Материал: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo

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Ref. p. 131]	3.1 Linear optics	83

3.1.3.3.2 Gauss-Laguerre beams (circular symmetry)

√

2 r2

Elp(r, ψ, z) = E0 exp {−i [kz − ϕlp(z)]}

Lpl

w(z)

w2(z)

× exp −

k x2

cos (lψ)

− i

sin (lψ)

w2(z)

2 R(z)

with

z : propagation direction,

r, ϕ : polar coordinates in the plane z-axis,

z0 =

πw02

: the Rayleigh distance (half depth of focus),

1 +

: the E00-beam 1/e2-intensity radius,

w(z) = w0

R(z) = z 1 +

: the radius of curvature of the wavefront at position z,

ϕlp = (2p + l + 1) arctan : Gouy’s phase, z0

Llp : Laguerre polynomial of degree p and order l [70Abr]:

(l + 1)(l + 2)

− (l + 2) ξ −

(ξ) = 1 , L1(ξ) = (l + 1)

−

ξ , L2(ξ) =

(ξ) =

(l + 3)(l + 2)(l + 1)

−

(l + 3)(l + 2)

ξ +

(l + 3)

−

. . . ,

∞

(l + p)!

d ξ ξl

exp(−ξ) Lpl (ξ) Lql (ξ) = δpq

(orthogonality relation) ,

p! : the factorial p.

(3.1.32)

(3.1.33)

–Two degenerate mode patterns are formed by the cosand sin-terms in (3.1.32).

–l = p = 0 means the rotational symmetrical Gaussian beam E00.

–The symmetry determines what system of Gauss-Laguerre polynomials or Gauss-Hermite polynomials is more appropriate for a wave ﬁeld development.

3.1.3.3.3 Cross-sectional shapes of the Gaussian modes

In Fig. 3.1.6 intensity distributions of Gauss-Hermite modes Emn are given (rectangular symmetry), in Fig. 3.1.7 intensity distributions of Gauss-Laguerre modes Epl (circular symmetry).

Landolt-B¨ornstein

New Series VIII/1A1

84		3.1.4 Di raction	[Ref. p. 131

Rectangular symmetry (Gauss-Hermite modes)
00	10	30
y	x
01	11	31

Fig. 3.1.6. Intensity distributions of Gauss-Hermite modes Emn. The two digits at each distribution are m and n.

Circular symmetry (Gauss-Laguerre modes)

Fig. 3.1.7. Intensity distributions of Gauss-Laguerre modes Epl. The two digits at each distribution are p and l. .

3.1.4 Di raction

Di raction of light by aperture rims or amplitude and phase modiﬁcations inside the aperture:

–Solutions of Maxwell’s equations taking into account the material properties of the aperture:

–special cases: exact solutions [99Bor, 86Sta],

–mostly: numerical solutions.

–Starting with a ﬁeld near the aperture with reasonable assumptions for this ﬁeld or its measurement: large variety of methods for di erent ranges of validity [99Bor, 86Sta, 61Hoe].

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 131]	3.1 Linear optics	85

3.1.4.1 Vector theory of di raction

–Vectorial generalization of Kirchho ’s theory: Given E and H in an aperture E and H in the volume by Stratton-Chu Green’s function representation [23Kot, 41Str, 86Sol, 91Ish].

–Two-dimensional problem and meridional incidence of light [61Hoe]: Separation of the polarizations E parallel and E perpendicular to the plane of incidence for half plane [99Bor], slit [99Bor], gratings [80Pet], and volume gratings [69Kog, 81Sol, 81Rus].

3.1.4.2 Scalar di raction theory

Two sources of scalar di raction theory are:

–Transition from vectorial theory to scalar theory: [99Bor, 86Sol]. The information about the polarization is lost.

–Mathematical formulation and generalization of Huygens’ principle: Each point on a wavefront may be regarded as a source of secondary waves, and the position of the wavefront at a later time is determined by the envelope of these secondary waves.

