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We demonstrate the generation of an optical frequency comb (OFC) with a Gaussian spectrum using a continuous-wave (CW) laser, based on spatial convolution of a slit and a periodically moving optical beam spot in a linear time-to-space mapping system. A CW optical beam is linearly mapped to a spatial signal using two sinusoidal electro-optic (EO) deflections and an OFC is extracted by inserting a narrow spatial slit in the Fourier-transform plane of a second EO deflector (EOD). The spectral shape of the OFC corresponds to the spatial beam profile in the near-field region of the second EOD, which can be manipulated by a spatial filter without spectral dispersers. In a proof-of-concept experiment, a 16.25-GHz-spaced, 240-GHz-wide Gaussian-envelope OFC (corresponding to 1.8 ps Gaussian pulse generation) was demonstrated.
©2010 Optical Society of America
1. Introduction
Optical frequency combs (OFCs) are important new light sources in various fields. They consist of a set of continuous-wave (CW) laser light sources separated by a constant frequency spacing with a fixed phase relationship. Recently, several techniques have been proposed for generating wide OFCs with a flat power envelope by imposing an electro-optic (EO) sinusoidal phase modulation on a CW laser [1–4]. Compared with OFCs generated by mode-locked lasers, OFCs generated by this technique are very stable. The OFC spacing is locked to the stable electrical signal. The carrier envelope phase can be directly accessed via the input CW laser frequency, which can be stabilized.
Techniques that can both generate and shape OFCs have attracted significant interest. Weiner et al. demonstrated optical arbitrary waveform processing with a temporal resolution of 1.65 ps based on line-by-line pulse shaping in which more than 100 spectral comb lines were individually manipulated using a high-resolution grating and a spatial filter [Fig. 1(a)] [5]. As an alternative technique, optical bandpass filters (OBPFs) have been used in applications in which full time spanning between input pulses is not required for shaped waveforms [Fig. 1(b)]. A highly stable picosecond Gaussian pulse train for optical communication systems has been generated using a Gaussian OBPF [6,7]. A flat-top gate pulse for nonlinear optical switches has been generated using a fiber Bragg grating [8]. OFC shaping using an OBPF is efficient and simple because it is not necessary to spatially disperse the frequency components of the input OFC; however, it is difficult to realize a tunable complex filter function.
In this paper, we propose a novel OFC generation technique in which a spatial filter functions as an OBPF. Figure 1(c) shows a schematic diagram of the proposed technique. A CW optical beam is deflected in the horizontal direction (y axis) by electro-optic deflector 1 (EOD1). It is then deflected in the vertical direction (x axis) by EOD2 to achieve a circular beam trajectory in the mapping plane (i.e., the Fourier-transform plane of the near field of EOD2) [9, 10]. The optical pulse train that forms the OFC is extracted through a narrow spatial slit placed in the mapping plane. The pulse shape corresponds to the beam profile in the mapping plane. The spectral shape of the generated OFC corresponds to the Fourier transform of the beam profile in the mapping plane; consequently, it corresponds to the spatial beam profile in the near-field region of EOD2. Because the spatial profile of the optical beam in the near-field region of EOD2 can be manipulated by the spatial filter, this filter functions as an OBPF. Thus, OFC shaping can be realized by a spatial filter (i.e., a shaped OFC is generated). In the conventional technique, the center frequency of the OBPF should be tuned together with that of the input CW laser. In our technique, the center position of the beam profile in the near-field region [x0 in Fig. 1(c)] corresponds to the carrier frequency of the input CW laser. Therefore, the shaped OFC can be tuned only by tuning the frequency of the input CW laser. Wavelength multiplexing of the shaped OFC can be realized using a wavelength division multiplexing (WDM) source as the input CW source, because the spatial filter acts as an OBPF for each input wavelength. These features of our technique will be useful for optical measurements and communication applications such as OTDM/WDM systems [11].
Fig. 1. Schematic diagram of (a) line-by-line shaping, (b) optical frequency comb shaping using an optical bandpass filter, and (c) the proposed technique. In (c), the laser beam is deflected in the horizontal direction (y axis) by EOD1 and then deflected in the vertical direction (x axis) by EOD2. In the mapping plane, a circular beam trajectory is realized by adjusting the relative deflection phase. OFC: optical frequency comb, OBPF: optical bandpass filter, EOD: electro-optic deflector, f: focal length of the Fourier-transform lens.
