SUPPLEMENTARY INFORMATION

Transcription

1 Supplementary Information Terahertz polarization pulse shaping with arbitrary field control Masaaki Sato,2, Takuya Higuchi 2,3,4, Natsuki Kanda 2,3,5, Kuniaki Konishi 2,6, Kosuke Yoshioka 2,4, Takayuki Suzuki,2,7, Kazuhiko Misawa,2,7, and Makoto Kuwata-Gonokami 2,3,4,6 Department of Applied Physics, Tokyo University of Agriculture and Technology, Naka-cho, Koganei, Tokyo , Japan 2 CREST, Japan Science and Technology Agency, Sanbancho Bldg., 5 Sanbancho, Chiyoda-ku, Tokyo 2-75, Japan 3 Department of Applied Physics, The University of Tokyo, 7-3- Hongo, Bunkyo-ku, Tokyo , Japan 4 Department of Physics, The University of Tokyo, 7-3- Hongo, Bunkyo-ku, Tokyo 3-33, Japan 5 RIKEN Advanced Science Institute, 2- Hirosawa, Wako, Saitama 35-98, Japan 6 Photon Science Center, The University of Tokyo, 7-3- Hongo, Bunkyo-ku, Tokyo , Japan 7 Interdisciplinary Reserch Unit in Photon-nano Science, Tokyo University of Agriculture and Technology Naka-cho, Koganei, Tokyo , Japan (Dated: June 7, 23) NATURE PHOTONICS

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3 Normalized α(ω) α(ω) Arg[α(Ω)] Frequency (THz) Arg[α(Ω)] (deg.) Supplemenraty Figure : Fourier form of the impulsive response, α(ω). The solid and dashed lines represent the absolute value and phase of α(ω), respectively.. Experimental data of impulsive response When an x-polarized laser pulse with an intensity profile of I x (t) propagates along the threefold axis of a GaP[] crystal, an x-polarized terahertz wave with an electric field waveform of Ex THz (t) = α(t τ)i x (τ)dτ is emitted. The impulse response function α(t) is obtained by the deconvolution of Ex THz (t) by I x (t). Using Wiener Khinchin theorem, the Fourier form of the impulse response function, α(ω) can be represented as fallows: α(ω) = E THz x (Ω)/I x (Ω), () where Ex THz (Ω) and I x (Ω) are the Fourier forms of Ex THz (t) and I x (t), respectively. For this analysis, we measured the intensity profile of the laser pulse I x (t) and the electric field waveform of the emitted terahertz wave Ex THz (t) by the sum-frequency-generation (SFG) cross-correlation method and the electro-optic (EO) sampling method, respectively. Supplementary Figure shows the experimentally obtained α(ω) of the GaP crystal. Arg[α(Ω)] is the argument, that is, the phase of the Fourier form α(ω). The bandwidth of α(ω) is mainly determined by the thickness of the crystal and the central wavelength of the incident laser through the phase-matching condition in GaP []. In the current study, a GaP crystal with thickness of 45 µm and a Ti:Sapphire laser with a central wavelength of 83 nm were employed. These conditions limited the bandwidth of α(ω) to be approximately.5 THz. NATURE PHOTONICS 3

4 2. Validity of one-to-one relation between time and frequency connected by instantaneous frequency or frequency dependent delay The instantaneous Stokes parameters (ISPs) of a laser pulse which is modulated for terahertz polarization shaping can be calculated by the following manner. For simplicity, let the intensity-spectrum I(ω) of the laser pulse be a Gaussian function as follows: [ ( ) ] 2 ω ω I(ω) =I exp, (2) ω where I and ω are the peak intensity and the central frequency of the spectrum, respectively. ω is the width of the spectrum, which is related to its full-width at half-maximum ω FWHM by ω = ω FWHM /(2 ln 2). By introducing a chirped modulation with a second-order spectral dispersion β in the phase of the laser pulse θ opt (ω) =β(ω ω ) 2 /2, the electric field waveform Ẽ(t) becomes Ẽ(t) = 2π Thus, the temporal intensity envelope I(t) becomes dω I(ω)e iθ opt(ω) e iωt. (3) I(t) = Ẽ(t) 2 [ ] I = ω 4 + β exp ω 2 t 2, (4) 2 ω 4 + β 2 and the instantaneous frequency of the laser pulse ω ins (t) becomes ω ins (t) d dt Arg[Ẽ(t)] βt = ω + ω 4 + β. (5) 2 Under a condition β ω 2, which can be easily achieved using ultrashort pulses (i.e., large ω), the intensity envelope I(t) and instantaneous frequency ω ins (t) can be written by their simple forms as follows: [ I(t) = I ( ) ] 2 t β exp ωβ ω ins (t) =ω + β t, (7) and the temporal duration of the intensity envelope t is defined as t ωβ. (6) Each frequency component appears with a temporal delay of t = β(ω ω ) in the time domain, 4 NATURE PHOTONICS

