1. Introduction

Optical arbitrary waveform shaping has been key technology in optical communications, sensing, microwave/millimeter-wave generation[1, 2], and so on[3]. Most of the shaping techniques are implemented by means of Fourier synthesis, in which amplitude and phase of Fourier components are independently manipulated in space using dispersive elements and spatial filter [4]. Manipulation of well-defined frequency comb which is generated by a deep electrooptic (EO) phase modulation of the continuous-wave (CW) laser [5, 6] is one of a promising technique because the center frequency can be controlled by input CW laser whereas the repetition frequency can be controlled independently by stable electric modulation signal. However, rather complicated techniques for generating flat spectral frequency comb [7, 8] as well as precise preparation of complicated spectral filter are required for high-fidelity arbitrary waveform generation by conventional Fourier synthesis scheme.

Recently, we proposed time-to-space mapping system with picosecond time resolution using ultrafast EO deflector (EOD) [9]. In our approach, an optical temporal signal stream is mapped into a spatial image signal by the deflection, therefore, the temporal properties of optical signal can be processed in the space directory. The spatially processed optical signal can be converted into the temporal signal by the space-to-time mapping, which is reverse process of the time-to-space mapping. Compared with conventional scheme of Fourier synthesis, proposed technique has significant advantages that the spatial filter for desired processing is simple and intuitive because the image of transmissivity of the spatial filter corresponds to desired temporal filter.

Previously, we shaped 1.6 ps pulse train from CW laser using narrow slit as a temporal filter based on the mapping system with single EOD[9]. The temporal resolution of the developed mapping system was time-dependent because of the nonlinearity of sinusoidal mapping function. Time slot in which the temporal resolution was lower than 2 ps was limited to be about 13 ps within the mapping half-period of about 36 ps. The developed system can be applied to intermittently processing within a specific temporal resolution. However continuous processing with constant temporal resolution is necessary for many applications.

In this paper, we propose a linear time-to-space mapping system based on two times EO sinusoidal beam deflection and confirm experimentally the basic operation. The direction of each deflection is adjusted to be mutually orthogonal with the relative deflection phase of π/2 rad so that the circular optical beam trajectory can be achieved. Continuous time-to-space mapping with constant temporal resolution can be achieved by the circular optical beam trajectory. In the proof-of-concept experiment, temporal resolution of 5 ps is roughly estimated from achieved resolvable spot number of the deflectors and the deflection period. The developed system is expected to be applied to not only the temporal processing applications such as modulated millimeter-wave generation with high-speed photodiode, but also the characterization of ultrafast optical signal, e.g. single-shot optical oscilloscope (streak camera) or optical spectrogram scope.

2. Basic concept of the linear time-to-space mapping

Figure 1(a) shows the schematic of proposed linear time-to-space mapping system. Input optical beam is sinusoidally deflected to the vertical direction (x-axis direction) by the first EOD and then deflected to the horizontal direction by the second EOD. As shown in Fig. 1(b), the trajectory of the optical beam at the Fourier transform plane of the output facet of the second EOD can be a circle by adjusting the relative phase of the deflection. We refer this Fourier transform plane (x ₁-y ₁ plane) as a mapping plane. W_d corresponds to an amplitude of the deflection and W_b is a half width of the beam spot at the mapping plane. In such a case, the temporal position of the input optical signal is linearly mapped into the spatial position along the circular beam trajectory. Because the optical beam spot moves along the circular trajectory with the constant angular velocity of ω_m=2πf_m, where f_m is the deflection frequency, constant temporal resolution (linear mapping) is realized.

For the analysis of the operation, we consider a CWGaussian beam. The deflected light field just after the second EOD is expressed as

where, w is the half beam width of the input beam, ω ₀ is the carrier angular frequency. Δθ_x(x) and Δθ_y(y) is the distribution function of the modulation index for the first and second EOD, respectively. ω_m is the deflection angular frequency and ϕ is the relative deflection phase. In the experiment, Δθ_x(x) and Δθ_y(y) should be

where, d is the half-width of each deflector, and Δθ_mx and Δθ_my are maximum modulation index.

In the analysis, we introduce an approximations of $\Delta {\theta }_x\left(x\right)=\frac{\Delta {\theta }_{\mathrm{mx}}}{d}x\left(-\infty \le x\le \infty \right)$ and $\Delta {\theta }_y\left(y\right)=\frac{\Delta {\theta }_{\mathrm{my}}}{d}y\left(-\infty \le y\le \infty \right)$ for analytical simplicity. This approximation is useful to understand the mapping operation qualitatively. Quantitative differences from the exact analysis is not so large for this approximation, because more than 90 % of optical power is confined within the deflector (-d≤x≤d, where d=w). However, the maximum modulation index within the optical beam is overestimated, therefore this approximation is not correct for temporal resolution analysis.

Using Fourier transform lens of focal length of f, the time-to-space mapped optical field at the mapping plane can be derived as

where, λ is the wavelength of the optical field.

