Advanced Laser and Photn Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Kenichi Ishikawa (石川顕一) http://ishiken.free.fr/english/lecture.html
[email protected]
Advanced Laser and Photon Science
レーザー・光量子科学特論E
4. High-harmonic generation
高次高調波発生 1
HARMONIC GENERATION
Process whereby integer multiples of the original light’s frequency (fundamental frequency) are generated.
高調波発生
crystal, gas
Linear optical effect 線形光学効果 ω Material response is linear in light intensity
結晶等
ω
Nonlinear optical effect € 結晶等 € 非線形光学効果 ω ,2 ,3 ,4 ,5 ,··· Nonlinear material response frequency conversion 物質の応答が、入射光強度に非線形に依存 波長変換
D = ε0 E + P
€
P = ε0 [ χ (1) E + χ (2) E 2 + χ (3) E 3 +!] 非線形分極 nonlinear 線形分極 linear polarization
€
€
2 ：2次高調波(2nd harmonic) 3 ：3次高調波(3rd harmonic)
.....
∂ 2D ∇ × ∇ × E = −µ 0 2 ∂t
€
2
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反転対称な物質では、偶数次はゼロ
Even-order components vanish for a medium with inversion symmetry E
-E
P(E)
P(-E) = -P(E)
P (E) =
0
P ( E) =
0
P (E) =
0
(1)
E+
(2)
E2 +
(3)
E3 +
(4)
E4 + · · ·
(1)
E+
(2)
E2
(3)
E3 +
(4)
E4
···
(1)
E
(2)
E2
(3)
E3
(4)
E4
···
(2)
= 0, 3
(4)
= 0, · · · 石川顕一
摂動論的高調波発生 PERTURBATIVE HARMONIC GENERATION 3rd harmonic ３次高調波発生の概念図 Ionization 電離
Transition matrix element 遷移行列要素 3
MTHG =
(3
h,i,f
+
Virtual level 仮想準位
!ω
!ω !ω
€
€ €
+
3!ω Ground state 基底状態
+
1
(
· D1h 1
1
1
(
1
1
· Dhi h )(2 1
· Dhi h )(2 1
3
· D1h
1
(
· D1h
1
1
· Dhi h )( 2 1
· D1h
1
· Dhi h )( 2 1 1
· Dij i )( 3
1
1
· Dij i )(
1 1
· Dij i )(
1 1
· Dj1 j)
· Dj1 j) 1
· Dj1 1
j)
· Dij 3 · Dj1 i )( 3 1 j)
€ 次数が高くなるほど、発生効率は減少。
order ↑
efficiency ↓ 4
石川顕一
摂動論的高調波発生 (PERTURBATIVE HARMONIC GENERATION) 3rd harmonic
5th harmonic
３次高調波
５次高調波 Ionization 電離
Ionization 電離 仮想準位 Virtual level
!ω Virtual level 仮想準位
!ω !ω
!ω
€ €
€
€
!ω !ω
€
3!ω
€ Ground state 基底状態 €
5!ω
!ω € !ω
Ground state 基底状態
€
次数が高くなるほど、発生効率は減少。 €
order ↑
efficiency ↓
5
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高次高調波発生 HIGH-HARMONIC GENERATION (HHG) discovered in 1987 Intense laser pulse
gas jet
harmonics of high orders
Highly nonlinear optical process in which the frequency of laser light is converted into its integer multiples. Harmonics of very high orders are generated. ! !
新しい極端紫外・軟エックス線光源として注目される。 New extreme ultraviolet (XUV) and soft X-ray source
6
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How high orders?
HARMONIC SPECTRUM 高調波スペクトル 041111-3
Wahlström et al., Phys. Rev. A 48, 4709 (1993)
Takahashi et al.
Takahashi et al., Appl. Phys. Lett. 93, 041111 (2008)
Harmonic intensity (arb. unit)
1015 W/cm2
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
FIG. 4. !Color online" Experimentally obtained harmonic spectra in Ar. Red and blue profile depict the spectra with #0 = 0.8 !m pump and #0 = 1.4 !m pump, respectively. Both HH spectra are normalized to the peak intensity. The laser focused intensity is adjusted to generate HH under a neutral condition for both wavelengths. The inset shows a measured two dimensional harmonic spectrum image driven by 1.4 !m pump.
