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All-optical phase correction scheme for few-cycle optical pulses has been proposed. The temporal phase structure of a phase-modulated pulse is recorded as spatial index modulation in a two-photon recording medium. This scheme drastically relaxes complicacy in stretcher design for few-cycle chirped pulse amplification (CPA) systems.
©2006 Optical Society of America
1. Introduction
An ultrashort pulse changes its temporal phase structure via group-delay dispersion (GDD). For few-cycle laser pulses especially, third- and higher-order dispersion seriously affects the pulse shape. Residual high-order dispersion critically limits the shortest-pulse duration in chirped pulse amplification (CPA) systems, typically to ~20 fs. The residual high-order dispersion must be compensated with an additional phase modulator to obtain shorter pulse duration [1–2]. In these cases, we have to accurately measure pulse duration and Fourier phase [3–4]. The repeated feedback control to the optical phase modulator is necessary for obtaining a transform-limited (TL) pulse.
The frequency-domain phase conjugator (FDPC) generates a time-reversed replica of an input pulse [5–7]. For example, the time-reversed replica of a phase-distorted pulse by a GDD element regenerates initial pulse shape after passing again through the same GDD element. The temporal phase distortion can be automatically compensated by the all-optical system. The spatial domain phase conjugation corresponding to a wave-vector-reversal is not the time reversal. In other words, the FDPC (the time reversal) means Fourier-phase conjugation of each angular frequency component [8]. For example, the photon-echo system [9] is one kind of FDPC, where very narrow homogeneous absorption lines in inhomogeneous broadening correspond to the frequency selective phase conjugator. The intrinsic absorption lines, however, tightly limit operation wavelength and laser pulse duration. The FDPC operation by the frequency resolved wave-mixing [5] or spectral holography [8] is a flexible method for the wavelength region where a pair of gratings is used for frequency separation. But the maximum pulse duration is limited by frequency resolution of the gratings. The four-wave-mixing in a χ(3) medium between chirped pulses [6] can be applied to generate a femtosecond pulse from a nanosecond pulse, but a grating-pair or some dispersive element is still required.
In this paper, we demonstrate FDPC operation in a far off-resonant material with no dispersive element. The instantaneous phase of a signal pulse (a frequency chirped pulse in Fig. 1) is temporally gated and is recorded in the space of a recording medium, as shown in Fig. 1. The phase structure of the signal pulse is recorded by using two-photon interference with a write (gate) pulse. A two-photon absorbing medium that changes index of refraction is used as the recording medium. The recorded index modulation acts as a programmable Bragg reflector. The -1st order diffraction of a TL pulse (read pulse) becomes the time-reversed replica of the signal pulse, i.e., a FDPC wave as shown in Fig. 1.
2. Two-photon gated recording and pulse regeneration by Bragg diffraction
2.1 Contrast ratio of two-photon interference
If we observe two-photon interference between two chirped pulses (autocorrelation), time-integrated interference fringes appear in a limited time window corresponding to their coherent time, as shown in Fig. 2(a). In contrast, when a TL pulse that has the same frequency bandwidth of the signal pulse is used as a gate pulse, the cross correlation between the gate and signal pulses has long-term fringes that remain for the duration of the signal pulse, as shown in Fig. 2(b). The phase information exceeding coherent time can be recorded as the second-order interference fringes. This is because the position of the interaction changes as a function of time, so the fringes are not overwritten.
The gate pulse has all frequency components of the signal. A part of them, corresponding to instantaneous frequency of the signal, makes interference fringes. Thus, the contrast of the fringes becomes degraded when the ratio of pulse duration becomes large. The contrast ratio has been numerically calculated for signal pulse duration and pulse energy, as shown in Fig. 3(a). The vertical and horizontal axes (RPW and RPE) are normalized by gate pulse duration and gate pulse energy, respectively. The optimum conditions lay on a slope of 2 in the contour plot. Figure 3(b) corresponds to a gradient of the ridge in Fig. 3(a) that is showing contrast ratio as a function of pulse duration. The contrast is given as the amplitude of the electric field, which is the function corresponding to the quartic root of the two-photon absorption, so this method is highly sensitive. For example, a gate pulse with a fraction (1/100) of the signal in energy is sufficient for obtaining a signal pulse with a duration that is 10,000 times longer.
