ISO/TR 14999-1:2005

TECHNICAL ISO/TR
REPORT 14999-1
First edition
2005-03-01
Optics and photonics — Interferometric
measurement of optical elements and
optical systems —
Part 1:
Terms, definitions and fundamental
relationships
Optique et photonique — Mesurage interférométrique de composants
et systèmes optiques —
Partie 1: Termes, définitions et relations fondamentales

Reference number
©
ISO 2005
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ii © ISO 2005 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Wave propagation and some topics on electromagnetic theory . 1
2.1 Parameters, symbols, units and constants, operators and computational procedures . 1
2.2 Maxwell's equations. 2
2.3 Electromagnetic fields in a medium. 3
2.4 Velocity of the wave. 3
2.5 Refractive index . 3
2.6 Scalar wave equation. 3
2.7 Amplitude, angular frequency, wavelength, wave number . 4
2.8 Complex notation, complex amplitude. 4
2.9 Irradiance . 5
2.10 Poynting vector . 5
2.11 Propagation of plane waves . 6
2.12 Propagation of spherical wave. 8
2.13 Propagation of waves with limited extent . 9
2.14 Propagation of aspherical waves . 10
3 General description of interference and different types of interferometers. 12
3.1 Interference between two waves . 12
3.2 Coherence. 14
3.3 Different arrangements of interference between two beams . 19
3.4 Characteristic features for interferometer structures . 25
4 Coupled ray-paths in interferometers. 33
4.1 Aperture stops and field stops; telecentric imaging. 33
4.2 Coupled ray-path. 34
4.3 Difference of coherent/incoherent optical imaging. 34
4.4 Principal layout of an interferometer . 35
4.5 Consequences of not properly imaging the test piece onto the detector . 39
5 Random and systematic error sources . 39
Annex A (informative) Visibility of fringes . 41
Bibliography . 42

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 14999-1 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee
SC 1, Fundamental standards.
ISO 14999 consists of the following parts, under the general title Optics and photonics — Interferometric
measurement of optical elements and optical systems:
 Part 1: Terms, definitions and fundamental relationships (Technical Report)
 Part 2: Measurement and evaluation techniques (Technical Report)
 Part 3: Calibration and validation of interferometric test equipment (Technical Report)
 Part 4: Interpretation and evaluation of tolerances specified by ISO 10110
iv © ISO 2005 – All rights reserved

Introduction
A series of International Standards on “Indications in technical drawings for the representation of optical
elements and optical systems” has been prepared by ISO/TC 172/SC 1, and published as ISO 10110 under
the title “Optics and photonics — Preparation of drawings for optical elements and systems”. When drafting
this series and especially its Part 5, Surface form tolerances, and Part 14, Wavefront deformation tolerance, it
became evident to the experts involved that additional complementary documentation is required to describe
how the necessary information on the conformance of the fabricated parts with the stated tolerances can be
demonstrated. Therefore, the responsible ISO Committee ISO/TC 172/SC 1 decided to prepare an
ISO Technical Report on Interferometric measurement of optical wavefronts and surface form of optical
elements.
When discussing the topics which had to be included into or excluded from such a Technical Report, it was
envisaged that it might be the first time, where an ISO Technical Report or Standard is prepared which deals
with wave-optics, i.e. which is based more in the field of physical optics than in the field of geometrical optics.
As a consequence, only fewer references than usual were available, which made the task more difficult.
Envisaging the situation, that the topic of interferometry has so far been left blank in ISO, it was the natural
wish to now be as comprehensive as possible. Therefore there was discussion, whether important techniques
such as interference microscopy (for characterizing the micro-roughness of optical parts), shearing
interferometry (e.g. for characterizing corrected optical systems), multiple beam interferometry, coherence
sensing techniques or phase conjugation techniques should be included or not. Other techniques, which are
related to the classical two beam interferometry, like holographic interferometry, Moiré techniques and
profilometry were also mentioned as well as Fourier transform spectroscopy or the polarization techniques,
which are mainly for microscopic interferometry.
In order to complement ISO 10110, the guideline adopted was to include what presently are common
techniques used for the purpose of characterizing the quality of optical parts. Decision was made to complete
a first Technical Report, and to then up-date it by supplementing new parts, as required. It is very likely that
more material will be added in the near future as more stringent tolerances (two orders of magnitude) for
optical parts and optical systems become mandatory when dealing with optics for the EUV range (wavelength
range 6 nm to 13 nm) for microlithography. Also, testing optics with EUV radiation (the same wavelength as
they are later used, e.g. at-wavelength testing) can be a new challenge, and is not covered by any current
standards.
This Technical Report should cover the need for qualifying optical parts and complete systems regarding the
wavefront error produced by them. Such errors have a distribution over the spatial frequency scale; in this
Technical Report only the low- and mid-frequency parts of this error-spectrum are covered, not the very high
end of the spectrum. These high-frequency errors can be measured only by microscopy, measurement of the
scattered light or by non-optical probing of the surface.
A similar statement can be made regarding the wavelength range of the radiation used for testing: ISO 14999
considers test methods with visible light as the typical case. In some cases, infrared radiation from CO -lasers
in the range of 10,6 µm is used for testing rough surfaces after grinding or ultraviolet radiation from excimer-
lasers in the range of 193 nm or 248 nm are used for at-wavelength testing of microlithography optics.
However, these are still rare cases, which are included in standards, that will not be dealt with in detail. The
wavelength range outside these borders is not covered.

