112

3.1.6 Geometrical optics

[Ref. p. 131

Element 1

Element 2

Element 3

Following

Lens

Air distance

System

elements

1

x1

2

x2

x

3

x4

4

x

n

M 1

3

n

M2

M3

Output plane

Input plane

Output plane

of element 1

= input plane

M = Mn 1... M3 M2 M1

of element 2

Fig. 3.1.34. Concatenation of di erent ray-transfer matrices for di erent types of sub-systems. Matrices known for systems before can be used to construct the matrix for a larger system containing the known systems. The sequence of the matrices is shown at the bottom of the figure.

3.1.6.2.1 Simple interfaces and optical elements with rotational symmetry

In Table 3.1.11 ABCD-matrices for simple interfaces and optical elements with rotational symmetry are listed.

3.1.6.2.2 Non-symmetrical optical systems

Rotational symmetry lacks and the axis is tilted due to the non-symmetrical optical system. In such a system, the central ray of imaging is called the basic ray. The optics in a narrow region around the basic ray is called parabasal optics [95Bas, Vol. 1, p. 1.47] as analogon to paraxial optics. For treatment of astigmatic pencils see [72Sta].

A special case of the non-symmetrical optical system is a system without torsion: Two orthogonal cases do not mix during propagation. Examples are di erent setups of spectroscopy and laser physics (ring resonators).

In Table 3.1.12 ABCD-matrices for non-symmetrical optical elements without torsion are listed.

3.1.6.2.3 Properties of a system

Properties of a system included in its ABCD-matrix are discussed in [75Ger, 96Ped, 05Hod, 05Gro1]. In Table 3.1.13 distances between cardinal elements of an optical system are listed, in Table 3.1.14 the meaning of the vanishing of di erent elements of the ABCD-matrix is depicted.

3.1.6.2.4 General parabolic systems without rotational symmetry

The generalization of the two-dimensional ray transfer after Fig. 3.1.33 to three dimensions [69Arn] is shown in Fig. 3.1.35. The ray in the input plane is characterized by two coordinates x1 and y1 of the piercing point P and two small (paraxial range) angles α1 and β1 .

The matrix S relates these parameters to the corresponding parameters in the output plane like in Fig. 3.1.33:

Landolt-B¨ornstein

New Series VIII/1A1

H, H : principal planes.

Unfolding of the mirror;

sign(r) > 0 , if the incident light sees a concave mirror surface.

Ref. p. 131]

3.1 Linear optics

113

Table 3.1.11. ABCD-matrices for simple interfaces and optical elements with rotational symmetry.

E ect

Figure

ABCD-matrix

Remark

Propagation

1 d

The rays propagate from I to O

d

0 1

within the same medium.

l

O

Spherical

1

0

Sign: r > 0 for convex surface seen

surface

r

n1 − n2

n1

by the propagating light.

n

n

1

2

n2 r

n2

I O

Plane		1	0
		0	n1
n1	n2		n2
I	O

Corresponds to a spherical surface with r ∞ .

Planar plate			0	n1		Contains two refractions.
	d			1
			1	n2 d
n1	n2	n1

IO

Thin lens		r1
		r2
	I	O
	n1	n2	n1
Thick lens
in air	r1>0	r2<0
	t
	n
	sH	s’H ’

IH H’ O

Spherical	r
mirror		substituted
		by
	I =O	I O

	Gradient-		n
	index lens	n1
			n1
	or
	thermal		t
	lens	I
			O

1

0

1

=

n2 − n1

1

1

,

1

1

f

n1

r1

− r2

−

f

air: n1 = 1 .

1

sH

d

1

= (n

1)

1

1

+

(n − 1)2 t

,

nsH

−

1f

f

−

r1 − r2

n r1 r2

−

1 +

(n

1) f t

f

f

sH =

−

−

, see (3.1.96) ,

n r2

s

H

=

−

(n − 1) f t

, see (3.1.97) ,

n r1

1 0 − 2r 1

C D

1

2γ t / n0 2γ ;

A B

A = cos

√2γ t ;

√

2

−

1

0

B = n

sin

√

−

D =

2γ t ;

√2γ t

C =

√2γ n /n

sin

;

n = n0 (1 γ x ) ;

cos

√

γ > 0 : higher

development

of

the

trigonometric

index on axis

functions for

√

t 1 simpli-

2γ

fications

Gradient optics:

see [02Gom, 05Gro1].