5 and hence, the azimuthal angle of each frequency component φ opt (ω) determines the timedependent azimuthal angle φ opt (t) as φ opt (t) φ opt (ω = ω ins (t)). The condition to validate above one-to-one relation between frequency and time is not limited to a simple linear chirp. We can extend it to general phases. For this purpose, we consider higher order dispersion β(ω), which is defined as β(ω) 2 θ ω 2 opt (ω). In this case, β(ω) ω 2 should be satisfied for all ω in order to introduce such one-to-one relation. 3. Laser parameters to control bandwidth and central frequency of terahertz radiation The bandwidth of the emitted terahertz wave is inversely proportional to t = ωβ, and hence to β, whereas 2γβ gives its central frequency, as discussed in the main text. Supplementary Figure 2 shows the parameters of a laser pulse that has a larger value of β than the one depicted in Fig. 2, and γ is also larger in order to maintain 2γβ to be the same. Such a pulse generates terahertz waves with a longer duration, and thus a shorter bandwidth. This achieved the control of the bandwidth of the terahertz waves shown in Figs. 3a and 3b in the main manuscript. On the other hand, 2γβ can be varied independently by keeping β and changing γ. For example, Supplementary Figure 3 shows the parameters of a pulse that has a smaller value of γ than that in Supplementary Fig. 2. In this way, the central frequency alone can be controlled, as shown in Figs. 3c and 3d in the main manuscript. NATURE PHOTONICS 5

6 Frequency (THz) a b c d ~ I(ω) t = τ(ω) ω= ω ins θ(ω) φ(ω) 365 5π π Intensity Phase (rad) π Intensity S & S 2 e f φ ins I ins S 2 η ins S Time (ps) π Azimuthal and Elliptical π Azimuthal Supplemenraty Figure 2: Laser pulse with a longer temporal duration. a. Intensity I(ω), b. instantaneous frequency, c. phase θ opt (ω), d. and azimuthal angle φ opt (ω). Parameters of resultant pulse are also presented: e. Instantaneous intensity, azimuthal angle and ellipticity, and d. ISPs of the laser pulse. 4. Laser parameters that introduce chirps to terahertz waves A frequency chirp can be introduced to the emitted terahertz wave by modulating the azimuthal angle of the laser pulse φ opt (ω) to a quadratic form as a function of frequency. In order to make a terahertz wave with a chirp constant of β THz, the laser parameters are to be set as follows: β2 φ opt (ω) = (ω ω ) 2, (8) 4β THz where ω is a constant value. Supplementary Figure 4 depicts the parameters of such a pulse. In the following, we confirm that such a modulated pulse indeed introduces the desired chirp to the terahertz wave. Using the relation ω ins (t) =ω + β t, the time-dependent 6 6 NATURE PHOTONICS

