High-Precision Measurement of Terahertz Frequency Using Unstabilized Femtosecond Laser

A method for terahertz frequency measurement by using unstabilized femtosecond laser is introduced. The frequency of repetition rate and beat signal are continuously measured by two frequency counters. The terahertz frequency can be calculated from the data with accuracy in the order of 10-10. Compare to the method by using stabilized femtosecond laser, the system is greatly simplified.


Introduction
Terahertz (THz) electromagnetic wave, lying at the boundary between microwave and the infrared, has emerged as a new method for spectroscopy, imaging, communication and other applications.Frequency is one of the most important physical quantities of EM waves.Precision measurement of THz frequency using THz comb was demonstrated by Yasui and co-workers [1][2].The repetition frequency of the femtosecond laser need to be stabilized with complicated control devices [3][4].And it is necessary to measure beat frequencies before and after shifting repetition frequency in order to determine the comb mode number that is nearest to the test THz wave.However, the use of stabilized femtosecond lasers may hinder its wide use.In this paper, we demonstrate a method for terahertz frequency measurement by using an unstabilized femtosecond laser.

Principles
A femtosecond laser contains a series of frequency combs with the frequency of _ , where r f represents the laser pulse repetition frequency and 0 f represents the carrier envelope phase shift.The photoconductive antenna generates broad band THz radiation excided by the femtosecond laser.0 f is offset during the difference-frequency processes.Thus, the THz frequency comb can be expressed as _  .Thus, the following relationship can be obtained: The frequency of the measured THz source can be expressed as:

Experimental Setup
The femtosecond laser used was a femtosecond fiber-laser manufactured by Toptica Photonics AG, with the output central wavelength of 1550 nm, the pulse width of 80 fs, and the pulse repetition frequency of 80 MHz approximately.After outputted from the femtosecond laser, the laser was divided into two beams by a fiberoptic coupler, one beam connected to the fiber-coupling THz photoconductive antenna to generate a THz frequency comb.The other laser beam was detected by a high-speed photodiode, which inputted the acquired electric pulse signal into the frequency counter I to measure repetition frequency of the femtosecond laser.
After the output signal from the frequency synthesizer was multiplied six times by the frequency multiplication module, a THz signal with the frequency range of 75-110 GHz was generated.The THz signal was irradiated on the surface of the photoconductive antenna, focused by silicon lens and then interacted with the THz frequency comb.The beat frequency signal obtained was amplified and inputted into the frequency counter II for measurement.Both frequency counters and the frequency synthesizer in the measurement system referred to the rubidium frequency standard (short-term stability: 2×10 - 11 , accuracy: 5×10 -11 ) so as to ensure accuracy of results of measurement.

Results and Discussions
As the first step, the variation of repetition frequency in 6h with the femtosecond laser in operation was continuously measured.The repetition frequency continuous reduced approximately 8 kHz in 6h, which was the result of a lengthened laser cavity due to the thermal expansion caused by continual temperature rise after the laser was switched on.As time went by, the laser gradually obtained thermal equilibrium.Accordingly, the reduction of repetition frequency was slowed down and relatively stabilized in 3h after the laser was switched on, which varied within a range of 100 Hz depending on changes in ambient temperature., the beat frequency showed the same variation trend with repetition frequency, separately corresponding to two conditions shown in Figure 1.The value of m was obtained with the following algorithm according to measured repetition frequency and beat frequency : Where, N stands for the total number of data and M stands for the selected data interval.In calculation, it was found that the smaller the value of M was, the bigger the error in calculation of m would be.When M>20, an accurate value of m could be obtained.The frequency of the measured THz source was calculated as follows: The frequency of the test source were set to be 91,546,980,000 Hz and 92,741,460,000 Hz respectively.The calculated frequency were 91,546,979,971 Hz and 92,741,459,985 Hz according to Eq. ( 4).The error are 29Hz and 15Hz, with precision 3.2×10 -10 and 1.6×10 -10 .
Comparing with the results of references, although the measurement precision of the method adopted in the study dropped from the order of 10 -11 to the order of 10 -10 , the measurement system was greatly simplified.Instead of carrying out complicated stabilization control over repetition frequency of the laser, it was only needed to simultaneously acquire repetition and beat frequencies using two frequency counters.This method will greatly expand the applicable scope of the frequency comb method in measuring THz frequency.
As shown in Figure 1, the frequency of the measured THz source can be expressed as THz r b f m f f = ⋅ ± , where b f represents the beat frequency of the measured THz source and THz frequency comb.In the equation, both r f and b f can be directly measured by using a frequency counter.The only unknown value is m, which must be obtained to determine the frequency of the measured THz source.

Figure 1 .
Figure 1.Principle of THz frequency measurement If interval of the THz frequency comb varies from r f to r r f f δ + , the beat frequency signal will change from b f to b

Figure 2 .
Figure 2. Results of continuous measurement of repetition frequency and beat frequency signal in the same period (a) Opposite variation trend (b) Same variation trend According to the principle of measurement, the variation of repetition frequency r f will result in corresponding change in the beat frequency b f .Figures 2 present the results of measurement of r f and b f in a continuous 40minute period.When THz r f m f > ⋅ , the beat frequency showed an opposite variation trend against repetition frequency; when THz r f m f < ⋅