800 nm, 1.6×1014 W/cm2
Only odd orders 奇数次のみ
TDSE Simulation 0
10
20 30 Harmonic order
40
50
800÷31= 26
was raised up to 26 mJ, a maximal output energy exceeding 7 mJ was achieved at the signal wavelength near 1.4 !m. Temporal characterization of amplified OPA pulses was performed using a single-shot autocorrelation !AC" technique. A typical AC trace is shown in the inset of Fig. 2. a Gaussian pulse shape, the pulse width of 1.4 !m nmAssuming pulse was evaluated to be 40 fs in full width at half maxi期待媒質は反転対称 mum !FWHM", the energy of which corresponds to the red 7 circles in Fig. 3. The solid red line depicts the Fourierfilled
gas is a medium of inversion symmetry
matching cond propagation ax the Ar harmon cutoff energy w spectrum drive magnitudes low measured HH significant cuto the 0.8 !m dr field generate higher energy This photon en predicted valu In conclu sources based monic beams. pulse width w 1.4 !m. Total #45% conver HH spectrum extension exce file is almost p is attractive no the kiloelectro ergy scaling o 1
M. Hentschel, R T. Brabec, P. Co
石川顕一
Even up to 1.6 keV, > 5000 orders almost x-ray!
Popmintchev et al., Science 336, 1287 (2012)
a new type of laser-based radiation source レーザーをベースにした新しいタイプの放射線源 8
石川顕一
Plateau and cutoff
プラトーとカットオフ
- remarkable feature of high-harmonic generation Wahlström et al., Phys. Rev. A 48, 4709 (1993) Harmonic intensity (arb. unit)
plateau cutoff
1015 W/cm2
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
800 nm, 1.6×1014 W/cm2
plateau cutoff
TDSE Simulation 0
10
20 30 Harmonic order
40
50
プラトー(plateau)：Efficiency does NOT decrease with increasing harmonic order. 次数が上がっても強度が落ちない。 カットオフ(cutoff)：Maximum energy of harmonic photons e2 E02 2 2 14 Up (eV) = = 9.3 10 I(W/cm ) (µm) Ec Ip + 3.17Up 2 4m ポンデロモーティブエネルギー ponderomotive energy
These features cannot be understood as perturbative harmonic generation. 摂動論的には解釈できない 9
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高次高調波発生のメカニズム
３ステップモデル 3-STEP MODEL Laser field レーザー電場 E(t) = E0 cos t recombination 再結合 → photon emission (HHG) 発光（高次高調波） electron
Semiclassical tunneling electron motion ionization 電場中の古典的加速 トンネル電離
Paul B. Corkum, Phys. Rev. Lett. 71, 1994 (1993) K. C. Kulander et al., in Super-Intense Laser-Atom Physics, NATO ASI Ser. B, Vol. 316, p. 95 (1993) Paul B. Corkum 10
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3-STEP MODEL OF HHG 高次高調波発生の３ステップモデル Ionization at t = t0 with vanishing initial velocity at origin 時刻 t0 でイオン化。原点に初速ゼロで出現 m¨ z=
Normalization 規格化 z=
E0 2
z(t ˙ 0) = 0
eE0 cos t
[(cos
cos
= t 0)
+(
0
z(t0 ) = 0
= t0 0 ) sin
0]
Ekin = 2Up (sin φ − sin φ0 )2
Recombination at φ = φret (φ0 ) , which satisfies z = 0 再衝突 Laser field レーザー電場 E(t) = E0 cos t recombination 再結合 → photon emission (HHG) 発光（高次高調波）
electron
tunneling トンネル電離 ionization 11
電場中の古典的加速 Semiclassical electron motion 石川顕一
TIME (PHASE) OF RECOMBINATION 再衝突時刻 (cos
z=0 (cos ) |
0
=
ret
cos
cos
0)
ret
cos
ret
0
+(
ret
0 ) sin
0
=0
0
phase of ionization vs phase of recombination イオン化時刻と再衝突時刻の関係 1.0
Phase of recombination (phi_r)
350
0.5
0.0
-0.5
-1.0 0
40
0
80
120
160
200
phi (degree)
240
280
320
250
2
200
