## Abstract

We demonstrate a time-to-space mapping of an optical signal with a picosecond time resolution based on an electrooptic beam deflection. A time axis of the optical signal is mapped into a spatial replica by the deflection. We theoretically derive a minimum time resolution of the time-to-space mapping and confirm it experimentally on the basis of the pulse width of the optical pulses picked out from the deflected beam through a narrow slit which acts as a temporal window. We have achieved the minimum time resolution of 1.6±0.2 ps.

©2006 Optical Society of America

## 1. Introduction

Not only generation but also control of ultrafast optical pulses has been key technology in many fields such as optical measurement, ultrafast photonic network, and so on. Various optical time-to-space mapping and processing techniques have been developed and demonstrated [1–10]. Most of them are based on spectral holography or nonlinear optics inside a pulse shaper like configuration. In those techniques, ultrafast optical temporal waveforms are transformed into their time-invariant spatial equivalents using disperser with Fourier transform lens and ultrashort reference pulse [9]. By spectral nonlinear optics method, picosecond-regime optical pulse sequence measurement was demonstrated by Ema and his coworkers [3] in the early 1990s and recently, real-time detection of femtosecond optical pulse sequences in the lightwave communications band was demonstrated by Weiner’s group [10].

On the other hand, manipulation of optical signals with picosecond to femtosecond time resolution by stable electrical signal is also important issue in ultrafast optoelectronics field. Pulse generation using an electrooptic phase modulator, followed by some type of dispersion control for pulse formation is one example of the electrical signal manipulation [11–13]. Because a temporal variation of optical signals is converted into a spatial variation through an optical beam deflection (time-to-space mapping), high speed electrooptic deflectors (EODs) together with conventional optics are shown to have potential for new time-space optical information processing applications, such as femtosecond optical oscilloscope, optical DEMUX, 1×N optical switch, and so on. In those applications, the time resolution of the system is inversely proportional to the spectral width of the sidebands generated by the EOD and should be comparable to the obtainable pulse width using the EOD.

Previously we demonstrated EODs that are driven by an X band standing wave along a microstrip line on an electro-optic (EO) crystal [14]. Using those EODs operating at repetition frequency of *f*_{rep}
=9.35 GHz, we also demonstrated optical pulse compression from continuous light wave to 9.35 GHz-16 ps with a diffraction grating. Recently we demonstrated a new type of traveling-wave EOD which operates at *f*_{rep}
=16.25 GHz with resolvable spot number of about 13 [15]. For the traveling-wave EOD, we used a quasi-velocity-matching (QVM) technique with periodic domain inversion [16]. The QVMtechnique compensates velocity mismatch between an optical group velocity and a microwave phase velocity in an EO crystal to realize a large modulation index by periodically inverting the sign of the EO coefficient of the crystal.

In this paper we demonstrate a time-to-space mapping of a continuous light wave with a picosecond time resolution using our traveling-wave EOD. We theoretically derive a minimum time resolution of the time-to-space mapping using sinusoidal deflector. The derived analytical expression for the minimum time resolution is confirmed experimentally on the basis of the pulse width of the optical pulses picked out from the deflected beam through a narrow slit which acts as a temporal window.

## 2. Time resolution of the time-to-space mapping

In this section, we will derive the analytical expression for a minimum time resolution of the time-to-space mapping using an EOD. To avoid unnecessary complexities, we employ the one-dimensional analysis.

#### 2.1. Basic configuration of the time-to-space mapping

We consider the operation of the sinusoidal EOD shown in Fig. 1(a). An incident continuous light wave from the left-hand side of the figure with amplitude distribution function *A*(*x*
_{0}) is modulated with a modulation index of Δ*θ*(*x*
_{0}) depending on the position *x*
_{0} within the optical beam cross section. The modulated light field just after the EOD is expressed as

where *f*
_{0} is the optical frequency and *f*_{m}
is the modulation frequency. When we assume the distribution function of the modulation index as Δ*θ*(*x*
_{0})=Δ*θ*_{m}*x*
_{0}/*d* for the sinusoidal deflection, the instantaneous phase shift of the output beam is given by

where *d* is the half width of the EOD, Δ*θ*_{m}
is the maximum modulation index. The time varying spatially distributed phase shift, Δ*ϕ*(*x*
_{0}, *t*), is shown in Fig. 1(b). Here, we assumed as Δ*ϕ*(*x*
_{0}, *t*)=0 for |*x*
_{0}|>*d*. The instantaneous optical frequency shift is then written by

Using Eq. (3), total spectral width of the deflected light field is approximately given by

As shown in Fig. 2, time-to-space mapping of the optical signal is realized by Fourier transforming the deflected beam. For the case that the focal length of the Fourier transform (FT) lens is *f*, the light field *E*
_{1}(*x*
_{1}, *t*) on *x*
_{1} axis is given by

where, λ is the wavelength of the input light wave.

