TERAHERTZ SPECTROSCOPY: Calibration tool aids terahertz researchers

Accurate and reliable calibration of terahertz time-domain spectrometers is essential to spectroscopic investigations in the frequency range from 0.1 to 3 THz. A "calibration suite" makes such adjustments routine and provides users with confidence in their measurements.

Aug 27th, 2009
Fig1a Naftaly

Accurate and reliable calibration of terahertz time-domain spectrometers is essential to spectroscopic investigations in the frequency range from 0.1 to 3 THz. A "calibration suite" makes such adjustments routine and provides users with confidence in their measurements.

MIRA NAFTALY AND RICHARD DUDLEY

The growth of research and applications at terahertz frequencies has led to time-domain spectrometers (TDS) emerging as a key measurement device for spectroscopic investigations in the frequency range from 0.1 to 3 THz. However, spectral measurements produced by terahertz TDS are only accurate and reliable if the spectrometers have been correctly calibrated. Until recently this wasn't easy to achieve. We have developed methods for calibrating the frequency and amplitude scales of a terahertz TDS, and are working towards development of a "calibration suite" for these systems.1, 2, 3

Calibrating frequency

Terahertz TDS measurements are made in the time domain using a pump-probe configuration. The time domain data is then converted to a frequency spectrum by applying the Fourier transform, calculated numerically using the Fast Fourier Transform (FFT) algorithm.

A terahertz TDS may suffer from several sources of frequency error, including delay line positioning errors and various sources of spatial beam distortion. Applying FFT to the data tends to amplify the effect of these errors and produces frequency uncertainties that are difficult to quantify. Although a formalism capable of quantifying such errors has been developed, in practice these techniques and calculations are difficult, and it remains far easier, quicker and more reliable in the context of a TDS to use a calibration standard.

Perhaps the most widely used means of frequency verification in the terahertz band is atmospheric water vapor, which possesses many strong narrow lines. However, although the frequencies of these lines are well known, their relative amplitudes vary with environmental conditions, such as atmospheric pressure and humidity. Moreover, many of the lines are doublets and triplets so require very high (subgigahertz) frequency resolution to define their peak maxima and profiles. The spacing of lines is particularly dense at frequencies above 2 THz, where the reduced signal-to-noise ratio and dynamic range of a typical TDS make accurate measurements more difficult.

We have developed a method for calibrating the frequency scale of a TDS system using etalon type interference, which provides a comb of peaks and troughs across the measurement band, together with a well-defined spectral profile. The test utilizes the occurrence of secondary peaks, or echoes, produced by multiple reflections in thin plane-parallel samples inserted in the terahertz beam. These echoes give rise to etalon-like oscillations in the calculated spectrum, where the peaks and troughs occur at regularly spaced frequency intervals determined by the optical thickness of the etalon.

For terahertz radiation two types of etalon should be considered: air-gap and solid wafer. Both have advantages and drawbacks. An air-gap etalon consists of two partially reflecting plates separated by a fixed-thickness spacer. It has the advantage that the refractive index is that of air and thus may be assumed to be unity. Frequency calibration using an air-gap etalon is therefore self-contained and potentially traceable. Such an etalon must, however, be carefully engineered and maintained to preserve its spacing to a high accuracy, and it may also be sensitive to temperature variations and atmospheric humidity. Uncoated silicon (Si) plates of sufficient thickness may provide a convenient and inexpensive engineering solution to a stable, rigid, and robust air-gap etalon (see Fig. 1). An alternative is a solid wafer made from a terahertz-transparent, high-refractive-index material, such as silicon. Such wafers are relatively insensitive to environmental factors and mechanically robust. Their main disadvantage is that the refractive index must be independently measured or otherwise known.

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FIGURE 1. An air-gap etalon consisting of uncoated silicon plates can be used for calibrating the TDS frequency scale. (Courtesy of NPL)

The simplest and most straightforward method of analyzing the spectral data for the purpose of frequency calibration is to note the frequencies of the peaks and troughs in the transmission spectrum of the etalon obtained via the usual FFT, then to compare these with the calculated peak/trough frequencies expected in an etalon. The results may be displayed by plotting the differences—that is, the frequency errors—as a function of the etalon peak/trough position. Such a plot would reveal any systematic frequency error, as well as the digitizing errors and the noise in the data. It also helps to identify the band over which frequency measurements are valid within a defined uncertainty (see Fig. 2).

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FIGURE 2. The actual frequencies of the peaks and troughs in the transmission spectrum of the etalon are compared with the calculated peak/trough frequencies expected in an etalon and the results are displayed by plotting the differences as a function of the etalon peak/trough position. Frequency errors of a TDS with incorrect delay show frequency scaling (left) are compared to frequency errors of a TDS with a correct frequency scale (right). (Courtesy of NPL)

A more accurate, although more elaborate and computation-intensive, analysis of the data can be obtained by applying a form of least-squares fitting, that is, by calculating the root-mean-square (RMS) difference between the model of etalon transmission and measured transmission, and then minimizing it by applying a frequency-scaling factor. This scaling factor identifies the true frequency calibration of the TDS being tested. It is also possible to perform the calibration separately for different frequency bands.

