optical metrology 3rd ed - kjell

Optical Metrology
Third Edition
Optical Metrology. Kjell J. G
˚
asvik
Copyright
 2002 John Wiley & Sons, Ltd.
ISBN: 0-470-84300-4
Optical Metrology
Third Edition
Kjell J. G
˚
asvik
Spectra Vision AS, Trondheim, Norway
Copyright  2002 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
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Contents
Preface to the Third Edition xi
1Basics 1
1.1 Introduction 1
1.2 Wave Motion. The Electromagnetic Spectrum 1
1.3 The Plane Wave. Light Rays 3
1.4 Phase Difference 4
1.5 Complex Notation. Complex Amplitude 5
1.6 Oblique Incidence of A Plane Wave 5
1.7 The Spherical Wave 7
1.8 The Intensity 8
1.9 Geometrical Optics 8
1.10 The Simple Convex (Positive) Lens 10
1.11 A Plane-Wave Set-Up 11
2 Gaussian Optics 15
2.1 Introduction 15

2.2 Refraction at a Spherical Surface 15
2.2.1 Examples 19
2.3 The General Image-Forming System 19
2.4 The Image-Formation Process 21
2.5 Reflection at a Spherical Surface 23
2.6 Aspheric Lenses 25
2.7 Stops and Apertures 26
2.8 Lens Aberrations. Computer Lens Design 28
2.9 Imaging and The Lens Formula 29
2.10 Standard Optical Systems 30
2.10.1 Afocal Systems. The Telescope 30
2.10.2 The Simple Magnifier 32
2.10.3 The Microscope 34
vi CONTENTS
3 Interference 37
3.1 Introduction 37
3.2 General Description 37
3.3 Coherence 38
3.4 Interference between two Plane Waves 41
3.4.1 Laser Doppler Velocimetry (LDV) 45
3.5 Interference between other Waves 46
3.6 Interferometry 49
3.6.1 Wavefront Division 50
3.6.2 Amplitude Division 51
3.6.3 The Dual-Frequency Michelson Interferometer 54
3.6.4 Heterodyne (Homodyne) Detection 55
3.7 Spatial and Temporal Coherence 56
3.8 Optical Coherence Tomography 61
4 Diffraction 67
4.1 Introduction 67

4.2 Diffraction from a Single Slit 67
4.3 Diffraction from a Grating 70
4.3.1 The Grating Equation. Amplitude Transmittance 70
4.3.2 The Spatial Frequency S pectrum 73
4.4 Fourier Optics 75
4.5 Optical Filtering 76
4.5.1 Practical Filtering Set-Ups 78
4.6 Physical Optics Description
of Image Formation 81
4.6.1 The Coherent Transfer Function 83
4.6.2 The Incoherent Transfer Function 85
4.6.3 The Depth of Focus 88
4.7 The Phase-Modulated Sinusoidal Grating 89
5 Light Sources and Detectors 99
5.1 Introduction 99
5.2 Radiometry. Photometry 99
5.2.1 Lambertian Surface 102
5.2.2 Blackbody Radiator 103
5.2.3 Examples 105
5.3 Incoherent Light Sources 108
5.4 Coherent Light Sources 109
5.4.1 Stimulated Emission 109
5.4.2 Gas Lasers 112
5.4.3 Liquid Lasers 114
5.4.4 Semiconductor Diode Lasers. Light Emitting Diodes 114
5.4.5 Solid-State Lasers 117
5.4.6 Other Lasers 119
CONTENTS vii
5.4.7 Enhancements of Laser Operation 119
5.4.8 Applications 122

