In recent years the traditional technique of ultrashort pulse generation based on phase reconstruction with dispersive elements has been improved. However, the measurement of the unknown phase distortion in the laser system and the lack of easy-to-handle dispersive elements for accurately controlling the phases of ultrabroadband radiation lead to difficulties in phase correction.
As in wave-front correction by a spatially phase-conjugate mirror [1
S. Kwong, G. A. Rakuljic, and A Yariv, “Real time image subtraction and exclusive or operation using a self-pumped phase conjugate mirror,” Appl. Phys. Lett.
48, 201 (1985). [CrossRef]
], temporal phase-distortion by group-delay dispersion (GDD) can be compensated by a frequency-domain phase-conjugate (FDPC) mirror, as shown in Fig. 1
. The FDPC wave is a time-reversed replica [2
D. A. B. Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett.
5, 300 (1980). [CrossRef] [PubMed]
] of the input wave, i.e., an opposite chirped pulse. The FDPC wave will reconstruct the initial phase structure after propagating through the same GDD medium. We can automatically compensate for the phase distortion of an ultrashort, broadband light pulse.
The generation of a time-reversed replica has been demonstrated in phase-sensitive hole burning in inhomogeneously broadened absorption medium, photon-echo devices [3
M. Mitsunaga and N. Uesugi, “248-bit optical data storage in Eu3+:YAlO3 by accumulated photon echoes,” Opt. Lett.
15, 195 (1990). [CrossRef] [PubMed]
] for microsecond or nanosecond pulse durations. Typically the systems use hyperfine absorption lines at a liquid-He temperature. The time-reversed replica, however, has some phase distortion arising from the material response. The pulse duration and maximum temporal resolution in the devices are limited by the linewidth of the inhomogeneous and homogeneous broadening of the photon-echo medium.
Fig. 1. Scheme for phase correction by phase-conjugate mirrors.
On the other hand, FDPC pulse generation in the picosecond or femtosecond region has been demonstrated with a spectral holographic technique [4
A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. of Quantum Electron.
28, 2251 (1992). [CrossRef]
]. Spectral phase information derived from interference between the input signal and a transform-limited (TL) reference pulse is encoded and is recorded as a spatial fringe pattern in a holographic plate in the 4-f arrangement. The recorded phase information can be reconstructed by irradiation of the reference pulse. Real-time generation of the time-reversed wave was demonstrated by wave mixing in a nonlinear crystal instead of the holographic plate [5
D. Marom, D. Panasenko, R. Rokitski, P. Sun, and Y. Fainman, “Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett.
25, 132 (2000). [CrossRef]
]. The crystal performs the encoding and decoding operation in the sequential process of sum-frequency generation followed by difference-frequency mixing. The crystal must be pumped at the sum frequency of two TL pulses that have pulse durations short enough to yield sufficient temporal resolution. The sum frequency of the TL pulses is more difficult to generate than the phase correction of the original pulse at the fundamental frequency. In addition, the time–space conversion in the 4-f system introduces space–time coupling [6
M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. of Quantum Electron.
32, 161 (1996). [CrossRef]
]. Unwanted additional pulses or lateral pulse distortion appears as a result of the space–time coupling.
In this paper we introduce a real-time FDPC pulse generation scheme that uses frequency mixing of prechirped pulses. This simple scheme uses a nonresonant χ
(3) Kerr medium with no spatially dispersive devices.
2. Frequency-domain phase-conjugate pulse generation with chirped pulses
Fig. 2. Generation of a time-reversed replica.
The key to generating the FDPC wave is phase-conjugate operation for Fourier components [7
A. M. Weiner “Comment on ‘Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves’,” Opt. Lett.
25, 1207 (2000). [CrossRef]
], i.e., Fourier phase conjugation. The Fourier components are separated by inhomogeneous broadening in the photon-echo system or the spatially dispersive elements in spectral holography. The phase-conjugate operation has two steps, encoding and decoding. The encoding operation extracts the phase-correlation function of the signal and the reference pulses, where the Fourier phase of each component is compared with the reference pulse. In the subsequent decoding process the time-reversed replica is reconstructed from the correlation function by use of another reference pulse. The encoding and decoding processes are essentially same, but the phase is temporally inverted relative to the reference pulse as shown in Fig. 2
. The time-reversed replica is seen as a mirror image of the signal pulse, where a pair of reference pulses plays the role of a temporal mirror. When the reference has a non-TL phase structure, the phase distortion in the reference pulses is added and appears in the reconstructed pulse as the same type of wave-front distortion that would result from an imperfect ordinary mirror. The non-TL distortion can be canceling out when a pair of temporally symmetric pulses is used as the reference. Instead of a pair of reference pulses, a single reference pulse can be used with a multistep process or with a two-photon process.
