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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