5.4.9 The Coherence Length of a Laser 123
5.5 Hologram Recording Media 125
5.5.1 Silver Halide Emulsions 125
5.5.2 Thermoplastic Film 126
5.5.3 Photopolymer Materials 127
5.6 Photoelectric Detectors 127
5.6.1 Photoconductors 128
5.6.2 Photodiodes 129
5.7 The CCD Camera 131
5.7.1 Operating Principles 131
5.7.2 Responsitivity 134
5.8 Sampling 135
5.8.1 Ideal Sampling 135
5.8.2 Non-Ideal Sampling 138
5.8.3 Aliasing 139
5.9 Signal Transfer 139
6 Holography 147
6.1 Introduction 147
6.2 The Holographic Process 147
6.3 An Alternative Description 150
6.4 Uncollimated Reference
and Reconstruction Waves 150
6.5 Diffraction Efficiency. The Phase
Hologram 153
6.6 Volume Holograms 154
6.7 Stability Requirements 156
6.8 Holographic I nterferometry 157
6.8.1 Double-Exposure I nterferometry 157
6.8.2 Real-Time Interferometry 157
6.8.3 Analysis of Interferograms 158

6.8.4 Localization of Interference Fringes 161
6.9 Holographic Vibration Analysis 165
6.10 Holographic I nterferometry
of Transparent Objects 168
7Moir
´
e M ethods. Triangulation 173
7.1 Introduction 173
7.2 Sinusoidal Gratings 173
7.3 Moir
´
e Between Two Angularly
Displaced Gratings 175
7.4 Measurement of In-Plane Deformation and Strains 175
7.4.1 Methods for Increasing the Sensitivity 177
viii CONTENTS
7.5 Measurement of Out-Of-Plane Deformations. Contouring 179
7.5.1 Shadow Moir
´
e 179
7.5.2 Projected Fringes 180
7.5.3 Vibration Analysis 186
7.5.4 Moir
´
e Technique by Means of Digital Image
Processing 188
7.6 Reflection Moir
´
e 189
7.7 Triangulation 190

8 Speckle Methods 193
8.1 Introduction 193
8.2 The Speckle Effect 193
8.3 Speckle Size 195
8.4 Speckle Photography 197
8.4.1 The Fourier Fringe Method 197
8.4.2 The Young Fringe Method 201
8.5 Speckle Correlation 203
8.6 Speckle-Shearing Interferometry 208
8.7 White-Light Speckle Photography 212
9 Photoelasticity and Polarized Light 217
9.1 Introduction 217
9.2 Polarized Light 217
9.3 Polarizing Filters 219
9.3.1 The Linear Polarizer 219
9.3.2 Retarders 221
9.4 Unpolarized Light 223
9.5 Reflection and Refraction
at an Interface 223
9.6 The Jones Matrix Formalism
of Polarized Light 227
9.7 Photoelasticity 230
9.7.1 Introduction 230
9.7.2 The Plane Polariscope 231
9.7.3 The Circular Polariscope 232
9.7.4 Detection of Isochromatics of Fractional
Order. Compensation 234
9.8 Holographic Photoelasticity 237
9.9 Three-Dimensional Photoelasticity 239
9.9.1 Introduction 239

9.9.2 The Frozen Stress Method 241
9.9.3 The Scattered Light Method 242
9.10 Ellipsometry 245
10 Digital Image Processing 249
10.1 Introduction 249
10.2 The Frame Grabber 249
CONTENTS ix
10.3 Digital Image Representation 251
10.4 Camera Calibration 251
10.4.1 Lens Distortion 252
10.4.2 Perspective Transformations 254
10.5 Image Processing 254
10.5.1 Contrast Stretching 255
10.5.2 Neighbourhood Operations. Convolution 256
10.5.3 Noise Suppression 257
10.5.4 Edge Detection 259
10.6 The Discrete Fourier Transform (DFT) and the FFT 262
11 Fringe Analysis 269
11.1 Introduction 269
11.2 Intensity-Based Analysis Methods 269
11.2.1 Introduction 269
11.2.2 Prior Knowledge 270
11.2.3 Fringe Tracking and Thinning 270
11.2.4 Fringe Location by Sub-Pixel Accuracy 273
11.3 Phase-Measurement Interferometry 276
11.3.1 Introduction 276
11.3.2 Principles of TPMI 276
11.3.3 Means of Phase Modulation 279
11.3.4 Different Techniques 279
11.3.5 Errors in TPMI Measurements 281

