Dressel.Gruner.Electrodynamics of Solids.2003

258	10 Spectroscopic principles

Fig. 10.4a; between the terahertz source and detector a standard optical arrangement of lenses or parabolic mirrors guides the radiation and focuses the beam onto the sample. The most suitable design of a terahertz radiation transmitter is a charged transmission line shorted by a laser pulse of a few femtoseconds duration. Fig. 10.4b shows a terahertz radiation source. The subpicosecond electric dipoles of micrometer size are created by photoconductive shorting of the charged transmission line with the femtosecond pulses from a dye or Ti–sapphire laser. The detection segment is designed similarly to the radiation source (Fig. 10.4c) via the transmission line; one side of the antenna is grounded and a current amplifier is connected across the antenna. During operation, the antenna is driven by the incoming terahertz radiation pulse which causes a time dependent voltage across the antenna gap. The induced voltage is measured by shorting the antenna gap with the femtosecond optical pulse in the detection beam and monitoring the collected charge (current) versus time delay of the detection laser pulses with respect to the excitation pulses [Ext89, Ext90]. If the photocarrier lifetime is much shorter than the terahertz pulse, the photoconductive switch acts as a sampling gate which samples the terahertz field. In principle, reflection measurements are also possible, although they are rarely conducted because the setup is more sensitive to alignment. Details of the experimental arrangement and the analysis can be found in [Nus98].

As an example of an experiment performed in the time domain, Fig. 10.5a shows the amplitude of the terahertz radiation transmitted through a thin niobium film deposited onto a quartz substrate at two different temperatures, above and below the superconducting transition temperature of niobium [Nus98]. Both the real and imaginary parts of the complex conductivity σˆ (ω) of the niobium film can be obtained directly from these terahertz waveforms without the use of the Kramers– Kronig relations. The frequency dependence of σ1(ω) and σ2(ω) normalized to the normal state value σn are shown in Fig. 10.5b for T = 4.7 K. These results may be compared with the analogous experiments performed in the frequency domain by the use of a Mach–Zehnder interferometer (see Fig. 14.5).

10.3 Fourier transform spectroscopy

The concept of a Fourier transform spectrometer is based on Michelson’s design of an interferometer in which a beam of monochromatic light is split into two approximately equal parts which follow different paths before being brought together again. The intensity of the recombined light is a function of the relative difference in path length between the two arms; i.e. the light shows an interference pattern. Recording the combined intensity as a function of the delay δ of one of these beams allows the spectral distribution of the light by Fourier transformation (Appendix A.1) to be recovered.

		10.3									Fourier transform spectroscopy																																				259
	1
u.)							(a)																											4.7 K										Nb
	0.5
(a.	0
Amplitude
	− 0.5												7 K
	−1
	−1.5
		0									5						10																	15													20
																						Time (ps)
																	Energy										hω (meV)
	1.5	0									1						2																	3													4
							(b)																													T = 4.7 K
n
σ / σ	1								σ2 / σn
Conductivity	0.5																																								σ1 / σn
	0
		0															0.5																														1.0
																	Frequency f (THz)

Fig. 10.5. (a) Terahertz transients transmitted through a thin niobium film (Tc ≈ 7 K) on quartz in the normal state (dashed line) and the superconducting state (solid line).

(b) Real and imaginary parts of the normalized complex conductivity, σ1(ω) and σ2(ω), of niobium in the superconducting state evaluated from the above data. The dashed line indicates the predictions by the BCS theory using the Mattis–Bardeen equations (7.4.20) (after [Nus98]).

Modern Fourier transform spectrometers mainly operate in the infrared spectral range (10–10 000 cm−1); however, Fourier transform spectrometers have been built in the microwave frequency range as well as in the visible spectral range. A large number of detailed and excellent monographs on Fourier transform spectroscopy are available [Bel72, Gri86, Gen98]; only a short description is given here.

260	10 Spectroscopic principles

Mirror 1

L	Mirror 2
	x
Source	L + δ /2
Beam-
splitter

Detector

Fig. 10.6. Basic outline of a Michelson interferometer. The beam of the source is divided by the beamsplitter to the fixed mirror 1 and the moving mirror 2, at distances L and L + δ/2, respectively. The recombined beam is focused onto the detector, which measures the intensity I as a function of displacement δ/2.

