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Supplementary Information

Petahertz optical drive with wide-bandgap semiconductor

Hiroki Mashiko1, Katsuya Oguri1, Tomohiko Yamaguchi1,2,

Akira Suda2, and Hideki Gotoh1

1NTT Basic Research Laboratories, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198,

Japan. 2Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 278-8510, Japan

Correspondence and requests for materials should be addressed to H.M.

([email protected])

Section S1: Experimental setup for transient absorption spectroscopy

A few-cycle pulse (7-fs duration and center photon energy of 1.65 eV) from a

Ti:sapphire laser was used for high-harmonic generation and as the pump-NIR pulse for the

transient absorption spectroscopy. Figure S1(a) shows schematic experimental setup. The

pump–probe system is also introduced in ref. [1, 2]. The output beam from a hollow-fiber

compressor is sent to an annular hole mirror (HM1), which splits the beam into the inner and

outer arms of a compact Mach–Zehnder interferometer. The inner beam (IAP arm) has 300-

µJ pulse energy, and it passes through a fused silica plate (FS1) with 1-mm thickness. The

FS1 gives group delay of approximately 5 ps, which avoids the temporal interference effect

of two NIR pulses in the high-harmonic generation process. The two quartz plates (Q1, 360-

µm-thick; Q2, 480-µm-thick) are the double optical gating (DOG) optics to generate the

isolated attosecond pulse (IAP)3. The designed temporal gate width in the DOG is less than

1.3 fs (half-cycle of NIR pulse) in this experiment. The piezoelectric transducer (PZT) has

position resolution of less than 1 nm. In the other arm, the pump-NIR pulse of outer beam

(NIR arm) is used for the transient absorption spectroscopy. The stability of the

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interferometer is monitored by a co-propagated continuous-wave laser (633-nm wavelength).

The timing jitter is 23-as at the root mean square over 12 h as shown in Fig. S1(b).

The two beams from the interferometer are sent through a 140-µm β-BaB2O4 (BBO)

crystal of final DOG optics and focused into a cell (2.5-mm lengths) filled with argon (Ar)

gas for high-harmonic generation. The generated IAP passes through a tin (Sn) filter (200-nm

thickness) to block the fundamental NIR driving pulse. The pump-NIR pulse (NIR arm) also

passes through the outer portion of the annular filter, a fused silica plate (FS2) with 1-mm

thickness. Since the FS2 gives group delay (approximately 5 ps) for the pump-NIR pulse,

now the IAP and the NIR pulse are temporally overlapped. A spherical focusing mirror has

two coatings: silicon carbide (SiC) for the IAP reflection at the center part and aluminum (Al)

for the pump-NIR pulse reflection at the outer part. Note that the second harmonics isn’t

generated from the pump-NIR pulse of outer beam (NIR arm) in the BBO crystal, because

the laser polarization direction is orthogonal to the crystal axis for the second harmonic

generation. Thus, the pump-NIR pulse has only one-color component on the GaN target. The

target intensity of the pump-NIR pulse is approximately 1×1010 W/cm2, which is estimated

from photoelectron energy shift with the intensity dependence of attosecond streak4.

In this experiment, we used epitaxially grown bulk GaN with the wurtzite (hexagonal)

structure [0001] as a target, which is prepared by NTT-AT Inc.5. The thin GaN target is

manufactured by combination of mechanical polishing and ion beam milling from millimeter-

thick bulk GaN. Commonly, the technology is used to sample process in transmission

electron microscopy. The GaN target has thickness graduation from a few nanometers to

several hundred micrometers. The target is mounted on ring holder equipped with linear

electronic actuator to select the proper thickness. It is installed in vacuum chamber. The

effective thickness of GaN is estimated from absorption cross-section6 using the IAP. After

the GaN target, the transmitted IAP is reflected by a second SiC mirror and sent to a vacuum




3

ultraviolet spectrometer equipped with a micro-channel plate and a cooled charge-coupled

device camera. The spectral resolution is 180 meV at 20.5-eV photon energy.