11.4 Spatial Phase-Measurement Methods 282
11.4.1 Multichannel Interferometer 282
11.4.2 Errors in Multichannel Interferometers 285
11.4.3 Spatial-Carrier Phase-Measurement Method 285
11.4.4 Errors in the Fourier Transform Method 287
11.4.5 Space Domain 289
11.5 Phase Unwrapping 290
11.5.1 Introduction 290
11.5.2 Phase-Unwrapping Techniques 292
11.5.3 Path-Dependent Methods 292
11.5.4 Path-Independent Methods 293
11.5.5 Temporal Phase Unwrapping 295
12 Computerized Optical Processes 297
12.1 Introduction 297
12.2 TV Holography (ESPI) 298
12.3 Digital Holography 301
12.4 Digital Speckle Photography 305
13 Fibre Optics in Metrology 307
13.1 Introduction 307
13.2 Light Propagation through Optical Fibres 307
13.3 Attenuation and Dispersion 310
x CONTENTS
13.4 Different Types of Fibres 313
13.5 Fibre-Optic Sensors 315
13.6 Fibre-Bragg Sensors 318
Appendices
A. Complex Numbers 325
B. Fourier Optics 327
B.1 The Fourier Transform 327
B.2 Some Functions and Their Transforms 329

B.3 Some Implications 332
C. Fourier Series 335
D. The Least-Squares Error Method 339
E. Semiconductor Devices 343
References and Further Reading 347
Index 355
Preface to the Third Edition
This edition of Optical Metrology contains a new chapter about computerized optical
processes, including digital holography and digital speckle photography. Chapter 2, on
Gaussian optics, and Chapter 5, on light sources and detectors, are greatly expanded
to include descriptions of standard imaging systems, light-emitting diodes and solid-state
detectors. Separate new sections on optical coherence tomography, speckle correlation, the
Fast Fourier Transform, temporal phase unwrapping and fibre Bragg sensors are included.
Finally, a new appendix about Fourier series is given. Solutions to the end-of-chapter
problems can be found at />Since the previous edition, the electronic camera has taken over more and more as the
recording medium. The word ‘digital’ is becoming a prefix to an increasing number of
techniques. I think this new edition reflects this trend.
It gives me great pleasure to acknowledge the many stimulating discussions with Pro-
fessor H.M. Pedersen at The Norwegian University of Science a nd Technology. Thanks
also to John Petter G
˚
asvik for designing many of the new figures.
1
Basics
1.1 INTRODUCTION
Before entering into the different techniques of optical metrology some basic terms and
definitions have to be established. Optical metrology is about light and therefore we must
develop a mathematical description of waves and wave propagation, introducing important
terms like wavelength, phase, phase fronts, rays, etc. The treatment is kept as simple as
possible, without going into complicated electromagnetic theory.

1.2 WAVE MOTION. THE ELECTROMAGNETIC
SPECTRUM
Figure 1.1 shows a snapshot of a harmonic wave that propagates in the z-direction. The
disturbance ψ(z,t) is given as
ψ(z,t) = U cos

2π

z
λ
− νt

+ δ

(1.1)
The argument of the cosine function is termed the phase and δ the phase constant. Other
parameters involved are
U = the amplitude
λ = the wavelength
ν = the frequency (the number of waves per unit time)
k = 2π/λ the wave number
The relation between the frequency and the wavelength is given by
λν = v(1.2)
where
v = the wave velocity
ψ(z,t) might represent the field in an electromagnetic wave for which we have
v = c = 3 × 10
8
m/s
Optical Metrology. Kjell J. G

˚
asvik
Copyright
 2002 John Wiley & Sons, Ltd.
ISBN: 0-470-84300-4
2 BASICS
z
y(
z
,
t
)
dl/2p
l
U
Figure 1.1 Harmonic wave
Table 1.1 The electromagnetic spectrum (From Young (1968))
The ratio of the speed c of an electromagnetic wave in vacuum to the speed v in a medium
is known a s the absolute index of refraction n of that medium
n =
c
v
(1.3)
The electromagnetic spectrum is given in Table 1.1.
THE PLANE WAVE. LIGHT RAYS 3
Although it does not really affect our argument, we shall mainly be concerned with
visible light where
λ = 400–700 nm (1 nm = 10
−9
m)

