Evolution of a complex-source-point spherical wave through a symmetric optical system with spherical aberration

Optik 122 (2011) 2094–2100

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Evolution of a complex-source-point spherical wave through a symmetric optical system with spherical aberration Soo Chang ∗ Department of Physics, Hannam University, 133 Ojungdong, Taejon 306-791, Republic of Korea

a r t i c l e

i n f o

Article history: Received 26 August 2010 Accepted 9 January 2011

PACS: 42.30.Va Keywords: Gaussian beam Spherical aberration Complex source point Beam quality factor

a b s t r a c t We discuss the evolution of a complex-source-point spherical wave (CSPSW) along the axis of a symmetric optical system with spherical aberration. The CSPSW is equivalent to the sum of all the higher order corrections to a paraxial Gaussian beam. First we formulate the diffracted CSPSW through spherical boundaries, where the terms of up to fourth order in aperture variables are taken into account. We then find a ray-optical solution for the diffracted beam that can be represented in terms of the coefficient of spherical aberration. Next we formulate the beam spot size, the far-field divergence angle, and the beam quality factor which are characteristic of the aberrated beam. For the case of a thick lens, we examine the effect of lens aberration on the beam parameters. The derived formulas can be used to determine the parameters of the Gaussian beam degraded by an arbitrary symmetric optical system with spherical aberration. © 2011 Elsevier GmbH. All rights reserved.

1. Introduction In earlier works [1,2], Siegman derived a simple formula for the degradation in the beam quality factor of a laser beam caused by the quartic phase aberration in an optical system. The derived formula could be used to evaluate the beam quality degradation in a spherically aberrated optical component, a thermally aberrated laser output window, or a divergent beam emerging from a high-index dielectric medium as in a wide-stripe, unstable-resonator diode laser. However, in case a strongly focused laser beam propagates in such an optical system that is made up of multiple refracting (or reflecting) surfaces separated by different spaces, we have to make all the higher order corrections to the laser beam, still more we need to trace the path of the beam in the optical system before determining a degree of quartic phase aberration. It is known that the Gaussian laser beam is equivalent paraxially to a spherical wave with a center at a complex location [3] and the sum of all the higher order corrections to the paraxial Gaussian beam is reduced to the so-called CSPSW(or complex-source-point spherical wave) [4]. However, the evolution of the CSPSW through a spherically aberrated optical system has not been examined, and it has not been used to determine the quality factor of a laser beam. In this paper, we investigate the evolution of the CSPSW along the axis of an arbitrary symmetric optical system with spherical

aberration. First we formulate the diffracted CSPSW through multiple spherical boundaries, where the terms of up to fourth order in aperture variables are taken into account. Then we find a rayoptical solution for the diffracted beam that can be represented in terms of the coefficient of spherical aberration. Next we also formulate the characteristic parameters of the aberrated beam, such as the beam spot size, the far-field divergence angle, and the beam quality factor. For the case of a thick curved-plano (or plano-curved) lens, we examine the effect of lens aberration on the beam parameters. We show that the beam parameters approach their paraxial values as the spot size of the incident beam decreases. We also find that the beam parameters depart fast from their paraxial values as the radius of curvature of the boundary decreases. The fourth order formulas derived here can be useful in determining the parameters of the Gaussian beam degraded by a rotationally symmetric optical system with spherical aberration, which consists of multiple refracting (or reflecting) surfaces separated by different spaces. 2. Paraxial formulas for a Gaussian beam Fig. 1(a) shows a paraxial Gaussian beam of vacuum wavelength  propagating along the z axis. The transverse amplitude profile for such a beam that is symmetric in x and y axes is given by



U1 (x1 , y1 , z1 ) = A1 exp ∗ Fax: +82 42 692 8313. E-mail address: [email protected] 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.01.003

−

x12 + y12 w12



,

(1)

where A1 is a constant, (x1 , y1 ) are the transverse coordinates within the waist plane located at z = z1 , and w1 is the minimun spot size of

S. Chang / Optik 122 (2011) 2094–2100

(x , y , z )

(x 1 , y 1 , z 1 )

n1

Beam B spot size

n1 U1 (x, y, z)  i (z − z1 )

z

w1

Paraxial Gaussian beam Paraxial P i l waist plane

b

(x 1 , y 1 , z 1 )

