1/14 Introduction of Gaussian Beam - Derivation of Laguerre-Gaussian mode - R. Saito Tohoku University Confocal Raman spectroscopy
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Introduction of Gaussian Beam- Derivation of Laguerre-Gaussian mode -
R. SaitoTohoku University
Confocal Raman spectroscopy
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Paraxial Helmholtz EquationWave Eq. for Electric field (we can obtain from the Maxwell Eq.)
When // ∝ , we get the Helmholtz Eq.
Propagating in the direction of :
Further if f changes slowly as a function of z compared with the wavelength
(Paraxial Helmholtz Equation)
=
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Solutions of Paraxial Helmholtz Eq.
= ⋅= ⋅ + ⋅= ⋅ + ⋅
= − + ⋅ − ( + )2∴
= − 1
= − 1 += − − +
∴ + 2 + = 0
(Paraxial Helmholtz Eq.)
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Gaussian beam Paraboloidal wave
(Paraxial Helmholtz Eq.)
is also a solution
Complex beam parameters : R(z), w(z) = ( − ) − = −
Intensity of the beam
Gaussian beam
= −tan=
1− =
= cos − tan = −= +
+ = 1+
/ = 1 1 + = ( ) =
amplitude phase
=
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Beam parametersRayleigh range on z-axis ( = 0) (Paraxial Helmholtz Eq.)
Beam radius: ( ), beam waist: 0 =
Intensity of the beam
Gaussian beam
=
amplitude Gouy’s phase
(0,0, )
Beam divergence: ∝ , divergence angle:
Conforcal parameter: 2 , depth of focus
0,0, = 0,0,02= 2
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Beam can not be focused < 2(Paraxial Helmholtz Eq.)
Intensity of the beam
Gaussian beam
=
amplitude Gouy’s phase
Conforcal parameter: 2 , depth of focus
0,0, = 0,0,02= 2
Uncertainty principles: Δ Δ > ℏ/2Δ ∼ , Δ ∼ ℎΔ = ℎ = ℎ
≡ Δp = ℎ ⋅ ℎ = ∼ < ⋅ < , ( < )We can not focus less than !
Radius of wavefront: ( )( )
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Higher order Gaussian BeamHermite-Gaussian mode
Lindlein N., Leuchs G. (2007) Wave Optics. pp152-155 In: Träger F. (eds)
Springer Handbook of Lasers and Optics. Springer Handbooks. Springer, New York, NY
Laguerre-Gaussian mode H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966)
Since I could not find the derivation of Laguerre-Gaussian mode, hereafter I solved by myself!!
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Laguerre-Gaussian mode H. Kogelnik, T. Li. Proc. IEEE 54, 1312 (1966)
Ψ + Ψ + 2i Ψ = 0 + = + 1 + 1Paraxial Helmholtz Eq. Cylindrical coordinate
Ψ = ( ) exp − + − ≡ ( , , )We assume the shape of Ψ
Ψ = 1 +Ψ = 1 + − +Ψ = −ℓΨ = − + − − 2
A B
C D E
F
G H I
1 × + 1 + 1 + 2i Ψ
= 1 + 1 − 2i + + 2 − ℓ + 2 + − = 0C A G D BB’ F H I E
− 1 ∝P. Helmholtz Eq. should not diverge for → ∞→ = 1 → ∞ → = −
B’
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Variable separation Ψ + Ψ + 2i Ψ = 0 Ψ = ( ) exp − + − ≡ ( , , )
1 × + 1 + 1 + 2i ΨC A G D BB’ F H I E
1 + 1 − + 2 − ℓ − 2 = 0C A GD BB’ F H
Analogous to Gaussian beam1 = 1 + ,
→ 1 = 1 − +→ − 1 = = 1 = +
( = 1)
square
derivatieve
∴ = 1 − , =Compare real and imaginary parts
( = 1)
1 − 2i + = 1 − 2i + 1 += 1 −
( = )
w z , , are functions of .
