Development of the pulse peak from peak-truncated Gaussian optical pulses in a serial array of high-$Q$ ring resonators

Development of the pulse peak from peak-truncated Gaussian optical pulses in a serial array of high- $Q$ ring resonators

Makoto Tomita, Taichi Sudo, Kota Yoshimura, and Daiki Sugio

Phys. Rev. A 102, 043507 – Published 9 October 2020

Abstract

We experimentally examined the propagation of a Gaussian-shaped temporal pulse through a serial array of ring resonators, truncating the pulse before its peak. We observed the process whereby a smooth pulse peak related to the Gaussian pulse emerged even when the incident pulse was truncated before its peak, as the pulse passed through an increasing number of ring resonators. The peak emerged when the negative group delay accumulated beyond the pulse truncation time point. This result demonstrated that the earlier portion of the input pulse envelope developed into a smooth Gaussian-shaped pulse peak in the superluminal pulse in good agreement with the analyses based on the instantaneous spectrum. We also discussed the propagation of truncated pulses in a slow light system, where the peak disappeared in the output pulse despite the peak in the input pulse.

Received 19 May 2020
Accepted 17 September 2020

DOI:https://doi.org/10.1103/PhysRevA.102.043507

Physics Subject Headings (PhySH)

Optics & lasers

Atomic, Molecular & Optical

Authors & Affiliations

Makoto Tomita, Taichi Sudo, Kota Yoshimura, and Daiki Sugio

Department of Physics, Faculty of Science, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka, 422–8529, Japan

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Vol. 102, Iss. 4 — October 2020

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Images

Figure 1
(a) Conceptual illustration of a serial array of ring resonators. $R$ is a ring resonator; $C$ is a coupler. (b) Schematic diagram of experimental setup. $FL$ , fiber laser; $LN, LiNb O_{3}$ modulator; $S$ , 2 × 2 fast optical switch; $C$ , coupler; $R$ , ring resonator; $RL$ , fiber recurrent loop; $D$ , detector; $FG$ , function generator; $PC$ , computer; $OS$ , oscilloscope. (c) Colored lines are peak-truncated Gaussian pulse profiles used in experiments. Truncation time is indicated by line 1, $t_{N A} = - 4 ns$ ; line 2, −8 ns; line 3, −19 ns; and line 4, −32 ns, respectively. Black line represents original Gaussian pulse.
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Figure 2
(a) Observed transmitted pulse profiles for the smooth Gaussian-shaped input pulse for different $N$ in undercoupled ring resonator (i.e., fast light system). Numbers of times that a pulse passed through the resonator were as follows: (a1) $N = 0$ (input pulse), (a2) 1, (a3) 2, (a4) 3, (a5) 4, and (a6) 5. (b)–(e) depict transmitted pulse profiles for peak-truncated input pulses for different $N$ (from top to bottom, $N = 0$ to 5 in all columns). Truncation times were (b) $t_{N A} = - 4 ns$ , (c) −8 ns, (d) −19 ns, and (e) −32 ns, respectively. All intensities were normalized with respect to maximum of smooth Gaussian pulse. Black downward arrows in (b5), (b6), and (c6) indicate emerged peaks. Dashed black inclined line in (a) shows position of pulse peak as a guide. Dashed black vertical line in (d) represents position of truncation times.
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Figure 3
Open circles represent advancement of pulse peak for the smooth Gaussian pulse as a function of $N$ in undercoupled ring resonator. Solid circles at $N = 3$ , 4, and 5 are peak positions observed with peak-truncated pulses. Truncation time was $t_{N A} = - 4 ns$ [corresponding to Figs. 2(b4), 2(b5), and 2(b6), respectively]. Solid squares at $N = 4$ and 5 are peak positions observed with peak-truncated pulses. Truncation time was $t_{N A} = - 8 ns$ [corresponding to Figs. 2(c5) and 2(c6)]. Open triangles represent position of pulse truncation point in output pulses as a function of $N$ observed with peak-truncated pulse, at truncation time of $t_{N A} = - 8 ns$ [corresponding to Fig. 2].
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Figure 4
Calculated temporal pulse profiles of transmitted pulses for different $N$ in undercoupled ring resonator (i.e., fast light system). Input pulses were (a) smooth Gaussian-shaped pulses and (b)–(e) peak-truncated pulses, corresponding to graphs in Fig. 2. All intensities were normalized with respect to maximum of smooth Gaussian pulse. Dashed black inclined line in (a) shows position of pulse peak as a guide. Dashed black vertical lines in (d) represent position of truncation time. Time origin $t = 0$ was set at pulse peak time.
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Figure 5
(a) Calculated transmitted pulse profiles for the smooth Gaussian-shaped input pulse for different numbers of resonators $N$ in overcoupled ring resonator (i.e., slow light system). Numbers of times that a pulse passed through the resonator were as follows: (a1) $N = 0$ (input pulse), (a2) 1, (a3) 2, (a4) 3, (a5) 4, and (a6) 5. (b)–(e) depict calculated transmitted pulse profiles for truncated input pulses for different $N$ (from top to bottom, $N = 0$ to 5 in all columns). Truncation times were (b) $t_{N A} = + 5 ns$ , (c) +20 ns, (d) +35 ns, and (e) +50 ns, respectively. Black downward arrows indicate peaks. Dashed black vertical lines in (d) represent positions of truncation times.
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Figure 6
(a) Observed transmitted pulse profiles for truncated input pulses in an overcoupled ring resonator (i.e., slow light system; $N = 1$ ). Curves 1 and 2 are the input and transmitted pulses, respectively. The truncation time was $t_{N A} = 15 ns$ . (b) Calculated temporal pulse profiles of transmitted pulses. Curves 1 and 2 are the input and transmitted pulses, respectively. The dotted lines indicate the input and transmitted temporal profiles, respectively, when the smooth Gaussian-shaped pulse was not truncated. All intensities were normalized with respect to the maximum of the input pulse. Black downward arrows indicate peaks.
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