Download The coherence length of black

Eur. J. Phys. 19 (1998) 245–249. Printed in the UK
PII: S0143-0807(98)86653-1
The coherence length of black-body
radiation
Axel Donges†
Fachhochschule und Berufskollegs NTA Prof. Dr Grübler, Seidenstrasse 12–35,
D-88316 Isny im Allgäu, Germany
Received 6 August 1997, in final form 20 February 1998
Abstract. We consider two-beam interference with black-body radiation. It is shown that the
coherence length lc of thermal radiation is given by the formula
lc T ∼
= 3.6 mm K.
The coherence length corresponds to the mean wavelength of the thermal radiation.
1. Two-beam interference with thermal radiation
We consider two-beam interference in a Michelson interferometer: a linear polarized wave is
split by a semi-transparent mirror into two identical waves. Both waves follow different paths
before they are again superposed. The difference of the optical paths is cτ , where c is the
velocity of light. The intensity of the recombined wave is then given by
�
�
I (τ )
1
�E(t) · E(t − τ )�
=
1+
(1)
I0
2
�[E(t)]2 �
where I0 is the intensity of the incident wave and E is the real disturbance of the wavefield
(e.g., a Cartesian component of the electric vector). The angular brackets denote the time
average, i.e.
�
� �T /2
f (t) dt .
(2)
�f (t)� = lim
T →∞
−T /2
Obviously interference effects are characterized by the so-called normalized self-coherence
function
�E(t) · E(t − τ )�
.
(3)
γ (τ ) =
�[E(t)]2 �
According to the Wiener–Kintchine theorem [1, 2] the normalized self-coherence function
γ (τ ) is related to the normalized power spectrum P (ν) of the wavefield E(t) by the Fourier
transform [3, p 400]
�∞
(4)
γ (τ ) = P (ν) cos(2πντ ) dν
0
† E-mail: [email protected]
0143-0807/98/030245+05$19.50
© 1998 IOP Publishing Ltd
245
246
A Donges
with
�∞
0
P (ν) dν = 1.
(5)
The power spectrum of black-body radiation is given by Planck’s well known formula [4],
which after normalization reads
15h4 ν 3
�
P (ν) =
(6)
(πkT )4 ehν/kT − 1]
where h and k are respectively Planck’s and Boltzmann’s constants and T the absolute
temperature of the light source. Thus we obtain
�
1�
I (τ )
= 1 + γ (τ )
(7a)
I0
2
with
�∞ 3
x cos(2πkT τ x/ h)
15
dx.
(7b)
γ (τ ) = 4
π
ex − 1
0
2. Discussion of the interference pattern
Figure 1 summarizes the result of numerical calculations of equations (7). It shows that as
the difference of optical path cτ is increased, the modulation depth of the interference pattern
decreases. When the time delay τ is of the order of or much greater than
h
(8)
τc ∼
=
4kT
interference effects are no longer appreciable. The time τc is known as the coherence time and
∼ hc
(9)
lc = cτc =
4kT
is the coherence length of thermal radiation. For example, the coherence lengths of sunlight
(T = 6 kK) and incandescent light with T = 3 kK are 0.6 µm and 1.2 µm, respectively.
Figure 1. Interference of two identical but phase shifted beams of thermal radiation.
The coherence length of black-body radiation
247
If we define the full width at half maximum (FWHM) of P (ν) (equation (6)) as the spectral
width �ν of the thermal radiation†, we obtain (by numerical calculation)
�ν = 4.3kT / h.
(10)
Using equation (8) gives the reciprocity relation‡
�ντc ∼
= 1.
(11)
For 2π kT τ/ h = 0.44 the intensity becomes I0 /2 (see figure 1), which corresponds (in the
case of monochromatic light) to an optical path difference of a quarter wavelength. Therefore
we can assign an effective wavelength of
λeff =
hc
3.6kT
(12)
to the thermal radiation. The effective wavelength λeff is approximately equal to the mean
wavelength of the thermal radiation
�∞
λu(λ, T ) dλ
hc
�λ� = �0 ∞
(13)
=
2.7kT
0 u(λ, T ) dλ
where u(λ, T ) is the spectral density of the black-body radiation.