Before the encoding operation, suppose that we give a large linear GDD to the signal and reference pulses. The encoding operation gives the same result independently of the given GDD because of correlation. In the slowly frequency-varying prechirped pulses, each Fourier component is already separated in time, so that no additional spatially dispersive element is required. When these two pulses overlap and propagate together in a χ
optical Kerr medium, the medium is driven by the instantaneous beat frequency. Then the Kerr medium plays the role of a frequency modulator; both upper and lower sidebands are generated as shown in Fig. 3(a)
. Because the time–frequency conversion was performed by the prechirping, the instantaneous beat frequency is proportional to the delay time. The lower sideband forms a mirror image of the signal pulse in the frequency domain.
Fig. 3. Energy and momentum conservation of four-wave mixing for (a) chirped pulses and (b) the phase-correction scheme. The lower sideband (red curve) traces a mirror image of the signal.
When the prechirped signal wave with cubic phase distortion (ϕs
) interacts with the linear chirped reference vR
with a delay time of Δτ
, the instantaneous frequency of the lower sideband vLSB
) traces the curve vLSB
. The delay time Δτ
must be much smaller than the pulse width of the chirped pulse. The mirror image has the opposite sign of the cubic distortion, as shown in Fig. 3(b)
, with a frequency shift of -2b
. The lower sideband reconstructs the time-reversed replica after a pass through a negative GDD system (it makes bt
→ 0), the same as the pulse compression in a chirped pulse amplification (CPA) system. After the pulse compression, the small frequency shift -2b
appears in the time domain as a time delay of 2Δτ
between the signal and the generated replica. In the time domain the scheme resembles a two-pulse photon echo.
The wave mixing discussed above is based on a two-photon pumped parametric process. The Kerr medium is used as a degenerative parametric amplifier that is pumped by the sum frequency of the reference pulse. The lower sideband (idler wave) has a instantaneous frequency of vi
, where vp
, and vi
are frequencies of the reference, the signal, and the idler, respectively. Suppose that we fix the frequency of the reference wave at vp
; the lower sideband traces the opposite frequency chirping. The flipped or mirror image at frequency vp
is not a simple spectral inversion. The signal and the idler always have a phase-conjugate relationship because of the parametric process. When the coherence length of the reference wave is sufficiently longer than the pulse duration of the signal wave, the system exactly detects the phase difference between the signal and the reference waves. Thus the generated idler is entirely phase conjugate with the signal wave. The generated idler has the opposite chirping; the initial phase structure can be reconstructed after the wave passes through the same GDD elements that originally gave the frequency chirp to the signal wave.
On the other hand, there are some limitations in this scheme. The parametric process has no energy storage mechanism. The amplitude of the signal (the pulse-shape envelope) is directly transferred to the replica. The pulse shape (which is given approximately by its spectral shape) must be an even function. In addition, the frequency chirp is flipped at the given frequency vp
, but the flipping is not temporal. The lower sideband vLSB
) traces the curve
) = vi
) = 2vp
) = vp
) shown in Fig. 4
. The cubic (or odd-order) phase distortion cannot be automatically compensated except when vp
is placed at the local minimum of the cubic distortion. Additional control of the cubic distortion is required for perfect phase correction. Controlling for cubic distortion, however, is much easier than in conventional techniques because the quadratic and quartic terms are automatically compensated.
is chosen to match the center frequency of the signal wave to avoid a large frequency shift. In this quasi-degenerate parametric process [8
R. P. Espindola, M. K. Udo, D. Y. Chu, S. L. Wu, and S. T. Ho, “Novel pulse-delayed scheme for degenerated optical parametric amplification in waveguides,” Opt. Lett.
