1. Introduction

Optical delay lines with programmable delays, fine resolution, and rapid scanning are necessary for a variety of applications such as optical coherence tomography, optical metrology, interferometry, broadband phase arrays and telecommunications [1–4]. We recently demonstrated an all-fiber, programmable, ultrafast optical delay line based on the concept of reversible frequency conversion using two matched electro-optic phase modulators as a time-prism pair, Fig. 1. This configuration allowed us to achieve a record scanning rate of 0.5 GHz (~1000 times faster than previous best efforts) without any mechanical moving parts [5]. One fundamental limitation in our previous work is that the dispersion used to obtain the necessary time delay also inevitably broadens the input pulse. Because the temporal width of the broadened pulse must not exceed the finite linear portion of the frequency-shifting phase profiles, the delay-to-pulse-width ratio of our previous work was limited to 2.4, even after careful optimization between dispersion, modulation frequency, and input pulse width. To overcome this limitation, one needs a scheme to produce a dispersive delay without dispersive pulse broadening. In this work, we create such a “delay-without-broadening” effect by using soliton propagation between time-prisms. Through theoretical analysis, we derive the fundamental relationships between the delay-to-pulse-width ratio and the input pulse width to show the enhanced performance using soliton propagation. We then demonstrate the concept experimentally to achieve an ultrafast optical delay line with a scan rate of 0.5 GHz, a delay range of 33.0 ps, no pre- and post-dispersion compensation, and a delay-to-pulse-width ratio of 6.0.

2. Theoretical analysis

The figure of merit for any delay line is the delay-to-pulse-width ratio which we define as R=Δτ/T_FWHM, where T_FWHM is the intensity full width at half maximum for the input pulse and Δτ is the maximum time delay at the output of the device, given by [5] as

Here Δλ is the wavelength shift imparted by the linear portion of the sinusoidal phase modulation drive, and D is the dispersion in ps/nm. This wavelength shift is determined by V _p-p, V_π, and T_m which are respectively the peak-to-peak drive voltage into the phase modulators, the voltage required to obtain a π phase shift, and the modulation period. For simplicity, we assume the input pulse has a Gaussian temporal profile and is chirp free in the following analysis. In our previous work, pre- and post-dispersion compensation are used in order to restore the original optical pulse width at the end of the device and to ensure symmetric operation of the time-prisms, Fig. 1. The optical pulse width at each phase modulator is therefore broadened from the original input pulse by an amount

where T_o is the intensity 1/e half-width of the Gaussian input pulse.

 

Fig. 1. Optical path of previous ultrafast delay line using linear propagation and pre- and post-dispersion compensation (pre- and post-comp). Evolution of the optical pulse and carrier frequency is shown above. The black curves represent the linear portion of the sinusoidal phase profile imposed by the time-prisms; PM: phase modulator.

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In order to have complete reversible frequency conversion, the broadened pulse plus the full delay is restricted to the linear portion (T_Lin) of the sinusoidal modulation drive,

The linear portion of the modulation drive depends on the fractional nonlinearity that can be tolerated in an experiment. Here we define the fractional nonlinearity, NL, as the ratio of the cubic term to the linear term in the Taylor expansion of the sinusoidal modulation. The fractional nonlinearity determines the amount of spectral and temporal distortion of output pulses and the linear portion for a given NL can be solved as T_Lin=βT_m, where $\beta =\frac{2}{\pi }\sqrt{\left(1.5\right)\mathit{NL}}$.

 

Fig 2. Delay-to-pulse-width ratio, R as a function of input pulse width for two contours of fractional nonlinearity, NL=5% (blue curves) and NL=30% (red curves). The linear propagation regime is shown by the dashed curve, and the soliton propagation regime is shown by the solid curves. The peak-to-peak drive voltage and modulation frequency are fixed to 5.24 V_π and 10 GHz respectively.

