7 . N on lin e ar S pe ctro s c opy Nonlinear effects => Sub - Doppler spectroscopy

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7 . N o n lin e ar S p e c t ro s c o py

Nonlinear effect s => Sub - Doppler spectroscopy

7 .1 Lin e ar an d N on lin e ar A b s orpti on

F or a plane lightw av e, E = E₀ cos ( t - k z ), (w here, = E_k - E_i ) pas ses thr ou gh a s ample, the pow er abs orbed in the v olum e d V = A dz

is giv en by

dP = A I _ik N dz [W ] (7.1a)

w here, N = [ Ni - ( gi/ gk) Nk]

T he r at e equ ation for the st ation ary population den sities of the n on degen er ate lev el |1> , |2> ,

dN₁/ d t = B₁₂ ( N₂ - N₁) - R₁N₁ + C₁ dN2/ d t = B12 ( N1 - N2 ) - R2N2 + C2

w here, R_iN_i : tot al relax ation r ate (including spont aneou s emis sion )

C_i =

k R_{k i}N_k + D_i : r epopulation r at e of the lev el |i>

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# Under st ation ary condition (dN / d t = 0),

T he un s atur ated population difference ( = 0) is

N⁰ = N ( = 0) = ( N⁰2 - N⁰1) = C₂R₁ - C₁R₂ R1R2

(7.4) T he s atur ated population difference ( 0) is

N = N ⁰

1 + B12 ( 1/ R1 + 1/ R2) = N ⁰

1 + S (7.5)

w here, S B₁₂

R^* : s atur ation par am et er w ith R^* = R1R2

R1 + R2

: m ean relax ation pr ob ability (7.6a )

T he pow er ab sorbed along the ab sorption length dz dP = - P 12 N⁰

1 + S dz = - P 12 N⁰

1 + B₁₂I / ( c R^*) dz (7.7)

# F or sm all inten sity I , S << 1,

dP - P ₁₂ N⁰ dz

and, the ab sorbed pow er is

P = P₀ ex p ( - ₁₂ N⁰z ) = P₀e^- ^z : Beer ' s law (7.9)

# F or strong int en sity I ,

dP = - P 12 N dz < - P 12 N⁰ dz

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7 .2 S atu rati on of Inh om o g en e ou s Lin e P rofile

Spectr al lin ew idth of homog eneou s line profile (Section 3.6),

s = ₀ 1 + S₀ ; S₀ = S ( ₀) (7.13)

7 .2 .1 H ole B urn in g

In the g aseou s s ample w ith a M ax w ell- Boltzm ann v elocity distribution , the abs orption cr os s section for a molecule w ith the v elocity component

v_z on a tr an sition |1 > |2 > is

12( , v_z) = ₀ ( / 2)²

( - ₀ - k v_z)² + ( / 2)² (7.14)

w here, ₀ = ( = ₀ + k v_z)

# F rom (7.5) and (3.85), the population den sities by a monochrom atic light w ith angular fr equency are

N₁( , v_z) = N⁰₁( v_z) - N⁰

1

[

⁽ ^- ⁰ ^{- k v}^S⁰⁽ ^z^{/ 2 )}⁾² ^{+ (}² ^s^{/ 2)}²

]

^{(7.15a )}

N2( , vz) = N⁰2( vz) + N⁰

2

[

⁽ ^- 0 - k v^S⁰⁽ _z^{/ 2 )})² + (² _s/ 2)²

]

^{(7.15b )}

w here, = 1 + 2 : hom ogeneou s w idth of the tr an sition

( ₁ ₂)^{- 1}

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# T he s atur at ed population differ ence is giv en by

N ( s , vz) = N⁰( vz)

[

^{1 -} ⁽ ^- ⁰ ^{- k v}^S⁰⁽ ^z^{/ 2)}⁾² ^{+ (}² ^s^{/ 2)}²

]

^(7.15c)

: B e nn e t H ole at v_z = ( - ₀) / k

w idth : _s = 1 + S₀

depth : N⁰( v_z) - N ( v_z) = N⁰( v_z) S₀

1 + S₀ (7.16)

# T he contribution t o the abs orption coefficient of the molecule w ith v elocity v_z t o v_z + d v_z :

d ( , v_z)

d v_z d v_z = N ( v_z) ( , v_z) d v_z (7.17)

T ot al ab sorption coefficient :

( ) = N ( v_z) ₁₂( , v_z) d v_z (7.18)

By (7.14) for ( , v_z) and (7.15) for N ( v_z),

( ) = N⁰ ₀

vp

e ^{- ( v}^z^{/ v}^p⁾

2

( - ₀ - k v_z)² + ( _s/ 2 )² d v_z (7.19) w here, N⁰ = N⁰( v_z) d v_z

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< Detection of the Bennet H ole by tw o las er beam s >