TECHNICAL REPORT ISO/TR 14999-1:2005(E)

Optics and photonics — Interferometric measurement of optical
elements and optical systems —
Part 1:
Terms, definitions and fundamental relationships
1 Scope
This part of ISO/TR 14999 gives terms, definitions and fundamental physical and technical relationships for
interferometric measurements of optical wavefronts and surface form of optical elements.
It explains why some principles of the construction and use of interferometers are important due to the wave
nature of the wavefronts to be measured.
Since all wavefronts with the exception of very extended plane waves do alter their shape when propagating,
this part of ISO/TR 14999 also includes some basic information about wave propagation.
In practice, interferometric measurements can be done and are done by use of various configurations; this
part of ISO/TR 14999 outlines the basic configurations for two-beam interference.
The mathematical formulation of optical waves by the concept of the complex amplitude as well as the basic
equations of two-beam interference are established to explain the principles of deriving the phase information
out of the measured intensity distribution, either in time or in space.
Both random and systematic errors may affect the results of interferometric measurements and error types to
be clearly differentiated are therefore described in this part of ISO/TR 14999.
2 Wave propagation and some topics on electromagnetic theory
2.1 Parameters, symbols, units and constants, operators and computational procedures
Basic parameters, symbols, units and constants are given in Table 1.
Operators and computational procedures are given in Table 2.
Table 1 — Parameters, symbols, units and constants
Parameters Symbols Recommended unit, constant
Electric field vector E V/m
Magnetic field vector H A/m
2 2
Electric displacement or electric flux density D C/m = As/m
2 2
Magnetic induction or magnetic flux density B T = Wb/m = Vs/m
a
Dielectric constant or permittivity ε F/m = As/Vm
−12
Dielectric constant in the vacuum ε 8,854 × 10 F/m
Relative dielectric constant (relative permittivity) ε 1
r
b
Magnetic permeability µ H/m = Vs/Am
−6
Magnetic permeability in the vacuum µ 1,257 × 10 H/m
Relative magnetic permeability µ 1
r
Velocity of the wave in the medium c m/s
Velocity of the wave in the vacuum c 2,997 924 58 × 10 m/s
Absolute refractive index n 1
a
Mathematical relationship: ε = ε ε .
0 r
b
Mathematical relationship: µ = µ µ .
0 r
Table 2 — Operators and computational procedures
Operator Definition/Computational procedures Name (type)

∂ ∂∂
∇ ,,
 Nabla (vector)
∂∂xy∂z

22 2

∂∂Ψ ΨΨ∂
≡++
22 2

∆Ψ ∂∂xy ∂z Laplacian (scalar)

