## The coherence length of black-body

## radiation

### Axel Donges

### †

Fachhochschule und Berufskollegs NTA Prof. Dr Gr¨ubler, Seidenstrasse 12–35, D-88316 Isny im Allg¨au, 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 lengthlcof thermal radiation is given by the formula

lcT∼_{= 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 iscτ, wherecis the velocity of light. The intensity of the recombined wave is then given by

I (τ )
I_{0} =
1
2
�
1+�E(t )·E(t−τ )�
�[E(t )]2_{�}
�
(1)
whereI_{0} is the intensity of the incident wave andEis the real disturbance of the waveﬁeld
(e.g., a Cartesian component of the electric vector). The angular brackets denote the time
average, i.e.
�f (t )_{� =} lim
T_{→∞}
�_{�}T /2
−T /2
f (t )dt
�
. (2)

Obviously interference effects are characterized by the so-called normalizedself-coherence function

γ (τ )_{=} �E(t )·E(t−τ )�

�[E(t )]2_{�} . (3)

According to theWiener–Kintchine theorem[1, 2] the normalized self-coherence function
γ (τ )is related to the normalizedpower spectrumP (ν)of the waveﬁeldE(t )by the Fourier
transform [3, p 400]
γ (τ )_{=}
∞
�
0
P (ν)cos(2π ντ )dν (4)
† E-mail: donges@server3.fh-isny.de
0143-0807/98/030245+05$19.50 © 1998 IOP Publishing Ltd 245

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

P (ν)_{=} 15h

4_{ν}3

(π kT )4�ehν/ kT _{−}_{1]} (6)

where h and k are respectively Planck’s and Boltzmann’s constants and T the absolute temperature of the light source. Thus we obtain

I (τ )
I0 =
1
2
�
1+γ (τ )� (7a)
with
γ (τ )_{=} 15
π4
∞
�
0
x3cos(2π kT τ x/ h)
ex_{−}_{1} dx. (7b)

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 pathcτ is increased, the modulation depth of the interference pattern decreases. When the time delayτ is of the order of or much greater than

τ_{c}∼_{=} h

4kT (8)

interference effects are no longer appreciable. The timeτ_{c}is known as thecoherence timeand
l_{c}_{=}cτ_{c}∼_{=} hc

4kT (9)

is thecoherence lengthof thermal radiation. For example, the coherence lengths of sunlight
(T _{=}6 kK) and incandescent light withT _{=}3 kK are 0.6µm and 1.2µm, respectively.

If we deﬁne the full width at half maximum (FWHM) ofP (ν)(equation (6)) as thespectral

width�νof 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 becomesI0/2 (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 wavelengthλ_{eff} is approximately equal to the mean
wavelength of the thermal radiation

�λ_{� =}
�_{∞}
0 λu(λ, T )dλ
�_{∞}
0 u(λ, T )dλ
= _{2}hc
.7kT (13)

whereu(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 ordermof 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_{= ±}1_{2}, 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 lengthl_{c}can
be calculated using equation (11)§ as 3µm. Because the spectral response of the human eye
reaches its maximum at�λ_{� ∼}_{=}550 nm,lc/_{�}λ_{� ∼}_{=}5.5. Therefore coloured fringes up to ﬁfth
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 = ��ν2P (ν)� − �νP (ν)�2_{, we have}_{�ν} _{=}

2.0kT / h, which is of the same order of magnitude.

‡ This is a special case of thereciprocity 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].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 Radius (a.u.) Intensity (a.u.) white light (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Radius (a.u.) Intensity (a.u.) quasi−monochromatic light (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 lengthl_{c}. 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:

Furthermore, the product of coherence timeτcand spectral width�νis 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]}
λmaxT _{=} hc

5.0k =2.9 mm K (15)

where λ_{max} is the wavelength of the black-body radiation at which the intensity per unit
wavelength of the spectrum reaches its maximum. Therefore

l_{c}∼_{=}λ_{eff} ∼_{= �}λ_{� ∼}_{=}λ_{max}. (16)
References

[1] Wiener N 1930Acta Math.55117 [2] Khintchine A 1934Math. Ann.109604

[3] Klein M V and Furtak T E 1988Optik(Berlin: Springer) [4] Mandl F 1971Statistical Physics(Chichester: Wiley) p 254

[5] Born M and Wolf E 1975Principles of Optics(Oxford: Pergamon) p 542

[6] Haferkorn H 1981Optik—Physikalisch technische Grundlagen und Anwendungen(Thun: Deutsch) p 132 [7] Willmott J C 1975Atomic Physics(Chichester: Wiley) p 57