高解析雷射光譜及其應用

Special topics course

Lecture speaker: Wang-Yau Cheng

2007/6

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大綱

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• 何謂高解析？
• 雷射線寬與譜線線寬的概念
• 幾個高解析雷射光譜應用之例子

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一般名稱 常用單位 常見應用 例

(解析度)

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Laser Spectroscopy cm^-1 (~30 GHz) 分子振動光譜 化學分析

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High Resolution GHz 微波光譜 Lamb Shift

Laser Spectroscopy 拉曼光譜 Four-wave Mixing

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Ultra-high Resolution < MHz 超精細光譜 Laser Cooling, BEC

Laser Spectroscopy Hyperfine Structure

Length Standard

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量測所使用的工具決定了物理的說法� -- J. S. Schwinger

Concept of Linewidth

Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser and comb laser

Linewidth of spectrum (The way matter feels light)

🡪 Lorentian linewidth

🡪 Doppler linewidth

🡪 Other linewidthes

Linewidth of light

1. Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser and comb laser

• Linewidth of spectrum (The way matter feels light)
• Lorentian linewidth
• Doppler linewidth
• Other linewidthes

Linewidth from uncertainty principle

(Well, then we don’t need any interpretation since uncertainty anywhere)

Linewidth from phase interrupted

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Linewidth from amplitude change

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Linewidth from superposition of different frequencies

Linewidth of light

Talk again about frequency?

1.What is perfect wave?

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2. What is “frequency”? What is phase?

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2. What is “instantaneous frequency”?

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3. What is “linewidth”?

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Linewidth from phase interrupted

Collisions broaden the frequency range of�light emission.

• A collision abruptly changes the phase of the sine-wave light emission.

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• Gases at atmospheric pressure have emission widths of ~ 1 GHz.
• Solids and liquids emit much broader ranges of frequencies.

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E(f)

f

~ 1/a

Linewidth!

Like damping OSC. 🡪 Lorentian

Linewidth from amplitude change

k

nk

E(x,t) = E

0

exp[(–

a

/2)x]exp[i(nkx–

ω

t)]

E(x,t) = E

0

exp[i(kx –

ω

t)]

Absorption depth = 1/a

λ/

n

Wavelength decreases

Single-exponential decay

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F(w)F(w)*= Lorentian

Linewidth from superposition of different frequencies

Indiv-

idual

waves

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Sum

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Envel-

ope

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Irrad-

iance:

1. Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser and comb laser

• Linewidth of spectrum (The way matter feels light)
• Lorentian linewidth
• Doppler linewidth
• Other linewidthes

Linewidth of CW laser

• Homogeneous and inhomogeneous broadening in gain medium

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• Cold cavity linewidth

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• Gain narrowing effect

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• Shawlow-Townes limit

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• Frequency-stabilized lasers

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homogenous

Homogenous and Inhomogeneous broadening

• Every individual emissions the same linewidth of light, but different individual might emission different frequency of light

Homogenous and Inhomogeneous broadening

Different individual emissions different frequency of light, we call it “inhomogeneous” (exists mostly in gas states, and special cases in solid states)

For example:

Cold cavity linewidth (1)

Note that:

FSR= C/2L

Etalon Transmittance vs. Thickness, Wavelength, or Angle

The transmittance varies significantly with thickness or wavelength.�We can also vary the incidence angle, which also affects δ.

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As the reflectance of each surface (r²) approaches 1, the widths of the high-transmission regions become very narrow.

Transmission maxima occur when:

2πL/λ = 2mπ

or:

The Etalon Free Spectral Range

λ_FSR

λ_FSR =

Free Spectral

Range

The Free Spectral Range is the wavelength range between transmission maxima.

Cold cavity linewidth (2)

The Linewidth δLW is a transmittance peak's full-width-half-max (FWHM).

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Setting δ equal to δ_LW/2 should yield T = 1/2:

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For δ << 1, we can make the small argument approx:

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The Finesse, F, is the ratio of the �Free Spectral Range and the Linewidth = F/FSR

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Substituting we have:

The Finesse is the number of wavelengths the interferometer can resolve.

δ = 2π corresponds to one FSR

Gain narrowing

Laser gain

Atomic linewidth

Siegman: lasers p,281

Home work: α 🡨🡪 χ” ?

