Terahertz frequency tunability has been studied in the past years [1]. This is motivated by many applications, such as terahertz communication [2], high-resolution terahertz spectroscopy [3], and radioastronomy [4]. One of the methods for generating a tunable terahertz signal is based on optical photomixing (also called optical beating or the optical heterodyne technique) [5]. The operating principle of photomixing is to combine two optical-frequency signals by using a photoconductive antenna or uni-traveling-carrier photodiode (UTC-PD). A dual-wavelength fiber laser, which may be made by different technical methods, can be used to irradiate the photomixer [6]. By changing the frequency interval of the two optical signals, the frequency of the terahertz signal generated from beating can be changed accordingly. However, wavelength tuning of the external cavity increases the complexity of a terahertz signal generator. In addition, due to incoherence of the input laser, it is difficult to obtain a terahertz signal with low noise. By using a fiber Bragg grating or injection-locked laser, two suitably spaced comb lines can be chosen from an optical-frequency-comb generator. With photomixing of the two tunable comb lines, a frequency-tunable terahertz signal can be generated [7, 8]; The disadvantage is that the intensity of each longitudinal mode within the central spectrum is roughly equal, and small, on both sides of the center of the spectrum. This means the energy falling into the frequency components for photomixing is very limited, and so the energy-utilization efficiency of the terahertz-signal-generation system is low. Our previously reported technique using the ±1-order Kelly sidebands of mode-locking lasers and photomixing to generate a terahertz signal has the advantages of good stability and high-intensity sidebands [9]. It is important to point out that our previous works [9, 10] were not particularly focused on optimizing the maximum of ∆λ_±1, but in this work, we focus on the dispersion dependence of the Kelly-sideband spacing; Then frequency-tunable terahertz signals can be generated by photomixing.

In this paper, we propose to use the ±1-order Kelly sidebands with dispersion-dependent spacing to generate a frequency-tunable terahertz signal. The principle of dispersion dependence of the Kelly sidebands is analyzed. A numerical model for simulating the dispersion dependence of Kelly sidebands is given. A dispersion-management mechanism introduced into a fiber-laser cavity is proposed to generate Kelly sidebands with dispersion-dependent spacing. Two kinds of mode-locking fiber lasers are designed to generate dispersion-dependent ±1-order Kelly sidebands, and the frequency spectrum of the generated tunable terahertz signals is worked out. Some conclusions are given at the end.

Kelly sidebands are an important characteristic of a soliton fiber laser. They result from dispersive waves emitted by perturbed optical solitons in an anomalous-dispersion laser cavity. To be specific, during many round trips the soliton resonantly couples to a copropagating dispersive wave. The quasi-matching between their relative phases results in constructive interference, with multiple pairs of sharp spectral peaks added to the soliton’s spectrum. An analytical model for the spacing ∆λ_±1 of the ±1-order Kelly sidebands is expressed as [11]

(1)$$\Delta {\lambda }_{\pm 1}=2\lambda {}_0\sqrt{\frac{2}{cDL}-0.0787\frac{{\lambda }_0^2}{{\left(c\tau \right)}^2}}$$

where λ₀ is the central wavelength of the laser, D is the average intracavity dispersion parameter, L is the cavity’s length, DL is the total dispersion of the laser cavity, τ is the pulse width (full width at half maximum), and c is the light speed. This model shows the inverse square-root dependence of ∆λ_±1 on DL of the cavity when the pulse width τ is constant. Moreover, the change of pulse width τ also influences ∆λ_±1. By changing the total dispersion of the laser cavity or the pulse width of the laser quantitatively, the spacing of the ±1-order Kelly sidebands can be changed, which means that the frequency of the generated terahertz signal can be tuned.

To determine the dependence of ∆λ_±1 on DL and τ, the relationship between DL and ∆λ_±1 for different values of the pulse width τ is plotted in Fig. 1. It is seen that the ∆λ_±1 curves in Fig. 1 exhibit similar variation trends with changing DL value of the laser cavity. With DL increasing from 10 to 40 ps/nm, ∆λ_±1 decreases from 2.5 to about 1 nm. In particular, when DL is less than 10 ps/nm, the four curves are approximately the same in variation (not shown in Fig. 1). This means that the change in pulse width does not contribute to Kelly sideband spacing when the total cavity dispersion is reduced to less than 10 ps/nm, say. In addition, for a given DL value, ∆λ_±1 increases with increasing pulse width τ, ∆λ_±1 as shown by the points in Fig. 1. However, the influence of pulse width τ on ∆λ_±1 is obviously less than that of the total dispersion DL on ∆λ_±1. For example, as pulse width τ increases from 4 to 20 ps when DL equals the maximum of 40 ps/nm in Fig. 1, the increase in ∆λ_±1 is only about 0.66 nm. Comparing changes in ∆λ_±1 introduced by changes in DL and τ, it can be seen that changes in DL can lead to a large change range of wavelength spacing ∆λ_±1. Therefore, this is suitably used to generate tunable terahertz signals over a wide range of frequencies. To simulate the dispersion-dependent Kelly sidebands of a soliton Er-doped fiber laser, an extended nonlinear Schrodinger equation (NLSE) including a set of laser parameters is used. The extended NLSE can be expressed as

