1. Introduction

With the aid of ultrashort laser pulses, THz technology has been developed with great progress. In past decades, many methods were proposed for generating and detecting the THz radiation and therefore many applications become realizable such as molecular recognition and material characterization [1–5]. With the aim of promoting these applications, having a simple and convenient method with a compact setup for measuring the spectral characteristic of THz waves is as important as developing a high-power and high-efficiency THz emitter with tunable wavelengths. For the spectral determination of the THz radiation, Fourier transform spectroscopy based on a Michelson Interferometer [6,7] and THz time-domain spectroscopy (THz-TDS) based on photoconductive (PC) [8–10] and electro-optic (EO) [11,12] samplings are the most commonly used methods. However, the construction of the previous spectral characterization setup is relatively complicated because of the invisibility of the THz waves and the long alignment distance. In addition, the setup of THz-TDS also has the disadvantage of lack of tunability, not to mention that THz-TDS requires another intensive femtosecond optical pulse and suffers from the limitation of the detection bandwidth of the photoconductive receiver or the EO crystal. Hence, a transmission-type Fabry-Perot interferometer (FPI) which can be easily aligned with a visible light has recently been proposed to perform the spectral analysis [13]. Taking advantage of its compact size and tunability, the Fabry Perot based spectrometer might be very attractive for many applications. On the other hand the complexity of its spectrum reconstruction rose due to the interferences of multiple beams.

In this letter, we propose and demonstrate a simple Fourier transform spectrometer based on a low-reflectivity Fabry-Perot spectrometer which has the merits of compact size and tunability as well as simple spectral reconstruction algorithm. In the case of low reflectivity, it will be shown that the interference is similar to the two-beam interference so that the spectrum can be easily reconstructed by taking a Fourier transform. This simple Fabry-Perot spectrometer could be one of the best candidates for the portable terahertz spectrum analyzer owing to its convenient features and the mature digital-signal-processing algorithm of fast Fourier transform in IC technology. For putting the concept into practice, we have demonstrated a simple spectrometer which successfully determines the emission spectra of a THz photonic transmitter [14–16] excited by an optical coherent control system [17]. With a simple algorithm for the spectral reconstruction and a compact portable structure with easy alignment, the demonstrated spectrometer will meet the needs of many important applications where a compact and convenient spectrometer is required in the frequency range from sub-THz wave up to even the visible light.

2. Theory and simulation results

For simplification, we take a symmetrical FPI consists of two identical parallel reflecting surfaces into consideration. The power transmission through the parallel surfaces for different spacing d is given by

where P_in is the power spectrum of the incident wave and T(d,f) is the Airy function that represents the power transmission of the FPI at a specific spacing and frequency f If the system is illuminated by a parallel plane wave of a single frequency f₀ with a magnitude P₀, this transmission function T(d,f₀) can be viewed as the normalized output power as a function of FPI spacing, given by

where H=4R/(1-R)² and R is the intensity reflection coefficient, n is the refractive index of medium between two surfaces, θ is the angle from normal incident, and c is the speed of light in vacuum. The magnitude ratio of the i-th to the (i-1)-th round-trip reflected wave at the output surface equals to R ² which represents a round-trip reflection coefficient. In traditional spectroscopic applications, the FPI uses large R to increase the finesse of the instrument. The value of finesse implies the sharpness of the Fabry-Perot mode, thus affects the spectral resolution. However, the frequency range that can be measured correctly is limited by the free spectral range of the FPI. To ease this limitation, we propose a design contrary to the traditional FPI by using partial reflective surfaces with low reflectivity. In the case of low reflectivity, all high-order round-trip reflected waves with sufficiently small magnitude are negligible so that the interference generated by a FPI can be approximated to the two-beam interference. For this reason, we may rewrite Eq. (2) as follows:

From Eq. (3) we can observe that the transmitted power is separated into a dc component and a one-sided cosine function. A perfect mapping relationship exists between the frequency of the cosine function and the original incident wave by a multiplying factor of 2ncosθ/c. Since a broadband signal can be taken as a combination of different frequency components, the transmitted power versus mirror spacing will be also a combination of cosine waves with different frequency. One can thus reconstruct the spectrum of the incident waves by using the algorithm of Fourier transform spectroscopy described as follows: (1) Remove the dc component by subtracting the mean value of the measured signal. (2) Mirror the one-sided signal about y-axis by letting P_out(-d) = P_out(d). (3) Take the Fourier transform of the signal and scaling the frequency axis by multiplying c/2ncosθ.

