Pulse shaping by a frequency filtering of a sawtooth phase-modulated cw laser

The spectrum of a cw field the phase of which experiences a periodic sawtooth modulation is analyzed. Two types of the sawtooth phase modulation are considered. One is created by combining many harmonics of the fundamental frequency. The second is produced by an electro-optic modulator fed by the relaxation oscillator, which generates a voltage slowly rising during charging of the energy storage device and dropping fast due to discharge by a short circuit. It is proposed to filter out the main spectral component of the sawtooth phase-modulated field. This filtering produces short pulses from the phase-modulated cw field. The duty cycle of this train is equal to the modulation period, while duration of the pulses depends on the rate of the phase drop. In the case of harmonics, this duration is $4N+2$ shorter than the period of the fundamental frequency, where $N$ is the number of the harmonics. In the case of the charge and discharge, duration of the pulses is close to $T_D/2$, where $T_D$ is a drop time of the phase in the discharge period. Depending on the modulation frequency, the proposed method is capable to produce pulses with duration ranging from nanoseconds to a fraction of a picosecond.

Figure 1

Time evolution of the sawtooth phase $\phi \left(t\right)$ (blue dotted line) and the phase ${\phi }_N\left(t\right)$, synthesized from $N=5$ harmonics (solid red line). Both are normalized to $\pi$. Horizontal black bars show limiting values of the phase change, $\pm \pi$.

Figure 2

Frequency content of the field with sawtooth phase modulation synthesized from $N=5$ harmonics. The amplitudes of the spectral components are shown by red (positive amplitudes) and blue (negative amplitudes) bars. Their maxima are linked by black solid lines for visualization.

Figure 3

(a) Time evolution of the phase $\psi \left(t\right)=n_m\mathrm{\Omega }t-{\phi }_N\left(t\right)$, which specifies the interference of the comb with the scattered field $E_{\text{sc}}\left(t\right)$. (b) Time dependence of the intensity of the pulses normalized to $I_0$. Solid red lines correspond to the sawtooth phase modulation with $N=10$, and blue dots correspond to $N=5$.

Figure 4

Time evolution of the phase $\psi \left(t\right)$ near one of the crossing points, i.e., at $t_k$ with $k=-1$, is shown by the red solid line. Its approximation by linear time dependence is shown by blue dots (see the text for details). The number of harmonics constituting the sawtooth phase modulation is $N=5$ (a) and 10 (b).

Figure 5

(a) Dependence of the maximum intensity of the pulse $I_{max}$ on the modulation index is shown by the red solid line and its approximation is shown by blue dots. (b) Dependence of the minimum intensity in the dark windows on $M$.

Figure 6

Time evolution of the phase ${\phi }_{RO}\left(t\right)$. The time scale is normalized to $T=T_R+T_D$. The parameters of the sawtooth phase modulation are related as $T_D=T_R/10$. The black horizontal bar corresponds to $2\pi$.

Figure 7

(a) Time evolution of the phase ${\psi }_{RO}\left(t\right)$ (in units of $\pi$) is shown by the red solid line. The sequence of pulses, generated by the frequency filtering, is shown by the blue dotted line. The field intensity is normalized to $I_0$. The time scale is in units of the period $T$. Both plots correspond to the case when $T_D=T_R/10$. (b) Zoom in on the area of the pulse formation around $t=T$. In both plots, horizontal black thin lines indicate the levels corresponding to the phase values equal to $(2k+1)\pi$.

Figure 8

(a) Sequence of pulses, generated by the frequency filtering, when $T_D=T_R/100$. The time scale is in units of the duration of the phase rising stage, $T_R$. (b) Zoom in on the area of the pulse formation around $t=T_R$.

Figure 9

Comparison of pulses produced by the phase modulation ${\phi }_{RO}\left(t\right)$ with $C$ (red solid line) and reduced coefficient $C$ (dotted blue line). Durations of the phase rising and dropping stages satisfy the relation $T_D=T_R/10$.

Figure 10

(a) Phase ${\phi }_{\mathrm{sh}}\left(t\right)$ evolution in time. (b) Evolution of phase ${\psi }_{\mathrm{sh}}\left(t\right)$ (blue dotted line) and pulse sequence produced by the filtering of the phase-modulated field (red solid line).