ν = (4.3–7.5) × 10
14
Hz
1.3 THE PLANE WAVE. LIGHT RAYS
Electromagnetic waves are not two dimensional as in Figure 1.1, but rather three-dimen-
sional waves. The simplest example of such waves is given in Figure 1.2 where a plane
wave that propagates in the direction of the k-vector is sketched. Points of equal phase
lie on parallel planes that are perpendicular to the propagation direction. Such planes are
called phase planes or phase fronts. In the figure, only some of the infinite number of
phase planes are drawn. Ideally, they should also have infinite extent.
Equation (1.1) describes a plane wave that propagates in the z-direction. (z = constant
gives equal phase for all x, y, i.e. planes that are normal to the z-direction.) In the general
case where a plane wave propagates in the direction of a unit vector n, the expression
describing the field at an arbitrary point with radius vector r = (x,y,z) is given by
ψ(x, y,z, t) = U cos[kn · r − 2πνt + δ] (1.4)
That the scalar product fulfilling the condition n · r = constant describes a plane which
is perpendicular to n is shown in the two-dimensional case in F igure 1.3. That this is
correct also in the three-dimensional case is easily proved.
0
y(
r
)
+
U
−
U
l
y = 0
y = 0
y = 0

y =
U
y = −
U
y =
U
k
k
Figure 1.2 The plane wave
4 BASICS
y
r
q
n
x
n
.
r = r cos q = const
Figure 1.3
Wavefront
Rays
Figure 1.4
Next we give the definition of light rays. They are directed lines that are everywhere
perpendicular to the phase planes. This is illustrated in Figure 1.4 where the cross-section
of a rather complicated wavefront is sketched and where some of the light rays perpen-
dicular to the wavefront are drawn.
1.4 PHASE DIFFERENCE
Let us for a moment turn back to the plane wave described by Equation (1.1). At two
points z
1

and z
2
along the propagation direction, the phases are φ
1
= kz
1
− 2πνt + δ and
φ
2
= kz
2
− 2πνt + δ respectively, and the phase difference
φ = φ
1
− φ
2
= k(z
1
− z
2
)(1.5)
Hence, we see that the phase difference between two points along the propagation direction
of a plane wave is equal to the geometrical path-length difference multiplied by the wave
number. This is generally true for any light ray. When the light passes a medium different
from air (vacuum), we have to multiply by the refractive index n of the medium, such that
optical path length = n × (geometrical path length)
phase difference = k × (optical path length)
OBLIQUE INCIDENCE OF A PLANE WAVE 5
1.5 COMPLEX NOTATION. COMPLEX AMPLITUDE
The e xpression in Equation (1.4) can be written in complex form as

ψ(x, y,z, t) = Re{Ue
i(φ−2πvt)
} (1.6a)
where
φ = kn · r + δ(1.6b)
is the spatial dependent phase. In Appendix A, some simple arithmetic rules for complex
numbers are given.
In the description of wave phenomena, the notation of Equation (1.6) is commonly
adopted and ‘Re’ is omitted because it is silently understood that the field is described
by the real part.
One advantage of such complex representation of the field is that the spatial and
temporal parts factorize:
ψ(x, y,z, t) = Ue
i(φ−2πνt)
= Ue
iφ
e
−i2πvt
(1.7)
In optical metrology (and in other branches of optics) one is most often interested in
the spatial distribution of the field. Since the temporal-dependent part is known for each
frequency component, we therefore can omit the factor e
−i2πvt
and only consider the
spatial complex amplitude
u = Ue
iφ
(1.8)
This expression describes not only a plane wave, but a general three-dimensional wave
where both the amplitude U and the phase φ may be functions of x, y and z.