In Table 3.1.4 di raction formulae with ﬁelds given near the di raction aperture are listed. Figures 3.1.8 and 3.1.9 are related to Table 3.1.4.

Remarks on the formulae of Table 3.1.4:

(3.1.37): Approximation of (3.1.34): Huygens’ principle with an additional directional factor (Fresnel).

(3.1.38): Approximation of (3.1.36): Huygens’ principle with a modiﬁed directional factor.

(3.1.39): Fresnel’s approximation (= paraxial approximation). The approximation conditions from (3.1.34) to (3.1.39) resp. (3.1.40) are explained in [96For, 86Sta, 87Ree].

Fresnel’s approximation: The condition NF(a/d)2/4 1 [91Sal] is valid for sharp-edged apertures A, but it is weakened for the transmission of Gaussian-beam-like ﬁelds [86Sie, p. 635] or Gaussian-like soft apertures. Fresnel’s approximation describes the propagation of the ﬁeld from plane z = 0 to plane z = z. This transformation can be cascaded to describe complex systems and is an often used tool in paraxial propagation of radiation (Sect. 3.1.4.5.2).

x’	Opaque screen
S (x’, y’, 0 )	Opaque screen
S (x’, y’, 0 )		x
		x
	y ’	Normal
		vector n
dx ’dy ’	r0	rSP P (x, y, z )
	z = 0
a		z
		pi
	A

b y

Diffracted field

E (x,y,z)

Fig. 3.1.8. Di raction at an aperture A with source

terms E(x , y , 0) and/or ∂z∂ E(x , y , z) z=0, respectively, and a or b the maximum radial distances

of source S or image point P , respectively. pi symbolizes di erent plane waves for (3.1.41)–(3.1.43).

Landolt-B¨ornstein

New Series VIII/1A1

B¨ornstein-Landolt

VIII/1A1 Series New

Table 3.1.4. Di raction formulae with ﬁelds given near the di raction aperture (rSP : see Fig. 3.1.8).

Integrals

Formula

Restrictions

Ref.

Rayleigh-

Sommerfeld of 1st kind

Rayleigh-

Sommerfeld of 2nd kind

Fresnel-Kirchho

RayleighSommerfeld

1st kind approx.

Fresnel-Kirchho approximation, refers to

Fig. 3.1.8

− 4π A

∂ z

rSP

ERS1(x, y, z) =

E(x

, y , 0)

∂

exp(−i krSP)

d x d y

ERS2(x, y, z) =

∂ E(x , y , z

)

z =0

exp(−ikrSP)

d x d y

− 2π

∂ z

rSP

EFK(x, y, z) =

[ERS1(x, y, z) + ERS2(x, y, z)]

i λ A

rSP

ERS1a(x, y, z) =

E(x , y , 0)

exp(−i krSP)

cos (n, rSP) d x d y

i λ A

rSP

EFKa(x, y, z) =

E(x , y , 0)

exp(−i krSP)

1 + cos (n, rSP)

d x

d y

(3.1.34)	rSP > λ0 ,	[99Bor]
(3.1.34)	plane aperture	[86Sta]
(3.1.35)	rSP > λ0 ,
(3.1.35)	plane aperture
(3.1.36)	rSP > λ0 ,
(3.1.36)	curved aperture
	curved aperture
(3.1.37)	rSP λ0
(3.1.38)	rSP λ0

Fresnel’s

i exp (−ikz)

(x − x )2 + (y

−

y )2

[99Bor]

EFre(x, y, z) =

E(x , y

, 0) exp

i π

d x

d y

(3.1.39) z

[96For]

approximation,

λd

−

λ z

[97For]

refers to

[87Ree]

Fig. 3.1.8	[86Sta]
	[86Sta]

	(continued)

ractionDi 4.1.3

131 .p .[Ref

B¨ornstein-Landolt

VIII/1A1 Series New

Table 3.1.4 continued.

Integrals

Formula

Restrictions

Ref.