Figs. 1(a) and 1(b) shows the simulated results of a transmission power as a function of FPI (R~10%) spacing and the reconstructed spectrum when illuminated by a Gaussian pulse with a central frequency of 500 GHz and a FWHM of 100 GHz. It can be observed from the simulated curve that the reconstructed power spectrum restores the Gaussian peak located at 500 GHz with the same FWHM as the incident wave. Nevertheless, the curve rises up slightly near the harmonic frequencies of the incident wave. The artifact to be found at the m-th harmonic frequency results from the interference between the i-th and (i+m)-th round-trip reflected wave. In view of the fact that the phase difference between the two reflected waves will reach 2π every λ/2m spacing, the reconstructed frequency is therefore equal to m times of the original one. Moreover, the interference occurs only when the relative delay of the two waves is less than the coherence length of them, which implies the bandwidth of the reconstructed signal. That is, in concern of the higher-order interferences, the bandwidth of the m-th artificial peak must be m times broader than that of the incident wave due to longer delay after several bounces. Though the artifacts may appear in company with the signal we want, the magnitudes of them are much lower than the true spectrum in our assumption of low reflectivity. Some simple characteristics for judging the significance of the m-th order interference can be expressed as follows:

 

Fig. 1. (a) Simulated transmission power as a function of FPI (R~10%) spacing (solid line) and the mirrored curves (dashed line) after removing the dc value and (b) the reconstructed spectrum when illuminated by a Gaussian pulse with a central frequency of 500 GHz and a FWHM of 100 GHz.

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Equation (4) represents the reconstructed frequencies from a frequency f₁ caused by m-th order interference. The amplitudes of these reconstructed components are shown in Eq. (5). Resulting from above two equations, a consequent FWHM broadening effect of the higher-order artifacts is expressed by Eq. (6). With these relationships, we may eliminate the artifacts higher than our tolerance iteratively and acquire the spectrum more accurately. For example, if a tolerance of 1% is acceptable, that is to say, we can completely ignore the third order interference when the intensity reflection coefficient is less than 17%. In this way, we can measure the spectrum over more than one octave.

3. Experimental setup

Intended for demonstrating the concept experimentally, we used an edge-coupled membrane photonic transmitter [18] to generate a quasi-continuous-wave THz pulse with a tunable central frequency. The micron-sized photonic transmitter was proved to have a record-high light-terahertz conversion efficiency [18] and thus is suitable for many real-world applications. Figure 2 shows the experimental setup of the THz generation and measurement system, where we used a mode-locked Ti:sapphire laser (Spectra Physics Tsunami) and an optical coherent control method to mimic the quasi-continuous-wave excitation. At first, a grating pair was placed after the laser source for stretching the 100 fs pulses with a repetition rate of 82 MHz and a central frequency of 850 nm into linearly chirped pulses of 8.7 ps. Then, the stretched pulses were guided into a Michelson interferometer to modulate the pulse envelope with a tunable beat frequency [17]. The beat frequency is proportional to the relative delay of two arms by a factor of 0.885 THz/ps (calibrated with an autocorrelator) with a lower limit of ~150 GHz due to the finite pulsewidth. At last, we used an 80x objective to couple the modulated pulses into the photonic transmitter. The radiated THz wave was collected by two off-axis parabolic reflectors with the low-reflectivity Fourier transform spectrometer located between them.