Section S2: Temporal characterization of IAP

The IAP is characterized with the attosecond streak method4. In our experiment, the

system configuration is similar to the above transient absorption spectroscopy. The

collinearly propagated IAP and NIR pulse are focused to the gas jet with krypton atoms (50-

µm interaction length; 740-mbar backing pressure). The estimated target intensity of the NIR

pulse is approximately 5×1010 W/cm2 in this measurement. The ionized photoelectrons

induced by the IAP are detected by a regular time-of-flight system. The resolution is 50 meV

at 7.5-eV photoelectron energy (ionization energy of a krypton atom7: 14 eV). To reconstruct

the temporal profile and phase of the IAP, we used frequency-resolved optical gating for

complete reconstruction of attosecond bursts (FROG-CRAB) method8. Figures S2(a) and (b)

show the experimental and retrieved FROG-CRAB traces. Figure S2(c) shows the

reconstructed temporal shape and phase of the IAP pulse. The duration is 660 as at the full

width at half maximum (FWHM). The IAP spectrum reconstructed by the FROG-CRAB

method (blue dashed line) agrees well with the measured spectrum (red solid line), as shown

in Fig. S2(d).

Section S3: Absorption spectrum and the definition

Figure S3(a) and (b) show the absorption spectra and optical density (OD) using only the

IAP. The thickness of GaN is estimated from absorption cross-section6. Here, we defined the

OD at laser frequency ω as OD(ω)=log[Iin(ω)/Iout(ω)], where Iin(ω) is the spectrum of the

input IAP. The Iout(ω) is the absorption spectrum with the GaN, and it also corresponds to the




4

transmitted spectrum from the target. The OD(ω) is proportional to regular absorption cross-

section9. Consequently, the OD(ω) monitors the spectral deviation with the GaN.

For the transient absorption spectroscopy, we select the GaN with 102-nm thickness. The

transient absorption spectrum at delay time τ between the IAP and NIR pulse is given by

ΔOD(ω,τ)=log[Iout(ω,τ)/INIR(ω,τ)], where Iout(ω,τ) is the absorption spectrum with the GaN

without NIR pulse. In the INIR(ω,τ), the NIR pulse is added. Consequently, the ΔOD(ω,τ)

monitors the spectral deviation with the NIR pulse.

Section S4: Multi-level model simulation based on the optical Bloch equation

We model the GaN electronic system for the analysis in the semi-classical

approximation, where the electromagnetic fields of the NIR pulse and IAP are described

classically, while the electronic structure in GaN is described quantum mechanically. A

schematic illustration for this system is shown in Fig. S4. The model system we assumed

consists of three states, a , b , and c , which mimic the VB, CB, and ionization

continuum state of GaN, respectively. This simple formulation is based on previous work10-13,

and we extend the previous two-level formalism to a three-level system. State a

corresponds to the ground state in this system. We assume that Eb=4.8 eV, which correspond

to the resonance condition that satisfies three-photon absorption of the NIR pulse. The

continuum state |𝑐𝑐 is treated as an ensemble of quasi-discrete states that are non-interacting

with each other. The energy Ec is assumed to be from 16 to 25.6 eV. The electromagnetic

fields of the NIR pulse and IAP are

ENIR t( ) = ANIR t( )cos ωNIRt +ϕ( ) , (S1)

EIAP t( ) = AIAP t −Δt( )cos ω IAP t −Δt( )"# $% , (S2)




5

where A(t) and ω(t) are the envelope function and the carrier frequency of the NIR pulse and

IAP, and ϕ is the carrier-envelope phase of NIR pulse. In addition, Δt is defined as a delay

between the NIR pulse and IAP. A negative value of Δt suggests that the IAP passes through

the GaN earlier than the NIR pulse. The NIR and IAP are linearly polarized in the parallel

polarization direction. We assume that the envelope function of ANIR(t) and AIAP(t) is a square

of the hyperbolic secant function with durations of 7 fs and 660 as at the FWHM,

respectively.