Fraunhofer

EFra(x, y, z) =

i exp (−i kz) p

E(x , y , 0) exp

i 2π

xx + yy

d x

d y

(3.1.40)

[99Bor]

far-ﬁeld

λ z

λd

[68Goo]

λz

approximation,

[96For]

refers to

with the additional phase term

[97For]

Fig. 3.1.8

[86Sta]

p =

λ z

for

λz

−

i π x

+ y

otherwise

exp

Plane-wave representation (also: angular-spectrum representation), refers to

Figs. 3.1.8 and 3.1.9

2-D Fourier transform (see remark on (3.1.40)) of the source distribution Es in plane z = 0:

rSP > λ0

[91Sal]

A0(fx, fy ) =

∞ ∞

Es(x , y , 0) exp

i 2 π (fxx

+ fy y

)

d x d y

(3.1.41)

[78Loh]

[97For]

[86Sta]

−∞ −∞

[99Bor]

propagation of plane waves with the spatial frequencies fx and fy along the z-direction by distance z:

exp {−i 2 π (fxx + fy y} exp −i 2 π (fxx + fy y +

1/λ2 − fx2 − fy2 z) ,

(3.1.42)

addition of plane waves at distance z:

−

E(x, y, z) 2

0 x y

−

x − y

+fy

<1/λ

A (f , f ) exp

i 2 π

f x + f y +

f 2 f 2 z

d f

(3.1.43)

1/λ2

equivalent to (3.1.34) [97For]

Far ﬁeld in the focal plane of a lens, refers to

Fig. 3.1.9

EP(x, y) = λf

∞ ∞

ES(x , y ) exp

i 2 π λf

+ λf y dx

i p

−∞ −∞

y2) (d

f )

p = exp i π

−

λ f 2

d, f λ

[91Sal]

(3.1.44)

(3.1.45)

131] .p .Ref

optics Linear 1.3

88			3.1.4 Di raction			[Ref. p. 131
x	Plane wave		x ’	Field ES (x’,y’)	Lens	x
				Plane wave		Convergent wave
						z
	x	= sin	1			Field EP (x,y)
	x	= sin	x
			x	d		f
x = 1/fx			z	d		f
x = 1/fx			z
a			b
Fig. 3.1.9. (a) Spatial frequencies of a plane wave with propagation direction Θx with respect to the
plane x = 0 (and Θy analogously) are fx and fy with Θx = sin−1(λfx) ≈ λfx and Θy = sin−1(λfy ) ≈ λfy
(≈: paraxial approximation). (b) Generation of the far ﬁeld in the focal plane of a lens: The Fourier
transformation (d = f ) is changed by an additional phase term for d = f with d: distance, f : focal length.

(3.1.40): Fraunhofer’s approximation

– Fresnel number :

NF = a2/λz .

(3.1.46)

–Validity of Fraunhofer’s approximation: NF 1 .

p = 1 (parabolic phase): the intensity of di racted light is the square of the modulus of the Fourier transform of E(x, y, 0) only.

– Additional condition with second Fresnel number NF = b2/λz 1 :

E(x, y, z) is the Fourier transform of E(x, y, 0) in dependence on the spatial frequencies fx ≈ (x/z)/λ ≈ Θx/λ and fy ≈ (y/z)/λ ≈ Θy /λ .

– Di erent conventions on the spatial Fourier transform F (fx) of a spatial distribution f (x) :

–The convention of the plane-wave structure exp(i kx − i ω t) is connected with the determination of F (fx) by

∞

F (fx) = d x f (x) e−i 2π fxx

−∞

[68Goo, 68Pap, 78Loh, 78Gas, 93Sto, 05Hod].

– The plane-wave structure exp(i ω t − i kx) can be combined with

∞

F (fx) = d x f (x) ei 2π fxx

−∞

[71Col, 73Men, 92Lug], but

∞

F (fx) = d x f (x) e−i 2π fxx

−∞

is deﬁned also in [88Kle, 91Sal, 95Wil, 96Ped].

Landolt-B¨ornstein

New Series VIII/1A1

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