3. Fringes with quasi-monochromatic and thermal light
If quasi-monochromatic light (lc � �λ�) is applied, interference patterns consist of a great
number of bright and dark quasi-monochromatic interference fringes. The order m of a bright
fringe is related to the optical path difference
� = m�λ�
(14)
(|m| = 0, 1, 2, 3, . . .) of the superposed light. However, if black-body radiation is used, the
coherence length is slightly less than the mean or effective wavelength. That is why no fringes
(modulation of intensity) of order |m| � 1 occur. So, in the case of unfiltered black-body
radiation, interference patterns consist only of a central maximum (m = 0) flanked by two
weak minima (m = ± 21 , see figure 1).
One should bear in mind that what is observed also depends on the spectral response of
the detector used. A case of practical importance is when the observation is visual. If we take
the FWHM of the spectral response of the human eye as 100 THz, the coherence length lc can
be calculated using equation (11)§ as 3 µm. Because the spectral response of the human eye
reaches its maximum at �λ� ∼
= 5.5. Therefore coloured fringes up to fifth
= 550 nm, lc /�λ� ∼
order are visible [6].
As an example figure 2 shows measured intensity distributions of Newton’s rings. The
fringes were generated with incandescent light and observed by a linear CCD array. Because
the spectral response of the detector used was similar to that of the human eye, about five
maxima occur (figure 2(a)). If the spectral bandwidth of the light is narrowed by a band-pass
filter, the number of fringes increases up to twelve (figure 2(b)).
�
† If we use another definition of ‘spectral width’, for example �ν = �ν 2 P (ν)� − �νP (ν)�2 , we have �ν =
2.0kT / h, which is of the same order of magnitude.
‡ This is a special case of the reciprocity inequality �ντc � 1/(4π ) [5].
§ In this paper we have derived equation (11) for black-body radiation. If the spectral width of the light is narrowed
by a filter (or detector), equation (11) still gives the correct order of magnitude for the coherence time [3, p 419].
248
A Donges
1
0.9
Intensity (a.u.)
0.8
0.7
0.6
0.5
0.4
white light
0
0.1
0.2
0.3
0.4
0.5
0.6
Radius (a.u.)
0.7
0.8
0.9
1
0.7
0.8
0.9
1
(a)
1
0.9
0.8
Intensity (a.u.)
0.7
0.6
0.5
0.4
0.3
quasi−monochromatic light
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Radius (a.u.)
(b)
Figure 2. Intensity distributions of Newton’s rings observed with a linear CCD array
(spectral response similar to the human eye—measured by L Engelhardt and R Krause,
NTA Isny). (a) White light: unfiltered incandescent light. (b) Quasi-monochromatic
light: band-pass filtered incandescent light (transmitted wavelengths: 560–595 nm).
4. Summary
Unfiltered black-body radiation is, roughly speaking, made up of stochastically independent
wave trains of finite length lc . This length is called the coherence length. It corresponds
approximately to the mean wavelength of the wavetrains. The coherence length is inversely
proportional to the temperature:
lc T ∼
= 3.6 mm K.
(9� )
The coherence length of black-body radiation
249
Furthermore, the product of coherence time τc and spectral width �ν is equal to unity. If the
spectral width of black-body radiation is narrowed by a filter or detector the coherence length
increases.
Finally, we mention that equation (9� ) is very similar to Wien’s displacement law [7]
hc
= 2.9 mm K
(15)
λmax T =
5.0k
where λmax is the wavelength of the black-body radiation at which the intensity per unit
wavelength of the spectrum reaches its maximum. Therefore
(16)
lc ∼
= λeff ∼
= �λ� ∼
= λmax .
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Wiener N 1930 Acta Math. 55 117
Khintchine A 1934 Math. Ann. 109 604
Klein M V and Furtak T E 1988 Optik (Berlin: Springer)
Mandl F 1971 Statistical Physics (Chichester: Wiley) p 254
Born M and Wolf E 1975 Principles of Optics (Oxford: Pergamon) p 542
Haferkorn H 1981 Optik—Physikalisch technische Grundlagen und Anwendungen (Thun: Deutsch) p 132
Willmott J C 1975 Atomic Physics (Chichester: Wiley) p 57