We demonstrated controlled shifting of the internal phase of broadband terahertz (THz) pulses. The internal phase of an ultrashort pulse is called the carrier envelope phase (CEP), which is an important parameter in the interaction of few-cycle light pulses and matter. Our CEP shifter utilizes the ultra-broadband feature of prism wave plates. We analytically derived the amount of CEP shift achievable by the CEP shifter using Jones matrixes. THz time-domain measurements clearly showed the shift of the CEP, and the results agreed well with the calculated values. The CEP shift was as high as , indicating that any CEP values can be chosen using our CEP shifter.
© 2016 Optical Society of America
Developments in high power terahertz (THz) sources have enabled researchers to induce THz high-field interactions with matter. Examples of the interactions that have been achieved include driving a ferroelectric soft mode in a perovskite-type crystal , spin control in antiferromagnets , and an insulator-metal phase transition in correlated-electron materials . In addition, high-energy THz pulses with energies of 0.9 mJ and electric fields up to 42 MV/cm have been demonstrated . From the viewpoint of intense light field interactions, intense THz pulses are few-cycle light pulses . A few-cycle pulse refers to a pulse that has a short pulse duration comparable to the laser field oscillation frequency. When using such ultrashort laser pulses to induce interactions with matter, the interaction depends not only on the intensity envelope, but also on the carrier phase [5–7]. The phase of the carrier frequency with respect to the peak of the pulse envelope is called the carrier envelope phase (CEP). Figure 1 shows a general illustration of the relationship between the pulse envelope, the CEP, and the CEP shift. In the optical region, a lot of light–matter interactions have CEP dependence, such as a collective electron motion in plasmas , optical poling , and nuclear reactions . Recently, generation of high harmonics by applying a multi-THz pulse to a crystalline solid was reported, and this phenomenon was strongly dependent on the CEP . Therefore, it is expected that CEP control techniques for THz pulses will soon become important. Normally, the CEP of ultrashort pulses has to be stabilized by using feedback techniques [12,13]. On the other hand, the CEP of THz pulses is mostly locked (except for a few examples, such as two-color laser plasma generation of THz radiation [14,15]); that is the electric field profile is the same for each shot. For example, the CEP of THz pulses generated via optical rectification is locked, because this process can be regarded as parametric three-wave mixing, similar to an optical parametric amplifier . Therefore, only a CEP shifter is required for controlling the CEP of broadband THz pulses.
The CEP of an ultrashort laser pulse is shifted when it passes through a dispersive element because the phase velocity and group velocity of the optical pulses are different. This method is used routinely for controlling the CEP of broadband infrared pulses. However, spectral dispersion in the THz region is not so large. For example, when a dispersive material with a thickness of 100 μm is used, the spectral dispersion needed to shift the CEP by is , which is very large for the THz region, corresponding to a refractive index difference of 4.8 between 0.5 and 2.5 THz. If a thicker material is used, relatively low-dispersion materials can be used; however, absorption will be an issue. In addition, by using such ultra-broadband pulses, distortion of the pulse-envelope and a simultaneous time delay have been found to occur . A CEP shifter adopting a pulse shaper composed of a liquid crystal spatial light modulator and a optical configuration was demonstrated for infrared femtosecond (fs) laser pulses . This type of CEP shifter can control the CEP independently of the time delay. However, such a pulse shaper has a restricted spectral range. It is probably difficult to realize a pulse shaper with a liquid crystal spatial light modulator for THz pulses because THz pulses have an ultra-broadband spectra. Recently, CEP-stable tunable THz-emission from a laser-induced plasma spark utilizing the dependence of the emitted THz pulse CEP on the fundamental laser CEP was demonstrated .
In this Letter, we demonstrate controlled shifting of the CEP of broadband THz pulses using a CEP shifter composed by suitably arranging several polarization optical components. A prism-type wave plate  plays an important role. The prism-type wave plate is a THz achromatic wave plate that covers a range of several THz. By rotating the half-wave plate, the CEP could be shifted. Since our CEP shifter controls the CEP of radiated THz pulses, it can be applied to any pulsed THz source. We analytically derived the relation between the rotation of the half-wave plate and the CEP shift. Time-domain measurement of THz pulses clearly showed the ability to achieve a controlled shift of the CEP.
