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Homework #1 Solutions for the Course of Photonics Engineering

1.

Verify that the peak irradiance I1of the first ringin the Airy pattern for far-field diffraction at a circular aperture

is such that I1/I(0)=0.0175. You must want to use the fact that

...2

1

!4!3

1

2

1

!3!2

1

2

1

!2!1

11

2

642

1 uuu

uuJ

Sol: The first ring (maximum) occurs for u=kaq/R=5.14.

Interpolating from the Bessel functions, J1(5.14)~-0.33954

Therefore,

0175.014.5

33954.022

0

22

1

u

uJ

II

2.

Plane waves impinge perpendicularly on a screen with a small circular hole in it. It is found that when viewedfrom some axial point P the hole uncovers 1/2 of the first half-period zone. What is the irradiance at P in termsof the irradiance there when the screen is removed?

Sol: The full first zone has a radiusaRq

2

22.11

Since area=q2, half the first zone corresponds toa

Rqq

22

22.1

2

1

22

0 AI A is for a plane wave.

0.01371016.2

4416.02

71016.2

71016.2

2

4

2

22.1

2

22.12

2

22

22.1

22

22.1

2

2

2

2

2

0

2

2

0

2

1

2

0

2

1

2

0

1

2

0

2

1

2

0

R

I

R

I

J

R

IJ

R

I

Ra

Rka

Ra

Rka

J

R

I

R

kaqR

kaqJ

R

II

3.

Find the spectrum of the frequency-modulated wave

qtBptAtf coscos when B is small.Sol: Hint:

00 ' tttftftf

Assume ptt 0

when the term of B2is neglected,

qtBptptqtBpt cossincoscoscos

Therefore, qtBptptAtf cossincos When

2

pp

and

2

qq

qpqpqpqpppiABA

F 42

4.

The function has the form g(t)cos(21t), where g(t)=a+bcos(2mt) is a real function. Find its spectrum.

Sol: dtebaG tim

2

12cos2cos

Since titititim eeandee mm 11 221222

12cos

2

12cos

mmmmba

G 111111

42