1. Introduction

Frequency-resolved optical gating (FROG) is a pulse characterization technique based on pulse spectrogram measurement in the time-frequency domain [1]. A variety of third-order (χ⁽³⁾) nonlinear optical processes have been used for the optical gating mechanism. χ⁽³⁾ FROG has the advantage of acquiring intuitive spectrograms, with which the order and the sign of the chromatic dispersion of optical pulses are easily determined. However, more than orders of magnitude higher pulse energy was required for χ⁽³⁾ FROG methods in comparison with high-sensitivity FROG methods based on second-order nonlinear processes [1].

High-sensitivity intuitive spectrogram measurement of weak optical pulses was recently performed using high-efficiency two-photon absorption (TPA), which is an imaginary part of χ⁽³⁾ nonlinear susceptibility, in a semiconductor crystal [2]. The TPA FROG was based on frequency-selective and phase-sensitive detection by a lock-in amplifier for elimination of transmission background of probe pulses, and hence a single-channel photoreceiver was used, since a multi-channel photoreceiver array was impractical for lock-in detection. The angle of the grating of the spectrometer was changed for wavelength scanning, and acquisition time of nearly a minute was required for the wavelength division of more than 60 pixels. In this paper, the real-time acquisition of TPA FROG spectrograms is demonstrated. Real-time TPA FROG is implemented through the introduction of synchronized wavelength-delay time scanning which is compatible with rapid data acquisition using a fast lock-in amplifier. The acquisition time is 1.8 s per spectrogram of 128×128 pixels.

There were also improvements in other techniques for characterization of weak optical pulses in the time-frequency domain. Sonograms of optical pulses were acquired by high-sensitivity two-photon absorption in a laser diode [3] or a photodiode [4]. Rapid scanning technique was developed for real-time sonogram measurement [4]. Real-time second-harmonic-generation FROG was implemented by line scanning of the gate beam in multiple shot measurement [5]. In comparison with these techniques, the TPA FROG has the advantage of the direct and rapid measurement of the magnitude and phase of optical pulses in the spectral domain without using pulse retrieval algorithms as described in Sect. 3.

2. Real-time acquisition of TPA spectrogram

Real-time spectrogram acquisition was performed using the TPA FROG apparatus illustrated in Fig. 1. Optical pulses under test were split into gate and probe pulses by a beam splitter. The beam of the gate pulses was chopped by an acousto-optic modulator for signal modulation, in order to lock the reference frequency and phase of a lock-in amplifier (Stanford Research Systems, Model SR844). Lock-in detection enabled the elimination of time-independent probe transmission background. Chopping rate was 170 kHz. Integration time constant in the lock-in amplifier of 0.1 ms or shorter was essential to the real-time acquisition of TPA FROG spectrograms. The medium used for ultrafast TPA optical gating was a 0.3-mm thick anti-reflection coated InP single crystal. Chromatic dispersion of the InP crystal did not cause pulse broadening significantly for the optical pulses having a spectral bandwidth of 50 nm or narrower at wavelengths of 1500 nm.

The incident angles of the gate and probe pulses were within 5 degree from the normal axis of the front surface. Transmission through optical elements such as the mirrors, the beam splitter and the acousto-optic modulator was almost independent of the angle of incident polarization (< 0.5 dB). The polarization dependence in TPA FROG measurement was limited by the crystal-axis dependence of TPA in the optical gating medium [6]. This was within a variation of 1.9 dB. The TPA FROG is useful for the measurement of optical pulses whose polarization changes significantly in systems such as long-haul optical fibers [7]. The four-wave mixing pulses generated by the spatial grating formation did not interfere with the transmitted probe pulses since the crossing angle between the probe and gate beams in the InP crystal was within a few degree and diffraction efficiency was negligibly small.

A novel spectral scanning mechanism was developed for real-time wavelength scanning synchronized with delay time scanning in the measurement scheme based on the lock-in detection. The main component of the wavelength scanning was a counter-rotating mirror (λ-mirror in Fig. 1) based on a Galvano scanner (Cambridge Technology, Model 6450). The λ-mirror was installed between the diffraction grating of an asymmetric spectrometer and an external slit. The asymmetric spectrometer has the forward and backward focus lengths of 12.5 mm and 25.0 mm, respectively [8]. The spectrum of the probe pulses after passing through the spectrometer was imaged on the external slit. The Galvano mirror was driven with a rump wave at a rate as high as 80 Hz. Delay time scan was repeated with a period of 1.6 s. The λ-mirror and the delay stage were controlled and synchronized by a personal computer. This synchronized wavelength-delay scanning is an analogue of video framing in which the fast horizontal and slow vertical axes correspond to the spectral and time axes, respectively. The TPA intensity in the time-frequency domain detected by the lock-in amplifier was processed by the computer and displayed as a function of wavelength (λ) and delay time (τ). The time required for the acquisition and display of a TPA FROG spectrogram of 128×128 pixels in size was 1.8 s. To our knowledge, this is the shortest acquisition time as reported for ultrafast time-resolved high-sensitivity spectroscopy based on the lock-in detection.

