Propagation of Gaussian beam in optical system with aberrations

Optik 114, No. 10 (2003) 437–440 Antonin Miks, Jiri Novak, Propagation of Gaussian beam in optical system with aberrations http://www.elsevier-deutschland.de/ijleo

437

International Journal for Light and Electron Optics

Propagation of Gaussian beam in optical system with aberrations Antonin Miks, Jiri Novak Department of Physics, Faculty of Civil Engineering, Czech Technical University in Prague, Tha´kurova 7, 166 29 Prague 6

Abstract: Our work describes a theory of propagation of axial and off-axis Gaussian beams in optical systems with aberrations. Furthermore, we obtained results for acceptable tolerances of aberrations of the optical system and a position of the optimal image plane. Key words: Optical systems – Gaussian beams – diffraction – aberration – optical tolerances

1. Introduction Lasers are used widely in various parts of science and engineering. In relation to an optical system, which transforms a laser beam, and appropriate detection and evaluation systems, one can design the optical instruments, which have higher technical parameters than other systems with classical light sources. The main goal of this work is to show an influence of aberrations of the optical system on axial and narrow offaxis Gaussian beams, which are transformed by the optical system, and to determine the conditions for an optimal compensation of aberrations of such systems. Several cases of the diffraction of Gaussian beams and the influence of aberrations on Gaussian beam propagation are described in [4–7].

2. Aberrations of optical system The ideal optical system transforms a spherical wavefront in an object space into a spherical wave-front in an image space. In the case of a real optical system, the wave-front exiting the optical system is not a spherical surface, but some general surface. The deviation of this real wave-front from the ideal spherical wave-front, measured in the normal direction to the wave-front, is called the wave aberration. The ray aberration is the deviation of the intersection of the ray with the image

plane from the ideal image point. The wave aberration Wðx; yÞ and ray aberrations dx, dy are related by the following equations [2, 3] dx ¼

R @Wðx; yÞ
; n @x

dy ¼

R @Wðx; yÞ
; n @y

ð1Þ

where R is the radius of the ideal spherical wave-front (reference wave-front), ðx; yÞ are the coordinates of the intersection of the ray with the ideal wave-front, and n is the index of refraction. In the case of axial point imaging, i.e. imaging with aperture rays, the wave aberration of the rotationally symmetric optical systems up to fifth order has the form [3] Wðx; yÞ ¼ W2 ðx2 þ y2 Þ þ W4 ðx2 þ y2 Þ2 þ W6 ðx2 þ y2 Þ3 ;

ð2Þ

where W2 is the coefficient of defocusing, W4 is the coefficient of the third order spherical aberration, and W6 is the coefficient of the fifth order spherical aberration.

3. Diffraction of the Gaussian beam The Gaussian beam is a beam, whose amplitude is given by [1, 3]
2  x
þ y
2 ; ð3Þ Uð
x; y
Þ ¼ exp  a2 where ð
x; y
Þ are the coordinates with respect to a centre of the beam, and a is a constant called usually the Gaussian beam radius. Consider now that ðx; yÞ are the coordinates in the exit pupil plane, and ðx0 ; y0 Þ are the coordinates of the centre of the beam. Then it holds x ¼ x
þ x0 ;

y ¼ y
þ y0 ;

ð4Þ

and x
2 þ y
2 ¼ ðx2 þ y2 Þ þ ðx20 þ y20 Þ  2ðxx0 þ yy0 Þ : ð5Þ

Received 3 July 2003; accepted 10 October 2003. Correspondence to: A. Miks Fax: ++420-233333226 E-mail: [email protected]

We can also use the polar coordinates x ¼ r sin j ; x0 ¼ r sin j0 ;

y ¼ r cos j ; y0 ¼ r cos j0 ; 0030-4026/03/114/10-437 $ 15.00/0

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Antonin Miks, Jiri Novak, Propagation of Gaussian beam in optical system with aberrations

and obtain 2

2

2

x
þ y
¼ r þ

r02

 2rr0 cos ðj  j0 Þ :

ð6Þ

For the simplicity, we limit our considerations to the case of Fraunhofer approximation of the diffraction integral. Then, we can calculate the field UðPÞ in some investigated point Pðx1 ; y1 ; z1 Þ of the area, which is closed by some surface S, on the basis of knowing the field UðMÞ on the surface S. Consider a spherical surface with the radius R, whose centre is situated in the x0 ; y
0 ; z
0 Þ in the image plane, and the point point P0 ð
M(x, y, z) lies on the surface S. We can write for the field in the investigated point Pðx1 ; y1 ; z1 Þ ðð i exp ðikRÞ UðMÞ exp ½ik0 WðMÞ UðPÞ ¼  l R S   nðx  x
0 Þ u þ nðy  y
0 Þ v  exp ik0 dS ; R (7) where we denoted u ¼ x1  x
0 , v ¼ y1  y
0 , and k ¼ k0 n is the wave number. We obtain using (3), (5), (6), and (7) ðð (8a) Uðu; vÞ ¼ K exp ðA þ B þ CÞ dS ; S

where partial derivatives are calculated at point ðx0 ; y0 Þ. Using (1), we obtain @W dx ¼n ; @x R

