3 . W idth s an d Profile s of S pe ctral Lin e s

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3 . W idt h s an d P ro file s o f S p e c t ra l Lin e s

ab sorption/ emis sion spectrum n ev e r s tric tly m on oc h rom atic

lin e profile ; spectr al distribution , I ( ) in the vicinity of 0

F W H M (lin e w i dth , h alf w i dth ) ; = | 2 - 1|, I ( 1) = I ( 1) = I ( 0) / 2

( = 2 , = - ( c/ ²) , | | = | | = | |)

3 .1 N atu ral Lin e w idth

ex cit ed electron => damped h arm onic oscillator (m , k, )

x ' ' + x ' + ²₀x = 0 (3.3)

w here, ²₀ = k/ m r eal solution ;

x ( t) = x0e - ( / 2 ) t

[ cos t + ( / 2 ) s in t] (3.4)

w here, = ( ²₀ - ²/ 4)^{1/ 2}

If ₀, ₀, x ( t) = x₀e - ( / 2 ) t

c o s ₀t (3.5)

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3 .1.1 Lore n t zi an Lin e P rof ile of th e E m itt e d R adi ati on F ourier tr an sform

x ( t) = 1

2 2 ⁰ A ( ) e^{i t}d (3.6)

A ( ) = 1

2 ^- x ( t) e ^{- i t}d t

= 1

2 ^- x₀( t) e - ( / 2 ) t

cos ( ₀t) e ^{- i t}dt (3.7)

= x0

8

(

ⁱ⁽ ^- ¹⁰^{) +} ^{/ 2} ⁺ ⁱ⁽ ⁺ ¹⁰^{) +} ^{/ 2}

)

^(3.8)

r eal int en sity , I(ω) ∝ A (ω)A^*(ω)

I ( - ₀) = C

( - ₀)² + ( / 2)² (3.9)

(norm alization 1) ;

0 L ( - ₀) d =

- L ( - ₀) d ( - ₀) = 1

=> C = I0 / 2 L ( - ₀) = 1

2 ( - 0)² + ( / 2 )² (3.10)

F W HM , _n = / 2 (3.11)

int en sity pr ofile, I ( ) = I0L ( - 0) (3.10a )

I ( ₀) = 2 I₀/ ( ), I₀ = I ( ) d

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(norm alization 2) ; I ( ₀) = I₀, F W H M 2 L^*( - 0) =

( - 0)² + ( )² (3.10b )

int en sity pr ofile, I ( ) = I₀L^*( - ₀)

0 I ( ) d =

- I ( x ) dx = I₀ (3.10d)

w here, x ( - 0) /

3 .1.2 R e l ation B et w e en Lin e w idth an d Lif e tim e (3.3)식의 양변에 m x '을 곱하면

m x ' ' x ' + m ²0x x ' = - m x '² (3.12)

=> ^d

d t

(

^ms x '² + m 2

2

0x²

)

⁼ ^{d W}d t = - m x '² (3.13)

(3.5)식의 해를 대입하면,

d W

d t = - m x²₀ ²₀ e ^- ^ts in² ₀t (3.14)

=> d W / d t = -

2 m x²₀ ²₀e ^- ^t (3.15)

decay time ; = 1/ 1/ A _i (2.6절의 결과)

=> 고 전 적 인 dam pin g f ac t or 는 Ein s tein A - c oefficient에 해 당 .

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3 .1.3 N atural Lin e w i dth of A b s orbin g T ran s ition s

A b s orption c o eff ic ie nt , ik( ) = ik( ) [ Ni - ( gi/ gk) Nk] (3.22) ( => _ik( ) = _ik( ) N_i for N_i N_k )

Int en sity I of a plane w av e pas sing through an ab sorbing m edium ;

dI = - I dz

=> I = I₀e ^- ⁽ ^{) z} (3.23)

Cl as s i c al f orc e d h arm on ic o s c ill at or m ode l ;

m x ' ' + bx ' + k x = q E₀ e^{i t} (3.24)

s olution

x = q E₀ e^{i t}

m ( ²₀ - ² + i ) , w her e = b/ m , ²₀ = k / m (3.25)

induced dipole m om ent

p = qx = q²E₀ e ^{i t}

m ( ²₀ - ² + i ) (3.26)

polarization

P = N p = ₀( - 1) E = ₀ E (3.28)

