Book chapter ; Quantum technology, quantum computing, quantum sensing, quantum cryptography

1. 16 Classical optics
when L1 = L2. Field maxima occur whenever
4π
λ
ΔL + Δφ = 2mπ , (2.38)
and minima when
4π
λ
ΔL + Δφ = (2m + 1)π , (2.39)
where m is again an integer. Thus as L2 is scanned, bright and dark
fringes appear at the output port with a period equal to λ/2. The inter-
ferometer thus forms a very sensitive device to measure differences in
the optical path lengths of the two arms.
A typical application of a Michelson interferometer is the measurement
of the refractive indices of dilute media such as gases. The interferometer
is configured with L1 ≈ L2, and an evacuated cell of length L in one
of the arms is then slowly filled with a gas of refractive index n. By
recording the shifting of the fringes at the output port as the gas is
introduced, the change of the relative path length between the two arms,
namely 2(n − 1)L, can be determined, and hence n.
2.3 Coherence
The discussion of the interference pattern produced by a Michelson inter-
ferometer in the previous section assumed that the phase shift between
the two interfering fields was determined only by the path difference
2ΔL between the arms. However, this is an idealized scenario that takes
no account of the frequency stability of the light. In realistic sources,
See Section 4.4 for a discussion of spec-
tral line broadening mechanisms.
the output contains a range Δω of angular frequencies, which leads to
the possibility that bright fringes for one frequency occur at the same
position as the dark fringes for another. Since this washes out the inter-
ference pattern, it is apparent that the frequency spread of the source
imposes practical limits on the maximum path difference that will give
observable fringes.
The property that describes the stability of the light is called the
coherence. Two types of coherence are generally distinguished:
Some authors use an alternative
nomenclature in which temporal coher-
ence is called longitudinal coherence
and spatial coherence is called trans-
verse coherence. A clear discussion
of spatial coherence may be found in
Brooker (2003) or Hecht (2002).
• temporal coherence,
• spatial coherence.
The discussion below is restricted to temporal coherence. The concept
of spatial coherence is discussed briefly in Section 6.1 in the context of
the Michelson stellar interferometer.
The temporal coherence of a light beam is quantified by its coherence
time τc. An analogous quantity called the coherence length Lc can
be obtained from:
Lc = cτc . (2.40)
The coherence time gives the time duration over which the phase of the
wave train remains stable. If we know the phase of the wave at some

2. 2.3 Coherence 17
position z at time t1, then the phase at the same position but at a
different time t2 will be known with a high degree of certainty when
|t2 − t1| τc, and with a very low degree when |t2 − t1| τc. An
equivalent way to state this is to say that if, at some time t we know the
phase of the wave at z1, then the phase at the same time at position z2
will be known with a high degree of certainty when |z2 − z1| Lc, and
with a very low degree when |z2 − z1| Lc. This means, for example,
that fringes will only be observed in a Michelson interferometer when
the path difference satisfies 2ΔL Lc.
Insight into the factors that determine the coherence time can be
obtained by considering the filtered light from a single spectral line
of a discharge lamp. Let us suppose that the spectral line is pressure-
broadened, so that its spectral width Δω is determined by the average
time τcollision between the atomic collisions. (See Section 4.4.3.) We This type of radiation is an example of
chaotic light. The name refers to the
randomness of the excitation and phase
interruption processes.
model the light as generated by an ensemble of atoms randomly excited
by the electrical discharge and then emitting a burst of radiation with
constant phase until randomly interrupted by a collision. It is obvious
that in this case the coherence time will be limited by τcollision. Further-
more, since τcollision also determines the width of the spectral line, it will
also be true that:
τc ≈
1
Δω
. (2.41)
The result in eqn 2.41 is in fact a general one and shows that the coher-
ence time is determined by the spectral width of the light. This clarifies
that a perfectly monochromatic source with Δω = 0 has an infinite
coherence time (perfect coherence), whereas the white light emitted by
a thermal source has a very short coherence time. A filtered spectral
line from a discharge lamp is an intermediate case, and is described as
partially coherent.
The derivation of eqn 2.41 for a gen-
eral case may be found, for example, in
Brooker (2003, §9.11).
The temporal coherence of light can be quantified more accurately by
the first-order correlation function g(1)
(τ) defined by:
In Chapter 6 we shall study the prop-
erties of the second-order correlation
function g(2)(τ). This correlation func-
tion is so-called because it characterizes
the properties of the optical inten-
sity, which is proportional to the sec-
ond power of the electric field. (cf.
eqn 2.28.)
g(1)
(τ) =
E∗
(t)E(t + τ)
|E(t)|2
. (2.42)
The symbol · · · used here indicates that we take the average over a
long time interval T:
E∗
(t)E(t + τ) =
1
T
T
E∗
(t)E(t + τ) dt . (2.43)
g(1)
(τ) is called the first-order correlation function because it is based
on the properties of the first power of the electric field. It is also called
the degree of first-order coherence.
Let us assume that the input field E(t) is quasi-monochromatic with
a centre frequency of ω0 so that it varies with time according to:
E(t) = E0 e−iω0t
eiφ(t)
. (2.44)