A THz temporal electric field is represented strictly by 1) indicates that the CEP is a frequency-independent phase offset of the spectral phase. This means that when the same phase shift is added to the whole spectrum, the CEP of broadband pulses is shifted. In other words, the change of CEP corresponds to the parallel translation of the phase spectrum up and down on the graph when taking the positive frequencies in the Fourier transform of the data.
Figure 2 shows a schematic diagram of the CEP shifter. The CEP shifter is composed of three optical elements: a first quarter-wave plate, a half-wave plate, and a second quarter-wave plate. The second quarter-wave plate can be substituted with a polarizer. In this arrangement, it is assumed that an input THz pulse is linearly polarized. The azimuth angle of the first quarter-wave plate is set to 45° relative to the polarization direction of the THz pulse. The half-wave plate and the second quarter-wave plate are rotatable; the azimuth angle of these wave plates are and , respectively. A qualitative explanation of the function of this series of optical components is as follows: the first quarter-wave plate converts the THz pulse into a circularly polarized pulse, the half-wave plate behaves as a rotator, which rotates the THz pulse azimuthally without deformation of the circularly polarized pulse shape, and the second quarter-wave plate converts the circularly polarized pulse into a linearly polarized pulse and determines the polarization direction of the output THz pulses.
This arrangement can be described with Jones matrices as follows: when we use a quarter-wave plate as the third optical component, the output THz pulse, , becomes2) is common to the whole spectrum. As mentioned above, the CEP can be shifted by adding the same amount of phase shift to the whole spectrum. Furthermore, this amount of phase shift corresponds to the amount of CEP shift. Therefore, the phase shift calculated using Jones matrices represents the amount of CEP shift.
Considering an initial phase, , an initial THz pulse, , that is linearly polarized is described by3) into Eq. (2), we have the Jones vector of the output THz pulse as follows: 4), the phase difference of the two orthogonal electric fields is , which indicates that the output pulse is linearly polarized. By calculating the polarization direction, , from Eq. (4), we have 6)].
Equations (5)–(7) indicate that the CEP depends on both and , and the polarization direction depends on only . In practice, first, the polarization direction is determined by rotating the third optical component (the second quarter-wave plate or the wire-grid (WG) polarizer), and then the CEP is changed by rotating the half-wave plate. Equation (6) indicates that the CEP, after passing through the CEP shifter, is shifted by an amount equivalent to twice the rotation angle of the half-wave plate.
In the experiment, we used prism wave plates  as THz achromatic wave plates. This type of wave plate utilizes the phase retardation due to total internal reflection. The amount of phase retardation depends only on the angle of incidence and the refractive index of the prism, so as long as the index does not vary significantly with frequency, the effect is achromatic. Our wave plate was made of silicon (Si), which has a high transparency for THz waves and has a flat refractive index characteristic across a broad THz spectral range up to 4.5 THz, according to published results . To use the prism wave plate with higher frequency THz pulses, such as the emission from plasma filaments  and GaP crystals , further characterization of the prism wave plate is needed. We designed the incident and exit light beams to be coaxial. Therefore, the prism wave plate can control the polarization state of light by rotating, like a conventional quartz wave plate.
As the third optical component, we chose a WG polarizer (see Fig. 2). The transmittance of a WG polarizer is 50% for circularly polarized light. In principle, a high transmittance is expected by using a wave plate as the third optical component. However, the prism wave plate made of Si had a high Fresnel reflection loss of about 50%. Therefore, the efficiency was almost the same as that of the WG polarizer. In addition, alignment of the WG polarizer was easier than the prism-type wave plate. Eventually, the power transmittance of our CEP shifter was about 10%. This low efficiency is a clear disadvantage of our CEP shifter. Research on anti-reflection coatings for broadband THz spectra showing no polarization dependence is required. A single-layer anti-reflection coating will suppress this loss to a certain degree . The azimuth angle of the WG polarizer was set to 0°. From Eq. (7), the polarization angle of the exit THz pulse also became 0°, which is the same as the polarization direction of the incident THz pulse.