Figure 1. Dependence of the ±1-order Kelly sideband spacing Δλ_±1 on total dispersion DL for different values of pulse width τ.

(2)$$\frac{\partial E}{\partial z}+\frac{\alpha }{2}E+\frac{i{\beta }_2}{2}\frac{{\partial }^{2}E}{{\partial }^{2}t}-\frac{g}{2}E-\frac{g}{2{\text{Ω}}_g^2}\frac{{\partial }^{2}E}{{\partial }^{2}t}=i\gamma {\left|E\right|}^2$$

where E is the electric field amplitude of the slowly varying pulse envelope, α is the attenuation constant, g is the gain coefficient of the Er-doped fiber (EDF), which is related to the small signal gain coefficient g₀, gain saturation energy E_sat, and optical field energy. g₀ is related to the fiber’s doping concentration, and E_sat is dependent on pump power (PP). γ is the nonlinear coefficient. β₂ is the group-velocity dispersion. Dispersion D and β₂ can be interconverted through the formula D = −2πc β₂/λ². Ω_g is the gain bandwidth. The symmetric split-step Fourier method is implemented in the simulation. It is noted that by changing the β₂ value of the laser cavity gradually, the relationship between the total dispersion and the spacing of the ±1-order Kelly sidebands can be obtained. In addition, if a discrete dispersion-management component such as a chirped fiber Bragg grating (CFBG) is added into the laser cavity, to simulate this situation the dispersion operator D^{^} used in the split-step Fourier method can be rewritten as D^{^} = iβ_CFBG ω²/2 in the frequency domain, where β_CFBG is the dispersion value of the CFBG.

According to our previous experimental and theoretical investigation, a mode-locking fiber laser generating stable and obvious Kelly sidebands should be designed carefully. Because a stable optical soliton including Kelly sidebands needs balance between dispersion and nonlinearity in a laser cavity, the variation of these two factors needs to be set within a reasonable range. Meanwhile, the dispersion and nonlinearity of a laser cavity are also related to the spacing and intensity of the ±1-order Kelly sidebands.

In order to generate ±1-order Kelly sidebands with widely tunable spacing, a new method that is a dispersion-management mechanism is introduced into the laser cavity, to change the total dispersion of the cavity over a large range. This new method has never been mentioned in our previous works [9, 10]. Two kinds of mode-locking fiber lasers with dispersion-management components are designed. The structures of the two lasers are shown in Fig. 2. The laser shown in Fig. 2(a) has a relatively small range of total-dispersion variation, and the other has large total dispersion. The dispersion-management components in these two fiber lasers are a dispersion-shifted fiber (DSF) and a CFBG respectively. By changing the dispersion of the dispersion-management components, the total dispersion of the two mode-locking fiber lasers can be changed to cover a range from 0.065 to 50 ps/nm. The simulation parameters for the two mode-locking fiber lasers are shown in Table 1. The dispersion of the EDF is 36 ps/(nm km), and that of the single-mode fiber (SMF) is 17 ps/(nm km). For the fiber laser in Fig. 2(a), by using an 11-m DSF with a different dispersion value, the total dispersion of the fiber laser can be changed over a range from 0.065 to 1.03 ps/nm. Moreover, in this fiber-laser structure, due to the absence of intracavity bandpass filtering the spacing of the ±1-order Kelly sidebands can be large. If there is a bandpass filter inside the laser cavity, the spacing of ±1-order Kelly sidebands is limited and the sideband strength is weakened. For the fiber laser in Fig. 2(b), thanks to the CFBG with a large dispersion value, the total dispersion of the fiber laser can change from 1.1 to 50 ps/nm continuously. To avoid the bandwidth-limiting effect of the CFBG, the bandwidth of the CFBG is reasonably set to 8 nm. In addition, the coupling ratio of the output coupler is 50%. For the semiconductor saturated absorbable mirror (SESAM) used in the structure to generate short laser pulses, the recovery time is 6 ps and the saturable fluence is 60 μJ/cm². The contrast of the SESAM (i.e. its nonlinear reflectivity change) is 14%. The unsaturable loss of the SESAM is 8%. It is noted that the optical circulators for routing the SESAM and CFBG have insertion loss to the light traveling in the cavity, but this loss is not included in the simulation, for simplicity. In practice the loss can be offset by using a high-gain EDF and high PP.

