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Aberrations of a Gaussian laser beam focused by a zone plate
Optik 124 (2013) 85–90
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Aberrations of a Gaussian laser beam focused by a zone plate 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 16 June 2011 Accepted 13 November 2011
PACS: 42.30.Va Keywords: Gaussian beam Seidel aberrations Beam quality factor Zone plate
a b s t r a c t We discuss the Seidel-type wavefront aberrations which degrade the quality of the Gaussian beam focused by a zone plate. First we examine the diffraction of a complex-source-point spherical wave through the zone plate, while the terms of up to fourth order in aperture variables are under consideration. Then we find an explicit formula for the diffracted beam that is represented in terms of Seidel-type wavefront aberrations. We also evaluate the effects of aberrations on the quality of the diffracted beam through the zone plate. If the axis of the incident beam is inclined slightly with respect to the optic axis, both spherical aberration and field curvature affect the beam quality in the sagittal section of the focal plane, whereas the beam quality in its meridian section is under the influence of all Seidel-type aberrations. The fourth order theory of propagation of a slightly inclined Gaussian beam as described here may be applied to evaluate the degradation in the quality of the Gaussian beam focused by a zone plate. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction A complex-source-point spherical wave (CSPSW) is known to be the sum of all the higher order corrections to the paraxial Gaussian beam [1,2]. In earlier works [3–5], we examined the evolution of the CSPSW through a rotationally symmetric system composed of a set of refracting or reflecting surfaces. We derived an explicit formula for the fourth order-corrected Gaussian beam that was given in terms of Seidel-type wavefront aberrations. Such formula was applied to evaluate the degradation in the quality of the Gaussian beam focused by the refracting or reflecting surfaces. On the other hand, a zone plate as a diffractive element is also capable of focusing light at a point much the same as an ordinary lens or mirror [6]. The zone plate is even useful in spectral regions where conventional optics are unavailable and for special applications in the visible spectrum such as optics for laser scanning [7,8]. The defects in the image of a point object formed by a zone plate have been studied in terms of third-order and chromatic aberrations [9]. However, the homogeneous spherical wave, emanating from a point source, has different properties from an inhomogeneous laser beam. The effects of aberrations on the quality of the laser beam focused by a zone plate have not been investigated. In this paper, we discuss the Seidel-type wavefront aberrations which degrade the quality of the Gaussian beam focused by a zone plate. First we examine the CSPSW diffracted through the zone plate, while the terms of up to fourth order in aperture variables
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are taken into account. Then we find a ray-optical solution for the diffracted beam that is expressed explicitly in terms of Seidel-type wavefront aberrations. If the Rayleigh range of the incident beam is large compared with the principal focal length of the zone plate, spherical aberration (or linear coma) depends upon the cube (or the square) of an inverse focal length, both linear astigmatism and field curvature are in inverse proportion to the focal length, and distortion is a function of the refractive indices of two media separated by the zone plate. Based on the derived formulas, we also evaluate the effects of aberrations on the quality of the diffracted beam through the zone plate. If the axis of the incident beam is inclined slightly with respect to the optic axis, both spherical aberration and field curvature affect the beam quality in the sagittal section of the focal plane, whereas the beam quality in its meridian section is under the influence of all Seidel-type aberrations. The fourth order theory of propagation of a slightly inclined Gaussian beam as described here may be useful in evaluating the quality of the Gaussian beam focused by a zone plate. 2. Propagation of a paraxial Gaussian beam through a Fresnel zone plate Suppose that a Gaussian beam of vacuum wavelength is incident obliquely upon a zone plate separating two media of refractive indices n and n . The zone plate consists of a set of concentric annuli, alternately transparent and opaque. The Gaussian beam, after being diffracted through the zone plate, converges to (or appears to diverge from) several foci corresponding to a certain finite number of diffracted orders. Assuming that the aperture stop is in contact with the zone plate, we take a Cartesian reference
86
S. Chang / Optik 124 (2013) 85–90
Transmission coefficient, (x,y)
a
Fig. 1. A Gaussian beam is incident obliquely upon a zone plate separating two media of refractive indices n and n . The zone plate consists of a set of concentric annuli, alternately transparent and opaque. The Gaussian beam diffracted through the zone plate converges to (or appears to diverge from) several foci corresponding to a certain finite number of diffracted orders. The Cartesian coordinate system is referenced to the center of the zone plate normal to the z-axis. The axis of the incident (or diffracted) beam makes an angle of u (or u ) with the zaxis. The incident (or diffracted) beam of Rayleigh range b (or bm ) is equivalent paraxially to the spherical wave originating from (0, y + ib sin u, z + ib cos u) (or (0, ym + ibm sin u , z m + ibm cos u )).
C (x, y, z) = exp(iknr − iωt), r
(1)
2
2 1/2
r = [x2 + (y − y − ib sin u) + (z − z − ib cos u) ]
.