7 Frequency (THz) a b c d ~ θ(ω) φ(ω) I(ω) t = τ(ω) ω= ω ins 365 5π π π Intensity Phase (rad) Azimuthal π Intensity S & S 2 e f φ ins I ins η ins S 2 S Time (ps) π Azimuthal and Elliptical Supplemenraty Figure 3: Laser pulse with a faster angular velocity of the azimuthal angle. a. Intensity I(ω), b. instantaneous frequency, c. phase θ opt (ω), d. and azimuthal angle φ opt (ω). Parameters of resultant pulse are also presented: e. Instantaneous intensity, azimuthal angle and ellipticity, and d. ISPs of the laser pulse. azimuthal angle of this laser pulse is given as φ opt (t) =φ opt (ω = ω ins (t)) = β2 4β THz (β t + ω ω ) 2, (9) and so, the time-dependent azimuthal angle of the emitted terahertz wave φ THz (t) is φ THz (t) = 2φ opt (t) = β2 2β THz (β t + ω ω ) 2. () Hence, the time-dependent frequency of the emitted terahertz wave is given by Ω THz (t) = d dt φ THz(t) =β THz {t + β(ω ω )}. () This means that the emitted terahertz wave indeed has a chirp constant of β THz and its central frequency is Ω THz = β THz β(ω ω ). Note that the temporal duration of the chirped terahertz wave is also t = ωβ. 7 NATURE PHOTONICS 7

8 385 a b c d Frequency (THz) 375 ~ I(ω) t = τ(ω) ω= ω ins θ(ω) φ(ω) 365 Intensity 75π π Phase (rad) π Intensity S & S 2 e f 6 S 3 I ins φ ins S 2 η ins 3 Time (ps) 6 π Azimuthal and Elliptical π Azimuthal Supplemenraty Figure 4: Laser pulse that generates a chirped terahertz wave. a. Intensity I(ω), b. instantaneous frequency, c. phase θ opt (ω), d. and azimuthal angle φ opt (ω). Parameters of resultant pulse are also presented: e. Instantaneous intensity, azimuthal angle and ellipticity, and d. ISPs of the laser pulse. 5. Experimental setup Supplementary Figure 5 shows the experimental setup for generating a polarizationshaped terahertz pulse and measuring its electric field vector. A laser pulse derived from a regeneratively amplified Ti:sapphire laser system is first divided into two beams by a beam sampler (BS). One of the two beams is used as a pump pulse for generating a terahertz wave, and the other is used as a probe pulse for the electro-optic (EO) sampling method. The pump pulse is directed to the optical polarization pulse shaper, which consists of a 4f setup, spatial light modulator (SLM), and quarter-wave plate (QWP). The phase θ opt (ω) and azimuthal angle φ opt (ω) of each frequency component of the pump pulse are independently modulated by this optical polarization pulse shaper. An incident horizontally ( x) polarized laser pulse is diffracted by a volume phase holographic grating (Wasatch Photonics, Inc.), 8 NATURE PHOTONICS

9 Laser system Probe SLM Polarizer Polarizer 45 On / Off GaP[] QWP PBS Balanced Detector BS QWP Grating THz Wave Polarization Shaped Pump N 2 purged GaP[] Filter Supplemenraty Figure 5: Experimental setup. The optical polarization pulse shaper consists of a 4f setup, spatial light modulator (SLM), and quarter-wave plate (QWP). A pump pulse shaped by the optical polarization pulse shaper is directed onto GaP[] in a closed box purged with dry nitrogen, and a controlled terahertz wave is emitted. A crystalline silicon filter transparent to the terahertz waves is used to block the pump pulses. With two wire grid polarizers, a certain polarization component of the emitted terahertz wave can be selected. We measured the electric field-waveforms of the and 45 polarization- components of the emitted terahertz wave by the electro-optic (EO) sampling method [3]. which has high diffraction efficiency with small deviations between s- and p-polarizations. Each frequency component is distributed to a pair of liquid crystal cells in the SLM. These cells independently control the phases θ 45 (ω) and θ 45 (ω) of their 45 and 45 components. After recombining them, a quarter-wave plate is placed so that y component has a phase factor of i relative to x component. Then the output is written as follows: E x(t) dω = E y (t) 2π e iωt Ẽ(ω) i eiθ45 (ω) 2 + e iθ 45 (ω) dω = 2π e iωt Ẽ(ω)e i θ 45 (ω)+θ 45 (ω) 2 cos(θ 45 (ω) θ 45 (ω)). (2) sin(θ 45 (ω) θ 45 (ω)) As a result, the average and difference of the phases of ω component, θ opt (ω) [θ 45 (ω)+ θ 45 (ω)]/2 and φ opt (ω) θ 45 (ω) θ 45 (ω), correspond to the phase and azimuthal angle NATURE PHOTONICS 9