TABLE 1. Parameters for the mode-locking fiber lasers.

Parameter		A	B
a)Fiber Length	LEDF (m)	1.5	1
	LHDF or LSMF (m)	11	5.8
b)Dispersion	DEDF [ps/(nm km)]	36
	DSMF [ps/(nm km)]	N/A	17
CFBG	c)BWCFBG (nm)	N/A	8
	d)RCFBG	N/A	1

Figure 2. Mode-locking fiber lasers for generating dispersion-dependent Kelly sidebands. The dispersion-management components in these two fiber lasers are (a) dispersion shifted fiber (DSF), and (b) chirped fiber Bragg grating (CFBG). WDM, wavelength division multiplexer; EDF, Erbium-doped fiber; SESAM, semiconductor saturable absorber mirror; SMF, single-mode fiber; OC, output coupler.

By changing the dispersion values of the DSF and CFBG gradually, optical solitons with Kelly sidebands of different spacings can be simulated numerically. The simulation results are shown by the red circles in Fig. 3(a). In particular, the Kelly sidebands with spacing ∆λ_±1 of greater than 7.4 nm are generated by the laser structure of Fig. 2(a), while the Kelly sidebands with spacing ∆λ_±1 of less than 7.4 nm are generated by the laser structure of Fig. 2(b). Then, using the model of Eq. (1) the spacing of the ±1-order Kelly sidebands can be calculated; The results are shown by the black squares in Fig. 3(a). There are similar variation trends for the results from the analytical calculation and numerical simulation. The pulse width of the soliton for each case is also recorded and shown in FIG. 3(b). It is shown that the pulse width τ increases with increasing DL. Besides, in the numerical simulation, increasing the total dispersion of the laser cavity requires increasing the PP of the laser accordingly, which is shown in Fig. 3(c). This is because a stable single soliton needs to balance the dispersion and nonlinearity of the laser cavity. With increasing DL the intracavity nonlinearity must be increased, which can be achieved by increasing the PP. Moreover, increasing the PP can also increase the intensities of the Kelly sidebands to a certain extent, which is useful for generating a terahertz signal with large signal-to-noise ratio (SNR).

Figure 3. Relationships between the total dispersion DL of the soliton mode-locking fiber laser and (a) spacing of the ±1-order Kelly sidebands Δλ_±1, (b) pulse width τ, and (c) pump power (PP). In (a), the red circles are the results of numerical simulation, while the black squares are analytical results. The marked points at different simulation conditions (i), (ii), (iii) and (iv) have their corresponding optical spectra, which are shown later in Fig. 4.

The four special optical spectra, including dispersion-dependent Kelly sidebands, labeled in Fig. 3(a) are shown in Fig. 4. According to the relationship between DL and ∆λ_±1, for cases (i) and (ii) a slight change in the dispersion of the DSF can create a large change in ∆λ_±1. For cases (ⅲ) and (iv), a large dispersion change leads to a small change in ∆λ_±1. The intensities of the Kelly sidebands with different spacings are different. Because the intensity of the Kelly sidebands depends not only on the dispersion but also on the nonlinearity of the laser cavity, this intensity can be enhanced by increasing the nonlinear coefficients of the laser cavity. Therefore, a CFBG with large dispersion is used in the laser. It can be seen clearly from Fig. 4. that stable Kelly sidebands appear on both sides of the spectra symmetrically. In the four figures, the spacing of ±1-order Kelly sidebands reduces from 15.6 to 1.04 nm when the total dispersion of the laser cavity increases from 0.24 to 50 ps/nm.

Figure 4. Optical spectra with different spacings of the Kelly sidebands. (a), (b), (c), and (d) are marked points (i), (ii), (iii), and (iv) in Fig. 3, respectively. The spacings of ±1-order Kelly sidebands shown in panels (a) to (d) are 15.6, 8.3, 3.2, and 1.04 nm.