Figure 1.5(a, b) shows examples of a cylindrical wave and a spherical wave, while in
Figure 1.5(c) a more complicated wavefront resulting from reflection from a rough surface
is sketched. Note that far away from the point source in Figure 1.5(b), the spherical
wave is nearly a plane wave over a small area. A point source at infinity, represents a
plane wave.
1.6 OBLIQUE INCIDENCE OF A PLANE WAVE
In optics, one is often interested in the amplitude and phase distribution of a wave over
fixed planes in space. Let us consider the simple case sketched in Figure 1.6 where a
plane wave falls obliquely on to a plane parallel to the xy-plane a distance z from it. The
wave propagates along the unit vector n which is lying in the xz-plane (defined as the
plane of incidence) and makes an angle θ to the z-axis. The components of the n-and
r-vectors are therefore
n = (sin θ,0, cos θ)
r = (x, y,z)
6 BASICS
(a)
(b)
(c)
Figure 1.5 ((a) and (b) from Hecht & Zajac (1974), Figures 2.16 and 2.17. Reprinted with
permission.)
y
z
n
q
x
Figure 1.6
THE SPHERICAL WAVE 7
These expressions put into Equation (1.6) (Re and temporal part omitted) give
u = Ue
ik(x sin θ +z cos θ)

(1.9a)
For z = 0(thexy-plane) this reduces to
u = Ue
ikx sin θ
(1.9b)
1.7 THE SPHERICAL WAVE
A spherical wave, illustrated in Figure 1.5(b), is a wave emitted by a point source. It
should be easily realized that the complex amplitude representing a spherical wave must
be of the form
u =
U
r
e
ikr
(1.10)
where r is the radial distance from the point source. We see that the phase of this wave is
constant for r = constant, i.e. the phase fronts are spheres centred at the point source. The
r in the denominator of Equation (1.10) expresses the fact that the a mplitude decreases
as the inverse of the distance from the point source.
Consider Figure 1.7 where a point source is lying in the x
0
, y
0
-plane at a point of
coordinates x
0
, y
0
. The field amplitude in a plane parallel to the x
0

y
0
-plane at a distance
z then will be given by Equation (1.10) with
r =

z
2
+ (x − x
0
)
2
+ (y − y
0
)
2
(1.11)
where x, y are the coordinates of the illuminated plane. This expression is, however, rather
cumbersome to work with. One therefore usually makes some approximations, the first
of which is to replace z for r in the denominator of Equation (1.10). This approximation
cannot be put into the exponent since the resulting error is multiplied by the very large
z
x
0
x
(
x
0
,
y

0
)
(
x
,
y
)
y
0
y
z
Figure 1.7
8 BASICS
number k. A convenient means for approximation of the phase is offered by a binomial
expansion of the square root, viz.
r = z

1 +

x − x
0
z

2
+

y − y
0
z


2
≈ z

1 +
1
2

x − x
0
z

2
+
1
2

y − y
0
z

2

(1.12)
where r is approximated by the two first terms of the expansion.
The complex field amplitude in the xy-plane resulting from a point source at x
0
, y
0
in
the x

0
y
0
-plane is therefore given by
u(x,y, z) =
U
z
e
ikz
e
i(k/2z)[(x−x
0
)
2
+(y−y
0
)
2
]
(1.13)
The approximations leading to this expression are called the Fresnel approximations. We
shall here not discuss the detailed conditi ons for its validity, but it is clear that (x − x
0
)
and (y − y
0
) must be much less than the distance z.
1.8 THE INTENSITY
With regard to the registration of light, we are faced with the fact that media for direct
recording of the field amplitude do not exist. The most common detectors (like the eye,

photodiodes, multiplication tubes, photographic film, etc.) register the irradiance (i.e. effect
per unit area) which is proportional to the field amplitude absolutely squared:
I =|u|
2
= U
2
(1.14)
This important quantity will hereafter be called the intensity.
We mention that the correct relation between U
2
and the irradiance is given by
I =
εv
2
U
2
(1.15)
where v is the wave velocity and ε is known as the electric permittivity of the medium.
In this book, we will need this relation only when calculating the transmittance a t an
interface (see Section 9.5).
1.9 GEOMETRICAL OPTICS
For completeness, we refer to the three laws of geometrical optics:
(1) Rectilinear propagation in a uniform, homogeneous medium.
(2) Reflection. On reflection from a mirror, the angle of reflection is equal to the angle of
incidence (see Figure 1.8). In this context we mention that on reflection (scattering)
from a rough surface (roughness >λ) the light will be scattered in all directions (see
Figure 1.9).
GEOMETRICAL OPTICS 9
qq
Figure 1.8 The law of reflection