The basic concept is based on a single-shot pulse shaping technique using the EOD [12] proposed by Kobayashi et al. They demonstrated pulse shaping using a spatial filter and a double-prism linear EOD driven by a 3.9 kV/2 ns ramp signal. The minimum pulse duration was 94 ps, which was limited by the slow deflection signal. The generated signal could not form an OFC because it uses single-shot pulse generation. By using two sinusoidal EO deflections, we realized a periodically moving optical beam spot with a constant velocity (linear time-to-space mapping), which is required for OFC generation.
In Section 2, we analyze our technique to determine the relationship between the temporal resolution of the linear time-to-space mapping system and the width of the generated OFC. In Section 3, we describe our experimental system. The experimental results and discussion are presented in Section 4.
2. Analysis of OFC generation
Figure 2 shows the OFC generation scheme based on spatial convolution of the spatial slit and the moving optical beam spot. In this scheme, optical pulses that form the OFC are extracted from a periodically moving beam spot by a spatial slit placed in the mapping plane. In the following analysis, we assume that the trajectory of the optical beam spot is circular, namely that the velocity of the moving spot is constant. When the spatial slit is sufficiently narrow compared with the beam spot size in the mapping plane, the spatial slit functions as a temporal impulse function. In this case, the position of the impulse function can simply be expressed by the position of the narrow slit, θ0. The power envelope of the OFC is not altered by the slit position, although the spectral phase changes with slit position. The spectral shape of the generated OFC corresponds to the Fourier transform of the beam profile in the mapping plane. Note that the basic operation mode of this system is Gaussian OFC generation using a Gaussian beam with no spatial filter (OBPF). In the following analysis, a Gaussian beam is assumed.
Figures 3(a) and 3(b) show a typical calculated power spectrum and spectral phase of a generated OFC, respectively. The deflection frequency is 16.25 GHz. The slit is assumed to be infinitely narrow. The OFC power envelope is Gaussian and there is no frequency chirping. The spectral phase depends on the pulse position (i.e., the position of the narrow slit placed in the mapping plane). When the pulse is extracted at θ0 = 0 rad, then all the spectral components are zero (red open circles). The blue closed circles are phases for θ0 = π/2 rad.
Realistically, the slit has a finite width. Figures 4(a) and 4(b) show the calculated pulse shape and the OFC envelope in the back Fourier-transform plane of the mapping plane, respectively. Here, ws and wb are the half-width of the slit and the Gaussian beam radius, respectively. The data are normalized to their peak values. The pulse shape changes from Gaussian to rectangular with increasing slit width, while the OFC envelope changes from Gaussian to the sinc function with increasing slit width.
Figure 5(a) shows the width of the pulses extracted through the slit as a function of the normalized slit width (ws/wb). The pulse width values (Δt) are normalized to the temporal resolution of the mapping system (Δτ) as Δt0 = Δt/Δτ. The pulse width approaches the temporal resolution of the system on narrowing the slit. We conclude that the pulse width will be almost the same as the temporal resolution of the mapping system, if the slit is narrower than 0.8wb. Figure 5(b) shows the time-frequency bandwidth product (TFB= Δt/Δν) as a function of the normalized slit width, where Δν is the FWHM of the OFC envelope. The broken line represents TFB=0.44 and it indicates that the generated Gaussian pulses are Fourier-transform limited. Based on these analyses we conclude that the relationship between the OFC width and the system temporal resolution for a Gaussian beam is ΔτΔν = 0.44, when ws/wb < 0.8.
Fig. 2. OFC generation scheme using a spatial slit in a linear time-to-space mapping system. The OFC is generated by convolution of the narrow spatial slit with the periodically moving optical beam spot.
Fig. 3. (a) Calculated power spectrum and (b) spectral phase of the OFC. The deflection frequency is assumed to be 16.25 GHz.
Fig. 4. (a) Typical calculated pulse shape and (b) OFC envelope. ws is the slit half-width and wb is the Gaussian beam radius.
If the diameter of the circular beam trajectory is constant, the temporal resolution of this system is determined by the spot size in the mapping plane. Therefore, if the slit is not sufficiently narrow (ws/wb > 0.8), the time delay within the slit is resolved, causing the pulse front to tilt.