The time-dependent Hamiltonian of the system within the dipole approximation is

Η̂ t( ) = Η̂ 0 + Η̂ dip t( ) , (S3)

Η̂ 0 =

Ea 0 00 Eb 00 0 Ec

!

"

####

$

%

&&&&

, (S4)

( )( ) ( )

( ) ( )( ) ( ) ⎟

⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

00

0ˆ

tEdtEdtEdtEdtEdtEd

tH

IAPcbIAPca

IAPbcNIRba

IAPacNIRab

dip , (S5)

where Η̂ 0 is the unperturbed Hamiltonian and dij is the dipole matrix element. To describe

the temporal evolution of each state by this Hamiltonian, we introduce the density matrix of

the system, ρ. The equation of motion for the density matrix is written as

∂ρ∂t

=1i

Η̂ t( ),ρ"#

$% . (S6)

We consider the eq. (S6) in the polarization direction of the NIR pulse and IAP, which

results in a one-dimensional problem. Here, we can translate this eq. (S6) into a formalism of

the optical Bloch equation, which is commonly used in analyzing laser-matter interactions.

The density matrix elements are substituted by three real quantities, u, v, and w, as follows.

u1 = ρba + ρab , v1 = −i ρba − ρab( ) (S7)




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u2 = ρca + ρac , v2 = −i ρca − ρac( ) (S8)

u3 = ρcb + ρbc , v3 = −i ρcb − ρbc( ) (S9)

w1 = ρbb − ρaa , w2 = ρcc − ρaa (S10)

When we consider the coupling between a and b states by the NIR pulse irradiation, it is

likely that the third-order interband polarization is dominantly induced by the three-photon

absorption of the NIR pulse. In order to incorporate the three-photon absorption into the

simulation, the multiphoton two-level model14 is used. The mathematical treatment describes

the NIR electromagnetic field as

ENIR t( ) ≅ E3ω t( ) = A3ω t( )cos 3ωt +ϕ( ) . (S11)

The optical Bloch equation for this system consists of the following eight differential

equations with the initial condition of u1(0)=u2(0)=u3(0)=0, v1(0)=v2(0)=v3(0)=0, and

w1(0)=w2(0)=0. The transition frequencies are defined as Ωba=Eb-Ea, Ωca=Ec-Ea, and Ωcb=Ec-

Eb.

∂u1∂t

= +Ωbav1 +dcaEIAP t( )

v3 +dcbEIAP t( )

v2 −γ1u1 (S12)

∂v1∂t

= −Ωbau1 +2dbaE3ω t( )

w1 −dcbEIAP t( )

u2 +dcaEIAP t( )

u3 −γ1v1 (S13)

∂u2∂t

= +Ωcav2 +dcbEIAP t( )

v1 −dbaE3ω t( )

v33 −γ2u2 (S14)

∂v2∂t

= −Ωcau2 +2dcaEIAP t( )

w2 −dcbEIAP t( )

u1 +dbaE3ω t( )

u33 −γ2v2 (S15)

∂u3∂t

= +Ωcbv3 −dcaEIAP t( )

v1 −dbaE3ω t( )

v2 −γ3u3 (S16)

∂v3∂t

= −Ωcbu3 +2dcbEIAP t( )

w2 −w1( )−dcaEIAP t( )

u1 +dbaE3ω t( )

u2 −γ3v3 (S17)




7

∂w1∂t

= −2dcaE3ω t( )

v1 +dcbEIAP t( )

v3 −dcaEIAP t( )

v2 −Γ1 w1 −w1 0( )#$ %& (S18)

∂w2∂t

= −2dcaEIAP t( )

v2 −dabE3ω t( )

v1 −dcbEIAP t( )

v3 −Γ 2 w2 −w2 0( )#$ %& (S19)

In these equations, we introduce phenomenological damping terms γ and Γ. We numerically

solve the Bloch equations simultaneously without employing the slowly varying envelope

approximation and the rotating wave approximation at each delay time. For each delay time,

we average the solution of u that is numerically solved for each energy of Ec. The same

averaging procedure is performed to v and w. Then, we evaluate the imaginary part of the

time-dependent dipole moment between the |𝑎𝑎 VB and the |𝑏𝑏 CB states and the |𝑐𝑐

continuum state d(t)=dcav2+dcbv3. After that, we obtain the absorption profile of the IAP via

the Fourier transform of d(t), which is proportional to the polarization P(ω) induced by the

IAP. In this manner, we calculate the absorption spectrum with and without the NIR

irradiation and then plot the differential ΔOD(ω,τ) as a function of the delay time.