# s atur ating pump laser : fix ed fr equen cy 1 w ith the w av e v ect or k1

# w eak probe las er : frequency tuning acros s the Voigt profile w ith the w av e v ector k2

# T he ab sorption coefficient for the pr obe laser :

s( 1 , 2) = N⁰ 0

v_p

e ^{- ( v}^z^{/ v}^p⁾

2

( ₀ - - k₂ v_z)² + ( / 2)²

[

^{1 -} ⁽ ⁰ ^- ^-^S⁰^k⁽¹^v^{/ 2)}^z⁾²²^{+ (} ^s^{/ 2)}²

]

^{d v}^z

(7.22)

=> _s( ₁ , ) = ⁰( )

[

^{1 -} ^{1 + S}^S⁰ ⁰ ⁽ ^- ⁽^{' )}²^{/ 2 )}^{+ (}² s/ 2 )²

]

^(7.23)

a s atur ation dips at the probe frequency ,

= ' = ₀ ( ₁ - ₀) k₁/ k₂ (+ : collinear , - : anticollinear ) h alfw idth : s = + s = [ 1 + ( 1 + S0)^{1 / 2}]

depth : ( ' ) = ⁰( ' ) - s( ' ) = ⁰( ' ) S₀

1 + S₀ ( 1 + 1 + S₀)

S₀ 2

0( ' ) for S₀ 1

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7 .2 .2 L am b D ip s

F or the count erpropag ating w av es w ith the s am e frequ ency , the s atur at ed population difference is giv en by

N ( v_z) = N⁰( v_z)

[

^{1 -} ⁽ ⁰ ^- ^{- k v}^S⁰⁽ ^z^{/ 2 )}⁾² ^{+ (}² ^s^{/ 2)}² ^- ⁽ ⁰ ^- ^{+ k v}^S⁰⁽ ^z^{/ 2)}⁾² ^{+ (}² ^s^{/ 2)}²

]

# T he s atur at ed ab s orption coefficient :

s( ) = N ( vz) [ ( 0 - - k vz) + ( 0 - + k vz) ] d vz

F or w eak field (S₀ 1),

s( ) = ⁰( )

[

^{1 -} ^S²⁰

(

^{1 +} ⁽ ^- ⁽0)^s²^{/ 2 )}+ (² _s/ 2 )²

)]

^(7.26a)

w here, s = 1 + S0 , S0 = S0( I , 0)

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F or strong field,

s( ) = ⁰( ) / 2

B

[

^{1 -}

(

^{2 (}A + B^- ⁰⁾

)

²

]

^{1 / 2}

(7.26b )

w here, A [ ( - ₀)² + ( / 2)²]^{1/ 2}

B [ ( - ₀)² + ( / 2)² ( 1 + 2 S ) ]^{1/ 2}

# s( 0) = ⁰( 0) / 1 + 2 S at the line cent er

= ⁰( ₀) / 1 + S for ( - ₀)

# m ax imum depth of the Lamp dip :

( - 0 s) - ( 0)

( 0) = 1

1 + S₀ - 1

1 + 2 S₀ - > m ax .

=> S₀ 1 .4

# If the probe w av e is v ery sm all (I₂ I₁),

s( ) = ⁰( )

[

^{1 -} ^S²⁰ ⁽ ^- ⁽⁰⁾²^s^{/ 2 )}^{+ (}² ^*^s ^{/ 2 )}²

]

^(7.27)

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7 .3 S atu rati on S pe c tro s c opy

T w o count erpropag ating las er beam s

7 .3 .1 E x perim e nt al S c h em e s

# Principal res on ance sign als

* Cros sov er r eson ance sign als occur simult aneou sly at the frequ ency ,

c = ^ca + c b

2 due t o the atom s w ith v elocity v_z = ^ca - c b

k

** Residu al Doppler linew idth due t o the cros sin g an gle.

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< Sen sitiv e v er sion of s atur ation spectros copy >

Optical isolat or , Du al pr obe beam s

ex ample)

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Int erm odulated fluores cence technique

# Pump and probe beam s are chopped at t w o different frequencies ,

I₁ = I₀( 1 + cos 2 f₁t) , I₂ = I₀( 1 + cos 2 f₂t)

F luores cence inten sity :

I_{f l} = C N_s( I₁ + I₂) (7.29)

w here, N_s = N⁰[ 1 - a ( I₁ + I₂) ]

=> I_{f l} = C [ N⁰ ( I₁ + I₂) - a N⁰( I₁ + I₂)²] (7.30)

* linear t erm s (f₁ , f₂ ) : norm al LIF w ith a Doppler - br oaden ed profile

* qu adr atic t erm s (f₁ + f₂ , f₁ - f₂ ) : s atur at ed sign als

=> detect only (f1 + f2) component by Lock - in detection

=> linear b ackgroun d is suppres sed !!