Fig. 2. Second-order interference between two-chirped pulses, i.e., autocorrelation function (a), in case of between a TL-gate pulse and a chirped pulse (b). TPA, two-photon absorber; Tp, pulse duration; Tc, coherent time.
Fig. 3. Fringe contrast ratio in the second-order cross correlation between the gate pulse and a frequency chirped pulse (a). Optimum possible contrast as a function of pulse duration (b). The Γ is a slope of the asymptote.
2.2 Diffraction efficiency of a few-cycle grating
The diffraction efficiency η diffraction of a grating that has coherent length L coherent is given by
The diffraction efficiency is proportional to the square of coherent length (i.e., number of grooves) and index modulation Δn, respectively. For example, a 1-cm-thick holographic volume grating written by visible monochromatic laser beams has approximately several tens of thousands of grooves. In contrast, a 10 fs laser pulse at 800 nm has only a four-cycle optical field. Thus, the number of grooves formed with interference of these pulses is eight. The reduction factor in diffraction efficiency for the 10 fs grating is smaller than one millionth. For compensation, the index modulation should be increased more than 1000 times.
In a case of a linearly chirped signal pulse ω(t)=ω 0 +αt, the written grating is also spatially chirped. The coherent length of a highly chirped grating is defined as the number of grooves up to π-phase shift. When we assume the bandwidth of signal is constant and instantaneous frequency is linearly chirped with the chirp parameter α , coherent length (tc: coherent time) of the written grating is proportional to the square root of the stretching factor, i.e.,
Thus, the diffraction efficiency of a linearly chirped grating is proportional to the stretched pulse duration. The numerical calculation for diffraction efficiency as a function of signal pulse width with an index modulation of Δn=0.01 is shown in Fig. 4. In the linearly chirped case, the bandwidth is fixed to be the same as that of a TL 10 fs pulse. The γ in Fig. 4 indicates the slope of the asymptote. For practical application, the diffraction efficiency should be limited to an appropriate value to prevent pulse distortion caused by light depletion. For example, the index modulation Δn=0.01 is suitable for experiments with a chirped pulse of several picoseconds.
Fig. 4. Calculated diffraction efficiency for monochromatic and linearly chirped grating as a function of pulse width. The index modulation is Δn=0.01.
2.3 Compensation of high-order dispersion in a CPA system
The conceptual designs of dispersion-free CPA systems are shown in Fig. 5. The basic concept is a round-trip, double pass configuration as shown in Fig. 5(a). Temporal phase distortion given by the first half is compensated by the second half with a time-reversed replica. The typical pulse duration after the stretcher, however, should be several hundreds of picoseconds. A thick FDPC medium (~10 cm) is required for this scheme. A GDD by the FDPC itself cannot be compensated (double of its own GDD will remain). In the second scheme, as shown in Fig. 5(b), the second-order dispersion (SOD) is mainly compensated by the commonly used stretcher and compressor. Only residual high-order dispersion (HOD) is recorded in the FDPC. The recorded FDPC device is relocated in front of the stretcher to regenerate a time-reversed replica of the phase distortion. The pulse duration broadened by HOD will be limited to within several picoseconds, and the requirement in thickness for the FDPC is drastically relaxed. The precompensation scheme has other advantages because the FDPC is placed under low intensity condition. The precompensation scheme prevents optical damage and reduces the size of the device. For example, the light intensity at the precompensator is kW/cm2 in contrast to typical write intensity, which is GW/cm2. Long-term stability can be expected because two-photon absorption at the precompensation is 10-12 times smaller than that at the recording. A typical CPA system has a system gain of 108-10 under saturated amplification. The output pulse energy after the high-gain and saturated amplifier will be insensitive to energy loss at the pre-compensation.
Fig. 5. Automatic phase correction scheme in a CPA system. (a) Round-trip double pass configuration, (b) Precompensation scheme coupled with a conventional CPA system.
A numerical simulation code has been developed to confirm that the high-order GDD is compensated separately [13]. The code includes the matrix method that is used to design multilayer dielectric mirrors. Each layer of thickness and index modulation is made by the two-photon interference between residual phase distortion and the gate pulse. In this calculation, a combination of an SF10 material stretcher and a grating pair compressor is used for generating a third-order GDD of 8.2 × 104 fs3. The initial pulse duration was set to 10 fs. The phase distortion, shown in Fig. 6(a), was recorded with a temporal window of 2.8 ps. The precompensated output became a near TL pulse as shown in Fig. 6(b). The insets in these figures show Fourier phases. The clipping with the bandwidth of 75 THz corresponds to the temporal window of recording pulse duration.