Paraxial path of complex ray

(0,0, 0 0 z 1 + ib1 ) Complex source point

the light disturbance reaching a point (x, y, z) in space from the waist plane can be formulated by applying a Fresnel–Kirchhoff diffraction integral [5]

Observing point

Paraxial waist plane

a

Observing point

(x , y , z )



z

z − z1

×

z − z 1 − ib1



∞



∞

dx1 dy1 U1 (x1 , y1 , z1 ) exp (ikn1 r1 ) , −∞

−∞

(3)

√ where n1 is the index of refraction of an optical medium, i(= −1) is the imaginary factor, and k( = 2/) is the magnitude of the wave vector in vacuum. For the sake of convenience, a time factor exp( − iωt) with an angular frequency of ω has been dropped in Eq. (3). If we solve the diffraction integral in Eq. (3) analytically, it is found that U1 (x, y, z) 

n1

− ib1

2095



C1 exp z − z1 − ib1

x 2 + y2 (z − z1 − ib1 ) + 2 (z − z1 − ib1 )

ikn1

 ,

(4)

where the Rayleigh range is defined by Equivalent CSPSW

Fig. 1. (a) A paraxial Gaussian beam of vacuum wavelength  propagating along the z axis. w1 is the minimun spot size of the beam and n1 is the index of refraction of the medium. (b) The Gaussain beam is equivalent paraxially to the spherical wave originating from a complex source point (0, 0, z1 + ib1 ), where b1 = n1 w12 /. The path of the complex ray is determined by Fermat’s principle.

the beam. If the distance of a point at the waist plane to an observing point (x, y, z) is given in a paraxial approximation by r1  (z − z1 ) +

(x − x1 )2 + (y − y1 )2 , 2 (z − z1 )

Spherical boundary

a

(2)

w 1'

n1 '

n1

R1

Paraxial path of complex ray

0

Another paraxial waist

n1 ' z 1' z 1 '+ ib1 '

x12 + y12

(x − x1 )2 + (y − y1 )2 + (z − z1 ) + 2 (z − z1 ) 2 (−ib1 )

n1 (z − z1 − ib1 ) ∂OPL − ib1 (z − z1 ) ∂y1

(x 1 ' , y 1 ' ) ib1 '

(0,0, z 1 '+ ib1 ')

(5)

and C1 = A1 ( − ib1 )exp (− kn1 b1 ). It should be noted here that the sign of n1 is positive (or negative) for the Gaussian beam propagating in the positive (or negative) z direction. We now investigate the propagation of a paraxial Gaussian beam from the standpoint of ray optics. The wave function in Eq. (4) is equivalent to the spherical wave with a center at a complex source point (0, 0, z1 + ib1 ), which is in the paraxial region where x2 + y2  | z − z1 − ib1 | 2 [3]. The spherical wave may be interpreted as a bundle of complex rays originating from the complex source point. If we look into the integrand in Eq. (3), the optical path length of a complex ray, reaching an observing point (x, y, z) from the waist plane, may be represented in a purely formal way by

∂OPL n1 (z − z1 − ib1 ) − ib1 (z − z1 ) ∂x1

z 1'

b (x , y ,0 )

,

 .

(6)

Applying Fermat’s principle to Eq. (6) [6], we find that the path of the complex ray should obey the formal equations

z

Beam spot size

Transmitted (or reflected) Gaussian beam



OPL  n1

(x 1 ' , y 1 ' )

0

n1 w12



Another paraxial i l waist i t

(x , y , z )

b1 =

z

Complex source point

Equivalent CSPSW Fig. 2. (a) A Gaussian beam transmitted through (or reflected at) a spherical boundary with a radius of curvature of R1 . w1 is the minimun spot size of the beam and n1 is the index of refraction of the medium. (b) The transmitted (or reflected) Gaussian beam is equivalent to the spherical wave originating from a complex source point (0, 0, z1 + ib1 ), where b1 = n1 w12 /. The path of the complex ray is determined by Fermat’s principle.





x1 +

ib1 x z − z1 − ib1

y1 +

ib1 y z − z1 − ib1



= 0,

(7)

= 0,

as shown in Fig. 1(b). By substituting Eq. (7) into Eq. (3) we can eliminate the integration variables to get the wave function in Eq. (4), except for the factors independent of x and y. We next examine the case in which the paraxial Gaussian beam in Eq. (4), after passing through (or being reflected at) a spherical boundary, propagates to another waist plane. As shown in Fig. 2(a), the origin of a coordinate system is situated at the point of intersection of the z axis with the spherical boundary. R1 denotes the radius of curvature of the boundary of which the sign is positive (or negative) when the center of curvature is on the right (or left)-hand side of the boundary. If z is much smaller in magnitude than R1 , an arbitrary point (x, y, z) at the boundary is constrained to obey the equation z

x 2 + y2 . 2R1

(8)