∇⋅∇ΨΨ= ∇
2.2 Maxwell's equations
Maxwell’s equations are the fundamentals for the electromagnetic wave propagation. Maxwell's equations for
an electromagnetic wave propagating in a medium which does not involve any charge or current and has
vanishing conductivity are expressed by:
∂B

∇× E + = 0

∂t

∂D

∇× H − = 0
(1)

∂t

∇⋅ D = 0 

∇⋅ B= 0

2 © ISO 2005 – All rights reserved

The mathematical relation between D and E as well as between B and H is given by:
D = ε E

(2)

B = µ H

in a linear medium.
2.3 Electromagnetic fields in a medium
For media in which the dielectric constant ε and magnetic permeability µ are uniform, Equation (1) gives the
following wave equations:

E
∂
E − εµ = 0
∇ 
∂t 
(3)

H
∂

H − εµ = 0
∇

∂
t 
2.4 Velocity of the wave
The velocity in an optically homogeneous and isotropic medium is given by:
c
c = (4)
εµ
Analogously, in a vacuum the velocity is given by:
c = (5)
ε µ
2.5 Refractive index
The ratio of the propagation velocities in vacuum and in the medium with ε and µ
c
n =  (6)
c
is called the refractive index of the medium or the absolute refractive index.
2.6 Scalar wave equation
As mentioned before, E and H are vectors. In many applications one deals with linearly polarized light, which
can be fully described by one vector component. Equation (3) then reduces to the scalar wave equation. In the
general form, the scalar wave equation may be written as:
1 ψ
∂
ψ − = 0 (7)
∇
∂
ct
Equation (7) is in conformity with a second-order differential equation. ψ is called the light disturbance.
The basic problem of light propagation is thus simply the determination of the manner in which a wave
propagates from one surface to another.
2.7 Amplitude, angular frequency, wavelength, wave number
We suppose a sinusoidal plane electromagnetic wave propagating in the z direction. The light disturbance ψ is
determined as a function of position z and time t
z
ψ()z,t = U cos ωδ (t − ) + (8)

c

where
U is the amplitude;
ω is the angular frequency;
δ is the phase constant of the wave.
The angular frequency ω is defined as 2πν , where ν is the frequency, i.e. the number of waves per unit time.
The wavelength λ is given from Equation (9):
2π v 2πc
λ = = (9)
ω n ω
The wave number k is defined as
2π
k = (10)
λ
In many applications of the concept of waves, as in diffraction or in interferometry, Equation (8) is used to
define “wavefronts”. In this case for a given position z the phase constant δ is a function of the lateral spatial
coordinates, e.g. δ = δ (x,y). This concept is useful, if δ = δ (x,y) is measured, as in the case of interferometric
measurements. Here δ = δ (x,y) is the phase-difference of two interfering waves and x and y are the
coordinates of the detector.
Sometimes it is more convenient to look for a “surface” in space z = z (x,y), where the value for the phase
δ = δ [x,y,z(x,y)] remains constant. Such a surface defines the “shape” of a wavefront, which can be spherical,
plane or aspherical, only to mention the most simple cases. This concept is used in 2.11 to 2.13 for discussing
the question of the propagation of waves. Here it is shown, that the “shape” of the wavefront changes with z,
with the only exception of a(n) (infinite) plane wave with constant amplitude.
In a more general case, also the amplitude U of the light stimulation might be a function of the lateral
coordinates x,y, e.g. U = U(x,y). If U is not constant with x,y, it is referred to as an “inhomogeneous wave”. In
practice, the variation of U is of minor importance, where the variation of δ is the quantity to be measured.
2.8 Complex notation, complex amplitude
The expression in Equation (8) can be written in complex form as
i(ωt)