C ¼ ik0 n

ðx  x
0 Þ u þ ðy  y
0 Þ v : R

Using the polar coordinates, we obtain a similar equation ÐÐ Uðr; wÞ ¼ K exp ðA þ B þ CÞ r dr dj ; (8b) S 2

 2rr0 cos ðj  j0 Þ ; a2 B ¼ ik0 Wðr; jÞ ; rr cos ðj  wÞ ; C ¼ ik0 n R where ðr; wÞ are the polar coordinates in the image plane. Equations (8) describe generally the diffraction of the Gaussian beam, which propagates through the optical system with arbitrary aberrations. Now, we will focus on several special and important cases for practice, namely the case of narrow off-axis beam and the narrow and aperture axial beam. A¼

r þ

r02

3.1. Narrow off-axis beam Consider that the aberrations of the optical system are small, then it holds @W @W x
þ y
; ð9Þ Wðx; yÞ ¼ Wðx0 ; y0 Þ þ @x @y

ð10Þ

By substituting previous relations into (8), we can write after a short modification ÐÐ exp ðFÞ dS ; ð11Þ Uðu; vÞ ¼ K1 S

where ðx  x0 Þ2 þ ðy  y0 Þ2 a2 n þ ik0 ½ðx  x0 Þ ðdx  uÞ þ ðy  y0 Þ ðdy  vÞ : R

F¼

3.2. Narrow axial beam In this case, we obtain the same equation as in the previous case, only x0 ¼ 0 and y0 ¼ 0.

3.3. Aperture axial beam From (8) we obtain for n ¼ 1 Uðr; wÞ ¼ K

where x2 þ y2 x20 þ y20 2ðxx0 þ yy0 Þ A¼  þ ; a2 a2 a2 B ¼ ik0 Wðx; yÞ ;

@W dy ¼n : @y R

2p ð ð1


2 r exp  2 exp ½ikWðr; jÞ a

0 0

 exp½ikrr cos ðj  wÞ r dr dj ; where r is the normalized radial distance, r is the radial distance in the image plane expressed as a multiple of 2c (c is the f-number of the optical system), and a is the Gaussian beam radius. The preceding equation can be modified after the integration with respect to j to the form
2 ð1 r UðrÞ ¼ C exp  2 exp ½ikWðrÞ J0 ðrrÞ r dr : a 0

If we use the following equation 1 ð1Þs x2s P
; J0 ðxÞ ¼ 2 2 s¼0 ðs!Þ we obtain UðrÞ ¼ C



1 ð1Þs r2s P
2 2 s¼0 ðs!Þ

ð1


2 r exp  2 exp ½ikWðrÞ rð2sþ1Þ dr : a

0

Assume now that the aberrations of the optical system are small. Then the function exp ½ikW can be expanded into a series exp ðikWÞ ¼ 1 þ ikðW2 r2 þ W4 r4 þ W6 r6 Þ k2 ðW22 r4 þ W42 r8 þ W62 r12 Þ 2  k2 ðW2 W4 r6 þ W2 W6 r8 þ W4 W6 r10 Þ ;



Antonin Miks, Jiri Novak, Propagation of Gaussian beam in optical system with aberrations

where we restricted only to the first two terms of the series. By substitution into the previous equation, we can write UðrÞ ¼ C

1 P s¼0

s

ð1Þ




2

ðs!Þ

r2s 2

½Uða; s; 0Þ þ ikUða; s; 1Þ W2

k2 k2 Uða; s; 2Þ W22  Uða; s; 4Þ W42  2 2 k2 Uða; s; 6Þ W62  k2 Uða; s; 3Þ W2 W4  2  k2 Uða; s; 5Þ W4 W6  k2 Uða; s; 4Þ W2 W6  ; (12) where Uða; s; nÞ ¼ exp




r2  2 a



where the coefficients A are given by the following formulas A2 ¼ ðU1 =U0 Þ2  U2 =U0 ; A4 ¼ ðU2 =U0 Þ2  U4 =U0 ; A6 ¼ ðU3 =U0 Þ2  U6 =U0 ;

þ ikUða; s; 2Þ W4 þ ikUða; s; 3ÞW6

ð1

439

A24 ¼ ðU1 U2 =U02  U3 =U0 Þ2 ; A46 ¼ ðU2 U3 =U02  U5 =U0 Þ2 ; A26 ¼ ðU1 U3 =U02  U4 =U0 Þ2 : Equation (14) gives us the so called Strehl definition for the optical system with the spherical aberration up to the fifth order. We will now deal with an influence of particular aberration of the optical system on the Strehl definition. a) Defocusing ðW4 ¼ W6 ¼ 0Þ

r2ðsþnÞþ1 dr :

I ¼ 1  k2 B2 W22 ;

0

The preceding integral can be calculated either from the following relation   @ sþn 1 exp ðaÞ 1 sþn ; a¼ 2 ; Uða; s; nÞ ¼ ð1Þ
sþn
@a 2a a

ð15Þ

where B2 ¼ A2. b) Third order spherical aberration ðW6 ¼ 0Þ I ¼ 1  k2 B4 W42 ;