10 of each frequency component, respectively: E x(t) dω = E y (t) 2π e iωt Ẽ(ω)e iθ opt(ω) cos φ opt(ω), (3) sin φ opt (ω) On the way from the pulse shaper to the sample, the light pulse experiences several optical elements. One may suspect if these elements may induce unwanted change in polarization state of the pulse. In our setup, the main elements that introduce such unwanted effects are mirrors to change the optical path. Namely, polarization dependent phase retardation by mirrors is obstacle. To minimize such effect, we employed metal mirror, not dielectric mirror. In addition, we put a quarter wave plate, whose fast axis is parallel to 45 degree angle, in between a pair of mirrors; this changes x-polarized component to y and y to x, which counterbalances the influences by these two mirrors. We employed four mirrors in total after the shaper, and the above procedure is repeated twice. As a result, the polarization state of the laser is conserved after reflection by these four mirrors. This was confirmed by taking ellipsometry spectra. For example, a circularly polarized light was first prepared, and we confirmed that the ellipticity was conserved within 5 percent even after these four mirrors. This deviation is small enough to be neglected for our purpose of terahertz vector pulse shaping. The polarization-shaped laser pulse is focused onto GaP [] and emits a terahertz wave whose time-dependent electric field vector can be controlled by laser pulse shaping. The emitted terahertz wave is delivered by parabolic mirrors and passes two wire grid polarizers (WGPs) that transmit and 45 -polarization components, respectively. Thereafter, the terahertz wave is focused onto an EO crystal GaP[]. The WGP that transmits the 45 - polarization component can be inserted and removed by a computer controlled translational stage so that electric field waveforms of the and 45 -polarization components can be measured by the EO sampling method [3]. We reconstructed the electric field vectors of the emitted terahertz waves by the waveforms of and 45 -polarization components. The path of the terahertz waves is purged with dry nitrogen to avoid absorption by water molecules. We estimated the maximum field amplitude available by optimizing experimental conditions from our proof-of-concept study. We assume to use a laser pulse with energy of mj and wavelength of.35 µm. This pulse energy is well below the damage threshold of any optical elements. Phase matching condition is satisfied for this wavelength and terahertz NATURE PHOTONICS

11 frequencies, which improves conversion efficiency. Indeed, there is a report on intense THz wave generation by using a GaP [] crystal [4] excited by.35 µm laser. According to this result, a THz pulse with energy of 3 nj was available by a excitation laser pulse with 2 µj. By increasing the beam size by 5 times while keeping laser power density (I.e., total pulse energy of mj), one can generate a THz pulse with energy of 5 nj, which corresponds to kv/cm. Note that when GaP [] is employed instead of [], the field amplitude is reduced by a factor of 2. So, THz pulses with field amplitude of 7 kv/cm are available by optimizing the experimental conditions from our proof-of-concept experiment. [] Casalbuoni, S., Sclarb, H., Schmidt, B., Schmëser, P., Steffen, B. & Winter, A. Numerical Studies on the Electro-Optic Detection of Femtosecond Electron Bunches. Phys. Rev. ST Accel. Beams, 7282 (28). [2] Selle, R., Nuernberger, P., Langhojeer, F., Dimler, F., Fechner, S., Gerber, G. & Brixner, T. Generation of Polarization-Shaped Ultraviolet Femtosecond Pulses. Opt. Lett. 33, 83 (28). [3] Kanda, N., Konishi, K. & Kuwata-Gonokami, M. Terahertz wave polarization rotation with double layered metal grating of complimentary chiral patterns. Opt. Express 5, 7 (27). [4] Hoffmann, M.C., Yeh, K.-L., Hebling, J., & Nelson, K.A. Efficient terahertz generation by optical rectification at 35 nm Opt. Express 5, 76 (27). NATURE PHOTONICS