 

Fig. 1. Illustration of apparatus for the real-time acquisition of TPA FROG spectrograms. BS: beam splitter. AOM: acousto-optic modulator. PC: personal computer. PD: photodiode.

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Fig. 2. Setup for real-time TPA FROG measurement of optical pulses with chromatic dispersion of variable magnitude and sign.

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To demonstrate real-time measurement of TPA spectrograms, femtosecond optical pulses from an optical parametric oscillator (OPO) were measured with variable chromatic dispersion in the measurement setup shown in Fig. 2. Variable chromatic dispersion was produced by a dispersive delay line consisting of a grating-lens dispersive telescope system including a liquid crystal spatial light modulator [9,10]. Second-order and third-order chromatic dispersion of variable sign and magnitude was produced in the dispersive delay line. The real-time acquisition of TPA FROG spectrograms of the OPO pulses is demonstrated as the movie images presented in Fig. 3. In this demonstration, the second-order chromatic dispersion added to the OPO pulses was changed in a stepwise manner with a period of 2 s by the dispersive delay line. The horizontal axis corresponds to wavelength from 1450 nm (left) to 1550 nm (right). The center wavelength is 1500 nm. The vertical axis represents delay time from top to bottom with a span of 4000 fs.

 

Fig. 3. Sample movie (1.6 MB) of the real-time acquisition of TPA FROG spectrograms of chirped femtosecond pulses with variable magnitude of second-order chromatic dispersion.

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3. Spectrogram analysis

Spectrogram acquired in TPA FROG measurement is generally defined as

where τ, ω, I_g(t) and E_p(t) are delay time, angular frequency, gate pulse intensity and probe pulse electric field, respectively. In Eq. (1), the optical pulses are shown as the time-frequency distribution of dark pulses in the time-independent background of the probe pulses. If lock-in detection is employed, gate-induced change in the transmitted probe intensity is detected as signal, and the time-independent transmission background is eliminated. Assuming I_g(t) ≪ 1, Eq. (1) is then reduced to [11]:

Equation (2) can be rewritten in the following form:

Here, ε_p(ω) and ι_g(ω) represent Fourier transform of E_p(t) and I_g(t). Constant factor 1/2π is omitted.

Unlike conventional FROG spectrograms, the TPA FROG spectrogram has good marginal properties, as explained below, on the basis of the mathematical treatment described in the literature [12]. Time marginal, which is obtained by integrating the distribution function S(τ, ω) of the TPA FROG spectrogram in Eq. (2) over frequency, leads to cross-correlation between the gate and probe pulse intensity. Time marginal in TPA FROG spectrogram is expressed as:

Frequency marginal obtained by integrating the distribution function S(τ, ω) in Eq. (3) over delay time provides the spectrum of the probe pulses as follows:

The relation ship in Eq. (5) was experimentally confirmed by comparing the frequency marginal of TPA FROG spectrogram with the pulse spectrum acquired independently using an optical spectrum analyzer. Average delay time <τ>_ω calculated using the distribution function in Eq. (3) is always equal to group delay time:

Here, τ is replaced with -id/dω and E (ω) = |E(ω)| exp(iΦ(ω)) with spectral phase ϕ(ω). In conventional FROG methods, on the other hand, this is only the case, if the gate window ι_g(ω) has a delta-functional form [8]. This indicates that the spectral intensity and the group delay time can be directly obtained in the TPA FROG measurement using the apparatus shown in Fig. 1, without using the FROG pulse retrieval algorithms [1,13].

TPA FROG spectrograms, spectral intensities and group delay times are presented in Fig. 4 for OPO pulses that are positively chirped, nearly zero chirped and negatively chirped, respectively. The traces for spectral intensity and group delay time were obtained from the frequency marginal in Eq. (5) and the average delay time in Eq. (6), respectively. The spectrograms and the spectral intensities are normalized with their peak values. The chirp of the OPO pulses is mainly limited by the second-order chromatic dispersion. There are some contributions due to higher-order dispersion terms. Similar measurements were performed for the optical pulses with positive and negative third-order chromatic dispersion as presented in Fig. 5. Group delay time in this case changes quadratically against wavelength.

 

Fig. 4. TPA FROG spectrograms and traces of spectral intensity (bottom) and group delay time (imposed on the spectrograms) of OPO pulses with negative, nearly zero and positive chromatic dispersion of the second order, respectively.

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Fig. 5. TPA FROG spectrograms and traces of spectral intensity and group delay time of OPO pulses with negative and positive third-order chromatic dispersion.

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4. Conclusion

Real-time acquisition technique for the intuitive spectrogram measurement has been developed using TPA in an InP crystal at 1500-nm wavelengths. The TPA FROG spectrogram was obtained by pump-probe transmission spectroscopy and led to the direct and rapid measurement of the spectral intensity and group delay time of ultrashort optical pulses without using the FROG pulse reconstruction algorithms. The real-time spectrogram measurement combined with the dispersive delay line will allow rapid feedback of chromatic dispersion and will serve as an active dispersion compensator in long-haul optical fiber transmission of ultrashort optical pulses since the TPA FROG has a high sensitivity for the spectrogram measurement of weak optical pulses with pulse energy of approximately 10 pJ.