2. (40%) Calculate and sketch the Fourier transform F (u) of the function

f(x) = sinc�x

b

� �cos

�2�x

�1

�+ cos

�2�x

�2

��:

Assume that the following condition holds:

1

b�

1

�1

;1

�2

;

���� 1�1

�1

�2

���� :

3. (30%) A very large observation screen (e.g., a blank piece of paper) is placed in

the path of a monochromatic light beam (wavelength �). A sinusoidal interference

pattern of the form

I(x) = I0

�1 + cos

2�x

�

�

is observed on the screen, where I0 is a constant with units of optical intensity,

� is a constant with units of distance, and x is a distance coordinate measured

on the observation screen.

3.a) Describe quantitatively two alternative optical �elds that could have led to

the same measurement on the observation screen.

3.b) Describe an experimental procedure by which we can determine which one

of the two alternative �elds is illuminating the observation screen.

GOOD LUCK

2

Goodman, problem 5-9 solution

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

2.710 Optics Fall '01

Final Examination Wednesday, December 19, 2001: 9am{Noon

Instructions

a) Use the paraxial approximation in all problems.

b) Use one-dimensional derivations unless it is clearly necessary to do otherwise.

c) If you need to make assumptions beyond those given in the problems, state them

clearly.

d) Label your sketches thoroughly and consistently with your derivations.

e) Keep Occam's razor in mind: \All else being equal, the simplest solution must

be the correct one."

1

1. (45%) We intend to use a spherical ball lens of radius R and refractive index n

as magni�er in an imaging system, as shown in Figure A. The refractive index

satis�es the relationship 1 < n < 4=3, and the medium surrounding the ball lens

is air (refractive index = 1).

1.a) Calculate the e�ective focal length (EFL) of the ball lens. Use the thick

lens model with appropriate parameters.

1.b) Locate the back focal length (BFL), the front focal length (FFL) and the

principal planes of the ball lens.

1.c) An object located at distance d to the left of the back surface of the ball

lens, as shown in Figure A, where

d = R4� 3n

4(n� 1):

Show that the object is one half (EFL) behind the principal plane, and use

this fact to �nd the location of the image plane.

1.d) Is the image real or virtual? Is it erect or inverted? What is the magni�-

cation?

1.e) Locate the aperture stop and calculate the numerical aperture (NA) of the

optical system of Figure A.

1.f) Sketch how a human observer using the optical system of Figure A as input

to her eye would form the �nal image of the object on her retina.

d

R

object ball-lens

Figure A

2

2. (20%) Consider the 4F optical system shown in Figure B, where lenses L1, L2

are identical with focal length f . A thin transparency with arbitrary transmission

function t(x) is placed at the input plane of the system, and illuminated with a

monochromatic, coherent plane wave at wavelength �, incident on-axis. At the

Fourier plane of the system we place the amplitude �lter shown in Figure C.

The �lter is opaque everywhere except over two thin stripes of width a, located

symmetrically around the y00 axis. The distance between the stripe centers is

x0 > a.

2.a) Which range of spatial frequencies must t(x) contain for the system to

transmit any light to its image plane?

2.b) Write an expression for the �eld at the image plane as convolution of t(x)

with the coherent impulse response of this system.

f f f f

inputplane

opticalaxis

Fourierplane

imageplane

L1 L2

t(x)

x’"xx

Figure B

x0

a

a

"y

x"

Figure C

3

3. (35%) The point-spread function (PSF) of a one-dimensional (1D) optical system

in response to monochromatic, spatially incoherent illumination is

~h(x) = sinc4�ucx

0; (1)

where x0 is the output plane coordinate.

3.a) Sketch the modulation transfer function (MTF) of this optical system with

as much detail as you can (but do not derive an explicit expression).

3.b) Sketch the MTF and PSF of the corresponding di�raction-limited optical

system, i.e. a di�raction-limited optical system which has the same spatial

bandwidth as the system described by (1).

3.c) An amplitude grating with transmission function as shown in Figure D is

illuminated with monochromatic, uniform, spatially incoherent light. Using

the modulated light as input, derive the response of the di�raction-limited

optical system.

3.d) Using the results of the above questions, discuss why the di�raction-limited

system is superior to the system of (1) for the purpose of imaging.

x

t(x)

0

1

......

uc

1

c2u1

Figure D (x is the input plane coordinate)

GOOD LUCK

4

2.710 Final exam fall ’01Solutions

G. Barbastathis & D. Gil

There are (

(iii)

2.c) What is the intensity measured at the observation plane?

2.d) Comparing your answers (a) and (c), how is T2 helpful in imaging the phaseobject T1?

(Note: the symbols a, b are defined in Figure B.)

3. Figure 3 below shows the schematic diagram of a simple grating spectrometer. Itconsists of a sinusoidal amplitude grating of period Λ and lateral size (aperture)a followed by a lens of focal length f and sufficiently large aperture. To analyzethis spectrometer, we will assume that it is illuminated from the left in spatiallycoherent fashion by two plane waves on–axis. One of the plane waves is atwavelength λ and the other is at wavelength λ + ∆λ, where |∆λ| ¿ λ. (Thetwo plane waves at different wavelengths are mutually incoherent.) Since the twocolors are diffracted by the grating to slightly different angles, the goal of thissystem is to produce two adjacent but sufficiently well separated bright spots atthe output plane, one for each color.

’x

outputplane

x

f f

aΛ

grating lens� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �

� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �

Figure 3

3.a) Estimate the minimum aperture size of the lens so that it does not impairthe operation of the spectrometer.

3.b) What is the maximum power efficiency that this spectrometer can achieve?

3.c) Show that a condition for the two color spots to be “sufficiently well” sep-arated is

λ

|∆λ| <a

Λ.

This result is often stated in spectroscopy books as follows: The resolvingpower of a grating spectrometer, defined as the ratio of the mean wavelengthλ to the spectral resolution |∆λ|, equals the number of periods in the grating.

3

2.c) What is the intensity measured at the observation plane?

2.d) Comparing your answers (a) and (c), how is T2 helpful in imaging the phaseobject T1?

2.e) Consider the limit b →∞. How does then answer (c) change? Is the largeraperture helpful in this case?

2.f) If a = 0.5cm and b →∞, is your answer (d) still valid? If yes, why? If not,what has gone wrong?

(Note: the symbols a, b are defined in Figure B.)

3. An infinite periodic square-wave grating with transmittivity as shown in Figure3A is placed at the input of the optical system of Figure 3B. Both lenses are pos-itive, F/1, and have focal length f . The grating is illuminated with monochro-matic, spatially coherent light of wavelength λ and intensity I0. The spatialperiod of the grating is X = 4λ. The element at the Fourier plane of the systemis a nonlinear transparency with the intensity transmission function shown inFigure 3C, where the threshold and saturating intensities Ithr = Isat = 0.1I0. Tocalculate the response of this system analytically, we need to make the paraxialapproximation; strictly speaking, that is questionable for F/1 optics, but we willfollow it nevertheless. An additional necessary assumption is discussed in thefirst question below.

3.a) To answer the second question, we need to neglect the Airy patterns formingat the Fourier plane and pretend they are uniform bright dots. Explain whythis assumption is justified and what effects it might have.

3.b) Derive and plot the intensity distribution at the output plane using theabove assumption.

X/2 X/2

t(x)

x

1

0

Figure 3A

3

ff

nonlineartransparency

inputplane

outputplane

illumination

f f

Figure 3B

out

I

I

I

I

in

thr

sat

Figure 3C

4