 

Fig. 1. Principle of the linear time-to-space mapping. (a) Schematic of the mapping system. The optical wave is sinusoidally deflected by first electrooptic deflector (EOD) to vertical direction (x-axis direction). The vertically deflected beam is then deflected to horizontal direction by second EOD. The temporal position of the input beam is mapped into the spatial position along the circular beam trajectory at the mapping plane (Fourier transform plane of the output facet of second EOD). (b) Beam trajectory at the mapping plane in the case of relative phase of ϕ=π/2 rad. The amplitude of two deflection is assumed to be same.

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Fig. 2. Optical beam trajectories at the mapping plane. Input beam is assumed to be CW The optical power is temporally integrated. (a) Δθ_mx=15 rad, Δθ_my=15 rad and ϕ=π/2 rad, (b) Δθ_mx=Δθ_my=15 rad and ϕ=0 rad, (c) Δθ_mx=Δθ_my=15 rad and ϕ=π/4 rad, and (d)Δθ_mx=15 rad, Δθ_my=10 rad, and ϕ=π/2 rad.

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Figure 2 shows the optical beam trajectories at the mapping plane. The CW input beam was assumed in the calculation. The calculations were carried out using equations (4)–(6). The optical intensity was temporally integrated. In the case of Δθ_mx=Δθ_my and ϕ=π/2 rad (Fig. 2(a)), we achieve uniform energy distribution along the circular trajectory because of the uniform temporal resolution of the mapping. Figure 2(b) is same situation as previous time-to-space mapping with single sinusoidal deflection[9]. The energy distribution is modulated along the direction of the deflection because of non-uniform velocity of the beam spot moving. For the relative deflection phase of ϕ=π/4 rad, an ellipse trajectory is achieved as shown in Fig. 2(c), though temporal resolution is not uniform. As shown in Fig. 2(d), the unbalance of the maximum modulation index for two deflection also results in non-uniform temporal resolution in spite of ϕ=π/2 rad. Balanced maximum modulation index with the relative deflection phase of ϕ=π/2 rad is necessary for linear time-to-space mapping.

3. Temporal resolution analysis

In this section, we will discuss the temporal resolution of the mapping system for the case of Δθ_mx=Δθ_my=Δθ_m with ϕ=π/2 rad. In the former section, we introduced the assumption that the infinite modulation index can be achieved at the tail of the Gaussian spatial mode profile. In the experiment, however, the maximum modulation index within the beam cross-section will be restricted. Distribution function of the modulation index for an actual device should be equations (2) and (3). In such a realistic case, the minimum temporal resolution for the sinusoidal time-to-space mapping system is given by [9]

where, T is the deflection period, N=W_d/W_b is the resolvable spot number of the deflector, and Δν is the total spectral width of the deflected light field. r=N_G/N_rec where N_G and N_rec are resolvable spot number for Gaussian beam and rectangular beam, respectively. The parameter r is a function of w/d and r=0.86 for w/d=1. The temporal resolution for the sinusoidal mapping system is a function of the deflection phase and Δτ_min can be achieved only at the specific deflection phase. On the other hand, constant temporal resolution expressed by equation (7) can be realized in the proposed linear mapping system. Normalized temporal resolution for the linear mapping system is expressed as

This temporal resolution corresponds to achievable minimum pulse width shaped from CW beam using infinitely narrow slit. For the maximum modulation index of 6.3 rad, the normalized temporal resolution of 0.51 can be realized.

The temporal resolution as a function of the maximum modulation index is shown in Fig.3. The right hand side axis expresses the scale for the deflection period of T=61.5 ps (f_m=16.25 GHz), which corresponds to an experimental condition.

The proposed system can also be applied to characterization tool for ultrafast optical signal, such as single-shot optical oscilloscope (streak camera) or single-shot spectrogram scope. The pulse width can be estimated from the ratio between the width of the mapped spatial image and the length of the circumference of the circular optical beam trajectory, which corresponds to the deflection period. In principle, the response time of the image acquisition device such as CCD camera will not influence the system temporal resolution. The system temporal resolution is determined by the deflection speed and the resolvable spot number of the deflectors.

 

Fig. 3. Temporal resolution as a function of the maximum modulation index. The right hand side axis expresses the scale for the deflection period of T=61.5 ps (f_m=16.25 GHz), which corresponds to an experimental condition.

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Fig. 4. Relation between the normalized actual input pulse width Δτ_i and normalized measured pulse width Δτ_m.(a) 0<Δτ_i<0.4 and (b) 0.4≤Δτ_i≤0.8.

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We calculated the relation between actual input pulse width to be measured and the pulse width measured by the linear time-to-space mapping system(Fig. 4). In the Fig. 4, each axis is normalized so that the normalized pulse width of 2π corresponds to the deflection period of T.

In the analysis, amplitude modulated optical pulse of

is assumed and mapped field pattern is calculated on the basis of the numerical Fresnel diffraction integral. Long dashed double-short dashed line is proportional line. In such a case, the system has infinite temporal resolution. In the figure, red and blue circles are numerically calculated relations for Δθ_m=15 rad and Δθ_m=20 rad, respectively.

