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Ref. p. 131]				3.1 Linear optics	91
x			x ’
			x ’
	y
a	r	~ sin	r	y ’
	d	~ sin
	d
		d		z
	Opaque screen			z

						Diffraction
			Circular			pattern
			aperture A
										Fig. 3.1.12. Di raction by a circular aperture.
	1.0										1.0
intensity/fieldNormalized	0.8									encircledNormalizedenergy	0.8
											0.8
	0.6	FWHM									0.2
	0.6
				Normalized intensity							0.6				1 st	2 nd		3 rd
	0.4			Normalized intensity											1 st	2 nd		3 rd
	0.4															dark ring
											0.4					dark ring
	0.2					Normalized field					0.4
						Normalized field

	0
0.2											0		0.4		0.8	1.0	1.2		1.4
a	0	0.2	0.4	0.6	0.8	1.0	1.2	1.4	1.6	b	0	0.2	0.4	0.6	0.8	1.0	1.2		1.4	1.6
a					ra/( d )					b					ra/( d )

Fig. 3.1.13. (a) Di racted ﬁeld and intensity. (b) Encircled energy in the di raction plane with a circular screen.

Table 3.1.6. Zeros and maxima of intensity for di raction by a circular aperture.

Number n	rna/(λd)	In/I0
0	0	1
FWHM	2 × 0.257	0.5
1	0.610	0
1	0.817	0.0175
2	1.117	0
2	1.340	0.00415
3	1.619	0
3	1.849	0.00160
4	2.121	0
4	2.355	0.00078

Field distribution:
	π a2		−i k	z +	kr2	2	J [2 π a r/(λ z)]
E(r, z) =		E0 exp					1	(3.1.49)
	i λ z				2z		2 π a r/(λ z)

with E0 the electric-ﬁeld amplitude and r the radius in the far-ﬁeld plane. Intensity:

I(r) = I(0, z) 2	12 π a r/(λ z)	2	(3.1.50)
		.
	J [2 π a r/(λ z)]

Landolt-B¨ornstein

New Series VIII/1A1

92	3.1.4 Di raction	[Ref. p. 131

3.1.4.4.2.1 Applications

Airy’s disc:

r1 Airy = 0.610 λ/ sin σ ,

(3.1.51)

1st-minimum radius of the intensity distribution in the focal plane of an aberration-free lens (Lommel 1885, Debye 1909, [86Sta, 99Bor]): Substitute in (3.1.50) a/z sin σ (numerical aperture = sinus of the intersection angle σ with optical axis in the focal point, generally: image point) and r = r1 Airy as above.

Annular aperture: obscuration of the central part in the circular aperture A of Fig. 3.1.12:

–Reduction of the central di raction maximum width by ≈ 20 %.

–Increase of secondary maximum by factor ≈ 7.

–See Bessel beams, Sect. 3.1.3.2.4, [05Hod].

3.1.4.4.3 Gratings
Grating equation:
sin α + sin β = m	λ	(3.1.52)
sin α + sin β = m	g	(3.1.52)
	g

with

α : angle of incidence (see Fig. 3.1.14), β : di raction angle,

g : grating constant (grating period, groove distance),

m : order of di raction. Convention [82Hut, p. 25] often used: If the di raction order is on the same side with the zero order (m = 0) as the grating normal: m > 0, otherwise m < 0. In Fig. 3.1.14, the directions of the +1st transmitted order and the grating normal (dashed and dotted lines) are on the same side of the 0th transmitted order. Therefore m = 1 > 0 .

Slit factor : represents the di raction by a single slit of the grating. Its form regulates the energy distribution between the di erent orders m [82Hut, 99Bor]. For the real phase and reﬂection gratings, it is substituted by the di raction e ciency curves in dependence on α or λ. There is an extreme diversity of cases. Catalogs of such curves: see [80Pet, 97Loe].

Theoretical spectral resolution of a grating:

Rtheor = λ/(∆ λ) = m N = W (sin α + sin β)/λ

(3.1.53)

1 st					Focused
1 st					orders
0 th					orders
0 th
1 st			1		1 st
		g			1 st

Reflected			Transmitted		0 th
Reflected			Transmitted		0 th
	orders		orders		1 st
1 st					1 st
0 th					Slit
1 st					factor
	Incident			f	Subsidiary
	Incident			f	maximum
	plane wave				maximum
	plane wave
	Grating		Lens		Focal
					plane

Fig. 3.1.14. Reﬂected and transmitted orders of

a grating, here with N = 4 slits. The far-ﬁeld distribution is visualized after focusing by an ideal

lens. Between the main maxima occur N − 2 sub-

sidiary maxima. The dashed envelope is the slit factor.