3. 18 Classical optics
On substituting into eqn 2.42 we then find that g(1)
(τ) is given by:
g(1)
(τ) = e−iω0τ
ei[φ(t+τ)−φ(t)]
. (2.45)
This means that the real part of g(1)
(τ) is an oscillatory function of τ
with a period of 2π/ω0. This rapid oscillatory variation produces the
fringe pattern in an interference experiment, and it is the variation of
the modulus of g(1)
(τ) due to the second factor in eqn 2.45 that contains
the information about the coherence of the light.
It is clear from eqn 2.42 that |g(1)
(0)| = 1 for all cases. For 0 τ
τc, we expect φ(t + τ) ≈ φ(t), and the value of |g(1)
(τ)| will remain
close to unity. As τ increases, |g(1)
(τ)| decreases due to the increased
probability of phase randomness. For τ τc, φ(t + τ) will be totally
uncorrelated with φ(t), and exp i[φ(t + τ) − φ(t)] will average to zero,
implying |g(1)
(τ)| = 0. Hence |g(1)
(τ)| drops from 1 to 0 over a time-scale
of order τc.
Light that has |g(1)(τ) = 1| for all
values of τ is said to be perfectly
coherent. Such idealized light has an
infinite coherence time and length. The
highly monochromatic light from a sin-
gle longitudinal mode laser is a fairly
good approximation to perfectly coher-
ent light for most practical purposes.
The detailed form of g(1)
(τ) for partially coherent light depends on the
type of spectral broadening that applies. For light with a Lorentzian
lineshape of half width Δω in angular frequency units, g(1)
(τ) is given by:
See Section 4.4 for a discussion of
spectral lineshapes. The derivation of
eqns 2.46–2.49 may be found, for exam-
ple, in Loudon (2000, §3.4).
g(1)
(τ) = e−iω0τ
exp (−|τ|/τc) , (2.46)
where
τc = 1/Δω . (2.47)
The equivalent formulae for a Gaussian lineshape are:
g(1)
(τ) = e−iω0τ
exp
−
π
2
τ
τc
2
, (2.48)
where
τc = (8π ln 2)1/2
/Δω . (2.49)
A typical variation of the real part of g(1)
(τ) with τ for Gaussian light
is shown in Fig. 2.5. The coherence time in this example has been set at
the artificially short value of 20 times the optical period.
Fig. 2.5 Typical variation of the real
part of the first-order correlation func-
tion g(1)(τ) as a function of time delay
τ for Gaussian light with a coherence
time of τc. The coherence time in this
example has been chosen to be 20 times
longer than the optical period.
The visibility of the fringes observed in an interference experiment
is defined as:
visibility =
Imax − Imin
Imax + Imin
, (2.50)
where Imax and Imin are the intensities recorded at the fringe maxima
and minima, respectively. It is qualitatively obvious that the visibility is
determined by the coherence of the light, and this point can be quantified
by deriving an explicit relationship between the visibility and the first-
order correlation function.
We consider again a Michelson interferometer and assume that we
have a light source with a constant average intensity, so that the fringe

4. 2.4 Nonlinear optics 19
pattern only depends on the time difference τ between the fields that
interfere rather than the absolute time. We can therefore write the
output field as: We have assumed a 50 : 50 power split-
ting ratio in eqn 2.51, which gives a
1/
√
2 amplitude combining ratio. We
have also assumed that the phase shift
Δφ introduced in eqn 2.37 is equal to
π. The path difference ΔL is related to
τ through τ = 2ΔL/c.
Eout
(t) =
1
√
2
(E(t) − E(t + τ)) . (2.51)
The time-averaged intensity observed at the output is proportional to
the average of the modulus squared of the field:
I(τ) ∝ Eout∗
(t)Eout
(t)
∝ (E∗
(t)E(t) + E∗
(t + τ)E(t + τ)
− E∗
(t)E(t + τ) − E∗
(t + τ)E(t))/2 . (2.52)
The constant nature of the source implies that the first and second terms
are identical. Furthermore, the third and fourth are complex conjugates
of each other. We therefore find:
I(τ) ∝ E∗
(t)E(t) − Re[E∗
(t)E(t + τ)] . (2.53)
We can then substitute from eqn 2.42 to find:
I(τ) ∝ E∗
(t)E(t)
1 − Re[g(1)
(τ)]
= I0
1 − Re[g(1)
(τ)]
, (2.54)
where I0 is the input intensity. Substitution into eqn 2.50 with
Imax/min = I0(1 ± |g(1)
(τ)|) readily leads to the final result that:
visibility = |g(1)
(τ)| . (2.55)
Hence the intensity observed at the output of a Michelson interferometer
as ΔL is scanned would, in fact, look like Fig. 2.5, with τ = 2ΔL/c.
A summary of the main points of this section may be found in
Table 2.1.
2.4 Nonlinear optics
2.4.1 The nonlinear susceptibility
The linear relationship between the electric polarization of a dielectric
medium and the electric field of a light wave implied by eqn 2.2 is an
Table 2.1 Coherence properties of light as quantified by the coherence time τc and the first-order correlation function g(1)(τ).
In the final column we assume |τ| 0.
Description of light Spectral width Coherence Coherence time |g(1)
(τ)|
Perfectly monochromatic 0 Perfect Infinite 1
Chaotic Δω Partial ∼ 1/Δω 1 |g(1)
(τ)| 0
Incoherent Effectively infinite None Effectively zero 0