△ν_L=自發放射量子雜訊造成之雷射線寬 hν=每個光子的能量

Po=雷射輸出功率

C =光速 n=雷射共振腔內介質的等效折射率

L=共振腔長

Rb=共振腔中高反射端面的反射率 Rf=共振腔中高反射端面的反射率

α=雷射內部光能量損失系數

nsp=自發放射因子，即每個模的自發放射率對同一模態的單一個光子的激發之比值

β=線寬增幅因子(linewidth enhancement factor)，即自發放射引起之折射係數實部變化對虛部變化之比值。

• Shawlow-Townes limit

An example for laser linewidth🡪 Extended cavity diode laser (ECDL)

Frequency stabilized lasers

Laser cavity

Light interacts with matter

Frequency-dependent error signal of laser

Feedback loop

Narrow linewidth light source

Cs 6S🡪 8S

F=3🡪 F=3

PZT lock

AOM lock

Off lock

Different transfer function to PZT and AOM

How to characterize the linewidth of a frequency-stabilized laser 🡪 Allan variance(1)

Laser frequency

time

time

Laser frequency

The two lasers have the same standard deviation on the frequency fluctuations while different Allan variance 🡪 which laser is more stable?

How to characterize the linewidth of a frequency-stabilized laser 🡪 Allan variance (2)

World record linewidth

1. Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser and comb laser

• Linewidth of spectrum (The way matter feels light)
• Lorentian linewidth
• Doppler linewidth
• Other linewidthes

Long vs. short pulses of light

• The uncertainty principle says that the product of the temporal

and spectral pulse widths is greater than ~1.

Long pulse

Short pulse

But a light bulb is also broadband.� ��What else is required to make an ultrashort pulse?

Answer: “Mode-locking”

Analogy to no-damping free induction process

• Locking the phases of the laser modes yields an ultrashort pulse.

W₁T-W₂T=2nπ, T=round trip time

Numerical simulation of mode-locking

Ultrafast lasers often have thousands of modes.

Therefore, frequency difference between modes, we call it repetition rate, that is: Δf = (n+1)c/2L – nc/2L= c/2L, we call it Δ latter in comb laser.

Standing wave criteria in laser cavity: L=nλ/2 , L=cavity length, n= integer

The concept of “repetition rate” comes from time-domain:

T=2L/c

Repetition rate Δ=1/T

Linewidth from mode-locked laser

Indiv-

idual

waves

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Sum

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Envel-

ope

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Irrad-

iance:

25 Femto-second

?

25 Femto-second

?

~40 nm

Femto-second comb laser -- Introduction

Δ = Comb spacing

δ = Comb offset from

harmonics of Δ

φ = Phase slip b/t carrier &

envelope each round trip

Phase velocity ≠ Group velocity

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, Science 288, 635 (2000).

Two kinds of combs

Self-reference by J. Hall

Reference to frequency standard by T. Hansch

Self-reference

Referring to frequency standard

• f_m is not sensitive to Δ (m~10⁶ ,n-m~10⁴ )

Cs stabilized CW laser

Slave mode-locked laser

Offset-frequency locking, δ

Ultra-stable RF source

Repetition rate locking, Δ

Progress in our group: instability of Δ: < 1 mHz. Δ f_n ~200 Hz.

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Cs standard from NIST

10 MHz time base

Global Positioning System (GPS)

Microstructure Optical Fiber �Continuum Generation

courtesy of Jinendra Ranka

fs comb after prism

Linewidth of optical frequency comb laser

• Linewidth of each mode: characterized by the conditions of frequency stabilization, typically smaller than 10 kHz

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• Bandwidth could even more than 90 nm

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🡪 could be considered as superposition of 10⁶ frequency-stabilized laser light sources, on the other words, superposition of 10⁶ coherent state photons.

Comb laser in IAMS🡪 the other approach

Motivations

1. Set up a frequency standard for the use of Ti:sapphire lasers

Rb 778 nm

Cs 822 nm

2. Possibly related to clock transition

3. More objective since 6S 🡪 8 S is not sensitive to magnetic field of the earth (Rb 778 is for 5S 🡪 5D)

4. Narrower linewidth than one photon transition

5. Potentially, could make a portable comb by referring to the Cs-stabilized diode laser

Linewidth of spectrum – the way matter feels light

1. Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser

• Linewidth of spectrum (The way matter feels light)
• Lorentian linewidth
• Doppler linewidth
• Other linewidthes

Lorentian linewidth

1.Lorentian comes from damping oscillation

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2. For examples

-- nature linewidth (nature broadening)

-- collision linewidth (pressure broadening)

-- nonlinear effect (power broadening)

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Lorentz model about dipole oscillator

1. Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser

• Linewidth of spectrum (The way matter feels light)
• Lorentian linewidth
• Doppler linewidth
• Other linewidthes

What is the Maxwell-Boltzmann Distribution?