(continued)

Landolt-B¨ornstein

New Series VIII/1A1

114

3.1.6 Geometrical optics

[Ref. p. 131

Table 3.1.11 continued.

E ect

Figure

ABCD-matrix

Remark

Gaussian

1

0

The amplitude transmission function

apodization,

i λ a

1

between I2 and O is

usable for

−

2 π

exp −a x /2 ,

q-parameter

λ : wavelength

x:

transverse coordinate

transfer

I O

of light

[86Sie, p. 787]

(Table 3.1.18)

Remark : Other treatments of the mirror see [86Sie, 98Sve, 75Ger].

Table 3.1.12. ABCD-matrices for non-symmetrical optical elements without torsion.

E ect

Figure

ABCD-matrix

Remark

Refraction

2

cos (θ1)

0

n1 sin (θ1) = n2 sin (θ2)

at a sphere

cos (θ2)

Tangential

1

∆ nt

n1 cos (θ2)

(Snell’s law)

r n2

n2 cos (θ1)

n2

(meridional)

n

1

∆ nt =

n2 cos (θ2) − n1 cos (θ1)

plane

cos (θ1) cos (θ2)

Sagittal
plane
n1	n2

1		0		∆ ns = n2 cos (θ2) − n1 cos (θ1)
	∆ ns		n1
	r n2		n2

	Rowland
	concave	2
	grating
		1
	(unfolded)
	Tangential	Radius of curvature r
	(meridional)
	plane

Sagittal plane

A B

,

C D

A = cos (θ1) ; cos (θ2)

B = 0 ;

C = − 2 cos (θ2) ; r t cos (θ1)

D = A .

0
12	1
− rs

Grating equation (3.1.52):

λ

sin (θ1) + sin (θ2) = m g ,

2 r cos2 (θ2)

r t = cos (θ1) + cos (θ2)

2 r

rs = cos (θ1) + cos (θ2) ,

general corrected holographical gratings: see [81Gue]

Spherical	Specialization of the
concave	Rowland grating to
mirror	g ∞ ,
	θ1 = θ2 .

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 131]	3.1 Linear optics	115

Table 3.1.13. Distances between cardinal elements of an optical system: F , F : objectand image-space focal points, respectively; H, H : objectand image-space principal points, respectively; I, O: input and output plane, respectively. The order of points determines the signs.

	Distance between					A, B, C, and D
	two points					for n1 = n2
							D
	I F
							C
						1
	F H					−
							C
							A
	O F					−
							C
							1
	H F
						− C

Table 3.1.14. The meaning of the vanishing of di erent elements of the ABCD-matrix.

Element	Figure								Remark
A = 0				0	B				x2 = B α1
				C D					Focusing of collimated light into the
	I						O		image-side focal plane.
B = 0				A 0					x2 = A x1
				C D					The input plane is imaged to the
							O		output plane (conjugated planes).
	I								A : magnification of imaging;
									appl.: calculation of image plane.
C = 0				A B					α2 = D α1
				0 D					Transformation of collimated light
	I						O		into collimated light.
									D : angular magnification;
									telescope (afocal system).
D = 0				A B					α2 = C x1
				C	0				Collimation of divergent pencil of
	I						O		rays.
									C : power of the element or system.

x2

Axx Axy Bxx Bxy

x1

y2

=

Ayx Ayy Byx Byy

y1

or

α

C

C

D

D

α

β2

Cxx

Cxy

Dxx

Dxy

β1

2

yx

yy

yx

yy

1

γ2		=	C D	γ1		= S	γ1		(3.1.99)
r2			A B	r1			r1

with the matrices A, B, C, D, and S given by comparison with the more detailed representations. Identities between the matrices, characteristic for the symplectic geometry (see Sect. 3.1.6.2.6),

are: A DT − B CT = I ; A BT = B AT ; C DT = D CT , and det		A B		=	n
	C D				n , where T means the

Landolt-B¨ornstein

New Series VIII/1A1