Fig. 6. Temporal phase distortion by HOD (a) and precompensated output (b) by the FDPC with a temporal window (thickness) of 2.8 ps. The insets are showing Fourier phases. Initial pulse duration is 10 fs.
3. Experiment
3.1 Experimental setup
The first FDPC experiment with sub-millijoule and 100 fs laser pulses has been demonstrated in semiconductor-doped colored glass with an index modulation of Δn=10-5 [10]. As mentioned before, an index modulation of Δn>10-2 at a far off-resonant wavelength is required for few-cycle experiments. The huge index modulation at the laser wavelength expected to be achieved by the bleaching of extremely strong absorption at the two-photon wavelength. Dye-doped thermoplastics having an absorption cross-section on the order of 10-16 cm2, which is approximately 1000 times larger than that of the colored glass, are used. This medium was used a thin film to avoid SPM and GDD. A 72-μm-thick vinyl acetate film containing C307 laser dye has an optical density of OD=11 at 400 nm.
10 fs pulses coming from a mode-locked laser were split into two beams and interacted at the focal plane of a microscope objective as shown in Fig. 7. The interaction region at the focal plane was 13 μm wide and 207 μm thick. The spatially averaged laser intensity at the focal plane was 3.7 GW/cm2. The laser pulses bounced 22 times between a pair of chirped mirrors to compensate a GDD of 1260 fs2 produced by the microscope objective. Direct monitoring of Fourier phase of the ultrashort pulses was carried out by using the frequency shearing interferometory (SPIDER) technique [11].
Fig. 7. Experimental setup for the two-photon gated writing with 10 fs laser pulses. OD: optical density of the sample. AP: aperture, BS: beam splitter, CC: retro reflector. CM: concave mirror for collimation. NA: numerical aperture of the microscope objective.
3.2 Diffraction efficiency and bandwidth
The change of refractive index due to the bleaching of dye was monitored in diffraction efficiency as shown in Fig. 8. The diffraction efficiency increased as a function of exposure time up to 10 minutes. The maximum diffraction efficiency of the 8-groove grating was 3.2 %. The diffraction efficiency corresponds to a change in refractive index of 0.03. This index change is the highest value written by two-photon process as far as we know. No high-order diffraction was observed, so the index modulation was effectively sinusoidal. No diffraction portion was observed when we stopped mode-lock. The influence by fundamental absorption was negligible. A spectral bandwidth of the grating was 150 nm centered at 800 nm, which is enough large for the 10 fs pulse experiment. A beam divergence was measured to be a diffraction-limited number.
Fig. 8. Diffraction efficiency as a function of exposure time. The change in refractive index was 0.03 after 10 minutes exposure.
Fig. 9. Fourier-phase conjugation and self-recompression. (a) Signal: positively chirped pulse by a glass plate, (b) phase-conjugated pulse, and (c) self-recompressed pulse after passing through the glass plate again.
3.3 Frequency-domain phase conjugation
A 3-mm-thick BK-7 glass plate was inserted into a single arm in Fig. 7, and the temporal phase structure of the chirped pulse was recorded in the film. The change of Fourier phase by the glass plate and the FDPC were monitored by the SPIDER, as shown in Fig. 9. An up-chirped laser pulse by the GDD in the glass plate was converted to a down-chirped pulse by the -1st order diffraction of the recorded gratings. The time-reversed replica became a TL-pulse after passing through the glass plate again. The Fourier phase in Figs. 9(a) and 9(b) include high-order GDD. The FDPC operation for a bandwidth exceeding 80 THz has been observed [12].
4. Summary
In conclusion, a precompensation scheme for eliminating phase distortion caused by high-order group-delay-dispersion has been proposed. This scheme drastically relaxes complicacy in stretcher design for few-cycle CPA systems. The frequency-domain phase conjugation and regeneration of a 10 fs laser pulse has been demonstrated by use of the two-photon written Bragg grating. The all-optically programmable Bragg grating has been directly written by 6 nJ IR pulses with an index change of 10-2. The bandwidth of the recorded grating was sufficient for few-cycle pulse shaping.
Acknowledgments
Part of this work is supported by a grant-in-aid for scientific research, and the 21st Century Center of Excellence (COE) program from the Ministry of Education, Culture, Science, Sports and Technology.
References and links
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