ψ()z,t = Re u( z)  (11)
e

and
i φ
uz() =U(z)e
(12)
z

φ =−ωδ+

c

4 © ISO 2005 – All rights reserved

where
u (z) is called the complex amplitude of the wave;
U is the spatial dependent modulus (e.g. amplitude);
φ is the spatial dependent phase.
This complex notation is much more convenient than the notation in Equation (8) with real quantities.
Nevertheless, only the real parts of Equations (11) and (12) have a physical meaning.
2.9 Irradiance
The irradiance Ι is given by the relationship
I=×EA∆t
where
Ι is the irradiance;
E is the energy;
A is the area;
∆t is the time interval.
A medium for direct recording of the field amplitude does not exist, since the frequency of the light stimulation
is too high to be resolved. So the most common detectors register the irradiance which is proportional to the
field amplitude absolutely squared:
I ∝|u =  (13)
| U
The correct relation between the complex amplitude and the irradiance is given by:
εc
0 ∗
I =  (14)
uu⋅


∗
where u is the conjugate complex amplitude.
2.10 Poynting vector
The vector S is known as the Poynting vector. S represents the amount of energy which crosses a unit area,
normal to the directions of E and H, per second
SE = × H (15)
S can be interpreted as the density of the energy flow. The magnitude of the Poynting vector is a measure of
the light intensity, and its direction represents the direction of propagation of the light.
The irradiance, I, is given by Equation (16) as follows:
I=⋅Sn (16)
where
n is the surface normal of the detector;
S is the time-averaged magnitude of the Poynting vector S.
2.11 Propagation of plane waves
Waves, which have a constant phase at a fixed time t over each of the planes normal to the direction of
propagation, are called plane waves (see Figure 1).
Let r (x,y,z) be a position vector of a point P in space and n (n ,n ,n ) the normal unit vector of the wavefront in
x y z
a fixed direction (see Figure 2). Wavefronts represent surfaces with constant phase. Any solution of
Equation (7) of the form
ψψ = ()rn ⋅ , t (17)
describes a plane wave since, at each instant of time, ψ is constant over each of the planes which are
perpendicular to the normal unit vector n:
rn ⋅ = κ (18)
where
κ is a constant.
The planes described by rn ⋅ = κ are defined as the wavefronts of the plane wave.
The disturbance ψ (x,y,z,t) of a harmonic plane wave that propagates in the n direction is given by
ψ()x, y, z,t = U cos(2π−νδt knr⋅ + ) (19)
The argument of the cosine function is termed the phase term; δ is the phase constant.
6 © ISO 2005 – All rights reserved

Figure 1 — Plane waves at the time t = 0

Figure 2 — Illustration of the condition r · n = κ
2.12 Propagation of spherical wave
A spherical wave, illustrated in Figure 3, is emitted by a point source O.

Figure 3 — Spherical waves
The complex amplitude representing a spherical wave should be of the form:
U
−ikr
u =  (20)
e
r
where r is the radial distance from the point source. The phase of this wave is constant for r equal to
a constant, i.e. the phase fronts are spherically centred at the point source O. The r in the denominator of
Equation (20) expresses the fact that the amplitude decreases as the inverse of the distance from the point
source.
Consider Figure 4, where a point source is lying in the x ,y -plane at a point of coordinates x ,y . The field
0 0 0 0
amplitude in a plane parallel to x ,y -plane at a distance z will then be given by Equation (20) with
0 0
2 2
()y −
r = + ()x − + y (21)
z x
where x, y are the coordinates of the illuminated plane. The approximation for the phase is carried out by a
binomial expansion of the square root; when r is approximated by the first two terms of the expansion, the
Fresnel approximation of the diffraction phenomena is obtained. In the amplitude factor [see Equation (21)], r
may be replaced by z, because (x – x ), (y – y ) << z. The complex amplitude of the field in the x,y-plane
0 0
resulting from a point source at x , y in the x ,y -plane is then given by:
0 0 0 0
2 2
U
()y−
−−iikz (k/2z) [()xx− + y ]
0 0
ux(), y, z =  (22)
ee
z
Figure 4 — A point source lying in x ,y -plane
0 0
8 © ISO 2005 – All rights reserved