The shift of the CEP was observed by using THz time-domain measurement. A change of the phase spectra directly reflects a shift of the CEP, as Eq. (1) indicates. We constructed a THz time-domain measurement system [25,26]. The light source for pump and probe THz pulses was a fs laser system (Micra, Coherent Inc.), which provided laser pulses with a central wavelength of 800 nm, a pulse duration of 30 fs, a repetition rate of 80 MHz, and an average power of 400 mW. ZnTe (111) crystals with thicknesses of 1 mm were used as a THz pulse emitter and detector. The fs laser pulse entered the emitter ZnTe crystal and was converted to a THz pulse. The THz pulse passed through the CEP shifter and was detected at the receiver ZnTe crystal. THz temporal waveforms were obtained via sampling the detected signal using a mechanical delay stage.
Figure 3 shows measured THz temporal waveforms. The azimuth angles of the half-wave plate were 0°, 45°, and 90°, respectively. The gray dotted line represents the pulse envelope. The shift of the CEP is clearly observed; only the internal phase changed, not the pulse envelope. In addition, there were no temporal delays among each pulse. From Eq. (6), when changes from 0° to 90°, the difference in the CEP is . Thus, the signs of the temporal waveforms when is 0° and 90° are opposite. Figure 4 shows the phase spectra obtained by the Fourier transformation of the temporal waveforms shown in Fig. 3. Three phase spectra had the same shape but offsets. The offsets were evaluated by taking the difference between the phase spectra of and . The offsets were about , as expected from Eq. (6). The largest error over ideal values (broken line in Fig. 4) is about , which is similar to the error with our prism wave plate . This error is caused by the misalignment of the prism wave plate, manufacturing errors of the prism wave plate, or the signal-to-noise ratio of our experimental system. The spectral shape was maintained, indicating that the CEP shifter had no spectral dispersion.
Finally, we investigated the dependence of the CEP shift on the half-wave plate rotation angle (Fig. 5). Red dots represent the amount of CEP shift () using a phase spectrum of 1 THz for every 5 degrees of half-wave plate rotation. We added offsets to the measured CEP shift so that the value at became , as Eq. (6) indicates; because the measured CEP shift was a relative value, we had to set a reference value. The CEP changed linearly when the half-wave plate was rotated. The solid line shows that the calculated values follow Eq. (6). The experimental values show very good agreement with the calculated values. The root-mean-square deviation between the measurements and the theoretical line is 0.066. The variable range of the CEP shift, that is, the range on the vertical axis, was from to , covering a range of . This shows that any CEP value could be achieved by using our CEP shifter.
In summary, we have developed and demonstrated a CEP shifter for broadband THz pulses. The CEP was shifted by adding equivalent phase shifts to the entire spectrum. The CEP shifter was composed of a suitable arrangement of three optical components: a first quarter-wave plate, a half-wave plate, and a second quarter-wave plate. The second quarter-wave plate can be replaced with a polarizer. From the Jones matrix calculations, we found that the CEP can be shifted by rotating the half-wave plate. The actual CEP shifter was composed of THz prism wave plates made of Si and a WG polarizer. We evaluated the CEP shifter by the THz time-domain measurement. Temporal waveforms clearly showed CEP shifts. By investigating the phase spectra, we found that the same phase offset was added to the entire spectrum by rotating the half-wave plate; this is evidence of a CEP shift. Furthermore, it turned out that the CEP shifted linearly when the half-wave plate was rotated. The amount of CEP shift was twice the rotating angle of the half-wave plate, as expected. The shift range of the CEP reaches , indicating that we can chose any CEP values deterministically by the rotation of the half-wave plate.
This CEP shifter will allow investigation of the CEP dependence of THz matter interactions. Discovery of new phenomena induced by intense THz pulses is also expected. In addition, our concept for shifting the CEP, that is, a combination of several-wave plates, is applicable not only to THz waves but to all electromagnetic waves as long as an achromatic wave plate exists. It should be emphasized that our results were obtained readily by using the ability to observe the phase of THz pulses. THz light waves that oscillate extremely slowly will become a suitable tool that will reveal new methods of controlling light.
We thank A. Hiruma, K. Yamamoto, and T. Hara for their encouragement.
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