1. Introduction

Efficient frequency conversion of continuous-wave (cw) laser radiation such as second harmonic generation and optical parametric oscillation, has been achieved using a technique that confines and intensifies the cw laser radiation in a high-finesse cavity [1]. Recently, the confinement of a femtosecond ultrashort pulse laser radiation in an enhancement cavity has also been extensively studied. The ability to achieve intracavity high-harmonic generation without the use of high-power amplifiers and to extend an optical frequency standard to the vacuum ultraviolet region by utilizing such intracavity high-harmonic generation makes such methods very attractive [2, 3]. On the other hand, a technique that confines nanosecond laser pulses in an enhancement cavity, an intermediate regime between the abovementioned cases, has been little discussed so far. In this paper, we demonstrate a technique that stabilizes a carrier frequency of a nanosecond laser pulse by reference to an external enhancement cavity, and simultaneously intensifies such nanosecond pulsed laser radiation by confining it in the enhancement cavity.

2. The concept for carrier-frequency stabilization and intensity-enhancement of a nanosecond laser pulse

 

Fig.1 schematic illustrating carrier-frequency stabilization and intensity-enhancement of a nanosecond pulse.

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In Figure 1, we illustrate the technique which stably confines and then intensifies a nanosecond laser pulse in an enhancement cavity. First, we introduce a portion of the cw laser radiation into an external enhancement cavity and lock the oscillation frequency on a resonance peak (ω₀) of the cavity. Next, we introduce this cw laser radiation into a nanosecond pulsed laser as a seed and induce an injection-locked pulsed oscillation. In this configuration, the carrier frequency of the nanosecond pulsed output coincides with the oscillation frequency ω₀ of the cw laser, that is, the carrier frequency is necessarily locked to the resonance peak of the enhancement cavity. Consequently, when the nanosecond pulsed output is introduced into the enhancement cavity, it is stably confined and then intensified in the cavity. As for an enhancement cavity, we can adopt either a standing-wave or a travelingwave type of cavity. In this paper, we discuss the standing-wave type cavity.

3. Analytical formula for a nanosecond pulsed electric-waveform introduced into an enhancement cavity

We put a pair of mirrors with a reflectivity, R (transmittance, T=1-R), at z=0 (mirror 0) and z=L (mirror L), respectively, to obtain a Fabry-Perot enhancement cavity. We introduce a single-frequency pulsed laser radiation, E(t, z)=E(t)exp{i(ωt-kz)}, propagating in the positive z direction, into this Fabry-Perot enhancement cavity, where E(t) is the envelope of the amplitude waveform. The reflected, transmitted, and intracavity pulsed electric field waveforms produced by the cavity can be calculated by summing up all the pulsed electric fields that arise after multiple reflections at the mirrors that compose the enhancement cavity, similar to the case when cw radiation is used. The formula thus derived, shown below, describes the reflected electric field waveform (z<0).

Here, c, k, and δ=2kL denote the speed of light, the wave number, and the phase change corresponding to one-cycle of propagation in the enhancement cavity, respectively. The first term represents a simple reflection from mirror 0, and the second term is the summation of all the pulsed electric waveforms propagating toward the negative z direction after multiple reflections inside the enhancement cavity, which takes into account the associated amplitude reduction, phase difference, and time delay from the multiple reflections.

Likewise, the transmitted electric field waveform (L<z) is obtained as follows.

The electric field waveform inside the enhancement cavity (0<z<L) consists of two components propagating toward the positive and negative z directions, with even and odd reflections on the mirrors, respectively. Summing up all the electric field waveforms by taking into account the phase changes on the reflections, we obtain the formula below.

When the carrier frequency, ω, satisfies the resonant condition of the enhancement cavity, this formula can be further reduced as follows, with the assumption that the length of the incident pulsed waveform in space is sufficiently longer than the enhancement-cavity length.

This expression shows that the electric field waveform inside the cavity is composed of both standing and traveling waves with an amplitude ratio of 2√R:(1-√R), similar to the case when the incident radiation is cw.

The above formula, Eqs. 1–4, was built in the time domain. The equivalent expression can also be obtained in the frequency domain, where the incident nanosecond pulse is decomposed into the Fourier components, that is, cw laser radiation with various frequencies. We found that both approaches could give us the same results numerically.