2.13 Propagation of waves with limited extent
The only type of wave that does not change its shape when propagating, is a plane wave with uniform
amplitude. Such a wave is infinite in both lateral coordinates. A spherical wave, which is either converging to,
or diverging from, its centre of curvature and which is defined on the complete solid angle of 4π is another
example of a very special wave. In this case, the phase is altered, but any phase-distribution is similar to the
adjacent one. Again, a uniform amplitude distribution should be demanded. These two special cases have
been discussed in the previous subclauses.
All other wavefronts, especially plane or spherical wavefronts, which contain a boundary due to any limiting
aperture, do alter their shape when travelling along their light path. One example is given in the next two
figures, where the well-known Cornu’s spiral is shown, which is a graphical representation of the solution of
the Fresnel’s integrals
v
2 
πv

xv= cos d
∫
2 
0 
(23)

v

πv
yv= sin d

∫

0 
x and y being the Cartesian coordinates of the plot and v being a parameter on the curve. Cornu’s spiral can
be used to visualize the complex amplitude of a plane wave diffracted at a knife-edge, as a most simple
example.
Look at Figures 5 and 6 simultaneously. At the point of the geometrical shadow, point 0 on the abscissa in
Figure 6 and the coordinate centre in Figure 5, the amplitude is dropped to 1/2 of the amplitude far away from
the edge in the illuminated region. So, the intensity is dropped to 1/4. The complex amplitude for that point is
visualized in Figure 5 by a straight line, joining points Z and 0. The point Z in Figure 5 remains the centre,
where all vectors originate. Now going further into the illuminated region, the parameter v grows and with it the
modulus of the complex amplitude, since the vectors grow larger. The extreme value is reached at b′, then
dropping until a first relative minimum is reached at c′. At the same time the phase of the wavefront changes,
since the direction of the vectors changes, when the point moves along the spiral. The gradients of the phase-
change take large values in the vicinity of extreme points of the spiral, like points b′, c′, d′ and so on and are
nearly zero when the line centred in Z is tangential to the Cornu’s spiral.
Figure 5 shows the phase-change ϕ that the diffracted wave undergoes between points b′ and B′.
Since the phase is the quantity measured with interferometers, it is clear that the result of such a
measurement is affected strongly by diffraction. That is one reason why precision measurements should be
avoided where Fresnel diffraction can occur on the wavefronts to be measured. The only way that precision
measurements can be performed is to carefully image any limiting aperture to the detector, where the two
wavefronts interfere with each other (Fraunhofer diffraction!). This is dealt with in detail in 4.5.
Figure 5 — Change of the phase-angle between the points b′′′′ and B′
for the Fresnel diffraction on a knife edge

Figure 6 — Modulus A of the complex amplitude of the Fresnel diffraction pattern on a knife-edge
2.14 Propagation of aspherical waves
Any general wave which has either a more or less complicated function for the modulus or for the phase of the
complex amplitude [(see Equations (11) and (12)], alters the values of both, modulus and phase, when
propagating. In principle, the new wavefront after some distance z (see Figure 4) can be calculated with the
10 © ISO 2005 – All rights reserved

help of the Fresnel-Huygens principle, calculating the diffraction integral over the given distribution of the
complex amplitude. For simplicity, this can be visualized with the help of the Huygens wavelets, where points
on adjacent wavefronts have constant distances, measured along “light rays”, see Figure 7.

