The coherence length of black-body radiation

Transcription

1 Eur. J. Phys. 19 (1998) Printed in the UK PII: S143-87(98) The coherence length of black-body radiation Axel Donges Fachhochschule und Berufskollegs NTA Prof. Dr Grübler, Seidenstrasse 12 35, D Isny im Allgäu, Germany Received 6 August 1997, in final form 2 February 1998 Abstract. We consider two-beam interference with black-body radiation. It is shown that the coherence length l c of thermal radiation is given by the formula l c 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) I = E(t) E(t τ) [E(t)] 2 where I 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 T /2 f (t) = lim f (t) dt. (2) T Obviously interference effects are characterized by the so-called normalized self-coherence function E(t) E(t τ) γ (τ) =. (3) [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 4] γ (τ) = [email protected] P (ν) cos(2πντ) dν (4) /98/3245+5$ IOP Publishing Ltd 245 (1)

2 246 A Donges with 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 15h 4 ν 3 P (ν) = (πkt ) 4 (6) e hν/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 I (τ) = γ (τ) (7a) I 2 with γ (τ) = 15 π 4 x 3 cos(2πkt τx/h) e x 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 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 τ c h = (8) 4kT interference effects are no longer appreciable. The time τ c is known as the coherence time and l c = cτ c hc = (9) 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.6 µm and 1.2 µm, respectively. Figure 1. Interference of two identical but phase shifted beams of thermal radiation.

3 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. (1) Using equation (8) gives the reciprocity relation ντ c = 1. (11) For 2πkT τ/h =.44 the intensity becomes I /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 λ eff = hc 3.6kT to the thermal radiation. The effective wavelength λ eff is approximately equal to the mean wavelength of the thermal radiation λu(λ, T ) dλ λ = u(λ, T ) dλ = hc (13) 2.7kT where u(λ, T ) is the spectral density of the black-body radiation. (12) 3. Fringes with quasi-monochromatic and thermal light If quasi-monochromatic light (l c 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 =, 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 = ) flanked by two weak minima (m = ± 1, see figure 1). 2 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 1 THz, the coherence length l c can be calculated using equation (11) as 3 µm. Because the spectral response of the human eye reaches its maximum at λ = 55 nm, l c / 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 = ν 2 P (ν) νp (ν) 2, we have ν = 2.kT /h, which is of the same order of magnitude. This is a special case of the reciprocity inequality ντ c 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].

4 248 A Donges Intensity (a.u.) white light Radius (a.u.) 1 (a) Intensity (a.u.) quasi monochromatic light 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: nm). 4. Summary Unfiltered black-body radiation is, roughly speaking, made up of stochastically independent wave trains of finite length l 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: l c T = 3.6 mm K. (9 )

5 The coherence length of black-body radiation 249 Furthermore, the product of coherence time τ c 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] λ max T = hc = 2.9 mm K (15) 5.k 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 = λeff = λ = λmax. (16) References [1] Wiener N 193 Acta Math [2] Khintchine A 1934 Math. Ann [3] Klein M V and Furtak T E 1988 Optik (Berlin: Springer) [4] Mandl F 1971 Statistical Physics (Chichester: Wiley) p 254 [5] Born M and Wolf E 1975 Principles of Optics (Oxford: Pergamon) p 542 [6] Haferkorn H 1981 Optik Physikalisch technische Grundlagen und Anwendungen (Thun: Deutsch) p 132 [7] Willmott J C 1975 Atomic Physics (Chichester: Wiley) p 57