Figure 1.9 Scattering from a rough surface
(3) Refraction. When light propagates from a medium of refractive index n
1
into a
medium of refractive index n
2
, the propagation direction changes according to
n
1
sin θ
1
= n
2
sin θ
2
(1.16)
where θ
1
is the angle of incidence and θ
2
is the angle of emergence (see Figure 1.10).
From Equation (1.16) we see that when n
1
>n
2
, we can have θ
2
= π/2. This occurs
for an angle of incidence called the critical angle given by
sin θ

1
=
n
2
n
1
(1.17)
This is called total internal reflection and will be treated in more detail in Section 9.5.
Finally, we also mention that for light reflected at the interface in Figure 1.10,
when n
1
<n
2
, the phase is changed by π.
q
1
q
2
n
1
n
2
Figure 1.10 The law of refraction
10 BASICS
1.10 THE SIMPLE CONVEX (POSITIVE) LENS
We shall here not go into the general theory of lenses, but just mention some of the more
important properties of a simple, convex, ideal lens. For more details, see Chapter 2 and
Section 4.6.
Figure 1.11 illustrates the imaging property of the lens. From an object point P
o

, light
rays are emitted in all directions. That this point is imaged means that all rays from P
o
which pass the lens aperture D intersect at an image point P
i
.
To find P
i
, it is sufficient to trace just two of these rays. Figure 1.12 shows three of
them. The distance b from the lens to the image plane is given by the lens formula
1
a
+
1
b
=
1
f
(1.18)
and the transversal magnification
m =
h
i
h
o
=
b
a
(1.19)
In Figure 1.13(a), the case of a point source lying on the optical axis forming a spherical

diverging wave that is converted to a converging wave and focuses onto a point on the
optical axis is illustrated. In Figure 1.13(b) the point source is lying on-axis at a distance
P
o
ab
ff
P
i
D
Figure 1.11
h
o
h
i
Figure 1.12
A PLANE-WAVE SET-UP 11
(a)
(b)
(c)
h
q
Figure 1.13
from the lens equal to the focal length f . We then get a plane wave that propagates along
the optical axis. In Figure 1.13(c) the point source is displaced along the focal plane a
distance h from the optical axis. We then get a plane wave propagating in a direction that
makes an angle θ to the optical axis where
tan θ = h/f (1.20)
1.11 A PLANE-WAVE SET-UP
Finally, we refer to Figure 1.14 which shows a commonly applied set-up to form a
uniform, expanded plane wave from a laser beam. The laser beam is a plane wave with

a small cross-section, typically 1 mm. To increase the cross-section, the beam is first
directed through lens L
1
, usually a microscope objective which is a lens of very short
focal length f
1
.AlensL
2
of greater diameter and longer focal length f
2
is placed as
shown in the figure. In the focal point of L
1
a small opening (a pinhole) of diameter
typically 10 µm is placed. In that way, light which does not fall at the focal point is
blocked. Such stray light is due to dust and impurities crossed by the laser beam on its
L
1
f
1
L
2
f
2
Figure 1.14 A plane wave set-up
12 BASICS
way via other optical elements (like mirrors, beamsplitters, etc.) and it causes the beam
not to be a perfect plane wave.
PROBLEMS
1.1 How many ‘yellow’ light waves (λ = 550 nm) will fit into a distance in space equal

to the thickness of a piece of paper (0.1 mm)? How far will the same number of
microwaves (ν = 10
10
Hz, i.e 10 GHz, and v = 3 × 10
8
m/s) extend?
1.2 Using the wave functions
ψ
1
= 4sin2π(0.2z − 3t)
ψ
2
=
sin(7z + 3.5t)
2.5
determine in each case (a) the frequency, (b) wavelength, (c) period, (d) amplitude,
(e) phase velocity and (f) direction of motion. Time is in seconds and z in metres.
1.3 Consider the plane electromagnetic wave (in SI units) given by the expressions
U
x
= 0, U
y
= exp i[2π × 10
14
(t − x/c) + π/2], and U
z
= 0.
What is the frequency, wavelength, direction of propagation, amplitude and phase
constant of the wave?
1.4 A plane, harmonic light wave has an electric field given by