Figure 7 — Wavefront and light rays normal to each other
The same is true for any wavefront, which is not a sphere, a so-called aspherical wavefront. Such a wavefront
can be characterized by the fact that the light rays do not come to a common centre point, but intersect along
a region, called the caustic region. In Figure 8, such a caustic wave is shown.
If an aspherical wavefront is to be measured in an interferometer, the caustic region should be avoided, since
the modulus of the complex amplitude varies greatly with the lateral coordinates and the correct mapping of
the wavefront is very difficult. Also, the steadiness of the wavefront is not given in that region, leading to more
difficulties. So, it is necessary to choose the region preceding the caustic for measurement of the phase-
distribution. From the discussions before it is clear that the result for the measurement, e.g. for the phase
distribution in the cross-section of the wavefront, where the measurement is carried out, depends very
strongly on the axial position which is chosen for this cross-section. Since for practical reasons it will not be
possible to locate the detector directly on that cross-section, the wavefront has to be imaged onto the detector.
The result for the measured phase distribution depends then very strongly on the position, where the detector
is placed in the region of the image of the wavefront under test. In other words, if the detector is shifted along
the optical axis, the conjugate plane to the detector is shifted as well along the optical axis and a different
cross-section of the aspherical wavefront is investigated. The result will be quite different, depending on the
amount of asphericity of the wavefront under test. Special care has to be taken in order to have the imaging
condition in the proper position.
The same is true for any deviation of a wavefront from the purely spherical or plane shape. If a surface is
tested in reflection, that surface will alter the shape of the phase-distribution on the impinging wavefront. This
phase-distribution of the wavefront is then measured, allowing to conclude from that result on the shape of the
surface. Since the wavefront will alter its shape when propagating some distance from the surface, it is a
requirement to properly image the surface under test onto the detector. This is especially necessary if the
surface has errors with high spatial frequencies which give rise to high gradients on the phase-distribution.
The shape of the wavefront can be altered significantly by moving more than a few micrometers away from
such a surface. False results will be obtained.
Key
1 caustic region
Figure 8 — Due to the strong spherical aberration of a
simple plano-convex lens, a caustic is produced
3 General description of interference and different types of interferometers
3.1 Interference between two waves
3.1.1 Intensity of two partial waves overlapping
Interference can occur when two or more waves overlap in space. Assume the simple case of two waves
described by:
iφ
uU= e (24)
iφ
uU= e (25)
12 © ISO 2005 – All rights reserved

When u and u overlap, the resulting field simply becomes the sum, i.e.
1 2
uu=+u (26)
The observable quantity is the intensity, which becomes:
I=|u =| + =U =U+U+2cUUos( −)
|| φ φ
uu
12 1 2 12
I= I +I +2cI Ios∆φ
12 12 (27)
where
∆−φ= φ φ (28)
3.1.2 Interference term of the intensity, destructive and constructive interference, mean value of the
intensity
The resulting intensity does not become the sum of the intensities (≠ I + I ) of the two partial waves. The
1 2
expression2cos∆φ is called the interference term.
II
a) Case 1:
∆πφ =m(2 +1)   for 0, m =1,2,.
cos∆φ = −1; I = I , i.e. the interference is destructive.
min
b) Case 2:
∆πφ = 2m   for m = 0,1,2,.
cos∆φ = 1; I = I , i.e. the interference is constructive.
max
The intensity distribution given in Equation (26) is sketched in Figure 9. The intensity varies between I and
max
I with a mean value equal to:
min
II+
max min
II==+I . (29)
m 12
For two waves of equal intensity, i.e. I = I = I , Equation (27) becomes
1 2 0
∆φ
I=2[I 1+cos ∆φ]=4I cos (30)
00 

where the intensity varies between 0 and 4 I .
Figure 9 — Intensity distribution in the x y-plane from interference between two plane waves
3.2 Coherence
3.2.1 Temporal coherence: bandwidth, coherence time, coherence length
According to electromagnetic theory, the atoms of a light source do not continuously send out waves.
Emission occurs through “wave trains”, and there is a relationship between the length of the wave train and
the spectral composition of the light emitted. The longer the wave train, the narrower the spectrum. At the
theoretical limit, an infinite wave train consisting of monochromatic radiation of frequency ν (the mean
frequency emitted) would be emitted.
The coherence between two wave fields at one point in space is termed temporal or longitudinal coherence.
One way of illustrating the light emitted by real sources is to describe it as sinusoidal wave trains of finite
length with randomly distributed phase differences between the individual trains.
A source in an interference experiment, e.g. the Michelson interferometer is divided into two partial waves of
equal amplitudes by a beam splitter where after the two waves are recombined to interfere after having
travelled different paths.
In Figure 10, two successive wave trains of the partial waves are illustrated. The two wave trains have equal
amplitude and length L , with an abrupt, arbitrary phase difference. Figure 10 a) shows the situation when the
c
two partial waves have travelled equal path lengths. Although the phase of the original wave fluctuates
randomly, the phase difference between the partial waves 1 and 2 remains constant in time. The resulting
intensity is therefore given by Equation (27). Figure 10 c) shows the situation when partial wave 2 has
travelled a path length L longer than partial wave 1. The head of the wave trains in partial wave 2 then
c
coincide with the tail of the corresponding wave trains in partial wave 1. The resulting instantaneous intensity
is still given by Equation (27), but now the phase difference fluctuates randomly as the successive wave trains
pass by. As a result, cos (∆φ) varies randomly between +1 and −1. When averaged over many wave trains,
cos (∆φ) therefore becomes zero and the resulting, observable intensity will be
I = I + I (31)
The length of the wave trains is called the coherence length l . Figure 10 b) shows an intermediate case
c
where partial wave 2 has travelled a geometrical path difference ∆l longer than partial wave 1, where
0 < ∆l < l . Averaged over many wave trains, the phase difference now varies randomly in a time period
c
proportional to τ = ∆l/c and remains constant in a time period proportional to τ − τ. If τ is the duration of each
c c
wave train and c is the speed of light, the coherence length is l = c τ, where τ is called the coherence time.
c
14 © ISO 2005 – All rights reserved