Calibration of linearity

The analysis and interpretation of spectroscopic data also rely crucially on the assumption that the measurement of amplitude is linear—that the recorded signal is proportional to the terahertz field across the entire dynamic range of the system. Factors such as beam distortion and dispersion may affect the amplitude linearity of a TDS system.

In practice, however, the linearity of terahertz TDS data acquisition is seldom tested and no published literature exists. It is important that the linearity of a TDS should not be assumed, but is experimentally verified.

The quantification of power/amplitude linearity of terahertz systems and detectors requires a measurement device that has a constant loss across the terahertz band. The loss must be capable of being varied accurately in reproducible steps spanning the dynamic range of the system being tested. Absorbing or scattering materials cannot be used because both absorption and scattering are strongly frequency-dependent. Beamsplitters, such as wire-grid polarizers and pellicles, have a constant loss over a large bandwidth, but are polarization-dependent. Moreover, the angle of a wire-grid polarizer must be controlled with subdegree accuracy in order to obtain the desired loss factor. This makes reproducible measurements difficult to achieve, especially in the high frequency part of the range, where they are arguably particularly important in spectroscopic studies.

The preferred solution is to use as "loss elements" a stack of optically flat silicon plates. High-resistivity float-zone silicon has negligible absorption and dispersion in the band of 0.1-6 THz. Therefore transmission loss through a silicon plate is due only to Fresnel reflection. The loss produced by a stack of plates separated by air gaps is multiplicative, to the power equal to the number of plates in the stack. The plates are mounted on a block designed so as to allow easy alignment of the plates parallel to each other and normal to the incident terahertz beam. The plates are robust and easy to handle, while the total length of the device is conveniently small to be easily used in a variety of optical setups (see Fig 3).

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FIGURE 3. Amplitude linearity of a TDS can be calibrated using a stack of optically flat silicon plates as loss elements. The plates are mounted on a block that allows easy alignment of the plates parallel to each other and normal to the incident beam. (Courtesy of NPL)

An important concern is the Fabry-Perot etalon effect whereby multiple reflections within a plate interfere to form a standing wave, resulting in frequency-dependent transmission that for a silicon plate varies between 1 and 0.35. However, the effect does not arise in the case of terahertz emitters activated by femtosecond lasers, as used in TDS, because their pulse length is of the order of 1 ps, which is shorter than the round-trip travel time within a thick plate, preventing the formation of standing waves.

In the absence of etalon interference, the single-pulse amplitude transmission factor through a stack of silicon plates is 0.7 for each plate—a loss of 0.3 per plate. Testing the linearity of a TDS can be done in two ways: by looking at the time-domain trace, or by considering particular frequencies in the calculated spectrum.

The first method verifies the linearity of the acquired time-domain signal, and is frequency-averaged. It is performed by plotting the amplitude of the time-domain peak maximum against the number of silicon plates in the beam path. For a linear system, a semi-log plot of the data will be linear with a slope of 0.7. The second method of testing a TDS involves calculating terahertz spectra and considering their amplitudes at chosen frequencies. The amplitude at each frequency is then plotted against the number of silicon plates in the beam path. As previously, the plots are expected to be linear with a slope of 0.7 (see Fig. 4).

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FIGURE 4. A misaligned TDS exhibits a nonlinear amplitude response, particularly at low frequencies (left). After realignment the same TDS exhibits a linear response (right). (Courtesy of NPL)

A calibration suite

Calibration of frequency and amplitude is an important aspect of spectroscopic measurements and is routinely practiced in optical and near-infrared instruments. Terahertz spectroscopy, however, tends to be viewed as a developing field, where strict calibration methodologies have not yet been widely adopted. This motivated our work on developing a suite of calibration techniques and standards for terahertz instruments, and in particular for TDS. NPL currently offers the frequency and linearity calibration kits with accompanying calibration certificates. In addition to frequency and amplitude linearity, we have designed and fabricated spatial and depth resolution masks for imaging systems. Work is also in progress on developing reflectivity standards to support the growing area of terahertz reflection spectroscopy

ACKNOWLEDGMENT
This work has been funded by the National Measurement Office of the U.K. Department for Business, Innovation and Skills.

REFERENCES
1. M. Naftaly, R. A. Dudley, J. R. Fletcher, F. Bernard, C. Thomson, Z.Tian, J. Opt. Soc. Am. B 26(7), p. 1357 (2009).

2. M. Naftaly, R. A. Dudley, Opt. Lett. 34(5), p. 674 (2009).

3. Z. Tian, R. A. Dudley, L. Jayes, R. S. Hutton, Infrared and Millimeter Waves, 2007, and the 2007 15th Int'l. Conf. on Terahertz Electronics, joint 32nd Int'l. Conf., Sept. 2-9 2007, p. 1014.

MIRA NAFTALY and RICHARD DUDLEY are Senior and Principal Research Scientists respectively at National Physical Laboratory (NPL), Hampton Road, Teddington, Middlesex TW11 0LW, UK; e-mail: mira.naftaly@npl.co.uk or Richard.Dudley@npl.co.uk; www.npl.co.uk/electromagnetics/terahertz/research

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