ð16Þ

where or from the recurrent formula Uða; s; nÞ ¼

W2 ¼ 

sþn exp ðaÞ Uða; s; n  1Þ  ; a 2a

1  exp ðaÞ Uða; 0; 0Þ ¼ : 2a Consider now a special case r ¼ 0. We obtain for the amplitude at the axis
Uð0Þ ¼ C U0 þ ikU1 W2 þ ikU2 W4 þ ikU3 W6 

2

2

k k k U2 W22  U4 W42  U6 W62 2 2 2 2

2

2

A224 : 4A2

W2 ¼

A24 A46  2A4 A26 W6 ¼ C1 W6 ; 4A2 A4  A224

ð17Þ

W4 ¼

A24 A26  2A2 A46 W6 ¼ C2 W6 ; 4A2 A4  A224

ð18Þ

I ¼ 1  k2 B6 W62 ;

ð19Þ

where B6 ¼ A2 C12 þ A24 C22 þ A6 þ A24 C1 C2 þ A26 C1 þ A46 C2 :



 k U3 W2 W4  k U4 W2 W6  k U5 W4 W6 ; (13) where the coefficients Un ¼ Uða; s; nÞ can be calculated from previous equations for different values a.

4. Optimal aberration compensation and aberration tolerances for axial Gaussian beam We can easily determine the intensity from (13) I ¼ UU * ¼ 1  k2 ðA2 W22 þ A4 W42 þ A6 W62 þ A24 W2 W4 þ A26 W2 W6 þ A46 W4 W6 Þ ;

B4 ¼ A4 

c) Fifth order spherical aberration

where

2

A24 W4 ; 2A2

ð14Þ

Equations (15), (16) and (19) describe the Strehl definition in the optimal image plane. Now, we will focus on the conditions for an optimal compensation of the spherical aberration of the optical system, and we will determine a position of the optimal image plane. The aberration coefficients can be expressed as [3] W2 ¼ 

s0 ; 8c2

W6 ¼ 

dsk ; 24ð1  q0 Þ c2

W4 ¼ 

3 W6 q0 ; 2 ð20Þ

where s0 is the defocusing, dsk is the spherical aberration for the maximal radius rk of the pupil of the optical system, c is the f-number, r0 is the radius for which

440

Antonin Miks, Jiri Novak, Propagation of Gaussian beam in optical system with aberrations

the spherical aberration is zero and q0 ¼ ðr0 =rk Þ2 . The wave aberration is then given by [3] s0 dsk 2q  3q0 2 q ; W ¼ 2 qþ 8c 48c2 1  q0 where q ¼ ðr=rk Þ2. According to (17) and (18), we obtain from (20) s0 ¼ C1

dsk ; 3ð1  q0 Þ

q0 ¼ 

2 C2 : 3

ð21Þ

Substituting q0 into the formula for s0, we can write s0 ¼ C1

dsk : 3 þ 2C2

ð22Þ

We can see from formulas (21) that both the position of the optimal image plane and the correction area depend on the Gaussian beam radius a. Let dI is an allowed change of the Strehl definition, then it holds for the tolerance of the spherical aberration sffiffiffiffiffiffi dI 12 ð1  q0 Þ c2 ð23Þ dsk  B6 p or similarly sffiffiffiffiffiffi dI ð3 þ 2C2 Þ c2 : dsk  1; 3 B6

ð24Þ

Equation (24) describes the tolerance condition for an acceptable size of spherical aberration of the optical system. As we can see it depends on the parameter a of the Gaussian beam.

5. Conclusion If we summarize the preceding results of our paper, we can conclude that aberrations of the optical system affect considerably the Gaussian beam that propagates through such optical system. In a general case, the beam is no longer the Gaussian beam. Its maximum is spatially shifted. The compensation of aberrations of the optical system, which transforms the Gaussian beam, is different from the compensation of aberrations of the classical optical system, and it can be also observed a distinct dependence on the Gaussian beam radius. Analogical results were obtained for acceptable tolerances for aberrations of the optical system and the position of the optimal image plane. Acknowledgement. This work has been supported by grant GACˇR 202/02/0314 and 103/03/P001.

References [1] Saleh B, Teich M: Fundamentals of photonics. J. Willey and Sons, New York 1991 [2] Born M, Wolf E: Principles of optics. Pergamon Press, Oxford 1968 [3] Miks A: Applied optics 10. Czech Technical University Press, Prague 2000 [4] Yoshida A, Asakura T: Electromagnetic field near the focus of Gaussian beams. Optik 41 (1974) 281–292 [5] Yoshida A, Asakura T: Electromagnetic in the focal plane of a coherent beam from a wide-angular annular-aperture system. Optik 40 (1974) 322–331 [6] Yoshida A, Asakura T: Diffraction patterns of off-axis gaussian beams in the presence of third-order spherical aberration in the optical system. Opt. Commun. 19 (1976) 387–392 [7] Yoshida A, Asakura T: Effects of aberrations on off-axis gaussian beams. Opt. Commun. 14 (1975) 211–214