10.3.1 Analysis

In order to discuss the principles of Fourier transform spectroscopy, the examination of a simple diagram of a Michelson interferometer shown in Fig. 10.6 is useful. Mirror 1 is fixed at a distance L from the beamsplitter, and the second mirror can be moved in the x direction. If mirror 2 is at a distance L ± δ/2 from the beamsplitter, then the difference in path length between the two arms is δ. If δ is an integer multiple of the wavelengths (δ = nλ, n = 1, 2, 3, . . .), we observe constructive interference and the signal at the detector is at a maximum; conversely, if δ = (2n + 1)λ/2, the beams interfere destructively and no light is detected. Hence the instrument measures I (δ), the intensity of the recombined beam as a function of optical path difference. In other words, the interferometer

10.3 Fourier transform spectroscopy

261

(a)

(b)

(c)
B (ν)		I (δ)
(d)
0	5000
Frequency ν (cm−1)		Displacement δ

Fig. 10.7. Different spectra recorded by the interferometer and their Fourier transforms:

(a) monochromatic light; (b) two frequencies; (c) Lorentzian peak; (d) typical spectrum in the mid-infrared range.

converts the frequency dependence of the spectrum B(ω) into a spatial dependence of the detected intensity I (δ). From this information, it is possible to reconstruct mathematically the source spectrum B(ω) no matter what form it has. In Fig. 10.7 we display such interference patterns: a single frequency obviously leads to a signal with a cos2 dependence of the path length difference; interferograms from non-monochromatic sources are more complicated.

262	10 Spectroscopic principles

The mathematical background of a Fourier transform spectrometer is laid out in Appendix A.1; here we apply it to the Michelson interferometer. We can write the electric field at the beamsplitter as

E(x, ν) dν = E0(ν) exp$i(2π xν − ωt)% dν ,

(10.3.1)

where ν = 1/λ = ω/2π c is the wavenumber of the radiation. Each of the two beams which reach the detector have undergone one reflection and one transmission at the beamsplitter, and therefore we can consider their amplitudes to be equal. If one light beam travels a distance 2L to the fixed mirror and the other a distance 2L + δ, then we can write the reconstructed field as

ER(δ, ν) dν = |rˆ||tˆ|E0(ν) exp$i(4π ν L − ωt)% + exp$i 2π ν(2L + δ) − ωt% dν,

where we have assumed that both beams have the same polarization; rˆ and tˆ are the complex reflection and transmission coefficients of the beamsplitter. For a given spectral range, the intensity is proportional to the complex square of the electric field (ER ER). The preceding equation then gives

I (δ, ν) dν E02(ν)[1 + cos{2π νδ}] dν ,

and the total intensity from all wavenumbers at a particular path difference δ is

		∞		E02(ν) 1 + cos$2π νδ% dν .
I (δ)
	0
This is usually written in a slightly different form:
	1			∞
I (δ) −			I (0)		E02(ν) cos	$	2π νδ	%	dν ,	(10.3.2)
		2
				0

often referred to as the interferogram. For a broadband source the intensity at infinite path difference I (∞) corresponds to the average intensity of the incoherent radiation which is exactly half the intensity obtained at equal paths: I (∞) = I (0)/2; the interferogram is actually the deviation from this value at infinite path difference. Finally, from Eq. (10.3.2) and using the fact that B(ν) ≈ E02(ν), we can use the inverse Fourier transformation to write

	∞	1
	B(ν) I (δ) −		I (0) cos	$	2π νδ	%	dδ .	(10.3.3)
		2
	0

Thus, I (δ) can be measured by the interferometer, and it is theoretically a simple task to perform the Fourier transform to arrive at B(ω), the power spectrum of the signal.

10.3 Fourier transform spectroscopy

263

		Interferogram	(a)
Intensity		200
0	1000	2000	3000
	Displacement δ (data points)

	1.0				(b)
	0.8
			Resolution
Intensity	0.6
			32 cm		−1
	0.4
	0.2
	0.0	0.2	0.4	0.6
	0.0
	Frequency ν (104 cm−1)

1.0			(c)
0.8
		Resolution
0.6
		2 cm−1
0.4
0.2
0.0	0.2	0.4	0.6
0.0
Frequency ν (104 cm−1)

	1.0							(d)
	0.8			Resolution
Intensity	0.6				0.2 cm−1
	0.4
	0.2
	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7
	0.0
			Frequency ν (104 cm−1)

Fig. 10.8. (a) The first 3500 points of an interferogram. The signal in the wings is amplified 200 times; (b) Fourier transform of the first 512 points of the interferogram in (a), corresponding to a resolution of 32 cm−1; (c) Fourier transform of 8196 points of the interferogram, corresponding to a resolution of 2 cm−1; (d) taking all 81 920 points leads to a resolution of 0.2 cm−1.