8

References

1. Mashiko, H., Bell, M. J., Beck, A. R., Abel, M. J., Nagel, P. M., Steiner, C. P., Robinson,

J., Neumark, D. M., and Leone, S. R., Tunable frequency-controlled isolated attosecond

pulses characterized by either 750 nm or 400 nm wavelength streak fields. Opt. Exp. 18,

25887-25895 (2010).

2. Mashiko, H., Bell, M. J., Beck, A. R., Neumark, D. M., and Leone. S. R., Frequency

tunable attosecond apparatus. Progress in Ultrafast Intense Laser Science X (PULSE X),

Springer series in chem. phys. 106, 49-63 (2014).

3. Gilbertson, S., Mashiko, H., Li, C., Khan, S. D., Shakya, M. M., Moon, E., and Chang, Z.,

A low-loss, robust setup for double optical gating of high harmonic generation. Appl. Phys.

Lett. 92, 071109 (2008).

4. Itatani, J., Quéré, F., Yudin, G. L., Ivanov, M. Y., Krausz, F., and Corkum, P. B.,

Attosecond streak camera. Phys. Rev. Lett. 88, 173903 (2002).

5. http://www.ntt-at.com/

6. Henke, B. L., Gullikson, E. M., and Davis, J. C., At. data nucl. data tables. 54, 181 (1993).

7. Kramida, A., Ralchenko, Y., Reader, J., and Nist ASD Team, NIST Atomic spectra

database (version 5.2). http://physics.nist.gov/asd (2014).

8. Mairesse Y., and Quéré, F., Frequency-resolved optical gating for complete reconstruction

of attosecond bursts. Phys. Rev. A 71, 0011401(R) (2005).

9. Ingle D. J., and Crouch, S. R., Spectrochemical analysis (Prentice Hall, 1988).

10. Mücke, O. D., Tritschler, T., Wegener, M., Morgner, U., and Kärtner, F. X., Carrier-wave

Rabi flopping in GaAs using 5 fs, 1012 W/cm2 pulses. Phys. Rev. Lett. 87, 057401 (2001).

11. Tritschler, T., Mücke, O. D., Wegener, M., Morgner, U., and Kärtner, F. X., Evidence for

third-harmonic generation in disguise of second-harmonic generation in extreme nonlinear

optics. Phys. Rev. Lett. 90, 217404 (2003).




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12. Tritschler, T., Mücke, O. D., and Wegener, M., Extreme nonlinear optics of two-level

systems. Phys. Rev. A 68, 033404 (2003).

13. Wegener, M., (2004). Extreme nonlinear optics in semiconductors. In: H. Kalt and M.

Hetterich eds. Optics of semiconductors and their nanostrauctures, pp. 171-188. Berlin

Heidelberg NewYork: Springer-Verlag.

14. Meystre, P. and Sargent. III. M., (1998). Elements of quantum optics. Berlin Heidelberg

NewYork: Springer-Verlag.




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Figure captions

Figure S1| Experimental setup and pump-probe stability. a, Experimental setup for

attosecond transient absorption spectroscopy. HM1 and HM2: annular hole mirrors. FS1:

fused silica plate. PZT: piezo-electronic transducer stage for delay control between IAP and

NIR pulse. DOG: double optical gating (DOG) optics (two quartz plates)3. BBO: β-BaB2O4

crystal of final DOG optics. IAP(Ar): IAP generation using argon gas. Sn on FS2: tin filter

mounted on annular fused silica plate. SiC: silicon carbide mirror. MCP: micro-channel plate.