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We use Eqs. (1–3) to solve for the delay-to-pulse-width ratio, R, as a function of input pulse width for linear and soliton propagation (Fig. 2). Each contour in Fig. 2 gives the upper bound for R for the corresponding NL. In the linear propagation regime (dashed curves), dispersive pulse broadening dominates for short pulses where the optical pulse bandwidth (Δλ_opt) is much greater than the wavelength shift produced by the phase modulator, i.e. Δλ_opt≫Δλ. The asymptotic behavior for short pulses at a given NL approaches

For long input pulse widths, Δλ _opt≪Δλ, pulse broadening is negligible (σ→1) and the asymptotic behavior for long pulses approaches

reaching zero for T_o=βT_m, the maximum input pulse width allowed for a given NL. For soliton propagation between time-prisms (solid curves), however, dispersive pulse broadening is completely balanced by self-phase modulation, i.e., σ=1in Eq. (3). Thus, the expression for R at any pulse width is given by Eq. (5). Although linear and solitonic propagation are the same in the long pulse limit, soliton propagation is far superior for short pulses (Fig. 2), eliminating the finite limit on R at short pulse widths. The enhanced ratio using soliton propagation between time-prisms can be understood intuitively in terms the spatial analogy, Fig. 3(a). It is well known that the analog to narrow-band dispersion in the time domain is paraxial diffraction in space [6]. Here we further extend this spatial-temporal analogy to include optical nonlinearity, where soliton propagation is just the temporal analog of spatial self-trapping. In the absence of self-trapping between a spatial prism pair (linear propagation), the spatially translated beam is also expanded by diffraction, requiring a balance between the input beam diameter and prism separation in order to have successful k̄ vector conversion. Compensating diffraction with self-focusing eliminates such a constraint and therefore improves performance.

 

Fig. 3. (a) A schematic of a spatial prism-pair demonstrating the space-time analogy [6,7]. (b) Experimental setup. The phase modulators (PM1) and (PM2) are each driven with a 10 GHz sinusoid whose amplitude is modulated by a 0.5 GHz sinusoid through an RF modulator. The operation of the time-prisms and the evolution of the carrier frequency are indicated. Both Δτ and the pulse width are small portions of the phase modulation period so that the phase modulation can be approximated by a linear phase ramp.

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3. Experimental setup and results

The experimental setup is shown in Fig. 3(b). The pulsed source (Calmar Optcom; PSL-10) emits a 5.5 ps FWHM (3dB bandwidth ~0.48 nm), chirp-free, 10 GHz pulse train at a center wavelength of 1549.3 nm. The pulse train is divided down to a 5 GHz repetition rate using a Mach-Zehnder LiNbO₃ modulator as a pulse picker. The carrier wavelength of the pulse train is then up-shifted by a time-prism, a 10 GHz sinusoidally driven LiNbO₃ phase modulator (PM1). An erbium doped fiber amplifier (EDFA) and a variable attenuator are used to adjust the optical power to satisfy the soliton condition for propagation in the following 2.88 km of SMF-28 fiber (total dispersion 49 ps/nm). After propagating as a soliton pulse train through the dispersive delay, the carrier wavelength is down-shifted back to its original value by a second time-prism (PM2), driven with the same sine wave, but π out of phase. The amount of wavelength shift, and therefore the amount of time delay, is controlled by adjusting the drive voltage into the phase modulators through a broadband RF modulator (Miteq; BMA0218LA1MD). Since V _p-p∝V _{RF_mod}, where V _{RF_mod} is the modulation input into the RF modulator, Δτ becomes a function of V_{RF _mod}, Eq. (1). Thus, ultrafast programmable time delays can be obtained by controlling the modulation waveform V _{RF_mod}.

To demonstrate the concept, we first verified soliton propagation by showing that for the proper input power, the spectrum and the pulse remain unchanged after propagating through the 2.88 km of fiber. For a 5.5 ps Sech pulse at 5 GHz repetition rate, we expected the fundamental soliton to be obtained between 18 to 19 dBm average input power into the SMF- 28. Figure 4 shows the various spectra and time traces for different propagation regimes. The blue dashed curves in Figs. 4(a) and 4(b) denote the original spectrum and time trace of the source before propagation. The spectral characteristics and pulse width at various input powers into the SMF fiber is typical for nonlinear propagation in the anomalous dispersion region and is in agreement with our numerical simulations. The 18.3-ps pulse width of the source as measured by the sampling scope is consistent with the impulse response of the oscilloscope (~17.0 ps) and a 5.5-ps input pulse. The spectral features in Fig. 4(a) are indicative of a small deviation from the ideal Sech input pulse. Figure 4(b) shows the contrast between soliton propagation and linear propagation (-10.0 dBm into the fiber).