2096

S. Chang / Optik 122 (2011) 2094–2100

a

paraxial waist

(x , y , z )

(x 1 ' , y 1 ' )

Second paraxial waist

(x 2 ' , y 2 ' )

0

n1 '

R1

R2

z 1' d 1'

First equivalent CSPSW CSPS W

b

Complex source point

0

z 2'

Fi t ttransmitt First itted d (or reflected) Gaussian beam

First boundary

z

n2 '

Firstt Fi paraxial waist

(x , y )

Paraxial path of complex ray Second S d paraxial waist

(x 1 ' , y 1 ' )

(x 2 ' , y 2 ' )

n1 ' ib1 '

(x , y )

Second Seco nd transmitted (or reflected) Gaussian beam

Second boundary

z 1 '+ ib b1 '

n2 '

z z 2'

z 1 '−d 1 '

z 1 '−d 1 '+ ib b1 ' d 1'

Complex source point

ib2 '

z 2 '+ ib2 ' Second equivalent CSPSW Second b boundary d

First d y b boundar

Fig. 3. (a) A Gaussian beam propagating to the second paraxial waist via the second boundary, and (b) the diagram of an equivalent complex-source-point spherical wave (CSPSW). The path of the complex ray is determined by Fermat’s principle.

Consequently, the incident Gaussian beam upon the spherical boundary takes the form of U1 (x, y, z) 

 ×

 ikn1

C1 exp −z1 − ib1

x 2 + y2 x 2 + y2 + (−z1 − ib1 ) + 2R1 2 (−z1 − ib1 )

b1 =

 ,

(9)

and the distance of a point (x, y, z) at the boundary to a point (x1 , y1 ) within another waist plane located at z = z1 is given in a paraxial approximation by



r1



z1

2



x1 − x + y1 − y x 2 + y2 − + 2z1 2R1

2 .

U1 (x1 , y1 , z1 )

n1

iz1

n1

+ ib1

=





dxdy U1 (x, y, z) exp ikn1 r1 ,

(11)

(12)

the analytic solution of Eq. (11) is reduced to a simple form





A1

exp

.

(14)

On the other hand, we can also derive the Gaussian function in Eq. (13) by using a ray-optical approach. If we regard the wave function in Eq. (11) as a bundle of complex rays, the optical path length of a complex ray, reaching another waist point (x1 , y1 , z1 ) via the spherical boundary, may be represented in a purely formal way by



OPL  n1

+ n1

x 2 + y2 x 2 + y2 + (−z1 − ib1 ) + 2R1 2 (−z1 − ib1 )



z1

2



x1 − x + y1 − y x 2 + y2 − + 2R1 2z1



2 

.

(15)

aperture

n − n1 n1 + 1 , R1 z1 + ib1

U1 (x1 , y1 , z1 )





where n1 is the index of refraction of the medium in which the transmitted (or reflected) beam propagates, and the area of aperture is assumed to be large enough to accept the paraxial beam. In particular, we should let n1 = −n1 when the spherical boundary is a reflecting surface. If we employ a new parameter b1 obeying z1

n1 w12

(10)

The light disturbance reaching another waist point (x1 , y1 , z1 ) via the spherical boundary can then be represented by

 

where A1 is the factor independent of x1 and y1 , and w1 is the minimum spot size of the transmitted (or reflected) Gaussian beam related to the new parameter

−

x12 + y12 w12



,

(13)

In this case, Fermat’s principle requires that the path of the complex ray should obey the formal equations as depicted in Fig. 2(b) ib1 n1 ∂OPL   ∂x z1 z1 + ib1 ib n ∂OPL  1 1    ∂y z1 z1 + ib1



x−

 y−

z1 + ib1 ib1

z1 + ib1 ib1



x1

= 0,

(16)

 y1

= 0,

in terms of the new parameter b1 . Eliminating x and y in Eq. (11) together with Eq. (16) leads us directly to the Gaussian function in Eq. (13).