All the molecules of a particular chemical, compound or element have the same mass, so their kinetic energy is only dependent on the speed of the particles.

Remember Kinetic Energy = ½mv²

In any particular mixture of moving molecules, the speed will vary a great deal, from very slow particles (low energy) to very fast particles (high energy). Most of the particles however will be moving at a speed very close to the average.

The Maxwell-Boltzmann distribution shows how the speeds (and hence the energies) of a mixture of moving particles varies at a particular temperature.

The Maxwell-Boltzmann Distribution

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Points to notice:

• No molecules at zero energy
• Few molecules at high energy
• No maximum energy value

For the reaction to occur, the particles involved need a minimum amount of energy - the Activation energy (E_ACT). If a particle is not in the shaded area, then it will not have the required energy so it will not be able to participate in the reaction.

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f_L= f₀(1+v/c)

f_L= f₀(1-v/c)

Δf_D =Doppler linewidth

1. Linewidth of light

🡪Where the linewidth originates

🡪Linewidth of CW laser

🡪Linewidth of mode-locked laser

• Linewidth of spectrum (The way matter feels light)
• Lorentian linewidth
• Doppler linewidth
• Other linewidthes

Other linewidth (broadening)

1.Voigt profile (Viogt linewidth)

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2. Transit broadening (sinc/Gaussian linewidth)

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3. Wave front curvature distortion

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Saturation broadening and lamb dip

Power shift

Power broadening

Linewidth for weak power (a<<1)

a = saturation parameter 🡪 laser power and dipole transition probability

dipole transition probability

laser power

Linewidth of two-photon transitions

a₁₁

a₂₂

a₁₁

a₁₁

a₂₂

a₂₂

a₁₂

a₂₁

a₁₂=a₂₁

a₁₁= saturation parameter by incident light

a₂₂= saturation parameter by reflecting back light

a₂₁= saturation parameter by both lights

Transit broadening

Transit broadening for Gaussian beam

Wavefront curvature broadening– correction of transit time broadening

R

r

x

r ~ waist~w

Δ ω_ω =Δf/T

T=transit time=w/v

大綱

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Part II: 高解析雷射光譜應用之例子

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• 一級長度標準之建立
• 原子核物理之應用
• 光纖通訊之頻標
• 精細結構常數 α 值之確認
• 超高解析雷射光譜及新長度標準 (光鐘 )
• 直接光梳光譜

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一級長度標準之建立 �

In 1983

Meter

is defined as light travels in vaccum

in 1/299792458

sec

0

1

meter

一級長度標準之建立

Iodine-stabilized

green He-Ne laser

Wavelength

standard

In 1992

雷射共振腔

碘分子超精細光譜

誤差訊號

回授控制電路

控制雷射腔長

40 Hz uncertainty per 5.5*10¹⁴ oscillation cycle

(for 30 sec. sampling)

一級長度標準之建立

一級長度標準之建立

λ/4

waveplate

PBS

6 cm I

2

cell

Mirror

( R=90% )

output

ω

3

ω

ω

Chart

Recorder

Heater

PZT

Screw

Photodiode

Function

Generator

Lock-in

Amplifier

f = 20

cm

PI Feedback

Controller

Faraday

Isolator

一級長度標準之建立

a serious

- R(12)26-0

b serious

- R(106)28-0

一級長度標準之建立

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Wang-Yau Cheng, Jow-Tsong Shy, JOSA B, 18, 363 (2001).

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W.-Y. Cheng, L. Chen, T. H. Yoon, J. L. Hall, and J.Ye,

Opt. Lett., 27, 571 (2002)

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超高解析雷射光譜及新長度標準

• 碘分子超精細光譜之重要性：

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• 從近紅外到可見光之最重要的光學尺

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• 非常狹窄的線寬(linewidth)適合於基礎分子特性研究。(molecule cooling, pre-dissociation, molecule recoil effect, ...)

