ has image focal distance $f_{1i}$ and $a_1$, $a_2$ are the radii of the lens apertures of $L_1$ and $L_2$.

(a) Let lens $L_1$ be convergent with $f_{1i}=2~\text{cm}$ and let the distance be $d=1~\text{cm}$. Furthermore, let $a_1=2~\text{cm}$ and $a_2=1~\text{cm}$. Determine by construction with a ruler the entrance pupil. Compute also its position and radius using the thin-lens imaging equation and derive the tangent of the angle that the marginal ray makes with the optical axis for an object on the optical axis at 4 cm to the left of $L_1$.

(b) Same question when $L_1$ is a divergent lens with $f_{1i}=-6~\text{cm}$, $d=3~\text{cm}$ and $a_1=a_2=1~\text{cm}$.

Problem 2.11 Diaphragm in a system of two thin lenses. Determine the entrance and exit pupils by ray construction, thin-lens imaging, and the matrix method. The situation is as shown in Figure 8. The focal distances of the two thin lenses are $f_1=10$ cm and $f_2=12$ cm and their distance $d = 6$ cm. Suppose that the aperture stop is as shown in Figure 8. It is at a distance of $1.5$ cm in front of the lens $L_1$ and has a diameter $D_a=5$ cm.

(a) Determine the position of the entrance pupil and its diameter $D_{e}$ by the mentioned three methods.

(b) Determine the position of the exit pupil and its diameter $D_{exit}$ by the mentioned three methods.

(c) Compute the $f$-number of the system.