S. Chang / Optik 122 (2011) 2094–2100

Nth boundary

Nth paraxial waist

(N-1)th paraxial waist

w N −1'

n N −1 '

wN '

nN '

Observing plane

(x N ' , y N ') R N +1 = ∞

RN

z N −1' −d N −1 '

2097

zN '

dN '  Fig. 4. The corrected Gaussian beam reaching a point (xN , yN ) within the observing  from the N th boundary. The N th paraxial waist is at plane located at a distance dN a distance zN from the N th boundary.

3. Fourth order correction to a paraxial Gaussian beam As mentioned above, the propagation of a Gaussian beam is equivalent paraxially to that of a complex-source-point spherical wave (CSPSW). The CSPSW is also known to be the sum of all the higher order corrections to the paraxial Gaussian beam in Eq. (4) [4]. Therefore, if the terms of up to fourth order in the transverse coordinates are taken into account, the Gaussian beam reaching a fixed point (x, y, z) from the waist plane at z = z1 should be represented by

 C1 exp U1 (x, y, z)  z − z1 − ib1

(z − z1 − ib1 )

ikn1



2

x 2 + y2 x 2 + y2 + − 3 2 (z − z1 − ib1 ) 8(z − z1 − ib1 )

a

 .

Incident paraxial i l waist

Fig. 6. (a) The beam spot size W2 , (b) the far-field divergence angle 2 , and (c) the beam quality factor Mx2 which are plotted as functions of w1 for the case of Fig. 5(a). We let z1 = 0, n1 = n2 = 1, n1 = 1.5, d1 = 5 mm, d2 = z2 , and  = 632.8 nm.

Second paraxial waist

R2 = ∞

R1

w1

w 2'

n2 '

n1 '

n1

Observing plane

z 2'

z1



Incident paraxial i l waist

R1 = ∞

n1

R2

n1 '

w 2'

n2 '

d 1'

,

(18)

 Observing plane

z 2'

z1

2

the incident Gaussian beam takes the form of

Second paraxial waist

w1

In case the corrected Gaussian beam is incident upon the first boundary with a radius of curvature of R1 , at which a point (x, y, z) is constrained to obey the fourth order equation x 2 + y2 x 2 + y2 + z 2R1 8R13

d 2'

d 1' b

(17)

d 2'

Fig. 5. (a) A Gaussian beam transmitted through a thick curved-plano lens, where R1 is varied but R2 = ∞, and (b) a Gaussian beam transmitted through a thick planocurved lens, where R1 = ∞ and R2 is varied.

C1 exp U1 (x, y, z)  −z1 − ib1



x 2 + y2 x 2 + y2 + + 2 (−z1 − ib1 ) 8R13

ikn1

(−z1 − ib1 ) +

2

 −

x 2 + y2

x 2 + y2 2R1

2

4R1 (z1 + ib1 )

 2

+

x 2 + y2

2 

8(z1 + ib1 )

3

(19) In addition, the distance of a point (x, y, z) at the first boundary to a point (x1 , y1 ) within another waist plane located at z = z1 is given in the fourth order appoximation by

.

2098

S. Chang / Optik 122 (2011) 2094–2100

where we have used the notation





S1 = Q13

1 n2 1



1

−

+ Q12

n21

1 R1





1 1 − n1 n1

,

(22)

in terms of a complex invariant Q1 =

z1

n1

+ ib1

−

n1 R1

=

n1 n1 − . R1 z1 + ib1

(23)

We note here that S1 and Q1 are analogous to the fourth order coefficient of spherical aberration and Abbe’s zero invariant in odrinary ray optics, respectively [6]. We next consider the case in which the Gaussian beam with fourth order corrections in Eq. (21) is incident upon the second boundary with a radius of curvature of R2 . The sign of R2 is positive (or negative) when the center of curvature is on the right (or left)hand side of the boundary. If the second boundary is situated at a distance d1 from the first boundary as shown in Fig. 3(a), the sign of d1 is positive (or negative) when the second boundary is on the right (or left)-hand side of the first boundary. Since an arbitrary point (x, y, z) at the second boundary is constrained to obey the fourth order equation



z

d1

x 2 + y2 x 2 + y2 + + 2R2 8R23

2

,

(24)

the distance of a point (x1 , y1 ) within the first paraxial waist to a point (x, y, z) at the second boundary is given by





x 2 + y2



−

2



x − x1 + y − y1 x 2 + y2   + 2R2 2 d − z

r2  d1 − z1 +

1

x − x1

2





+ y − y1

4R2 d1 − z1

2 

1



−

2

2



+

x 2 + y2

2

8R23

x − x1

2





+ y − y1

8 d1 − z1

2 2

3

. (25)