In the calculation, we assumed Gaussian spatial beam profile. In such a case, the instrumental function of our system becomes Gaussian function having the width of Δτ₀. When the Gaussian temporal pulse profile is assumed, the measured pulse profile will be a convolution of the Gaussian instrumental function and the Gaussian temporal pulse profile. Consequently, relations between the actual pulse width and measured one can be approximated by following equation.

Solid lines in the Fig. 4 are calculated using (10). Results obtained using this simple equation (red and blue solid lines) agree well with results obtained using numerical Fresnel diffraction integral (red and blue circles).

4. Experiment

Figure 5 shows experimental setup. A 514.5 nm CW Ar laser is used as a light source. Two quasi-velocity-matched EODs (QVM-EODs) are driven at the deflection frequency of 16.25 GHz (T=61.5 ps). The QVM-EODs are fabricated by use of simple domain-engineering processes in a z-cut LiTaO₃ electro-optic crystal with a thickness of 0.5 mm and a length of 30 mm. A silver microstrip line with a width of 0.5 mm (d=0.25 mm) is evaporated on the each crystal to guide the modulating microwave. Details of the QVM-EOD are described in the literature [10].

The beam width at the input of each EOD is set to be 0.5 mm, therefor d=w. The image of the output plane of the EOD1 is transferred to the EOD2 through the Fourier transform mirror of the radius of 2 m and the Fourier transform lens of the focal length of 600 mm. The output image of the EOD1 is rotated by 90 degrees with the dove prism so that the direction of the deflection of two EODs becomes mutually orthogonal. The relative deflection phase is controlled by the optical path length between two EODs. The modulation power was fed to two EODs through 50:50 divider and adjusted by 1-dB-step attenuator so as to attain modulation index matching (Δθ_mx=Δθ_my). The optical beam polarization is controlled by the λ/2 plate. The CCD is placed at the Fourier plane of the output plane of the EOD2 for observation of the beam trajectory.

A 16.25 GHz pulsed magnetron is used as a modulating microwave source. The duration of the operation is 1µs, therefore, 1 µs pulse train is generated by acousto-optic deflector (AOD) for elimination of the unmodulated component. Mapped images are observed by the CCD camera synchronized with the pulse train. The repetition frequency of the observation is set to be 3 Hz. CW modulation source and synchronized CCD camera or shaping element which operate at 16.25 GHz should be used for continuous processing. However, the refresh rate of the CCD camera or the shaping element is much lower than 16.25 GHz. Therefore, current system requires that the time-slot in which we can process the optical signal must be less than the temporal period of the deflection.

 

Fig. 5. Experimental setup. AOD:acousto-optic deflector, Att. variable attenuator.

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Fig. 6. Optical beam trajectories observed by the CCD camera. The relative phase of the deflection was set to be about (a)0 rad, (b)π/4 rad, (c)π/2 rad, and (d) -π/4 rad, respectively.

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Figure 6 shows the typical images of the deflected beam trajectory observed by the CCD camera. The relative deflection phase was set to be (a)0 rad, (b)π/4 rad, (c)π/2 rad, and (d) -π/4 rad, respectively. We confirmed that the shape of the trajectory was changed by the relative deflection phase. In this experiment, the deflection period was 61.5 ps, therefore fluctuations of the optical path length between two EODs could be negligible and stable trajectories were observed.

Almost circular trajectory was obtained for the relative deflection phase of π/2 rad (Fig. 6 (c)). From the observed deflected image, the resolvable spot number was measured as at least N=4. The temporal resolution of this mapping system has been roughly estimated as about 5 ps or less from equation (7). This temporal resolution corresponds to the normalized temporal resolution of 0.51. Therefore, the modulation index Δθ_m is estimated from the equation (8) as Δθ_m=6.3 rad. Previously, we achieved 20.8 rad of the modulation index by 40-mm-long EOD [10] which corresponds to the temporal mapping resolution of about 1.5 ps for f_m=16.25 GHz. Relative less temporal resolution of current system is attributed mainly to modulation power splitting and shorter device length.

As shown in Fig. 6 (c), the achieved beam trajectory for ϕ=π/2 rad slightly deviated from the circle because of the mismatching of the modulation index. Modulation efficiencies for two QVM-EODs having same modulation electrode length are usually different. One of the reason can be that the phase velocity of the modulation microwave is slightly different in each substrate. Therefore, it is necessary for current system to achieve modulation index matching by adjusting the modulation power using continuously variable attenuator. This system complexity and modulation power splitting problem can be solved by a single-chip type mapping device. Two QVM-EODs can be fabricated in single substrate with “U” shaped modulation electrode.

5. Conclusion

We have proposed the time-to-space mapping system, which is based on two times sinusoidal EO beam deflection. Through two times sinusoidal deflection with proper relative deflection phase, circular trajectory can be achieved and as a result linear mapping from the temporal signal to the spatial signal is realized. The temporal resolution of the mapping system has been analyzed and the relation between actual input pulse width and mapped pulse width has been derived. The basic operation has been confirmed by proof-of-concept experiment using two QVM-EODs and 5 ps of temporal resolution has been demonstrated.