21, 710 (1996). [CrossRef] [PubMed]
] the phase-matching condition is satisfied for both forward and backward geometrical arrangements. In particular, the backward four-wave-mixing arrangement shown in Fig. 5
, which is commonly used for the spatial phase-conjugate mirror, may make it possible to combine the FD and spatial-domain (SD) phase-conjugate mirror roles. When the nonlinear medium used in a conventional SD phase-conjugate mirror (for example, a photorefractive crystal) is replaced with a nonresonant Kerr medium, transient moving gratings generated in the Kerr medium can correct the frequency chirping and wave-front distortion simultaneously. The SD-FDPC mirror that is installed as a folding mirror of a two-pass laser amplifier could possibly be used to improve the beam quality and the GDD compensation in a high-energy or high-average-power ultrashort laser system.
Fig. 4. Opposite-chirped-pulse generation with a constant-frequency reference. The even-order phase distortions are compensated by changing the sign of the chirp.
Quasi-degenerate phase-matching condition in the backward arrangement of the scheme of Fig. 3(b)
. The idler wave satisfies both the spatial and the frequency phase-conjugation relation. The inset shows the same phase-matching condition with closed wave vectors.
3. Experimental setup
The first demonstration of FDPC wave generation for the scheme of Fig. 3(a)
was carried out with two chirped pulses. A CPA Ti:Al2
laser system producing a 100-fs pulse with an energy of 5 mJ at 10 Hz was used. A chirped pulse from a stretcher with a pulse duration of 130 ps is divided into two pulses and is temporally separated by optical delay lines as shown in Fig. 6
. The two pulses are coaxially injected to a regenerative amplifier (RA). Transmissive optics in the RA, such as polarizers, a Pockels cell, and the laser crystal are used as the Kerr media. An output pulse from the RA is recompressed by a grating-pair compressor.
The phase relation between the injected and the generated pulses is monitored by second-harmonic generation (SHG) frequency-resolved optical gating (FROG) and phase-
= 420 fs is shown in Fig. 7
. This trace was taken with 10,000 shots with no averaging. Each fringe in this trace corresponds to single shot. The recompressed output pulse had four separated pulses; we mark them with symbols p1 through p4, where p2 and p3 are the injected signal and reference pulses and p1 and p4 are the generated upper and lower sideband pulses, respectively. The autocorration trace of this pulse train should have three terms. At a delay time of zero, the sum of the autocorrelation of each pulse appears as
, where Ei
denotes the electric field of the pulse pi
. At the input-pulse separation delay time (420 fs), the SH-signal corresponds to the cross correlation function between adjacent pulses
and the other correlation function centered at 2 Δ τ
is showing cross correlation between every other pulse. In other words, I
exhibits a phase relation between the injected pulse and the generated FDPC pulse.
Fig. 6. Experimental setup for FDPC pulse generation with two prechirped pulses.
had a contrast ratio close to the theoretical value of 8 to 1. In contrast, the function I
had no interference fringes. In addition, the envelope of I
is the same as that of the intensity cross correlation that is given by integration for the delay time. A phase shift over the pulse width of 100 fs that is due to the frequency difference of -2b
= 20 GHz is negligible (2×10-3
rad). If the generated pulse has random phase fluctuation, some scattered data points or a coherent spike should appear in the function I
because of the single-shot measurements. We have confirmed the phase relation between the pulses with the FROG trace shown in Fig. 8
. No spectral broadening was observed at the delay of 840 fs. The result demonstrates that the generated pulse has an entirely fixed phase relation to the injected pulse. Thus, the lack of interference in the cross correlation indicates that the generated pulse satisfies a phase-conjugate relation to the signal pulse, i.e., the time-reversed replica.
A scheme for FDPC pulse generation by frequency mixing with a prechirped pulse has been proposed. A femtosecond time-reversed replica has been generated by the temporal-frequency conversion technique. The schemes, based on two-photon-pumped optical parametric process, can be applied for any nonresonant χ
(3) media, but have some limitations owing to the non-energy-storing interaction.
Part of this work is supported by a grand-in-aid for scientific research from the Ministry of Education, Culture, Sports, Science and Technology.
Fig. 7. Phase sensitive autocorration of the output pulse train. Each fringe was obtained by a single-shot technique.
Fig. 8. SHG-FROG trace of the output pulse train. The shading shows the SH intensity.