#### 2.2. Resolvable spot number of the EOD

The quality of the deflected beam trajectory is one of the important factors that should be considered for ideal time-to-space mapping. The deflected beam trajectories at the Fourier plane calculated using Eq. (5) are shown in Fig. 3. The amplitude distribution function of the input laser beam in Fig. 3(a) and (b) is assumed to be *A*(*x*
_{0})=exp[-(*x*
_{0}/*w*)^{2}]. The relative beam parameters *p*≡*w*/*d* in Fig. 3(a) and (b) are assumed as *p*=0.5 and *p*=1, respectively. It is obvious that the quality of the deflected beam trajectory decreases with an increase of the relative beam parameter *p*, because of the interference between the diffracted non-mapped components (|*x*
_{0}|>*d*) and deflected mapped components (|*x*
_{0}|≤*d*). In the case of *p*≪1, we can achieve high quality deflection pattern. In that case, Eq. (5) can be approximated by

From Eq. (6), the amplitude of the deflection and the beam spot size (FWHM) in the Fourier plane are estimated to be *λf*Δ*θ*_{m}
/(*πd*) and
$\frac{\sqrt{2\mathrm{ln}2}\lambda f}{\left(\pi w\right)}$
, respectively. Therefore the resolvable spot number ${N}_{G}^{\mathit{\text{app}}}$
defined by the ratio of the full amplitude of deflection to the beam spot size is given by

This equation is valid for *p*≪1 (roughly *p*>0.5).

Of course, a high quality deflected beam trajectory can be achieved by using rectangular beam profile (beam width of 2*d*) as an input beam. In this case, the deflected beam pattern in Fourier plane can be written as

and is shown in Fig. 3(c). The full amplitude of the deflection and the beam spot size (FWHM) in the Fourier plane are found from calculation to be *λf*Δ*θ*_{m}
/(*πd*) and 1.39 *fλ*/(*πd*), respectively. Therefore the resolvable spot number *N*_{rec}
for the rectangular beam profile is given by

Equation (9) represents that the maximum resolvable spot number depends only on the maximum modulation index, Δ*θ*_{m}
. To achieve higher resolvable spot number, high modulation index is required of the EOD.

In the experiment, we usually employ Gaussian beam together with a slit (width of 2*d*) placed just after the EOD. Figure 4 shows the calculated relation between parameters *p*≡*w*/*d* and *r*≡*N*_{G}
/*N*_{rec}
, where *N*_{G}
is the resolvable spot number for the Gaussian beam profile with the slit. Solid line is numerically calculated. Dotted line is calculated using Eq. (7) and Eq. (9). For the case of roughly *p*<0.5, *N*_{G}
is approximated by ${N}_{G}^{\mathit{\text{app}}}$
and can be calculated directly using Eq. (7). For the case of *p*≥0.5, *N*_{G}
should be estimated using Eq. (9) and Fig. 4. For the fixed EOD width, it is obvious that the resolvable spot number decreases with a decrease of the parameter *p*, because the diffraction angle of the beam also increases with a decrease of the Gaussian half-width *w*. If we take a large *w* together with a fixed slit width of 2*d*, the distribution function of the beam at the output plane of the EOD will closely resemble an ideal rectangle, though the loss of the optical energy at the slit becomes large. If we take a Gaussian beam with the beam parameter *p*≪1, the loss of the optical energy become small, though the resolvable spot number becomes small.

#### 2.3. Time resolution

Time resolution of the time-to-space mapping is related to the deflection speed and the resolvable spot number of the deflection. For the sinusoidal deflection, minimum time resolution is achieved at *x*
_{1}=0 in the Fourier plane. Spatial position of *x*
_{1}=0 is corresponds to the temporal position of *t*=±*T*/2, where *T* is the deflection period.