The result is an interference pattern according to Equation (27) but with a reduced contrast. To account for
this loss of contrast, Equation (27) can be written as
I = + +2(| γτφ) |cos∆ (32)
II II
12 12
with γ (τ) = complex degree of coherence or coherence function, where |γ (τ)| means the absolute value of γ (τ).

a)
b)
c)
Key
1,2 partial waves
Figure 10 — Two successive wave trains of the partial waves
3.2.2 Coherence function, contrast
To see clearly that this quantity is related to the contrast of the pattern, we introduce the definition of contrast
or visibility K:
−
II
max min
K = (33)
+
II
max min
The contrast is a function of τ :
2(II γ τ)
K = (34)
I+I
For two waves of equal intensity, I = I , and Equation (34) becomes
1 2
K = γ()τ (35)
Three cases are identifiable:
γ (0) = 1 (36)
γ() =0 (37)
τ
c
0( uuγτ) 1 (38)
Equations (36) and (37) represent the two limiting cases of complete coherence and incoherence respectively,
while inequality (38) represents partial coherence.
Of more interest is to know the value of l where γ (τ) = 0.
c
In the case of a two-frequency laser this happens when
c
lc== τ (39)
cc
∆ν
where ∆ν is the spectral bandwidth, the difference between the two frequencies.
It can be shown that this relationship applies to any light source with a frequency distribution of width ∆ν. l is
c
termed the coherence length and τ the coherence time.
c
Sources of finite spectral width will emit wave trains of finite length. This is verified by the relation
c∆λ
∆ν = (40)
λ
where
∆λ is the bandwidth of the radiation;
λ is the mean wavelength for the bandwidth of the radiation.
EXAMPLE 1 For a mercury vapour lamp with an interference filter with the width of the spectral lines ∆λ = 10 nm by
the wavelength λ = 546 nm the coherence length is:
λ l
0c −13
l=≈λτ30 µm  and == 10 s (41)
c0 c
∆λ c

EXAMPLE 2 For a Helium-Neon-laser with λ = 632,8 nm and ∆ν = 14 MHz [half width of the distribution I (ν)], all
1/2 ν
vibrational modes are included, and the coherence length is:
−7
lc=≈ττ20 m and = = 0,7×10 s (42)
cc c
∆ν
3.2.3 Spatial coherence
It is also possible to measure the coherence of a wave field at two points separated laterally in space. This
phenomenon is called spatial or transverse coherence and can be analysed by the classical Young's double slit
(or pinhole) experiment.
16 © ISO 2005 – All rights reserved

The incident wavefront is divided by passing through two small holes at P and P in a screen S . The
1 2 1
emerging spherical wavefronts from P and P will interfere, and the resulting interference pattern is observed
1 2
on the screen S (see Figure 11).
Key
1 source
2 double slit S
3 screen S
Figure 11 — Young’s double slit interference experiment
The geometric path length difference ∆l of the light reaching an arbitrary point x on S from P and P is found
2 1 2
from Figure 11. When the distance z between S and S is much greater than the distance D between P
1 2 1
and P
D
<< 1 (43)
z
then a good approximation is obtained as
∆lx
= and, after rearranging, gives
D z
D
∆=lx (44)
z
The phase difference therefore becomes
22ππD
∆∆φ = l = x (45)
λλz
which can now be inserted into the general expression for the resulting intensity distribution. Equation (27)
gives