CCD: cooled charge-coupled device. b, Stability (red filled circle) of the interferometer

measured with continuous-wave laser (633-nm wavelength) over 12 h.

Figure S2| Temporal characterization of IAP based on attosecond streak. a, Measured

and b, reconstructed FROG-CRAB traces using krypton atoms. c, Reconstructed temporal

profile (red solid line) and phase (blue dashed line). The duration is 660 as at the FWHM

(pink arrow). d, Reconstructed spectrum (red solid line) and phase (blue dotted line). For

comparison, the measured spectrum (green dashed line) without the streak field of the NIR

pulse is also shown.

Figure S3| Absorption spectra with GaN. a, Thickness dependence of absorption spectra

(color solid lines) monitored by only the IAP. b, OD is estimated from (a). The OD value is

proportional to regular absorption cross-section9. For the transient absorption spectroscopy,

we select the GaN with 102-nm thickness (light blue solid line).

Figure S4| Scheme of a direct gap semiconductor in the first Brillouin zone. VB: valance

band (orange shaded area). CB: conduction band (green shaded area). CS: continuum state

(purple shaded area). K: wave vector. E: energy, a: lattice constant. Eg: bandgap energy (pink

b

Time (hour)0 2 4 6 8 10 12

Tim

ing

jitte

r (as

)

0

150

100

50

-50

-100

-150

Inte

rfer

omet

er

disp

lace

men

t (nm

)

20

10

0

-10

-20

Timing jitter = 23 as at RMS

FS1

Sn on FS2

GratingMCP CCD

IAP

PZT

IAP arm

NIR

arm

SiC mirror

SiC mirrorDOG

GaN target

BBO

HM1

HM2

a




10

Figure captions






















10

Figure captions






















Pho

toel

ectr

on e

nerg

y (e

V)

a

4

6

8

10

-10 -5 0 5 10Delay (fs)

1

0 Cou

nt (a

rb. u

.)

1

0 Cou

nt (a

rb. u

.)

Pho

toel

ectr

on e

nerg

y (e

V)

4

6

8

10

-10 -5 0 5 10Delay (fs)

660 as

-2 -1 0 1 2Time (fs)

-1

0

1

2

3

4

Pha

se (r

ad)

Inte

nsity

(arb

. u.)

1.2

1

0.8

0.6

0.4

0.2

0

Inte

nsity

(arb

. u.)

1.2

1

0.8

0.6

0.4

0.2

0

Pha

se (r

ad)

-1

0

1

2

-217 18 19 20 21 22 23 24

Photon energy (eV)

c

b d




10

Figure captions





















(purple shaded area). K: wave vector. E: energy, a: lattice constant. Eg: bandgap energy (pink 10

Figure captions






















11

arrow). At each K, the optical interband transition resembles that of a two-level system with

transition energy Ωba (red arrow). Ωca and Ωcb (blue arrows) are the transition energies from

the VB and CB to CS. Near the center of the first Brillouin zone (K=0), the bands are nearly

parabolic and the effective mass approximation can be employed.

17Photon energy (eV)

Inte

nsity

(rel

ativ

e) 1

0.2

0

1

0.2

0O

D

0.4

0.6

0.8

1.2

18 19 20 21 22 23 24

0.4

0.6

0.8

17Photon energy (eV)

18 19 20 21 22 23 24

a b

without GaN1 nm

12 nm 39 nm58 nm

79 nm24 nm 102 nm

130 nm167 nm




10

Figure captions






















11

arrow). At each K, the optical interband transition resembles that of a two-level system with

transition energy Ωba (red arrow). Ωca and Ωcb (blue arrows) are the transition energies from

the VB and CB to CS. Near the center of the first Brillouin zone (K=0), the bands are nearly

parabolic and the effective mass approximation can be employed.

K-π/a +π/a0

E

〉c

〉b

〉a

CB

VB

CS

Ωcb

ΩbaEg

Ωca


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