The peak power of the generated pulse is equal to the input CW laser power. Therefore, there is a trade-off between the OFC width and the optical output/input average power efficiency, η. The relation between the efficiency and the width of the Fourier-transform-limited Gaussian OFC is , where T is the deflection period. Note that the optical power efficiency is the same as that for an OFC generator based on a conventional Fabry–Perot EO modulator (FP-EOM) [13], although the modified FP-EOM makes it possible to convert the input CW optical power into the OFC with near 100 % efficiency [14].
Fig. 5. (a) Calculated normalized pulse width and (b) time-frequency bandwidth product as a function of the normalized slit width. TFB: Time-frequency bandwidth product.
3. Experimental system
Fig. 6. (a) Schematic diagram of the single-chip quasi-velocity-matched (QVM) EODs. The asymmetric U-shaped microstrip line is the modulation electrode. The ratio of the interaction lengths of the two EODs is set to 1:0.73 to realize the same resolvable spot number. (b) Linear time-to-space mapping system using single-chip QVM-EODs. F.T. lens: Fourier-transform lens.
A linearly moving optical beam spot can be realized by a linear time-to-space mapping system [10] with two EODs. For linear time-to-space mapping operation, the resolvable spot numbers of the two EODs should be equal. In the former system, we used two individual quasi-velocity-matched (QVM) EODs [9], which were driven by the same modulation source [10]. The modulation power was divided using a 50:50 power splitter. Since the modulation efficiencies of the two EODs were not equal, the divided modulation powers for both EODs were adjusted using variable attenuators to achieve the same resolvable spot numbers. This degradation in the modulation efficiency limited the temporal resolution of the system.
Figure 6(a) schematically shows the newly developed device. We refer to this device as a single-chip QVM-EOD. This device imparts two benefits to the time-to-space mapping system: (1) the modulation power does not have to be separated for two EODs, so that the modulation power can be used efficiently, which enhances the temporal resolution, and (2) linear time-to-space mapping can be achieved with good reproducibility by appropriately setting the length ratios of the two branches of the U-shaped microstrip line. The domain inversion for two QVM-EODs is performed in the same Z-cut stoichiometric LiTaO3 substrate. The substrate has a length of 40 mm and a thickness of 0.5 mm. The domain inversion period for the QVM condition is about 2.8 mm. The U-shaped Ag microstrip line was evaporated onto the surface of the domain-inverted substrate to guide the modulating microwave. The microstrip line was 0.5 mm wide. We measured the modulation efficiencies of the two EODs and found that the ratio of the interaction length of the EODs should be 1:0.73 to realize the same resolvable spot numbers. This difference is mainly attributed to losses in the microstrip lines. The lengths of the microstrip lines are set to 35.5 mm and 25.9 mm.
Fig. 7. Beam trajectories observed in the mapping plane. The relative deflection phase ϕ is set to (a) 0 rad, (b) π/4 rad, (c) π/2 rad, and (d) -π/4 rad.
Figure 6(b) shows the linear time-to-space mapping system using fabricated single-chip QVM-EODs. The optical beam is sinusoidally deflected in the horizontal direction by EOD1. The image in the output plane of EOD1 is transferred to EOD2 by two lenses. 90-degree rotation of the deflection direction is realized using a dove prism. The relative deflection phase (ϕ) is controlled by adjusting the optical path length between the two EODs.
4. Experimental results and discussion
Figure 7 shows typical images of the deflected beam trajectory observed by a CCD camera placed in the mapping plane. In the experiment, we used a 514.5-nm CW Ar laser as the light source. The linewidth was about 10 MHz and the frequency drift was less than 100 MHz. The deflection frequency was 16.25 GHz. The modulation source was a pulsed magnetron with a pulse duration of 1 μs. The relative deflection phases were set to (a) ϕ = 0 rad, (b) ϕ = π/4 rad, (c) ϕ = π/2 rad, and (d) ϕ = -π/4 rad, respectively. As shown in Fig. 7(c), we achieved a circular beam trajectory.
Fig. 8. (a) Typical circular trajectory. The temporal resolution can be calculated from the diameter of the circular trajectory (2Wd) and the beam spot size (2Wb) in the mapping plane. (b) Achieved temporal resolution as a function of the square root of the modulation power. Solid line is theoretical curve fitted to the data.