W2 ,

2 ,

Fig. 7. (a) The beam spot size (b) the far-field divergence angle and (c) the beam quality factor Mx2 which are plotted as functions of w1 for the case of Fig. 5(b). Other parameters are the same as those in Fig. 6.

r1  z1 −

 +

x2

+ y2

2R1

x 2 + y2



 +

x1 − x

2



+ y1 − y

2z1

x1 − x

2



+ y1 − y

2 −

2 

 −

4R1 z12



x 2 + y2

x1 − x

2



+ y1 − y 8z13

U2 (x, y, z) 



2

8R13

Therefore, the incident Gaussian beam upon the second boundary can be formulated as



n1





i d1 − z1



∞



−∞

∞

−∞

dx1 dy1 U1 (x1 , y1 , z1 ) exp

× ikn1 r2 .

(26)

If we solve the diffraction integral in Eq. (26) in a ray-optical approximation, where the path of the complex ray obeys the paraxial equations

2 2 .

x1 =

(20)

−ib1

d1 − z1 − ib1

y1 =

x,

−ib1

d1 − z1 − ib1

as shown in Fig. 3(b), it is found that By inserting Eqs. (19) and (20) into Eq. (11), the light disturbance reaching another waist plane via the first boundary can then be determined, while the fourth order correction is made to the transmitted (or reflected) Gaussian beam. To find the fourth order terms of spherical aberration, we have solved Eq. (11) in a ray-optical approximation, where the path of the complex ray obeys Eqs. (12) and (16). As a result, the light amplitude at a point (x1 , y1 , z1 ) within another paraxial waist is given by

U1 (x1 , y1 , z1 )  A1 exp −



−

 2 2

ik x12 + y1 8

x12 + y12 w12

 −

2 2



 4

ib1

S1 ,

exp



d1 − z1 − ib1





4w12 b2 1



x 2 + y2

−

2



4R2 d1 − z1 − ib1



(21)

d1 − z1 − ib1

ikn1

(27)

x 2 + y2 x 2 + y2 x 2 + y2 + + +     2R2 8R23 2 d1 − z1 − ib1 −

x12 + y1

  4

z1 + ib1

U2 (x, y, z) 



C1

y,

ik x2 + y2 8

2





2 − 

4

z1 − d1 + ib1

2

2

8 d1 − z1 − ib1

z1 + ib1

where C1 is a constant.

x 2 + y2





3



4 S1 ,

(28)

S. Chang / Optik 122 (2011) 2094–2100

We now suppose that the Gaussian beam with fourth order corrections in Eq. (28), after passing through (or being reflected at) the second boundary, reaches the second paraxial waist. If the waist plane is located at a distance z2 from the second boundary as shown in Fig. 3(a), the distance of a point (x, y, z) at the second boundary to a point (x2 , y2 ) within the second paraxial waist is given by



r2

z2





2



x2 − x + y2 − y x 2 + y2 − + 2R2 2z2

x 2 + y2



+

x2 − x

2 

+ y2 − y

2



−

2 



−

4R2 z22

x 2 + y2

2

x2 − x

2

n2



+ y2 − y 8z23

 

2 2



(29)



n1

=

z1 − d1 + ib1

+

n2 − n1 R2

,

z2 + ib2 ib2

x2 ,

z2 + ib2

y=

ib2



 −

x22 + y22

2

y2 ,

(32)

×

 S2 +



x22 + y22



−

4w22 b2 2

−

w22

ik x22 + y22

2 

z2 + ib2



8

z1 + ib1



4 S1  4

ib2

4

4

n2 w22 

(34)

,

and the coefficient of spherical aberration at the second boundary is also defined by



S2 =

Q23

1 n2 2

−

1



2

n1

1 + Q22 R2



1 1 −  n2 n1



,

(35)

Q2 =

z2 + ib2

−

n2 R2

=

n1

2 + y2 xN N  zN

 + ib − dN N



3

N 

8

⎛ Sj ⎝

N  m=j

⎞⎫ ⎬ ⎠ ,  − d + ib 4 ⎭ (zm m m)  (zm

+ ibm )

4

(37)

 , y ) denote the transverse coordiwhere AN is a constant, and (xN N nates within the observing plane. In addition, zj is the distance of

the j th boundary to the j th paraxial waist, dj is the distance of the j

th boundary to the (j + 1) th boundary, and bj is the Rayleigh range related to the j th paraxial waist size wj by bj =

nj wj2 

.