264	10 Spectroscopic principles

The integral in Eq. (10.3.3) is infinite, but for obvious reasons the interferogram is measured only over a finite mirror displacement (δ/2) ≤ (δmax/2). The problem of truncating the interferogram is solved by choosing an appropriate apodization, instead of cutting it abruptly off.1 As demonstrated in Fig. 10.8, this restriction limits the resolution to ν = 1/δmax – as can be easily seen from the properties of the Fourier transformation explained in Appendix A.1 – but it can also lead to errors in the calculated spectrum depending on the extrapolation used. In addition, an interferogram is in general not measured continuously but at discrete points, which might lead to problems like picket-fence effects and aliasing, i.e. the replication of the original spectrum and its mirror image on the frequency axis. However, this allows one to use fast computational algorithms such as the fast Fourier transformation [Bel72, Gri86].

10.3.2 Methods

One limitation of the spectral range of Fourier transform spectrometers is the availability of broadband sources. For a typical black-body radiation source, the peak of the intensity is typically somewhat below the visible. The spectral power is small above and well below this frequency and falls to zero as ω → 0. In general, three different sources are used to cover the range from the far-infrared up to the visible (Fig. 10.9). A set of filters, beamsplitters, windows, and detectors are necessary to cover a wide spectral range. At the low frequency end, in the extreme far-infrared, standing waves between the various optical components are of importance and diffraction effects call for dimensions of the optical components to be larger than a few centimeters. The limitation at higher frequencies is given by the mechanical and thermal stability of the setup and by the accuracy of the mirror motion; this is – as a rule – limited to a fraction of a micrometer. In practice, interferometers are used in two different ways depending on the scanning mode of the moving mirror. For a slow scanning interferometer or a stepped scan interferometer, on the one hand, the light is modulated with a mechanical chopper and lock-in detection is employed. The advantage of a stepped scan interferometer is that it can accumulate a low signal at one position of the mirror over a long period of time; it also can be used for time dependent experiments. A rapid scan interferometer, on the other hand, does not use chopped light because the fast mirror movement itself modulates the source radiation at audio frequency; here the velocity of the mirror is limited by the response time of the detector used. During the experiment, the recombined light from the interferometer is reflected off or transmitted through a sample before being focused onto the detector. Any

1The interferogram is usually multiplied by a function, the apodizing function, which removes false sidelobes introduced into transformed spectra because of the finite optical path displacement.

10.3	Fourier transform spectroscopy	265
		Sources
	Transmission
Detectors	unit

Reflection	Interferometer
unit

Fig. 10.9. Optical layout of a modified Bruker IFS 113v Fourier transform interferometer. The radiation is selected from three sources, guided through an aperture and a filter to the Michelson interferometer with six beamsplitters to select. A switching chamber allows transmission or reflection measurements of the sample inside the cryostat. The light is detected by one of six detectors.

arrangement used in optical measurements can also be utilized in combination with Fourier transform spectroscopy; in the following chapter we discuss the measurement configurations in detail.

A major disadvantage of the standard Fourier transform measurement – common to other optical techniques such as grating spectrometers – is that in general only one parameter is measured.2 As discussed above, this means that the Kramers– Kronig relations must be employed in order to obtain the complex optical parameters such as Nˆ (ω) = n(ω) + ik(ω) or ˆ(ω) or σˆ (ω).

In Fig. 10.10 experimental data obtained by a Fourier transform spectrometer are shown as an example. The polarized optical reflectivity (E a) of Sr14Cu24O41 was measured at room temperature and at T = 5 K over a wide spectral range. The highly anisotropic material becomes progressively insulating when the temperature decreases, as can be seen by the drop in low frequency reflectivity. Above 50 cm−1 a large number of well pronounced phonon modes dominate the spectra; they become sharper as the temperature decreases. In order to obtain the optical conductivity σ1(ω) via Eq. (11.1.1b), the data are extrapolated by a Hagen–Rubens

2In order to measure both components, the sample must be placed in one of the active arms of the interferometer, e.g. replacing the fixed mirror in the case of a highly reflective sample or in front of a mirror in the case of dielectric samples. This arrangement is also called dispersive Fourier transform spectroscopy.