4. Numerical calculations of the reflected, transmitted, and intracavity pulsed electric field intensity waveforms

The characteristics of a nanosecond single-frequency pulse introduced into the enhancement cavity are displayed in Fig. 2. The curves are calculated based on Eqs. 1–4. The cavity length and the reflectivity of the mirrors are set to be 75.0 mm (FSR: 2.00 GHz) and 0.9875 (Finesse: 250), respectively, in accordance with the experimental parameters, which are discussed in the f ollowing section. We also assume a Gaussian incident pulse with a duration of 28.3 ns (FWHM).

 

Fig. 2. Calculated electric field intensity waveforms of a nanosecond single-frequency pulse introduced into an enhancement cavity. a: incident (black dotted curve), reflected (red solid curve), and transmitted (green solid curve) waveforms; b: intracavity waveform (black solid curve) and transmitted waveform with a multiplication factor of $\frac{\mathbf{4}}{T}$ (green bold curve, T=0.0125 is the transmission of the cavity mirror). The temporal duration of the incident Gaussian pulse is 28.3 ns. The FSR and finesse of the enhancement cavity are 2.00 GHz (cavity length: 75.0 mm) and 250, respectively.

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The black, red, and green curves in Fig. 2a depict the incident, reflected, and transmitted electric intensity waveforms, respectively. It is apparent that the reflected waveform exhibits two peaks. The cause of this characteristic waveform is described as follows.

(I) When the radiation intensity in the cavity is almost negligible, only the simple reflection from the mirror 0 is observed.

(II) The incident radiation that is transmitted through mirror 0 is reflected repeatedly inside the enhancement cavity, and thus the radiation intensity in the cavity is gradually accumulated, which is also simultaneously transmitted back and forward through mirror 0 and mirror L, respectively. Since the reflected and transmitted radiations back at mirror 0 have opposite phases, destructive interference occurs between the two types of radiation. The gradual accumulation of the radiation intensity in the cavity reaches a value such that the radiation transmitting back through mirror 0 is comparable in amplitude to the simple reflection component at mirror 0 and the radiation vanishes completely when they have the same amplitude. At this point, the radiation transmitted forward through the enhancement cavity exhibits a maximum.

(III) Conversely, when the incident pulsed waveform nearly passes through the mirror 0, only the radiation transmitted back through the mirror 0 is remained, giving rise to the second peak in the reflected electric field waveform. In this region, the accumulated radiation in the enhancement cavity is emitted with equal intensities in both the forward and backward directions. In addition, both radiations decay exponentially with the cavity lifetime.

The electric field intensity waveform inside the enhancement cavity is shown in Fig. 2b. The peak intensity is enhanced by 96.1 times due to the confinement of the incident nanosecond pulse in the cavity. The dotted curve shows the transmitted intensity waveform multiplied by $\frac{\mathbf{4}}{T}\left(=320\right)$, where the transmission T is 0.0125 for the cavity mirrors. It turns out that the shape of intracavity intensity waveform coincides exactly with that of the transmitted intensity waveform. In addition, the both intensities are related to the mirror transmission T.

The relationship that the intracavity radiation intensity is connected to that of the transmitted radiation by a factor of $\frac{\mathbf{4}}{T}$, can be understood as follows. The amplitude of the forward propagating radiation in the cavity is $\frac{\mathbf{1}}{\sqrt{T}}$ times larger than that of the transmitted radiation. Furthermore, backward propagating radiation of nearly the same amplitude also exists and produces a standing wave inside the cavity, as discussed in section 2. The radiation amplitude at the antinodes of the standing wave in the cavity is thus $\frac{\mathbf{2}}{\sqrt{T}}$ times as large as that of the transmitted radiation. Hence, the intracavity radiation intensity is ${\left(\frac{\mathbf{2}}{\sqrt{T}}\right)}^{\mathbf{2}}=\frac{\mathbf{4}}{T}$ times greater than the intensity of the transmitted radiation.

Generally, it is extremely difficult to directly measure radiation intensities inside a cavity. However, the transmitted radiation can be measured with sufficient precision. Figure 2b clearly shows that the pulsed intensity waveform confined in the cavity can be obtained quantitatively by measuring the transmitted intensity waveform.

5. System for carrier-frequency stabilization and intensity-enhancement of nanosecond laser pulse

 

Fig. 3. System for carrier-frequency stabilization and intensity enhancement of nanosecond laser pulses. Blue shading: system for locking the oscillation frequency, ω ₀, of the external-cavity controlled laser diode (ECLD) to the enhancement cavity. Yellow shading: system for confining the nanosecond laser pulse in the enhancement cavity. EOM: electro-optic modulator, FG: function generator, PD: photo diode, DBM: double balanced mixer, PZT: PZT actuator.