D
I()x = 2I 1+cos 2π x (46)
0
λz


We get interference fringes parallel to the y-axis with a spatial period a:
λz
a = (47)
D
which decreases as the distance between P and P increases.
1 2
The distance l =λ/θ between P and P for which interference fringes disappear is called the spatial
c 1 2
coherence length. The spatial coherence length l is inversely proportional to the diameter of the aperture θ in
c
analogy with the temporal coherence length, which is inversely proportional to the spectral width.
The condition [Inequality (43)] therefore becomes:
λ
Dl<< and l = (48)
cc
θ
Suppose the fringes on S are formed, when the double slit is illuminated by a point source located at Q , on
2 2
the symmetry axis between P and P . From symmetry considerations constructive interference takes place
1 2
on S at x = 0, since the optical path length from Q to that point on S is equal in both cases, with the light
2 2 2
passing either through P or through P . At x = 0, ∆φ = 0 and therefore a bright fringe with a fringe order
1 2
number of zero (the zero-order fringe). As already stated, adjacent to that zero-order fringe are further bright
fringes at spatial distances given by Equation (47), which may be numbered with the help of order numbers
m = + 1, + 2, . and m = –1, –2, ., depending on the sign convention for ∆φ.
Now suppose the point source Q is shifted by distance L to the position Q . Since the optical path difference
2 1
from Q to P gets shorter and the optical path difference from Q to P gets longer, the zero-order fringe on
1 1 1 2
S is shifted down, as shown in Figure 11. All other fringes are shifted together with the zero-order fringe by
the same amount. If Q is shifted laterally a distance L, then the geometric path length difference ∆l is equal to
DL/R, as can be concluded from the quite similar consideration of Equation (44).
If the distance L is chosen such that the fringes are shifted exactly by an amount of one fringe period, the
intensity at x = 0 has undergone one full period and the geometric path length difference ∆l is equal to λ.
When the intensity would be integrated during the shifting of the point source from position Q to position Q ,
2 1
a mean intensity 2I would be measured. The same is true for all other points on the screen.
If the light source has a spatial extent equal to L, the integration over all points of the light source should be
carried out in space rather than in time but with the same result, i.e. no fringes are visible on S . Therefore for
a light source of extent L, the spatial coherence is zero. From this consideration, it can be concluded that the
spatial coherence function can be found by integrating the equation for the intensity of the two-beam
interference, with the source being the region of integration. In the case of a slit-like source of width l, a sinc-
function results, having its first zero position at l = L. In the general case, the van Cittert-Zernike theorem
states that the degree of coherence between two points is the modulus of the scaled and normalized Fourier
transform of the source intensity distribution.
The limits between “coherent” and “incoherent” are somewhat arbitrary. In the discussion of the source of extent L
a value was chosen where the variation of the path length difference ∆l equals 1 λ. In this case, no fringes are
visible at all. If it still desirable to see fringes, but on a lower contrast level, take the limit for the source size more
stringently. It is the convention to define the limit for the spatial coherence such that the path length difference is
∆l = λ/2.
Since R >> D, D/R might be replaced by the angle ε, under which the two points P and P appear when
1 2
observed from the source. Equally, since R >> L, L/R may be replaced by θ.
By assuming an extended source with incoherent light, the condition for spatial coherence is given by:
DL λ λλ
uu       LDεθ       u (49)
R22 2
18 © ISO 2005 – All rights reserved

3.3 Different arrangements of interference between two beams
3.3.1 General
In order to produce interference phenomena, arrangements called interferometers are used. Most
interferometers consist of the following elements:
a) light source;
b) means for shaping the waves emerging from the light source;
c) element for dividing the light into two partial waves;
d) two different propagation paths, where the partial waves experience different phase-contributions;
e) element for combining the partial waves;
f) means for imaging the waves onto the detector;
g) detector for
...