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

Внимание! Если размещение файла нарушает Ваши авторские права, то обязательно сообщите нам

Ref. p. 131] 3.1 Linear optics 119

Table 3.1.16. Third-order spherical aberration and coma for a thin plano-convex lens [76Jen, p. 152], [88Kle, p. 185], [87Nau, p. 109] in comparison with the di raction-limited resolution for a plane wave or Gaussian illumination.

Figures

Formulae

Lens equation (3.1.95) with t = 0 ,

a −∞ , r2 −∞ , f = f ,

which is modiﬁed outside Sect. 3.1.6.1:

(3.1.101)

s ’t

= (n − 1)

s ’tc

∆sl

−

n3 − 2 n2 + 2

2 ,

(3.1.102)

s’lc

2 n (n − 1)2

s’l

(3.1.103)

∆st

= ∆sl

Fig. 3.1.37. Spherical aberration at a plano-convex

plane of least confusion [87Nau, 99Pau, 99Bor],

[01I , p. 214]:

lens. IP: paraxial image plane, LC: least confusion.

∆slc

≈ 0.8 ∆sl ,

(3.1.104)

∆stc

≈ 0.4 ∆st .

(3.1.105)

Gaussian weights of the illumination change the geo-

metric optical position of least confusion [01Mah].

x = θ f ,

(3.1.106)

		a	a =		n2 − n − 1 h
h	x ’	2a	a =	−	n2 − n − 1 h
	x ’		f		2n (n	−	1)	f
			f		2n (n		1)	f
			with
		z	θ : angle of incidence.
			θ : angle of incidence.
f	IP
f

θ(3.1.107)

Fig. 3.1.38. Coma at a plano-convex lens.

nmed

h
	s ’	s ’
	s ’	tb
	tg
f	f

a b

Fig. 3.1.39. Di raction-limited resolution for (a) a Gaussian beam with waist h (1/e2-intensity level) in the object-side focal plane, (b) a plane wave at circular stop with radius h.

∆stg	=		λ		(3.1.108)
				,

		π nmed(h/f )
∆stb	= 0.61		λ
			nmed sin σ

	≈ 0.61		λ		(3.1.109)

			nmed (h/f )

with

λ : wavelength [m], h : zonal height [m], f : focal length [m],

nmed : refractive index of the image space.

Landolt-B¨ornstein

New Series VIII/1A1

120	3.1.7 Beam propagation in optical systems	[Ref. p. 131

3.1.7 Beam propagation in optical systems

Paraxial propagation of light in a system given by its ABCD-matrix can be calculated

–for (coherent) Gaussian beams by q-parameter propagation (Sect. 3.1.7.2),

–for general ﬁeld distributions by Collins integral (Sect. 3.1.7.4),

–for second-order moments of the electric ﬁeld by propagation of the Wigner distribution in Chap. 2.2 (beam characterization).

3.1.7.1 Beam classiﬁcation

In Table 3.1.17 various types of beams are listed.

3.1.7.2Gaussian beam: complex q-parameter and its ABCD-transformation

3.1.7.2.1 Stigmatic and simple astigmatic beams

3.1.7.2.1.1 Fundamental Mode

–Stigmatic beam and rotational-symmetric system:

both longitudinal cross sections are treated equally,

–Simple astigmatic beam and elements with a symmetry plane:

two di erent sets of ABCD-matrices for the tangential and sagittal cut (see Table 3.1.12).

The introduction of the complex q-parameter [66Kog1, 66Kog2]

1	=	1		−	i λ
qx(z)	=	Rx(z)		−	π wx(z)2

formalizes the x-part of the fundamental-mode equation (3.1.31)

−

kx2

(x, z) =

w0x

exp

− i

wx(z)

wx(z)2

2Rx(z)

to the simple complex shape

(x, z) =

exp

−

kx2

1 + i z0

qx(z)

(3.1.110)

(3.1.111)

(3.1.112)

In Fig. 3.1.40 the transfer of a ﬁeld distribution by an optical system given by its ABCD-matrix is shown. In Table 3.1.18 the q-parameter transfer for stigmatic and simple astigmatic beams is given.

Landolt-B¨ornstein

New Series VIII/1A1

B¨ornstein-Landolt

VIII/1A1 Series New

Table 3.1.17. Types of beams.

Beam	Generated by	Beam type is characterized Examples	References with practical
[69Arn, 05Hod]		by the shape of the matrix S	example calculations
		(3.1.99)

Stigmatic	x		Fundamental-mode laser
	x
	θx
	w0x
	w0y	θy	z
y
Simple astigmatic			Semiconductor lasers
	x		or:
			or:
			Anamorphic optical system
z2	z1	z	(f.e. cylindrical lens) in com-
z2	z1		bination	with	a	stigmatic
y			bination	with	a	stigmatic
y			beam
			beam
General astigmatic			General	rotation		of	an
			anamorphic		optics		in
			relation	with	a	simple
			astigmatic beam

Axx = Ayy ; Axy = Ayx = 0 ,

and the same for B, C, D

Axx = Ayy ; Ayx = Axy = 0

(no mixing of both orthogonal planes),

and the analog for B, C, D

TE00-mode handling in laser	[75Ger, 86Sie, 91Sal],
applications	[96Yar, 01I , 05Hod],
	see Sect. 3.1.7.2.1

