## Abstract

An acousto-optically Q-switched frequency-shifted feedback (FSF) diode-pumped laser with a Nd:YVO_{4} crystal is demonstrated. The laser emits more than 1W average power with multi-ten ns duration at up to 50 kHz repetition rate, and peak power higher than 1 kW is achieved at frequency of 20 kHz. This kind of pulsed laser has special value for some applications with outstanding features of broadband continuous spectrum and chirping frequency in the oscillation spectrum. Simulation results of the FSF laser through rate equations in Q-switched regime are also presented.

©2007 Optical Society of America

## 1. Introduction

Well-defined temporal and spatial coherency and discrete and narrow longitudinal mode structures are the essential properties of the common laser. Another kind of laser with a cavity closed via the first-order diffracted beam of an acousto-optic modulator (AOM) is referred to as frequency-shifted feedback laser, which exhibits broadband continuous spectrum and chirping frequency in its oscillation spectrum. The intracavity light field experiences a fixed amount of frequency shift on each round trip, and the oscillation spectrum of this kind of laser becomes drastically different from that of a conventional laser.

The frequency-shifted feedback cavity was proposed for mode locking and electronic tuning in the 1970s, but the broadband continuous spectrum characteristic of this kind of laser
was not discovered until Kowaski *et al*. reported a modelock-like pulsed output without Fabry–Perot frequency structure observed in 1988 [1]. This kind of laser was regarded as no-mode structure by Littler *et al*. and was called as modeless laser [2–4]. In 1993, Balle *et al*. conducted an experiment by using a Fabry-Perot interferometer with sufficiently low frequency-shifting rate (2.4×10^{14} Hz/s), and it turned out that the output spectrum of a FSF laser consists of frequency components and the chirping is stepwise [5]. Recently, H. Ito and his coworkers demonstrated a diode-pumped Nd:YVO_{4} FSF laser [6–8]; their experimental results revealed that the oscillation spectrum of an FSF laser consists of a kind of longitudinal mode structure, and it is chirped across the oscillation bandwidth.

The above-mentioned lasers were all operated in continuous-wave or modelock-like modes, no experiment was conducted with a Q-switch. In this work, we demonstrate the feasibility of a pulsed, diode-pumped Nd:YVO_{4} laser with an acousto-optical Q-switch (AOQ) and based on frequency-shifted feedback mechanism. No laser of this type has been constructed before, to our knowledge. Due to the prominent feature of its spectrum, that is, its broadband continuous spectrum and chirping frequency, this kind of pulsed laser is suitable as seed source for some laser amplifiers, especially a fiber one in which nonlinear process, such as stimulated Brillouin scattering (SBS), is prominent and harmful.

## 2. Experimental configuration

The experimental arrangement is shown schematically in Fig. 1, where the first-order diffracted beam of the AOM was fed back while the zero-order beam was coupled out as useful output. A Nd:YVO_{4} crystal pumped by a fiber-coupled diode was employed as the gain medium. Its dimensions were 3×3×5 mm and the doping concentration was 1.0 at. %; the left surface was coated with AR film at 808 nm and 1064 nm; and the right surface was coated with 808 nm HR film and 1064 nm AR film. The crystal was placed in a copper heat sink and a temperature controlling unit was used to control its temperature. The pumping LD, which was capable of emitting 10 W pumping light with wavelength of 808 nm, was coupled with a fiber bundle with total diameter of 400 μm and the numerical aperture of 0.37. Its working temperature was controlled at about 25°C by another temperature controlling unit. An AOM driven at frequency of 70 MHz was inserted into the cavity constructed by the full reflecting mirror M_{2} and M_{1}, which was coated with 1064 nm HR film and 808 nm HT film. The AOM had the maximal diffraction efficiency of about 75% and worked in Bragg diffraction mode. An AOQ was used to Q-switch the laser; its highest diffraction efficiency was 80% and the working frequency tuning range was 10k–150 kHz.