 

Fig. 4. (a) Spectra corresponding to different amounts of average power into the 2.88 km of SMF-28. The average power and corresponding temporal pulse widths are written on the respective curves. The dashed blue curve corresponds to the original spectrum. All spectra are taken at 0.05 nm resolution bandwidth (RBW). (b) Oscilloscope time traces for different propagation regimes. The dashed blue curve corresponds to the time trace of the input pulse.

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Fig. 5. Evolution of optical spectra through the ultrafast delay-line using soliton propagation: original spectra (dashed blue curves); after PM1 (solid black curves); after PM2 (solid red curves). All spectra were taken at 0.05 nm RBW.

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To demonstrate ultrafast scanning capability of our delay line and the improved performance using soliton propagation, we modulate the 10 GHz RF sine wave with a 0.5 GHz sinusoid. The maximum RF output into the wavelength up-shifting phase modulator was approximately 5.24V_π, corresponding to a maximum wavelength shift of 0.66 nm. Figure 5 shows the evolution of the optical spectrum through the ultrafast delay-line: 1) before phase-modulation; 2) after wavelength up-shifting (after PM1); and 3) after wavelength down-shifting (after PM2).

By using soliton propagation between time-prisms, dispersive broadening no longer occurs, gaining an additional 10 ps in time delay. The improved performance is demonstrated by the excellent match between the original soliton spectrum (blue dashed curve) and the spectrum after wavelength down-shifting (solid red curve) in Fig. 5. The broadened spectrum after wavelength up-shifting (solid black curve) represents the superposition of the spectra of five pulse trains, whose center wavelengths are successively up-shifted. We also note that much of the oscillatory structure in the down-shifted spectrum has vanished. This agrees with simulation results and is due to the fact that a good portion of the dispersive non-solitonic waves have been shifted out of band by the time-prisms.

 

Fig. 6. (a) Oscilloscope time trace demonstrating rapid scanning of delays. The dashed grid lines represent the original position of pulses with no phase modulation. (b) Delay Δτ as a function time obtained from (a). The solid curve is the 0.5 GHz modulation. The impulse response of the scope is ~17.0 ps.

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Scanning of the optical delay in the soliton propagation regime is shown by the oscilloscope time trace in Fig. 6(a). The corresponding delays are plotted in 6(b), with the 0.5 GHz modulation (V _{RF_mod}) plotted on top of the measured data points. The excellent agreement in Fig. 6(b) demonstrates the precise mapping of the modulation input to the time delay. Here the measured maximum time delay is 33.0 ps, giving a delay-to-pulse-width ratio of 6.0. This matches well to what is expected given the 0.66 nm wavelength shift and 49 ps/nm dispersion. While additional measurements on the spectral phase will uniquely match the pulse shape before and after the delay, the nearly identical pulse spectra and time traces are very good indications that the output pulse train is simply a delayed replica of the input.

4. Conclusion

Through theoretical analysis and experiment, we show that the performance of an ultrafast optical delay line using a time-prism pair is significantly improved when solition propagation is used between time-prisms. The enhancement becomes most dramatic in the case of short pulses where the optical bandwidth greatly exceeds the frequency shift of the time-prisms. For the purposes of demonstration, we show ultrafast scanning for an input pulse width of 5.5 ps and obtain a delay-to-pulse-width ratio of 6.0. Although standard SMF was used to achieve the dispersive delay in our experiment, other fiber types, such as those with much smaller mode field diameters can be used to reduce the power requirement for solitons, making our technique applicable for a sub-picosecond pulse train.