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 131]	3.1 Linear optics	93

with

N : number of grooves of the grating, W : width of the grating,

α, β : see (3.1.52).

Real resolution contains theoretical resolution and the aberrations of the optical elements for collimation and focusing of the grating-di racted plane waves or by the aberrations of the concave gratings with imaging properties. [87Chr, 82Hut].

Holographical gratings [82Hut] show lower disturbations than mechanically produced gratings (application: external laser resonators).

Blazed gratings di ract light into an order m wanted with more than 60–90 % over one octave of wavelengths [80Pet, 82Hut, 97Loe].

Volume gratings: [81Sol, 81Rus].

Mountings of spectral devices: [82Hut].

3.1.4.5 Fresnel’s di raction ﬁgures

Fresnel’s approximation is given in (3.1.39) in Table 3.1.4.

3.1.4.5.1 Fresnel’s di raction on a slit

In Fig. 3.1.15 Fresnel’s di raction pattern of a slit with width 2a is shown.

NF = 0.5

NF = 3.5

NF = 1.0

NF = 4.0

NF = 1.5

NF = 4.5

NF = 2.0

NF = 5.0

NF = 2.5

NF = 10

NF = 3.0

NF = 20

Fig. 3.1.15. Fresnel’s di raction pattern of a slit with width 2a

(see Fig. 3.1.10 with b ∞).

Fresnel’s number NF = a2/(λz) is the essential parameter to characterize the transition from farﬁeld (Fraunhofer) approximation (NF < 0.2 . . . 0.5) to near-ﬁeld (Fresnel) approximation (NF > 0.5). NF = 0.5 : one central maxi-

mum only, NF = 3 : three maxima, NF = N : N maxima. Hard-edge

di raction results in a ripple in the

near ﬁeld, which can be avoided by soft apertures, for instance

Gaussian-like [86Sie] (apodization in optics [99Bor]). Figure after [86Sie, p. 721].

Landolt-B¨ornstein

New Series VIII/1A1

94	3.1.4 Di raction	[Ref. p. 131

3.1.4.5.2 Fresnel’s di raction through lens systems (paraxial di raction)

Given: a system of lenses and the ﬁeld distribution E(x, y) to be propagated.

The sequence of steps easily taken is:

–

Given: E(x, y) in the plane z = 0. Required: the ﬁeld in the plane z = z. Solution: (3.1.39).

–

Given: E(x, y) in the plane z = 0 and near to this plane a lens. Required: the ﬁeld in the plane

z = z. Solution: modiﬁcation of (3.1.39) by an additional factor L(x , y ) to:

Fre

(x, y, z) =

i exp {−i kz}

E(x , y , 0) L(x , y )

λd

× exp −i π

−

)

2 + (y

−

d x d y ,

(3.1.54)

λ z

2 + y

L(x , y ) = p(x , y ) exp {−i kn tL} exp

i k(x

(3.1.55)

with

n : refractive index of the lens, tL : thickness of the lens,

f : focal length of the lens,

p(x , y ) : amplitude part, which can describe a marginal aperture or a Gaussian apodization.

A general complex function L(x , y ) can model di ractive optical elements.

Cases of integration:

–No transversal limitations (without stops) and quadratic arguments of the exponential functions due to analytical results. The Collins integral is the closed form of such a calculation (see Sect. 3.1.7.4).

–One stop (ﬁnite integration limits): The result includes the error function [70Abr].

–Two and more ﬁnite integration limits are not useful. Then, (commercial) numerical ﬁeld propagation programs through systems should be consulted.

Examples: [68Goo, 91Sal, 71Col, 85Iiz, 92Lug, 68Pap].

The Beam Propagation Method (BPM) in integrated optics (many “inﬁnitely thin lenses”) is the generalization of this method [95Mae, 91Spl, 99Lau, 98Hec].

3.1.4.6 Fourier optics and di ractive optics

Fourier optics results from the transformation of the temporal frequency methods of electrical engineering to spatial frequency methods in optics, see Figs. 3.1.9, 3.1.10 and (3.1.41), (3.1.43), (3.1.44).

References: principles of Fourier optics: [68Goo, 78Loh, 83Ste, 85Iiz, 89Ars, 93Sto, 98Hec, 99Lau], ﬁltering: [92Lug], ﬁltering in connection with holography: [96Har, 71Col], noise suppression: [91Wyr].

Example 3.1.5. Spatial spectral ﬁltering

In Fig. 3.1.16 low-pass ﬁltering of a laser beam with a four-f -setup is shown.

Landolt-B¨ornstein

New Series VIII/1A1