## 3. Experimental results

Figure 2 demonstrates the output power and pulse duration changed with the pulse frequency rate when pumped at 10 W and 7.5 W, respectively. The output power given here was the sum of two zero-order beams. We achieved the highest pulse rate of 50 kHz in our work. The output power and frequency rate were both limited by the diffraction efficiency of the AOM, in which the highest value was only 75%. This resulted in the large loss of the laser, and for the same reason, the pulse duration changed notably with the frequency rate.

Figure 3 shows the oscilloscope trace of an output pulse with duration of about 70 ns at 50 kHz repetition rate and 10 W of pumping power. The width stability in RMS was better than 6% in this case. The inset demonstrates the corresponding spectrum curve achieved by an optical spectrum analyzer (OSA). It is confirmed that the spectrum outline of the pulsed FSF laser has an approximate Gauss-type smooth envelope and no spikes occur. The spectrum width is 0.11 nm with center wavelength at 1064.343 nm, which deviates from the center wavelength of a common Nd: YVO_{4} laser about 0.04 nm, that is 10 GHz in frequency.

A scanning Fabry–Perot (FP) interferometer was used to analyze the spectrum composition of the laser output, the free spectrum range (FSR) of the FP interferometer was 4 GHz (corresponding to scanning voltage 56 V, which was marked in the following figure), and the finesse was about 100. The output of the interferometer was then observed by an oscilloscope, which is shown in Fig. 4. Figure 4(a) (left) shows the result pumped at 10 W with a pulse frequency rate of 10 kHz. The lower curve is the magnitude FFT of the pulse waveform, a prominent component of the frequency 10 kHz, and its multiple can be seen clearly. In this presentation, the horizontal scale is 20 kHz/div and the vertical scale is 20 dB/div. For comparison, the scanning output of a common pulsed laser is also shown in Fig. 4(b) (right). It is clear that some stable spectrum components exist. So the conclusion can be reached that the spectrum of a FSF laser comprises more components than a common one. This dissimilarity gives an indirect proof for the broadband continuous spectrum and chirping frequency for this kind of laser.

## 4. Simulation results

#### 4.1 Rate equations treatment

Similar to Ref. [9], the basic thought for treating the frequency-shifted feedback laser with rate equations is to discretize the spectrum and time, respectively. The gain spectrum of laser crystal is divided into many bands in units of round trip frequency shift. The number of photons in a single band satisfies the ordinary differential rate equation describing the evolution of the photon density, while the inversion number is described by another differential equation. The frequency-shift process is carried out in mathematics by storing the result of a specified band with its neighboring band. With these equations, we are able to calculate the evolution of the photon number in a cavity and some transient processes, but we do not have access to the characteristics concerning the phase because the coherency property is neglected. The following gives the handling of a frequency-shifted feedback laser based on a four-level system.

The equation describing the evolution of photon number *M _{q}* (q is an integer to identify each band) in a single band is given by

The first item in the right-hand side expresses the net stimulated emission, where the constant *B _{q}* denotes the frequency-related stimulated emission coefficient and is in direct ratio with emission cross-section

*σ*, given by

_{q}*B*=

_{q}*σ*

_{q}*c*/

*V*. The second item in the right-hand side expresses the intracavity loss. Here,

_{m}*τ*is the round trip time,

_{r}*γ*is the modulated loss introduced by AOQ, and

_{AOQ}*γ*is given by

*γ*=-ln[(1-

*l*)(1-

_{c}*l*)]. The third item represents the spontaneous emission; constant

_{o}*A*stands for the frequency-related spontaneous emission coefficient and is determined by

_{q}*A*=

_{q}*B*.

_{q}τ_{r}The other parameters are: *c* is the light velocity in a vacuum; *V _{m}* is the mode volume;

*σ*is the stimulated emission cross-section;

_{q}*l*is the constant loss except for useful output, such as absorption and diffraction loss,

_{c}*et al.*; and

*l*is the useful output. The total intracavity photon number is given by $M={\displaystyle \sum _{q}}{M}_{q}$. Spontaneous emission plays an important role in the evolution, its modeling may be based on random process or just a steady rate.