2 + 2 ≡ ( )1 + 1 − − ℓ = − ( )
Variable separationBB’ H
× ≡ ( )∴ + − − ℓ − = 0
C A GD F
= 1 − 4
= 1 + 1 − 2i + + 2 − ℓ + 2 + − = 0
A G D
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Laguerre differential Eq. Ψ + Ψ + 2i Ψ = 0 Ψ = ( ) exp − + − ≡ ( , , )
( ) = ℓ ( )We assume the shape of
= 2 , = 2 , = 2 = = 4 2 = 2
= = = 2
= = 12 2 = 2
= + 4
+ 4 + − 2 2 − ℓ − =0
+ 8 − − ℓ − =0
= ℓ ′+ℓ ℓ , = ℓ + ℓ ℓ + ℓ2 ℓ2 − 1 ℓ
8 ℓ + 8ℓ ℓ + 2ℓ ℓ − 2 ℓ
+ 8− ℓ+ℓ2
ℓ − 2ℓ − ℓ = 0
∴ + − − ℓ − = 0
÷ 8 ℓ+ ℓ + ℓ ℓ − 2 1 + 1 − +
ℓ2 − ℓ4 − 8 = 0+ (ℓ +1 − ) − ℓ2 − 8 = 0
− ℓ2 − 8 ≡+ (ℓ +1 − ) + = 0
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Laguerre differential Eq. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966)
≡ − ℓ − ≡: integer→ = ℓ ( )
Associated Laguerre polynomials: ℓ
Rodrigues's Formula
Generating function
Recursion formula
ℓ = 1ℓ = 1 + ℓ −
ℓ = + 1 + ℓ − ℓ − ( + ℓ) ℓ+ 1
ℓ = ℓ! ( ℓ)
ℓ = 1 − ℓ
ℓ = 0 Laguerre polynomials =Recursion formula
Ψ + Ψ + 2i Ψ = 0Ψ = ( ) exp − + −
( ) = ℓ ℓ ( )
≡ = + 4ℓ
= 1 → = + ≡ −= 2
Laguerre differential Eq.
Step1
Step2
Step3
ℓ + (ℓ +1 − ) ℓ + ℓ = 0
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Compare with Gaussian beam
≡ − ℓ − : integer
Ψ + Ψ + 2i Ψ = 0 Ψ = ( ) exp − + − ≡ ( , , )
( ) = ℓ ℓ ( )≡ = + 4ℓ
2 + 2 ≡ = + 4ℓ → 1 = − + 2ℓ
= 1 → ≡ −
1 = 1 + 2 = − (4 + 2ℓ)
Gaussian beam
= 1 + 2 + 4 + 2ℓ = 1 + ( + 2ℓ + 2)
= 1 + 4 + 2ℓ + 2 = + + ( + ℓ + 1)( + )
= = log +2 + + ℓ + 1
( + ) = arctan ≡ = 2 +exp − = 1 + × exp − + ℓ + 1
BB’ H 1( ) =
= , = ,=
tan = −( + ) = 12 log +
÷
ℓ + (ℓ +1 − ) ℓ + ℓ = 0
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Solution of Laguerre Gauss mode
− ℓ − ≡ : integerΨ + Ψ + 2i Ψ = 0
Gaussian beam1( ) =
= 1 + , = 1 + , ≡ 2
= 1 + = ( )
Ψ = ( ) exp − + −( ) = ℓ ℓ = 2 ℓ
ℓ 2 ( )
exp = exp − expexp − = 1 + × exp − + ℓ + 1
ℓ ( ) + (ℓ +1 − ) ℓ ( ) + ℓ ( ) = 0Ψ = ( ) 2 ℓ
ℓ 2 ( ) exp − exp − − + ℓ + 1= 2
= 1 + , ≡ 2 ,= 1 + , ≡ arctan
= 2
Paraxial Helmholtz Eq.
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Comparison with previous solutions for Laguerre Gauss modes
− ℓ − ≡ : integerΨ + Ψ + 2i Ψ = 0
ℓ ( ) + (ℓ +1 − ) ℓ ( ) + ℓ ( ) = 0Ψ = ( ) 2 ℓ
ℓ 2 ( ) exp − exp − − + ℓ + 1= 2
= 1 + , ≡ 2 ,= 1 + , ≡ arctan
Gaussian beam = ℓ = 0, ( ) = 11( ) =
= 1 + , = 1 + , ≡ 2
=
1 + = ( )
H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966)
Ψ + Ψ − 2i Ψ = 0
Present results
W. Paufler et al, J. Opt 21 094001 (2019)
Lindlein N., Leuchs G. (2007) Wave Optics. In: Träger F. (eds) Springer Handbook of Lasers and Optics. Springer Handbooks. Springer, NY