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Figure 3 is a schematic of a system that stabilizes the carrier-frequency and enhances the intensity of a nanosecond laser pulse. The whole system consists of two parts. The blue shaded part was used to stabilize an oscillation frequency of the external-cavity-controlled laser diode (ECLD), employed as a master oscillator, to the enhancement cavity. The yellow shaded part served to confine a single-frequency nanosecond laser pulse, generated by means of an injection-locking technique, in the enhancement cavity. Here, we applied the Pound-Drever-Hall (PDH) method [4] to stabilize the oscillation-frequency of the ECLD. The PDH method is especially useful in cases where an oscillation frequency of a laser needs to be stabilized to a relatively high-finesse cavity.

Herein we describe the operation of the whole system. First, we divided the output (ω ₀) from the ECLD by a beam splitter, one of which was then introduced into an electro-optic modulator (EOM) to be phase-modulated at 20 MHz. We then introduced this phase-modulated radiation into the enhancement cavity (cavity length of 76 mm, free-spectral-range (FSR) of 2.0 GHz, finesse of 250, and the reflectivity of the mirrors was 0.9875) after passing it through an isolator. We detected the reflected radiation phase-sensitively at 20 MHz with a function generator as a local oscillator and produced an error signal that crossed zero at the resonant condition [4]. This error signal was fed back to the ECLD and its oscillation frequency was stabilized to the enhancement cavity. The other ECLD output from the beam splitter was injected into the nanosecond pulsed Ti:sapphire (Ti:s) laser as a seed. The Ti:s laser adjusted its cavity-length so that the seed could be matched to be resonant to the Ti:s laser cavity [5, 6]. A second harmonic output of a Q-switched Nd:YAG laser (repetition rate: 10 Hz) was then introduced into the Ti:s laser cavity as a pump to drive a nanosecond pulsed oscillation. The pulsed output energy was completely concentrated into radiation that was at the frequency of the injected seed, thus achieving an injection-locked single-frequency nanosecond pulsed oscillation.

Finally, this injection-locked pulsed output was introduced into the enhancement cavity. To measure the pulsed intensity waveforms, we employed biplanar phototubes (HAMAMATSU Photonics K. K., R1328U, rise time: 60 ps) and a fast digital oscilloscope (Tektronix, DPO7254, bandwidth: 2.5 GHz). We carefully adjusted the time origins of the incident, reflected, and transmitted pulsed waveforms and monitored all three simultaneously.

6. Experimental measurement of the reflected and transmitted intensity waveforms

 

Fig. 4. Intensity waveforms of the incident (gray dotted), reflected (red soild), and transmitted (green solid) radiations. The waveforms are normalized by the peak intensity of the incident laser pulse. The temporal duration of the incident laser pulse was set to be 28.3 ns at FWHM. The FSR and the finesse of the enhancement cavity employed were 2.0 GHz (cavity length of 76 mm) and 250, respectively. The peak intensity of the transmitted waveform was 0.26. The inset shows a 10 shot superposed transmitted waveform (the incident laser pulse duration was 20.7 ns for this data).

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The stability of the oscillation frequency of the ECLD locked onto the enhancement cavity was estimated, based on the resultant error signal. It was confirmed that the ECLD oscillation showed long-term stability with a frequency fluctuation less than ± 470 kHz.

We injected this ECLD cw output as a seed to generate an injection-locked nanosecond laser pulse. By controlling the pump energy introduced into the Ti:s laser, we varied the temporal duration of the injection-locked pulsed outputs from 12.5 to 40.3 ns [7]. We introduced these injection-locked nanosecond laser pulses into the enhancement cavity, and measured the incident, reflected and transmitted intensity waveforms precisely against the respective pulse durations. The typical intensity-waveforms as observed are displayed in Fig. 4 (the parameters used for the incident laser pulse and the cavity were same as those assumed in the calculation in Fig. 2). The black, red, and green curves are the incident, reflected, and transmitted intensity waveforms, respectively. The reflected waveform exhibited two peaks while the transmitted waveform reached a maximum at the minimum between these two peaks. Both the reflected and transmitted intensities then decayed with the same envelope. The peak intensity of the transmitted waveform normalized by the incident waveform was 0.26, whereas the expected intensity was calculated to be 0.30. The experimental data, including the intensities, matched the features in the theoretical calculation plotted in Fig. 2 extremely well.