266	10 Spectroscopic principles

Energy hω (meV)

	1.0		100	101	102
			(a)	Sr14Cu24O41
	0.8			E a
			300 K
R
	0.6
Reflectivity
	0.4
			5 K
	0.2
	0.0
		2	(b)
)	10
1
−			300 K
cm
	10	1
1
−
(Ω
1	100
σ
Conductivity	10−1
	10−2		5 K

10−3	100	1000
10

Frequency ν (cm−1)

Fig. 10.10. (a) Frequency dependent reflectivity R(ω) of Sr14Cu24O41 measured at two different temperatures for the electric field E oriented parallel to the a axis. (b) Optical conductivity σ1(ω) of Sr14Cu24O41 obtained by the Kramers–Kronig analysis of the reflection data [Gor00].

behavior (5.1.17) – for a metal as in the case of 300 K – or constant reflectivity

– for an insulator as in the case of 5 K – in the limit ω → 0 and by assuming a smooth decrease of the reflectivity with the functional dependence of R ω−2 for

ω → ∞.

Further reading

267

References

[Bel72] R.J. Bell, Introductory Fourier Transform Spectroscopy (Academic Press, New York, 1972)

[Boh95] R. Bohmer,¨ B. Schiener, J. Hemberger, and R.V. Chamberlin, Z. Phys. B 99, 91 (1995)

[Dav70] S.P. Davis, Diffraction Grating Spectrographs (Holt Rinehart & Winston, New York, 1970)

[Ext89] M. van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989)

[Ext90] M. van Exter and D. Grischkowsky, Phys. Rev. B 41, 12 140 (1990)

[Fel79] Yu.D. Fel’dman, Yu.F. Zuev, and V.M. Valitov, Instrum. Exp. Techn., 22, 611 (1979)

[Gem73] M.J.C. van Gemert, Philips Res. Rep. 28, 530 (1973)

[Gen98] L. Genzel, Far-Infrared Fourier Transform Spectroscopy, in: Millimeter and Submillimeter Wave Spectroscopy of Solids, edited by G. Gruner¨ (Springer-Verlag, Berlin, 1998), p. 169

[Gri86] P.R. Griffiths and J.A. de Haseth, Fourier Transform Infrared Spectrometry (John Wiley & Sons, New York, 1986)

[Gor00] B. Gorshunov, P. Haas, and M. Dressel, unpublished

[Hec92] J. Hecht, The Laser Guidebook, 2nd edition (McGraw-Hill, New York, 1992)

[Hil69] N.E. Hill, W.E. Vaughan, A.H. Price, and M. Davies, Dielectric Properties and

Molecular Behaviour (Van Nostrand Reinhold, London, 1969) [Hyd70] P.J. Hyde, Proc. IEE 117, 1891 (1970)

[Key80] Keyes, R.J., ed., Optical and Infrared Detectors, 2nd edition, Topics in Applied Physics 19 (Springer-Verlag, Berlin, 1980)

[Nus98] M.C. Nuss and J. Orenstein, Terahertz Time-Domain Spectroscopy, in:

Millimeter and Submillimeter Wave Spectroscopy of Solids, edited by G. Gruner¨ (Springer-Verlag, Berlin, 1998)

[Sug72] A. Suggett, in: Dielectric and Related Molecular Processes, Vol. I (Chemical Society, London, 1970), p. 100

Further reading

[Cha71] G.W. Chantry, Submillimetre Spectroscopy (Academic Press, London, 1971)

[Fly87] M. O’Flynn and E. Moriarty, Linear Systems: Time Domain and Transform Analysis (Harper & Row, New York, 1987)

[Han01] P.Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, and X.C. Zhang, J. Appl. Phys. 89, 2357 (2001)

[Kaa80] U. Kaatze and K. Giese, J. Phys. E: Sci. Instrum. 13, 133 (1980)

[Mac87] J.R. Macdonald, Impedance Spectroscopy (John Wiley & Sons, New York, 1987)

[Mil86] E.K. Miller, ed., Time-Domain Measurements in Electromagnetic (Van Nostrand Reinhold, New York, 1986)

[Mit96] D.M. Mittleman, R.H. Jacobsen, and M.C. Nuss, IEEE J. Sec. Topics Quantum Electron. 2, 679 (1996)