_{o}Another equation describes the changing rate of inversion due to pumping, spontaneous emission, and stimulated emission, which reads

where *P _{a}* is the valid power absorbed by the gain medium,

*h*is the Planck constant,

*ν*is the photon frequency of pumping, and

_{p}*τ*is the lifetime of spontaneous emission.

_{f}The realization of frequency shifting is to replace the photon number *M _{q}* in band q with the photon number

*M*

_{q+1}in band q+1 after the roundtrip time. By solving the set of rate equations and the process of frequency shifting, we are able to obtain the intracavity photon number, the inversion number evolving with time, and the distribution of photon number on the spectrum. With the photon density in the cavity, the real-time power output is given by

where *ν _{l}* is the frequency of the output laser. To compute the average of the real-time power over a period of time, we have the average power as

where Δ*t* is the calculating step time. The effective gain of the FSF laser is given by *G _{q}* =

*B*-

_{q}Nτ_{R}*γ*, and the spectrum range of

*G*≤ 0 defines the spectrum width Δ

_{q}*ν*for the amplification process.

_{g}#### 4.2 Q-switching results

Based on the model described above, we simulated the FSF laser corresponding to our experimental laser. The correlative parameters are given in Table 1.

Figure 5 shows the calculated results for the Q-switched working state. The related parameters are as follows: the frequency rate is 20 kHz, the diffraction efficiency of the AOQ is 80%, the rise and fall times are both 100 ns, and the interval for oscillation is 500 ns. The Q-switch was opened at 49.9 μs and the pulse was built at around 50.2 μs, so the build-up time of the pulse shows a difference of about 300 ns. The contour plot for the time-resolved spectral distribution of photon density is shown in Fig. 5(a), and the inset is a 3-D plot of the same pulse. The output pulse is given in Fig. 5(b), and the inset is the modulated loss of AOQ.

The pulse width was calculated as 37 ns. Figure 5(c) gives the evolution of the inverse number in the laser. The frequency shift and spectrum width are displayed separately in Figs. 5(d) and 5(e). It must be emphasized that the result given in Fig. 5(d) is the instantaneous frequency shift from the gain curve center of the peak power. For a specific spectrum component, the frequency chirping value is the chirping rate multiplied by the pulse duration, which is about 2.6 GHz in this case.

The simulated results are helpful in understanding the Q-switched FSF laser. By comparing the simulated results and the experimental ones, it is found that the pulse width and the spectrum width are in accord with each other, to some extent. It also validated that the coupling efficiency of pumping was rather poor in our laser.

#### 5. Conclusions

We have successfully demonstrated an all solid-state Q-switched pulsed laser with a Nd:YVO_{4} crystal based on frequency-shifted feedback mechanism. The outstanding features of this kind of laser centralize on its spectrum. Frequency chirping is the intrinsic characteristic brought by the FSF mechanism; the chirping may be lethal for an optical process needing stable frequency, but it is significant for a situation in which a stable spectrum should be destroyed. For a quasi-continuous spectrum, all spectrum components in an output field may exist for a pulse with an appropriate duration. Taking the laser implemented in this work as an example, the chirping rate is 7×10^{16} Hz/s, the longitudinal mode interval is 500 MHz, and for a pulse with a duration of 50 ns the chirping range is calculated as 3.5 GHz, which is much wider than a longitudinal mode interval. In addition, FSF mechanism exerts influence on the width of the output spectrum, which was measured as 0.11 nm or so and exhibits weak dependence on pumping level. The simulated results also confirmed these properties.

To further investigate this kind of laser, a better quality AOM with higher Bragg diffraction efficiency is expected. A Q-switched pulsed laser with a FSF cavity may serve as a seed laser where the quasi-continuous spectrum and chirping frequency are preferential.

## References and links

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