The observed reflected and transmitted waveforms were stable for a long term. The inset displays the transmitted intensity waveforms superimposed by ten random shots (the pulse duration was 20.7 ns for this data). It turns out that the transmitted intensity waveforms were reproduced well.

Our primary concern is the intracavity intensity waveform. As confirmed in Fig. 2, this can be deduced quantitatively from the transmitted waveform. Multiplying the peak intensity, 0.26, by $\frac{4}{T}\left(T=0.0125\right)$, we obtain an enhancement factor of 83.2 (theoretical enhancement factor: 96.1).

Through the above measurements, it was confirmed that the stable intensity enhancement of the nanosecond laser pulse in the enhancement cavity was achieved by both the stabilization of the seed frequency to the enhancement cavity and the production of a nanosecond laser pulse that had the same carrier-frequency as the seed using an injection-locking technique.

7. Enhancement of a nanosecond laser pulse in cavity as a function of pulse duration

 

Fig. 5. Enhancement factor of a nanosecond laser pulse in the cavity as a function of the pulse duration of the nanosecond pulse. The solid curve shows calculated enhancement factor. The red circles denote the observed enhancement factors. The dotted curve is the enhancement factor obtained with the boundary condition in which the incident pulsed energy is kept constant.

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Figure 5 shows the relationship between the enhancement factor in the cavity and the pulse duration of the incident nanosecond laser pulse. The solid curve shows the enhancement factor calculated based on Eq. 3. The enhancement factor increases with the temporal duration of the incident nanosecond pulse. The maximum value of the enhancement is given by $\mathbf{320}=\frac{\mathbf{4}}{T}$, which corresponds to the enhancement factor when a single-frequency cw radiation is used. The green arrow indicates the temporal duration of 60 ns, which corresponds to the condition such that the spectral width of a pulse is equal to the full width (8 MHz) of the resonance profile of the enhancement cavity. When the temporal duration exceeds 60 ns, the enhancement factor asymptotically approaches the maximum value. These features are physically understandable well. The enhancement factors denoted with red circles in the same graph are estimated from the observed transmitted intensity waveforms. The observed enhancement factors increases, consistent with the calculated solid curve, although the pulse duration was experimentally limited to 40 ns. The greatest enhancement factor observed was 120 (predicted: 134), obtained at the longest pulse duration of 40 ns.

The solid curve in Fig. 5 illustrates the behavior of enhancement factors, defined in this case as the ratios of the radiation intensities confined in the cavity to the intensities of the incident pulses. In actual laser systems, however, the energy of the incident pulse is limited. Hence, if the primary purpose is to maximize the radiation intensity in the cavity, a longer pulse duration is not necessarily more desirable. The dotted curve in Fig. 5 represents the enhancement factor calculated with a different condition, in which the incident pulse energy is kept constant. Using this condition, an optimum incident pulse duration exists for a given finesse of an enhancement cavity in order to achieve maximum radiation intensity in the cavity. As shown by the dotted curve, the optimum pulse duration is 32 ns for the present enhancement cavity. In other words, this implies that optimum enhancement cavity can be designed for a given nanosecond pulse duration to maximize the radiation intensity in the cavity. To achieve optimum enhancement, the width of the resonance profile of the enhancement cavity should be $\frac{\mathbf{1}}{1.7}$ times as large as a spectral width of the nanosecond pulse.

8. Summary

We have demonstrated the stable confinement and intensity enhancement of a nanosecond laser pulse in an enhancement cavity. By utilizing a nanosecond single-frequency pulse generated by an injection-locked, nanosecond pulsed Ti:s laser that employed an ECLD as a master oscillator, we have studied the nanosecond laser pulse confinement in an enhancement cavity with a finesse of 250. By measuring both the incident waveform and the reflected and transmitted waveforms from the cavity, we have confirmed that a stable intensity enhancement of 120 was achieved for a 40-ns incident pulse. The experimental results obtained were consistent with theoretical predictions. If we combine the present technique with a current technique of a high-power LD-driven nanosecond pulsed laser, a unique high-intensity excitation condition [8], completely different from those realized by using either cw or femtosecond lasers, will be achieved at repetition rates greater than 1 kHz. The excitation conditions offered by such a source will be attractive for various studies, such as the coherent control of molecule. The present technique is also closely related to the so-called “slow light” phenomenon [9] and would indeed be interesting from the viewpoint of such work [10].