In the case of *p*≪1, the minimum time resolution Δ${\tau}_{G\mathit{:}\mathit{\text{min}}}^{\mathit{\text{app}}}$
can be calculated from Eq. (6). The profile of the electric field of the continuous light wave which is time-to-space mapped at *x*
_{1}=0 can be written as

From this equation, we achieve

In the case of rectangular beam profile, we achieve

In any case including Gaussian beam profile with the slit, the minimum time resolution can be calculated using resolvable spot number and modulation frequency and expressed as

For the rectangular case, the minimum time resolution is expressed as

where Δ*ν* is the total spectral width of the deflected light field given by Eq. (4). The minimum time resolution is inversely proportional to the spectral width of the sidebands generated by the EOD. The minimum time resolution is equal to the interval time required to the deflected beam spot for passing over the point *x*
_{1}=0. Experimentally, we can evaluate the minimum time resolution on the basis of the pulse width of the pulse train which extracted through the narrow enough spatial slit placed at *x*
_{1}=0.

Figure 5(a) schematically shows the time slot of the time-to-space mapping using sinusoidal deflector. Figure 5(b) shows the relation between the normalized time slot, Δ*t*_{slot}
/*T*, and the normalized time resolution, Δ*τ*_{max}
/Δ*τ*_{min}
, where Δ*τ*_{min}
is the minimum time resolution obtained at the position *x*
_{1}=0 and Δ*τ*_{max}
is the maximum time resolution obtained within the time slot. As shown in Fig. 5(b), the maximum time resolution and the time slot are in the relation of the trade-off because of the sinusoidal deflection.

## 3. Experimental evaluation of the minimum time resolution

We experimentally evaluate the minimum time resolution based on the pulse width of the optical pulses picked out through a narrow slit which acts as a temporal window. Optical pulse generation can also be considered to be one of the applications of the time-to-space mapping. Figure 6 shows the experimental setup. A 514.5 nm continuous wave Ar laser and 16.25 GHz pulsed magnetron (pulse width of 1 *µ*s, repetition rate of 1 kHz) are used as a light source and as modulating microwave source, respectively. The traveling-wave EOD is fabricated by use of simple domain-engineering processes in a LiTaO_{3} electro-optic crystal. We perform the periodic domain inversion by applying a high-voltage pulse on a z-cut LiTaO_{3} crystal with a thickness of 0.5 mm and a length of 40 mm. A silver microstrip line with a width of 0.5 mm (equal to 2*d*) is evaporated on the crystal to guide the modulating microwave. Details of the EOD are described in reference Ref. [15].

In the experiment, incident beam was gated by an acousto-optic switch to block the light beam while the magnetron was inoperative. The half width of the input Gaussian laser beam was *w*=0.255mm, therefore the beam parameter is *p*=1.02. From Fig. 4, the parameter *r* for *p*=1.02 is estimated to be *r*
_{p=1.02}=0.87. A slit3 (width of 0.5 mm) was placed just after the EOD to shut out the non-mapped components. The pulses extracted through the narrow slit were observed by the streak camera (Hamamatsu: C5948). Simultaneously with the observation by the streak camera, we measured the spectrum of the extracted pulse train to estimate Δ*θ*_{m}
. The sideband components of the pulse train were spatially separated by a 5-cm-width grating with 2400 lines/mm, and observed with a CCD camera placed at the focal plane of a 2-m-radius FT mirror.

Figure 7(a) shows typical streak image of generated pulse train. The repetition frequency of the pulse train is 32.5 GHz (=2*f*_{m}
). Figure 7(b) shows the relation between Δ*θ*_{m}
and pulse width. The solid line is calculated using Eq. (9) and Eq. (12) with *r*
_{p=1.02}=0.87. The open circles indicate experimental results estimated from the measured intensity profiles of the streak image. The experimental results roughly agree with the theoretical calculation for the minimum time resolution of the time-to-space mapping. The achieved minimum pulse width was 1.6±0.2 ps for the maximum modulation index of 18.9 rad. The modulation power was about 2 kW. These experimental results indicate that the minimum time resolution of the time-to-space mapping using our traveling-wave EOD is about 1.6 ps. From Fig. 5(b), the time slot for the time-to-space mapping with the maximum time resolution of less than 2 ps is estimated to be 12.6 ps. In our case, the maximum modulation index was limited only by a discharge on the microstrip line. By making the crystal longer, we can improve the minimum time resolution. Additionally, a thinner crystal will improve the modulation efficiency.

## 4. Conclusion

We have demonstrated a time-to-space mapping of a continuous wave laser with picosecond time resolution using a traveling-wave EOD. The analytical expression for the minimum time resolution has been derived and experimentally confirmed based on the pulse width of the optical pulses picked out through a narrow slit which acts as a temporal window. The achieved minimum time resolution has been 1.6±0.2 ps for the maximum modulation index of 18.9 rad.

## Acknowledgments

This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research on Priority Areas, 18040005, 2006 and Young Scientists (B), 18760040, 2006.

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