超高解析雷射光譜及新長度標準

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• 碘分子超精細光譜
• H_hyperfine= H_neq+H_sr+H_tss+H_sss

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=

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=

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~ 10⁹+10⁹+5000+500 Hz

the hyperfine constant eq. Q, C, d, d could be determined by measuring the frequency intervals among the hyperfine lines

2

1

2

1

2

1

:

6

1

I

I

m

B

m

i

B

i

m

E

Q

∙

∙

∙

+

−

−

∇

−

δ

R(12) 26-0 R(106) 28-0

DeqQ (MHz) 1917.62(7) 1914.452 (14)

DC (kHz) 59.1(2) 70.268 (8)

Dd (kHz) -35 (2) -34.6(6)

Dd(kHz) -15(3) -6.7(9)

standard 8.1 6.8

deviation (kHz)

The lower level constants are following: eqQ"=-2452.5837 MHz, C"=3.162 kHz, d"=1.58 kHz and d"=3.66 kHz. Here, DeqQ=eqQ'-eqQ"=high level-low level, the rest constants Dc, Dd, Dd may be deduced by analogy.

由實驗值計算得出之超精細常數

超高解析雷射光譜及新長度標準�在 JILA 之工作

Wavelength: 507 nm

Time constant: 5 ms, linewidth: 100 kHz

uncertainty could be below 1 Hz per second

(二) 原子核物理之應用���Wang-Yau Cheng, Jow-Tsong Shy, Apply Physics B: 70, 1 (2000).

原子核物理之應用

• Normal Mass Shift (ν_n)

ν_t: 雷射頻率

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M_1,2: 同位素質量

• Specific Mass Shift (ν_s)

A: 質子、電子重量比

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原子核物理之應用

原子核物理之應用

＊本實驗證實SMS J-dependence 之說法

• 結論：(for specific mass shift)

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• 對Ne 3 s₂ --> 2p₁₀ 之躍遷 (綠光)
• 本實驗所得為：-2535 MHz

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• 對 Ne 3s2 --> 2p4 之躍遷(紅光)
• 其他實驗所得為: -2919MHz

結論：

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• 測量了二種同位素雷射之絕對頻率

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• 得出 specific mass shift

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• 與high finesse Fabry-Perot etalon 配合可從事需同時具有長、短期穩定度之實驗

原子核物理之應用

(三) 光纖通訊之頻標��������W.-Y. Cheng, J. –S. Shy, T. Lin, C. –C. Chou,�“Molecular Iodine Spectra and Laser Stabilization by Frequency-Doubled 1534 nm Diode Laser” �JJAP 44, 3055-3058 (2005)

光纖通訊之頻標

光頻率高達

10¹⁴ Hz !!!

如何定頻標?

光纖通訊之頻標

過去定頻標的問題：

• 可吸收此波段之原子分子太少
• 即使有原子分子可於此波段躍遷，範圍仍過小(only few GHz)，不敷未來使用

使用碘分子超精細光譜的好處：

• 頻標密集
• 超精細躍遷線寬小

問題：

• 須二倍頻或三倍頻才有可吸收的譜線

光纖通訊之頻標

Isolator

1/4

λ

Plate

I

2

Cell

PBS

Photodiode

1550 nm

Diode Laser

EDFA

Lock-in

Amplifier

PI

Controller

Fiber

Coupler

Polarization

Controller

QPM PPLN

Waveguide

Temperature

Controller

Schematic diagram of the iodine-stabilized 1550 nm diode laser system. PBS: polarizing beam splitter.

Function

Generator

(三) 精細結構常數 α 值之確認���� .Gulin Wu, Wang-Yau Cheng, Jow-Tsong Shy

精細結構常數 a 值之確認

a 到底是否為 universal constant, 須利用各種互不相關之物理定律來驗證

精細結構常數 a 值之確認

S. Chu’s laboratory, Appl. Phys. B, 217 (1994)

精細結構常數 α 值之確認

精細結構常數 a 值之確認

本實驗：α^-1=137.0360144(27)

其他實驗：

• α^-1=137.03599958(13)
• by electron annomalous magnetic moment (T. Kinoshita, 2000)
• α^-1=137.0360108(52)
• by neutron interferometry (Kruger et al., 1995)
• α^-1=137.0359979(32)
• by quantum Hall effect (Cage et al., 1989)
• α^-1=137.0359770(77)
• by ac Josephson effect (Williams et al. , 1989)