(38)

In Eq. (37) we have also used the notation



Sj =



1

Qj3

n2 j

−

1 + Qj2 Rj

1 n2 j−1



1 1 −  nj nj−1

 ,

(39)

Qj =

nj

zj + ibj

−

nj Rj

=

nj−1

  zj−1 − dj−1 + ibj−1

−

nj−1 Rj

,

(40)

where nj is the index of refraction of the medium in which the j th transmitted (or reflected) beam propagates, and Rj is the radius of curvature of the j th boundary. If the j th boundary is a reflecting element (or mirror), we should let nj = −nj−1 . The sign of Rj is positive (or negative) when the center of curvature is on the right (or left)-hand side of the j th boundary. It is worthy of notice here that if all bj ’s in Eq. (37) get to zero, we can obtain the field for such a spherically aberrated image that is conjugate to a real source point in ordinary ray optics.

z1 − d1 + ib1

−

n1 R2

.

 from As shown in Eq. (37), the light amplitude at a distance dN the N th boundary departs from an ideal Gaussian beam, while fourth order corrections are made to the beam. One method of quantifying how much the corrected beam departs from a Gaussian is the so-called M2 (or beam quality) factor [7]. The M2 factor is related to the second moments of the irradiance distribution of the beam in the near and far fields. The far-field angular spread of the beam in Eq. (37) can be represented by a Fourier integral

 

VN (fx , fy )=





       dxN dyN UN (xN , yN , dN ) exp −i2 fx xN + fy yN



,

(41)

in terms of the second complex invariant n2

2



(33)

where A2 is a constant, w2 is the paraxial waist radius of the second transmitted (or reflected) beam related to the new parameter b2 =

−

4. Beam parameters with fourth order corrections ,

z1 − d1 + ib1

2

iknN

in terms of the j th complex invariant

as shown in Fig. 3(b). Thus we have U2 (x2 , y2 , z2 )  A2 exp

2 + y2 ik xN N



2

 − d + ib 8 zN N N

(31)

the diffraction integral in Eq. (30) can be solved again in a rayoptical approximation, where the path of the complex ray obeys the paraxial equations x=



2 + y2 xN N

j=1

aperture

is the index of refraction of the medium in which the where second transmitted (or reflected) beam propagates, and the area of aperture is assumed to be large enough to accept the incident beam. In particular, we should let n2 = −n1 for the case of a reflected beam. If a new parameter b2 is defined by z2 + ib2



.

n2

n2

   UN (xN , yN , dN )  AN exp

−

dxdy U2 (x, y, z) exp ikn2 r2 , (30)

iz2





Consequently, the light amplitude at the second paraxial waist can be represented by U2 (x2 , y2 , z2 ) 

from the N th boundary (Fig. 4)

+

8R23

2099

(36)

On the other hand, Eq. (33) may be generalized to the case of N transmitting (or reflecting) boundaries. Following the same pro cedures as above, we can find the light amplitude at a distance dN

where fx (or fy ) denotes the component of spatial frequency in the x (or y) direction. If we now define the second moments of the irradiance distribution in space and in spatial frequency by 2 xN

!

"" =

#

#2

 dy x2 #U  (x , y , d )# dxN N N N N N N

""

#

#2 ,

 dy #U  (x , y , d )# dxN N N N N N

(42)

2100

S. Chang / Optik 122 (2011) 2094–2100

and

# #2 ! dfx dfy fx2 #VN (fx , fy )# 2 fx = " " # #2 dfx dfy #V  (fx , fy )# ""

"" =

N

#

#2

 dy #∂U  (x , y , d )/∂x # dxN N N N N N N

""

#

#2

 dy #U  (x , y , d )# dxN N N N N N

,

(43)

respectively, the beam quality factor in the x direction is then given by Mx2 = 4

$

2 xN

!

!

fx2 .

(44)

2 (or f 2 ) with y2 (or f 2 ) in Eq. (44) also leads us to the Replacing xN x y N beam quality factor in the y direction. The beam quality factors at  = z  ) are found to approach unity the N th paraxial waist (i.e., dN N (i.e., Mx2 = My2 = 1), when fourth order corrections in Eq. (37) are negligible.  In addition, the spot size of the corrected beam at a distance dN from the N th boundary and its far-field divergence angle may be defined by

$

!