Fig. 9. (a) Generated typical OFC lines. Solid curve is Gaussian function fitted to the OFC lines. (b) Beam profile in the mapping plane. Hollow circles are experimental data obtained by profiling the image obtained using a CCD camera. The solid curve is the Gaussian function fitted to the data.
Figure 8(a) shows a typical circular beam trajectory and its cross-sectional intensity profile. The normalized temporal resolution of the mapping system can be calculated by [10]
where 2Wb is the FWHM of the beam in the mapping plane, and 2Wd is the diameter of the circular trajectory. The deflection period was T = 61.5 ps. Figure 8(b) shows the achieved temporal resolution as a function of the square root of the modulation power. Solid circles are experimental data obtained from circular trajectories using Eq. (1) and the solid curve is the theoretical curve fitted to the data. The obtained data agree very well with the theoretical curve. We achieved a maximum temporal resolution of about 1.8 ps. In this case, the modulation power was estimated to be 1.8 kW and most of it was dissipated in the device. The deflection angle and the resolvable spot number of each EOD were approximately ±0.001 rad and 11, respectively. Using the linear time-to-space mapping system with a maximum temporal resolution of 1.8 ps, we generated an OFC by inserting a narrow spatial slit in the mapping plane. The normalized slit width was ws/wb ≃ 03. A Gaussian beam was used. The spectrum of the generated OFC was measured using a homemade spectrometer using a 50-mm-wide diffraction grating. The theoretical spectral resolution was about 5 GHz, but the actual resolution was lower than this theoretical value due to misalignment.
Fig. 10. OFC width as a function of the square root of the modulation power. Hollow circles are experimental data. Solid line is calculated from the temporal resolution of the system using ΔτΔν = 0.44.
Figure 9(a) shows observed typical OFC lines. The deflection frequency was 16.25 GHz and the OFC spacing was 16.25 GHz. The modulation power was 1.8 kW. Figure 9(b) shows the beam profile in the mapping plane. Hollow circles are experimental data obtained by profiling the image obtained using a CCD camera. Each data point corresponds to a single camera pixel. The solid curve is a Gaussian function fitted to the data. The generated OFC envelope agrees well with the Fourier transform of the beam profile in the mapping plane.
We measured the OFC width at several modulation power levels. Figure 10 shows the OFC width as a function of the square root of the modulation power. The OFC spreads without changing the Gaussian shape. The experimental data were obtained from fitted Gaussian function. The solid line is obtained from the system’s temporal resolution curve shown in Fig. 8(b) using the relation ΔτΔν= 0.44. The measured Gaussian OFC width agrees well with the solid line. The numerically derived relation of ΔτΔν = 0.44 has been experimentally confirmed. The width of the generated OFC at a modulation power of 1.8 kW is about 240 GHz, which corresponds to a Gaussian pulse width of 1.8 ps. The achieved pulse is as short as those from other sources, such as mode-locked lasers used in current 160-Gbit/s OTDM experiments [16,17].
Our technique can be used to generate shaped pulses at high repetition rates. In our technique, the temporal pulse position can be controlled by the spatial slit position. Therefore, by placing the slit array along the circular beam trajectory in the mapping plane, we can generate an N-channel pulse train with a fixed relative delay time, where N ≃ (πWd)/(2Wb). Multiplexing of these channels can be realized by simply inserting a Fourier-transform lens behind the mapping plane. The current experimental results correspond to 17 channels of 16.25 Gbit/s (>270 Gbit/s).
5. Conclusion
We have proposed an OFC generation technique in which the OFC shape is the Fourier transform of the spatial beam profile in the mapping plane. We demonstrated Gaussian OFC generation using a Gaussian beam. The relationship between the OFC width (Δν) and the temporal resolution (Δτ) of the mapping system has been numerically derived and experimentally confirmed as ΔτΔν = 0.44 for a Gaussian beam, when ws/wb < 0.8. By increasing the modulation power, we confirmed that the spectral width spreads without changing the OFC shape. Generation of 16.25-GHz-spaced, 240-GHz-wide Gaussian OFC has been demonstrated.
Acknowledgments
This research was partially supported by a Grant-in-Aid for Young Scientists (B), 21760039, 2009, a Grant from the Global COE Program, ”Center for Electronic Devices Innovation,” from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and a Grant-in-Aid for Scientific Research on Priority Areas, 19023006, 2008.
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