To explore the maximal spacing of the ±1-order Kelly sidebands, the total dispersion DL of the laser cavity is further reduced. When the dispersion of the DSF is 4 ps/nm and the other parameters are the same as those in Table 1, the calculated optical spectrum is shown in Fig. 5(a) and ∆λ_±1 is 24.2 nm, which is slightly larger than the spacing of the Kelly sidebands in [12]. On this basis, if the dispersion of the DSF is reduced further to 1 ps/nm and the total dispersion DL of the laser cavity is 0.065 ps/nm accordingly, the PP should be decreased to 24.6 mW to avoid pulse splitting. The calculated optical spectrum is shown by the dotted line in Fig. 5(b). It can be seen that in this case ∆λ_±1 is 29.5 nm, which is similar to the spacing in [13]. It is also found that with decreasing the total dispersion DL, the intensity of the ±1-order Kelly sidebands is reduced accordingly, and excessive changes in DL can cause sideband instability. Thus there is a trade-off consideration among the parameters of the spacing, intensity, and stability of the ±1-order Kelly sidebands. To keep the sidebands with maximal spacing while having high intensity, the method mentioned in [10] is used. The calculated optical spectrum is shown by the solid line in Fig. 5(b). It can be seen that intensity of the Kelly sidebands with maximal spacing is increased by 10 dB, which is beneficial for practical applications. It is important to point out that the spacing of the Kelly sidebands is calculated to be as large as 29.5 nm, which is believed to approach the maximum. Combined with Fig. 4, it is found that the total variation range of ∆λ_±1 is about 28.46 nm.

Figure 5. Optical spectra with large spacing of the Kelly sidebands. The spacing of the ±1-order Kelly sidebands shown in (a) and (b) are 24.2 nm and 29.5 nm respectively. In (b), the solid line represents the enhanced-sideband spectrum, while the dotted line represents the unenhanced-sideband spectrum.

The photomixing process can be described mathematically. The optical components of the peaks of the ±1-order Kelly sidebands are written as E₁(t) = |E₁|exp[j(ω₁t + φ₁)] and E₂(t) = |E₂|exp[j(ω₂t + φ₂)] respectively, where ω is the angular frequency and φ is the initial phase. The intensity of the terahertz wave generated by photomixing two optical components is I_mix = |E₁|² + |E₂|² + 2|E₁||E₂|cos[(ω₁ − ω₂)t + (φ₁ − φ₂)]. It can be seen that the photomixing process produces a component whose frequency is equal to the difference between the two optical component frequencies, ω_mix = ω₁ − ω₂. This means that the frequency of the generated terahertz signal is determined by the frequency difference of the two laser wavelengths, while the intensity of the terahertz signal is related to the intensities of the lasers. Based on the principle of photomixing and by using the optical spectrum, the whole optoelectronic spectrum can be calculated and derived. By using the optical-spectra data of Fig. 4, calculations are carried out to obtain the frequency spectra of the generated terahertz signals. The results are shown in Fig. 6.

Figure 6. Frequency spectra of the terahertz signals generated by photomixing the ±1-order Kelly sidebands with dispersion-dependent spacing in the laser spectrum. The terahertz frequency spectra from (a)–(d) are calculated using the optical-spectra data shown in Figs. 4(a)–4(d) respectively.

Each frequency spectrum is normalized to its own maximum. In addition, the frequency axis is limited to a suitable range for showing the generated terahertz signal clearly. It can be seen that the ±1-order Kelly sidebands with large spacing correspond to a high-frequency terahertz signal, and vice versa. To be specific, the optical spectrum with 15.6-nm spacing of ±1-order Kelly sidebands shown in Fig. 4(a) can generate the signal of 1.48 THz shown in Fig. 6(a). In our study, the minimum frequency of the generated signal is 0.09 THz, which corresponds to Kelly sidebands with a spacing of 1.04 nm. Similarly, using the spectra data of Fig. 5 in the photomixing operation, the generated THz signals are shown in Fig. 7. It can be seen that the higher frequency is 2.27 THz. The total frequency-variation range of the generated signal is about 2.18 THz.

Figure 7. Frequency spectra of the terahertz signals calculated using the optical-spectra data shown in Fig. 5. The terahertz signals generated by photomixing the ±1-order Kelly sidebands in (a) and (b) are 1.86 and 2.27 THz respectively.

We have investigated the dispersion-dependent spacing performance of the ±1-order Kelly sidebands of mode-locking fiber lasers for frequency-tunable terahertz signal generation. According to the model of the Kelly sidebands, the inverse square-root relationship between the spacing of the ±1-order Kelly sidebands and the total dispersion of the laser cavity has been found. Two kinds of fiber lasers using different dispersion-management components were designed, for generating dispersion-dependent ±1-order Kelly sidebands with large spacing. By changing the dispersion of the intracavity components, ±1-order Kelly sidebands with a large tuning range of 28.46 nm were obtained. Accordingly, the frequency of the generated THz signals could be changed from 0.09 to 2.27 THz.

The authors declare no conflicts of interest.

All data generated or analyzed during this study are included in this published article.

The author(s) received no financial support for the research, authorship, and/or publication of this article.