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Complex refr activ e index By (3.26), (3.28) & n = ^{1/ 2}

n² = 1 + N q²

0m ( ²₀ - ² + i ) (3.30)

If n 1, n² - 1 = ( n + 1) ( n - 1) 2 ( n - 1)

=> n = 1 + N q²

2 0m ( ²0 - ² + i ) n ' - i (3.31, 32)

Plane w av e pas sing through the m edium , k = k₀n

E = E₀e ^{i (} ^{t -} ^{k z )} = E₀ e ^{- k}⁰ ^z e ^{i (} ^{t -} ^k⁰^{n ' z )} (3.33) Int en sity pas sing thr ou gh the medium , I E E^* ;

I = I₀ e ^{- 2} ^k⁰^z (3.34)

A b s orption c o eff ic ie nt / R e frac tiv e in dex ;

Comparing (3.34) w ith (3.23), = 2 k₀ = 4 / ₀ (3.35)

= N q² ₀

c ₀m ( ²₀ - ²)² + ( )² (3.36a)

n ' = 1 + N q² 2 ₀m

2

0 - ²

( ²0 - ²)² + ( )² (3.37a)

; Kr am er s - Kr onig diper sion r elation

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In n ear reson ance, | ₀ - | ₀,

( ) = N e² 4 ₀m c

/ 2

( ₀ - )² + ( / 2 )²

n ' = 1 + N e² 4 0m 0

0 -

( ₀ - )² + ( / 2 )² (3.36a, 37b )

ex 1) N a D¹ line ; 3P 3/ 2 (τ= 16 n s ) - > 3S 1/ 2 : _n = 1/ 2 = 10 MHz ex 2) M olecular vibr ation al line ; typically τ= 10^{- 3} s : _n = 160 Hz ex 3) F orbidden lin e ; 2S (τ= 1 s ) - > 1S : n = 0 . 15 Hz

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3 .2 D opple r W idth

is due t o the th erm al m oti on of the ab sorbing/ emitting m olecules (low pr es sure g as s ample )

1) emitting m olecule w ith a v elocity v = { v_x, v_y, v_z}

detecting fr equen cy : _e = ₀ + k v (3.38)

2) abs orbing m olecule w ith a v elocity v = { v_x, v_y, v_z}

w av e frequency in the fr am e of the m oving m olecule : ' = - k v

abs orption frequ ency : a = ₀ + k v (3.39a ) If w e choose k = {0 , 0 , k_z}, _a = ₀( 1 + v_z/ c ) (3.39b )

M ax w ell v elocity distribution ;

ni ( vz) d vz = N_i

v_p e ^{- ( v}^z^{/ v}^p⁾

2

d vz (3.40)

w here, N_i = n_i ( v_z) d v_z : t ot al den sity of molecule in lev el Ei .

v_p = ( 2 k T / m )^{1/ 2} : m ost pr ob able v elocity

(3.39b ) => d v_z = ( c/ ₀) d ni( ) d = Ni c

0 v_p ex p

[

^-

(

^{c (} ⁰^-^v^p ⁰⁾

)

²

]

^d ^(3.41)

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Int en sity pr ofile :

I ( ) = I0 ex p

[

^-

(

^{c (} ⁰^-^v^p ⁰⁾

)

²

]

^d ^(3.42)

: Gau s sian (Doppler ) pr ofile

D = 2 ln 2 0vp/ c =

(

c⁰

)

8 k T ln 2 / m (3.43)

: Doppler w idth

I ( ) = I0 e x p

(

^- ⁽^{0 . 3 6}^- ⁰⁾D²²

)

^(3.44)

ex 1) Lym an line (H at om , 2P - > 1S ) : D = 5 . 6 10⁹ Hz ex 2) N a D - line (3P - > 3S ) : _D = 1 .7 10⁹ Hz

ex 3) CO2 vibr ation al line : D = 5 . 6 10⁷ Hz

< Lore n t zi an an d Gau s s i an P rof ile s >

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< V oi g t profile >

Not all m olecules w ith a v elocity emit/ ab sorb at the s am e frequency

=> Doppler spectrum cannot be strictly r epresented by a pur e Gau s sian

=> frequency r espon se of these m olecules w ith the s am e v elocity is r epresented by a Lorent zian profile

I ( ) = I₀ n ( ' ) L ( - ' ) d ' (3.45)

by (3.10), (3.41)

I ( ) = C

0

e x p { - [ ( c / v_p) ( ₀- ' ) / ₀]² }

( - ' )²+ ( / 2 )² d ' (3.46)

w here, C = N_i c 2 vp

3 / 2 0

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3 .3 Colli s ion B ro aden in g of S pe c tral Lin e s

Energy lev el shift due t o the int er action betw een A and B, w hich depends on the electron configur ation s of A and B .