 

Fig. 2. Experimental setup of the THz excitation and measurement system. The wavelength of the THz wave was tuned by changing the relative delay τ of the Michelson interferometer. The spectrum of the radiated wave was determined by measuring the transmitted power for different FPI spacing d.

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Figure 3(a) shows a prototype of the Fourier transform spectrometer which was made of two sapphire backed metal meshes as the partial reflective mirrors. The metal mesh was composite of square holes arranged in a square lattice, lithographically fabricated on the 2-inch double polished sapphire wafer with 20nm-thick Ti. The lattice constant is 90 μm and each square hole has the size of 63μm × 63μm. Figure 3(b) shows the mesh pattern photographed with a microscope and a CCD camera (The white part is metal and the dark part is the substrate). The measured reflectivity of the mesh in the sub-THz to THz range is about 10 % which is suitable for the terahertz Fourier transform spectrometer. In the construction of the FPI, we mounted the meshes on an aluminum ring which was fixed by an ordinary mirror mount for adjusting the tilt of the meshes. One of them was carried by a high resolution motorized translation stage (Newport MFN25CC) to control the spacing of the interferometer. For checking the parallelism of two meshes, the interferometer was illuminated by a pulsed laser of normal incident. Subsequently, we used a laser spectrometer to verify the fringes of the spectrum caused by the overlap of multiple transmitted beams at different spacing. The fringe will become more obvious when two meshes are more close to parallel. THz wave passing through the Fourier transform spectrometer was detected by an Infrared Laboratories liquid helium cooled silicon composite bolometer. The signal is obtained by measuring the amplified output of the bolometer using lock-in amplifier for noise reduction.

 

Fig. 3 (a) A prototype of the Fabry Perot based Fourier transform spectrometer. (b) The photographed mesh pattern under a microscope. The white part is metal and the dark part is the substrate.

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4. Measurement results

Figure 4 shows the measured transmitted power for different mirror spacing and the reconstructed radiation spectra when the optical beating frequency was tuned to ~460 GHz and ~520 GHz. The measured THz spectrum shows a peak radiation frequency of 442 GHz and 517 GHz which are close to the optical excitation frequency of the photonic transmitter, respectively. It can be observed from Fig. 4(b) that an obvious dip appears in the spectrum around 485 GHz. We attribute this peak to the absorption of the atmosphere since the system is exposed to the free space directly [19]. The experimental result reveals that the radiation from the photonic transmitter is quasi-continuous-wave, following our expectation. Just like the ordinary Fourier transform spectrometer, the spectral resolution of the system is determined by the maximum spacing of the interferometer while the highest frequency that can be reconstructed is on the other hand limited by the spatial resolution of the FPI. For our simple and compact system, a short traveling range of 2.5 cm corresponds to a spectral resolution of 6 GHz. Higher spectral resolution can be achieved with longer traveling distance between two parallel plates. On the other hand, with a step-size shorter than 0.1 μm, the highest resolvable frequency can be up to the range of infrared light. As a result, this type of simple and compact spectrometer could have many convenient applications in the wide frequency range in addition to the THz region.

 

Fig. 4 The reconstructed spectra and the measured FPI traces after removing the dc values (insets) when the optical coherent control frequencies are (a) ~460 GHz and (b) ~520GHz.

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5. Conclusion

In conclusion, we have demonstrated a compact Fourier transform spectrometer with a low-reflectivity Fabry-Perot interferometer. We theoretically derived an algorithm and discussed its performance as a function of the reflectivity qualitatively. A prototype of the Fourier transform spectrometer which was made of two sapphire backed metal meshes as the partial reflective mirrors was demonstrated and was applied to the determination of the photonic transmitter emission spectra. With a broadband characterization capability and a simple spectral reconstruction algorithm, this instrument can provide a convenient and easy spectral measurement solution to many THz applications.

The authors gratefully acknowledge financial support from National Science Council of Taiwan (Grant No. 93-2215-E-002-040) and NTU Center for Genomic Medicine.