The first optical function generator

The electron is a major constituent of all matter in the universe. The movement of electrons in matter and their transition from one energy state to another in atoms, molecules, or materials are the foundation of many everyday physical phenomena, such as electrical current flow, LEDs, lasers, solar cells, superconductivity, photosynthesis, and elementary chemical reactions. Consequently, if we can control the microscopic behavior of the electron, we can govern many of the functions and activities enabled by its dynamical motion. From elementary physics we learned that electromagnetic fields act on electrons to direct their motion. An electron could transit through an atom or a molecule in a time of less than a few femtoseconds to several attoseconds where a femtosecond is 10⁻¹⁵ of a second, and an attosecond is one thousand times faster (10⁻¹⁸ of a second).

To alter the electron's path during this time, the shape of the electric field waveform will also have to be able to change. The instrument used to make electric field waveforms is the function generator or waveform synthesizer. In the laboratory, a function generator uses electronic circuits to create periodic electric waveforms. However, the fastest electronic circuits to date have a speed of only up to several hundred gigahertz. They are too slow to change the fields on the femtosecond timescale. By using light fields that oscillate with a rate of about one cycle per femtosecond, they are able to accomplish the task of generating waveforms whose shape can vary within a femtosecond. A device that uses this optical approach to making waveforms is an optical function generator.

Figure 1. Pictorial demonstration of the synthesis of specific waveforms by the coherent superposition of the first five harmonics of a fundamental frequency. fs: Femtosecond. ω: Angular frequency.

Figure 2. Schematic of the experimental setup to synthesize waveforms using harmonics generated by coherent modulation of the H ₂ molecule. ω₁and ω₂ designate the frequencies of the inputting laser pulse to the H ₂ cell. AM and PM are liquid crystal spatial light modulators. BBO: Beta-barium borate. (Reprinted from Chan et al.¹)

Figure 3. Experimental data obtained for (A) subcycle cosine, (B) subcycle sine, and (C) square waveforms. The black square points are measured shaper-assisted linear cross-correlation²data, and the solid red curves are numerical simulation. arb.: Arbitrary. (Reprinted from Chan et al.¹)

An optical waveform operates on the theory of Fourier synthesis. According to Fourier transform theory, electromagnetic fields of arbitrary shape can be synthesized by setting the frequency, amplitude, and phase of different sinusoidal waves to values required by the specific shape. In the optical regime these synthesized fields can then vary in the femtosecond and/or subfemtosecond timescale.³ As shown in Figure 1, Fourier synthesis can be enacted using a coherent spectrum composed of a comb of harmonic frequencies that span more than one octave. There are a number of ways to create this comb. One of the most effective is to use molecular modulation, first proposed by Stephen Harris and Alexei Sokolov at Stanford University.² In molecular modulation, the coherent motion of a molecular ensemble is taken by two lasers to its maximum value by driving the molecules near one of its Raman resonances. The laser excitation is slightly detuned from the resonance to avoid any absorption loss by the molecules. At this point, the refractive index of the molecular medium varies at the frequency of the coherent motion. The varying refractive index in turn modulates the frequency of the incident beam and produces sidebands with high efficiency.

In our laboratory, we started with an intense IR laser tuned to near the vibrational Raman frequency of the hydrogen molecules and induced strong coherent motion of the molecules using the IR laser and its second harmonic. These molecules then modulated the frequency of the incident IR laser beam to generate a new optical beam that contained a coherent comb of frequencies comprising several harmonics of the laser frequency.⁴ We next used liquid crystal spatial light modulators to adjust the spectral amplitudes to predetermined values and to align the phase of the harmonics. Figure 2 shows the experimental arrangement for a comb consisting of the first to fifth harmonics.

At this point, with the beam of harmonic comb frequencies overlapped in space and in time, they constitute an optical waveform of a predetermined shape.¹ We verified the shapes by cross-correlation in which the required reference pulse was extracted from the main pulse using a pulse-shaping technique. We call this self-correlation method approach shaper-assisted linear cross-correlation (SALC²). A few of the waveforms synthesized with five harmonics and verified by SALC² are the sawtooth, square, subcycle cosine, and subcycle sine (see Figure 3). These measurements are a defining step to ultimately achieving a fully functional arbitrary optical waveform generator. The results signify achievement of a new ultrafast light source that, with further development, could find application in nanoelectronics, nanomaterials, ultrafast electronics, and the control of chemical reactions.

While molecular modulation is most effective in producing a large number of harmonics in a single setting, other methods are available to produce a harmonic frequency comb if the number of harmonics is less. For example, up to five harmonics are commercially available for the neodymium-doped yttrium aluminum garnet and titanium sapphire lasers. An intriguing possibility is the use of quasi-phase-matched harmonic generation. These engineered crystals can be employed to generate multiple harmonics by cascaded quasi-phase-matched nonlinear mixing in a single crystal.⁵ We are working on realizing a portable optical function generator by combining such a crystal with a compact light modulator.