–	ring lasers,	[05Hod, 99Gao, 86Sie],
–	lasers, including dispersive	see Sect. 3.1.7.2.1
	elements (dye-lasers),

–tolerance calculations for resonators and beamguiding optics

General case	Transformation of higher-	[05Hod, 99Gao],
	order radiation modes	see Sect. 3.1.7.2.2

131] .p .Ref

optics Linear 1.3

121

122				3.1.7 Beam propagation in optical systems		[Ref. p. 131


		Optical system
Field	characterized by its				Field
Field					Field
u I		A B	- matrix		uO
		C D
					Fig. 3.1.40. Transfer of a ﬁeld distribution by an optical system
Input plane
Input plane				Output plane given by its ABCD-matrix.
Table 3.1.18. q-parameter transfer for stigmatic and simple astigmatic beams.

Given					Propagated ﬁeld

– Gaussian beam in the input plane:

u I(x, z) = exp −i kx2 . (3.1.113) 2q I x

–ABCD-matrix of the optical system (see Tables 3.1.11 and 3.1.12).

–Starting point:

R I = 1/Re (1/q I) ,

w I = 1/ −π Im (1/q I)/λ .

– Transformation of the q-parameter:

q O x =

Aq I x + B

(3.1.114)

Cq I x + D

– Field in the output plane:

u O(x, z) = exp −i

kx2

(3.1.115)

2q O x

with the real parameters of the output beam [96Gro]:

– beam radius:

w O x = w I x

(3.1.116)

π w2

A +

R I x

I x

λB

– curvature radius of the wavefront:

R O x =

A + R I x

π w2I x

λB

A + R I x

C + R I x + D

π w2I x

λB

(3.1.117)

Example 3.1.15. Given: the waist of a Gaussian beam

u I = exp −	x2	= exp −i	kx2
	w02		2q1

with q1 = i z0 in comparison with (3.1.110).

Asked: free-space propagation along the distance z with the ABCD-matrix 1 z 0 1

Solution:

q2 = Aq1 + B = z + i z0

Cq1 + D

and

U O = A +

−1/2

exp −i

kx2

= 1 +

−1/2

exp −i

kx2

2q2

i z0

2(z + i z0)

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 131]	3.1 Linear optics	123

3.1.7.2.1.2 Higher-order Hermite-Gaussian beams in simple astigmatic beams

Treatment of the x- or y-component of Hermite-Gaussian-beams after (3.1.27): The complex q-

parameter transformation is treated as above, the fundamental mode part is given as above, the

√

new beam radius for the Hermite polynom of order m, Hm( 2 x/w Ix) is calculated from the new q-parameter and the phase is derived from it, too [70Col].

For complex Hermite-Gaussian beams: see [86Sie].

3.1.7.2.2 General astigmatic beam

In Table 3.1.19 the Q−1-matrix transfer for general astigmatic beams is given. The matrix Q−1 is the matrix scheme of inverses of q-parameters and no inverted matrix [96Gro].

Table 3.1.19. Q−1-matrix transfer for general astigmatic beams.

Given	Propagated ﬁeld

– General Gaussian beam in the input plane:

	k		1 r
U I(r) = exp −i		r Q−I		, (3.1.118)
	2

r (x, y) the transverse position vector

perpendicular to the propagation axis z .

– Q−I 1-matrix:

			1	1
Q−1	=		qxx		qxy	(3.1.119)
I			1	1
		qxy			qyy

with qxx, qxy , qyy complex terms describing the general amplitudeand phasedistribution of U I , and

r Q−1 r =	x2	+ 2	xy	+	y2	. (3.1.120)

I	qxx	qxy		qyy

–S-matrix of the optical system (see Table 3.1.15) with

A B

S = C D

– Transformation of the Q−I 1-matrix to its output value:

Q−O1 = C + D Q−I	1 A + B Q−I 1 −1 ,			(3.1.121)
see [88Sim, 96Gro, 05Hod].
– Field in the output plane:

		k
U O(r) = exp −i		2	r Q−O1 r .	(3.1.122)

after (3.1.99).

Example 3.1.16. Transformation of a simple astigmatic Gaussian beam (no mixing between x and y)

	!	1	qyy "
I		0
with a θ-rotated cylindrical lens to a general astigmatic beam: We start with Q−1 =		qxx	0
				1 .

The rotation of an x-aligned cylindrical lens, given as Scyl in Table 3.1.15, is performed by multiplying ﬁrst Scyl with the rotation matrix R of Table 3.1.15, and then the product with the inverse of R is:

Landolt-B¨ornstein

New Series VIII/1A1

Источник: https://files.student-it.ru/previewfile/280942

Смотрите также:




_1966(~1
_files_materials_5053_prog.ntr
_Антонян Ю.М., Сексуальные преступления. Чикатило и другие
_РГР № 1 (2)
- Габариты и масса 27
!Фармакология препаратов для терапии заболеваний дыхательных путей
'Образ–Я' и его развитие средствами диалога