3 .3 .1 P h en om en ol og ic al D e s cription

tr an sition frequency

El as ti c / In e l as tic c olli s ion s

- Elastic collision (ph ase- perturbing collision ) :

damped oscillat or의 진폭 변화 없이 ph ase만 변화 - Inelastic collision (quen ching collision ) :

damped oscillat or의 진폭을 변화

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In e l as tic c olli s i on al bro ade nin g (pre s s u re bro ade n in g ) T ot al tr an sition pr ob ability A i

w here,

: pres sur e broadening Int en sity pr ofile :

3 .3 .2 T h e oreti c al T re atm en t of El as tic Colli s ion (정 리 숙 제 )

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3 .3 .4 Colli s ion al N arrow in g of Lin e s (D ic k e n arro w in g )

ex cit ed st at e의 수명이 평균충돌시간보다 긴 경우(IR, μW 영역)

=> 탄성충돌에 의해 원자평균속도가 줄어들고, 이로인해 Doppler broadening 이 작아질수 있다.

=> Doppler br oadening이 pr es sure broadening보다 클 때 발생.

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3 .4 T ran s it - T im e B ro ade nin g

광과의 interaction time ( T = d / v)이 원자의 여기준위 수명보다 짧은 경우 (여기준위 수명이 긴 원자보다는 분자에서 dominent )

damped oscillat or에서 damping t erm e ^- ^{t / 2}를 무시할 때 (slow damping ), 광과의 interaction은 유한시간 T 까지만 지속되므로

F W HM : _T = 5 . 6 / T

* T 인 경우 : ^v

d

광이 Gaus sian인 경우,

를 이용하면,

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< B ro ade nin g by w av e - f ront c u rv atu re >

r² = R² - ( R - x )² = > x = r²/ 2 R , ( x << R )

* R r²/ 이면 이므로 broadening 무시 가능.

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3 .5 H om o g en e ou s an d In h om og e n e ou s Lin e B ro ade nin g

H om ogeneou s broadening : 같은 에너지 준위의 모든 원자에 대해 일정 주파수에서의 transition probability가 동일

ex ) n atur al br oadenin g , inelstic collision broadening

Inhom ogen eou s broadening : 같은 에너지 준위의 개개 원자에 대해 일정 주파수에서의 transition probability가 서로 다름

ex ) Doppler br oadenin g , v elocity ch ange collision

3 .6 S atu rati on an d P ow e r B ro ade nin g

Str on g r adiation에 의한 population의 변화로 유발

3 .6 .1 S atu ration of Le v e l P opul ation by Opti c al P um pin g T w o- lev el sy stem , N₁ + N₂ = N

R at e equ ation

F rom st ation ary con dition , dN_i/ d t = 0

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P , N₁ N₂, 0

P = 0 (no r adiation field)

* If the spont aneou s emis sion is the only relax ation m ech anism

=> R₁ = 0 an d R₂ = A _{2 1}

** Pump r ate, P = ₁₂( ) I ( ) / S = 2 ₁₂I ( )

A 12

(3.80b )

T he s atur ated ab sorption coefficient , ( ) = 12 N

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3 .6 .2 S atu ration B ro ade n in g of H om o g en e ou s Lin e P rofile

Ab sorbed pow er per unit v olum e

; for m onochrom atic w av e

w e can introdu ce a fr equen cy - depen dent s atur ation par am et er , S

F rom (3.83), (숙제)

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< A b s orption Co ef fic ie n t >

Ab sorbed pow er per unit v olum e, ^{d W}¹²

d t

Int en sity decrease per cm , dI = - _sI 이므로

* = 0 => s( 0) = 1 1 + S0

( 0) : ( 1 + S0)^{- 1} 만큼 감소.

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3 .6 .3 P o w e r B ro ade nin g

str on g field에 대한 식(2.79)으로부터

|b > 준위가 평균 relax ation con st ant 로 자발방출된다면, 평균 밀도 확률은

* linew idth : _s = 1 + S

; pow er broadening