WN = 2

2 , xN

and N =

2

# # #n #

$

(45)

!

fx2 ,

(46)

N

respectively. That is why the paraxial approximations of Eqs. (45) and (46) can be reduced to the relations  WN = wN ,

(47)

and N

=



 # # = N , #n # w N N

(48)

 = z  , respectively. at dN N As an example, by applying Eq. (37) to Eqs. (44)–(46), we have numerically evaluated the parameters of the corrected Gaussian beam. We first examine the Gaussian beam transmitted through a thick curved-plano lens in Fig. 5(a), where R1 is varied but R2 = ∞. Fig. 6 shows (a) the beam spot size W2 , (b) the far-field divergence angle 2 , and (c) the beam quality factor Mx2 as functions of the minimum spot size of the incident Gaussian beam w1 , where we let z1 = 0, n1 = n2 = 1, n1 = 1.5, d1 = 5 mm, d2 = z2 , and  = 632.8 nm. We see that W2  w2 , 2  2 , and Mx2  1 for small values of w1 . The spot size W2 changes slowly with w1 , while the divergence angle 2 increases rapidly with increasing w1 . Consequently, the beam quality factor Mx2 depends mostly upon 2 . The beam parameters depart fast from their paraxial values as the radius of curvature R1 decreases. We next evaluate the parameters of the Gaussian beam transmitted through a thick plano-curved lens in Fig. 5(b), where R1 = ∞ and R2 is varied. Fig. 7 also shows (a) the beam spot size W2 , (b) the far-field divergence angle 2 , and (c) the beam quality factor Mx2 as

functions of the minimum spot size of the incident Gaussian beam w1 . Other parameters are the same as those in Fig. 6. By comparing the corresponding parameters in Figs. 6 and 7, we find that the case of Fig. 5(b) has been more affected by the fourth order terms of spherical aberration than the case of Fig. 5(a). In general, the fourth order formulas in Eq. (37) and Eqs. (44)–(46) can be used to determine the parameters of the Gaussian beam passing through (or being reflected at) a rotationally symmetric optical system with spherical aberration, which is composed of multiple surfaces separated by different spaces. 5. Conclusions In this paper, we have investigated the evolution of a complexsource-point spherical wave (CSPSW) along the axis of an arbitrary symmetric optical system with spherical aberration. The CSPSW is equivalent to the sum of all the higher order corrections to a paraxial Gaussian beam. First we have formulated the diffracted CSPSW through multiple spherical boundaries, while the terms of up to fourth order in aperture variables are taken into account. Then we have found a ray-optical solution for the diffracted beam that can be represented in terms of the coefficient of spherical aberration. Next we have formulated the characteristic parameters of the aberrated beam, such as the beam spot size, the far-field divergence angle, and the beam quality factor. For the case of a thick curved-plano (or plano-curved) lens, we have examined the effect of lens aberration on the parameters of the transmitted Gaussian beam. We have shown that the beam parameters approach their paraxial values as the spot size of the incident beam decreases. We have also found that he beam parameters depart fast from their paraxial values as the radius of curvature of the boundary decreases. The fourth order formulas derived here can be useful in determining the parameters of the Gaussian beam degraded by a rotationally symmetric optical system with spherical aberration, which is made up of multiple refracting (or reflecting) surfaces separated by different spaces. Acknowledgment The author thanks C.S. Lim for his helpful discussion in preparation for this article. References [1] A.E. Siegman, Analysis of laser beam quality degradation caused by quartic phase aberrations, Appl. Opt. 32 (1993) 5893–5901. [2] J.A. Ruff, A.E. Siegman, Measurement of beam quality degradation due to spherical aberration in a simple lens, Opt. Quantum Electron. 26 (1994) 629–632. [3] G.A. Deschamps, Gaussian beam as a bundle of complex rays, Electron. Lett. 7 (1971) 684–685. [4] M. Couture, P.A. Belanger, From Gaussian beam to compex-source-point wave, Phys. Rev. A24 (1981) 355–359. [5] M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford, 1980, pp. 378–383. [6] W.T. Welford, Aberrations of Optical Systems, Adam Hilger, Bristol, 1986, pp. 15–16, 130–140. [7] A.E. Siegman, New developments in laser resonators, Proc. SPIE 1224 (1990) 2–14.