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7 .3 .3 Intrac av ity S aturati on S p e c tro s c opy

Las er output pow er

P_L ( )

{

^G( ^- ¹ ^- ⁰^{( )}

[

^{1 -} ^S²⁰

(

^{1 +} ⁽ ^- ⁽⁰⁾^s²^{/ 2)}^{+ (}² ^s^{/ 2)}²

)]}

^(7.31)

A ² + B + C + D

( - ₀)² + ( _s/ 2 )² (7.32)

Deriv ativ es of the laser pow er :

PL^{( 1)}( ) = 2 A + B - 2 D ( - ₀) [ ( - ₀)² + ( _s/ 2)²]²

P_L^{( 2 )}( ) = 2 A + 6 D ( - ₀)² - 2 D ( _s/ 2)²

[ ( - ₀)² + ( _s/ 2)²]³ (7.33)

P_L^{( 3 )}( ) = 24 D ( - ₀) [ ( - ₀)² - ( _s/ 2)²]

[ ( - ₀)² + ( _s/ 2)²]⁴

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* B ro ad b ac k g rou n d di s appe ars f or th e h ig h e r deriv ativ e s !!

ex ample)

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7 .3 .4 L am b - D ip F re qu en c y S t abiliz ation of L as ers

application ex ample)

* fr equen cy offset locking

ex ample)

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7 .4 P ol ari z ation S pe c tro s c opy

ab sorption spectr os copy : abs orption ch ange

polarization spectroscopy : refr activ e index ch ang e

7 .4 .1 B as ic P rin c iple pump laser light - >

n onuniform population and unequ al s atur ation of the M - sublev els - >

birefrin gent for the incident probe las er light - >

plane of polarization dir ection of the pr obe beam is slightly rot at ed

0, pump beam : v_z + v_z = ( ₀ - ) / k / k

probe beam : = - ( ₀ - ) / k / k

probe beam is n ot influenced by the pump beam

= ₀, pr obe beam ex periences a birefring ence !!

: an alogou s to the s atur ation spectr os copy

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7 .4 .2 Lin e P rofil e s of P ol ariz ati on S ig n al s

Incident pr obe light w av e

After a path L through the pumped region of the s ample, the t w o component s are

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Con sidering the index of refr action of the w indow , n_w = b_r + i b_i , probe w av e behind the exit w indow ,

If the tr an smis sion ax is of the an alyzer P² is tilt ed by a sm all angle , the trn smitted amplitu de becom es E_t = E_x s in + E_y cos .

If L 1, L k 1, and b 1 (pr actical), the tr an smitt ed amplitude is

E_t = E₀ e^{i t}ex p [ i ( nL + b_r) / c - 1

2 L - b_i] ( + ) (7.37)

# det ect ed sign al :

S ( ) I_t( ) = c ₀E_tE^*_t

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w here, I0 : incident inten sity

' = + / ( 2 c) b_r

# ab s orption coefiicient :

# refr activ e index : (by Kr am er s - Kr onig relation )

# con st ant b ackgroun d :

t erm : imperfection of polar ozer (extinction r atio) Glan - T hom son , Glan - Laser polarizer < 10^{- 6}

' term : finite uncros sing angle bet w een the poly zer s b t erm : birefringence of the cell w indow

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F or = 0, the sign al becom es a Lor entzian ( b_i _i : optimization !) F or b_i, the sign al becom es a pure disper sion

ex ample)

* s atur ation spectr os copy

* polarization spectroscopy

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Similarly , for the linearly polarized pump beam , the sign al is giv en by

# 0 => Lorentzian

7 .4 .4 S e n s itiv ity of P ol ariz ati on S p e c tro s c opy T he amplitude of the disper sion sign al

cf) s atur ation spectroscopy : full inten sity det ection

polarization spectroscopy : detect only the ch ang e => m ore s en sitiv e

In the abs ence of w indow birefring ence (b_r = b_i = 0 ' = ), the sign al- to- nois e r atio for the disper sion sign al becom es

w here, I₁/ a : m ean noise of the incident probe w av e

# m ax imum r atio (w hen ² = ) :

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T he m aximum sign al- t o- noise r atio for the Lor entzian sign al ( ' = 0), from (7.41)

If bi 4

0L ,

cf) In s atur ation spectroscopy , ^S

N = 1

2 a 0L S0

=> enh an cem ent fact or : C^*JJ₁/ ( 8 ) for the optimized Lorentzian

C^*_JJ₁/ ( 4 ) for the optimized disper sion

7 .4 .5 A dv ant ag e s of P ol ari z ation S pe c tro s c opy High res olution

T he s en sitivity is 2- 3 order s of m agnitude larger th an th at of s atur ation spectros copy (But , intr acavity arr an gem ent is impos sible) T he pos sibility of distinguishin g betw een P , Q, R lines

T he disper sion profile of the sign al can be obt ained

=> laser fr equen cy st abilization w ithout any fr equency m odulation