U
z
= U
0
exp i

π10
15

t −
x
0.65c

while travelling in a piece of glass. Find
(a) the frequency of the light,
(b) its wavelength,
(c) the index of refraction of the glass.
1.5 Imagine that we have a non-absorbing glass plate of index n and thickness z which
stands between a source and an observer.
(a) If the unobstructed wave (without the plate present) is U
u
= U
0
exp iω(t − z/c),
(ω = 2πν) show that with the plate in place the observer sees a wave
U
p
= U
0
exp iω


t −
(n − 1)z
c
−
z
c

(b) Show that if either n ≈ 1orz is very small, then
U
p
= U
u
+
ω(n − 1)z
c
U
u
e
−iπ/2
The second term on the right may be interpreted as the fi eld arising from the oscil-
lating molecules in the glass plate.
PROBLEMS 13
1.6 Show that the optical path, defined as the sum of the products of the various indices
times the thicknesses of media traversed by a beam, that is,

i
n
i
x

i
, is equivalent
to the length of the path in vacuum which would take the same time for that beam
to travel.
1.7 Write down an equation describing a sinusoidal plane wave in three dimensions with
wavelength λ, velocity v, propagating in the following directions:
(a) +z-axis
(b) Along the line x = y, z = 0
(c) Perpendicular to the planes x + y + z = const.
1.8 Show that the rays from a point source S that are reflected by a plane mirror appear
to be coming from the image point S

. Locate S

.
1.9 Consider Figure P1.1. Calculate the deviation  produced by the plane parallel slab
as a function of n
1
, n
2
, t, θ .
1.10 The deviation angle δ gives the total deviation of a ray incident onto a prism, see
Figure P 1.2. It is given by δ = δ
1
+ δ
2
. Minimum deviation occurs when δ
1
= δ
2

.
(a) Show that in this case δ
m
, the value of δ, obeys the equation
n
2
n
1
=
sin
1
2
(α + δ
m
)
sin
1
2
α
(b) Find δ
m
for α = 60
◦
and n
2
/n
1
= 1.69.
1.11 (a) Starting with Snell’s law prove that the vector refraction equation has the form
n

2
k
2
− n
1
k
1
= (n
2
cos θ
2
− n
1
cos θ
1
)u
n
q
n
1
n
1
n
2
t
∆
Figure P1.1
14 BASICS
a
n

1
n
2
n
1
d
2
d
1
d
Figure P1.2
where k
1
, k
2
are unit propagation vectors and u
n
is the surface normal pointing
from the incident to the transmitting medium.
(b) In the same way, derive a vector expression equivalent to the law of reflection.
2
Gaussian Optics
2.1 INTRODUCTION
Lenses are an important part of most optical systems. Good results in optical measure-
ments often rely on the best selection of lenses. In this chapter we develop the relations
governing the passage of light rays through imaging elements on the basis of the paraxial
approximation using matrix algebra. We also mention the aberrations occurring when rays
deviate from this ideal Gaussian behaviour. Finally we go through some of the standard
imaging systems.
2.2 REFRACTION AT A SPHERICAL SURFACE

Consider Figure 2.1 where we have a sphere of radius R centred at C and with refractive
index n

. The sphere is surrounded by a medium of refractive index n. A light ray making
an angle α with the z-axis is incident on the sphere at a point A at height x above the
z-axis. The ray is incident on a plane which is normal to the radius R and the angle of
incidence θ is the angle between the ray and the radius from C. The angle of refraction
is θ

and the refracted ray is making an angle α

with the z-axis. By introducing the
auxiliary angle φ we have the following relations:
φ = θ

− α

(2.1a)
φ = θ − α (2.1b)
sin φ =
x
R
(2.1c)
n sin θ = n

sin θ

(2.1d)
The last equation follows from Snell’s law of refraction. By assuming the angles to be
small we have sin φ ≈ φ,sinθ ≈ θ,sinθ


≈ θ

and by combining Equations (2.1) we get
the relation
α

=
n − n

n

R
x +
n
n

α =−
P
n

x +
n
n

α(2.2)
Optical Metrology. Kjell J. G
˚
asvik
Copyright

 2002 John Wiley & Sons, Ltd.
ISBN: 0-470-84300-4