(2)
In the above the branch of r should be chosen such that its real part is equal to (z − z) when it is large, and the Rayleigh range of the incident beam is defined by b = nw02 / in terms of the waist radius w0 . When |u| 1 and x2 + (y − y)2 (z − z)2 + b2 , the Fresnel approximation allows the wave function in Eq. (1) to be rewritten as
C z − z − ib
ikn z − z − ib +
× exp
(x, y) =
x2 + (y − y − ibu)2 2(z − z − ib)
,
bm exp[−ikm(x, y)],
(4)
n 2 (x + y2 ), 2f
0
for m = ±1, otherwise,
or y/(2f /n')
1/2
or y/(2f /n')
2
3
2
3
1/2
1.0
0.5
0.0
-3
-2
-1
0
x/(2f /n')
1 1/2
Fig. 2. Transmission coefficient of a zone plate plotted as a function of position in the radial direction: (a) a sinusoidal (or Gabor) zone plate and (b) a binary (or Fresnel) zone plate.
for a sinusoidal (or Gabor) zone plate as illustrated in Fig. 2(a) or bm =
⎧ ⎨ 1/2 ⎩
for m = 0, (7)
sin(m/2)/m
for odd m,
0
for even m( = / 0),
for a binary (or Fresnel) zone plate as plotted in Fig. 2(b). If the distance from any point (x, y) in an opening of the zone plate to the observation point (x , y , z ) is in the quadric approximation written as r z +
(x − x)2 + (y − y)2 , 2z
1 (x , y , z ) iz
(8)
∞
× exp(ikn r ) =
dxdy
(x, y, 0)(x, y)
aperture bm Um (x , y , z ),
(9)
m=−∞
where the aperture size is assumed to be large enough to accept the (x , y , z ) stands for the mth-order paraxial Gaussian beam, and Um component of the diffracted beam. Solving the diffraction integral (9) analytically, we have
Cm
exp ikn
x
(5)
2
+ (y − ym − ibm u )2
2(z − z m − ibm )
,
(10)
is the factor independent of x and y , and the paraxial where Cm parameters are defined by
⎧ ⎨ 1/2 for m = 0, 1/4
1
1/2
b
(x , y , z ) Um
with a principal focal length of f, and the coefficients are of
⎩
0
-1
where the phase function becomes
bm =
-2
x/(2f /n')
(3)
m=−∞
(x, y) =
-3
the diffracted Gaussian beam through the zone plate may be evaluated in the Fresnel–Kirchhoff diffraction integral [10].
where we have dropped a time factor exp(− iωt). If the sign of n is positive (or negative), the wave function in Eq. (3) represents the Gaussian beam proceeding in the positive (or negative) z direction. The amplitude transmittance of a zone plate may also be expressed in general as follows: ∞
0.0
√
where C is the normalization constant, i(= −1) is the imaginary symbol, k(= 2/) is the magnitude of the wave vector in vacuum, ω is the angular frequency of the light, and
(x, y, z)
0.5
Transmission coefficient, (x,y)
system with origin at the center of the zone plate in which the z-axis corresponds to the optic axis (Fig. 1). The axis of the incident (or diffracted) beam placed at the yz plane makes an angle of u (or u ) with the z-axis. The angles of inclination u and u are positive when measured clockwise from the beam axis to the zaxis. If the center of the incident beam has coordinates (0, y, z) such that y = z tan u, the complex amplitude of the beam may be represented by the spherical wave originating from a complex source point (0, y + ib sin u, z + ib cos u) [1,2]
1.0
(6)
n
z m + ibm
n u nu,
n n , + (f/m) z + ib
(11)
S. Chang / Optik 124 (2013) 85–90
ym z m u .
(11) zm
In the above denotes the distance from the zone plate to the mth-order focus measured in the z direction, ym is the height of the beam axis at the mth-order focus, and bm is the Rayleigh range of the mth-order component of the diffracted beam, related to the waist radius wm 2 n wm .
bm =
(12)
The wave function in Eq. (10) describes the Gaussian beam converging to (or diverging from) the mth-order focus if m = / 0, and the incident beam if m = 0. This function is equivalent paraxially to the spherical wave radiating to a complex image point (0, ym + ibm sin u , z m + ibm cos u ) as shown in Fig. 1. The sign of n is positive (or negative) when the diffracted beam proceeds in the positive (or negative) z direction. Therefore, if the zone plate is a reflecting surface, we have to put n = − n. It also follows from Eqs. (6) to (7) that a sinusoidal (or Gabor) zone plate produces the foci of orders m = ± 1, while a binary (or Fresnel) zone plate generates the foci of odd orders m = ± 1, ±3, ±5, etc. Further if both b and bm are equal to zero, the first of Eq. (11) reduces to the Gaussian formula for a thin lens of focal length (f/m) in geometrical optics [6]. In this case, the function (10) represents the spherical wave centered at the image point (0, ym , z m ) which is conjugate to the object point (0, y, z). On the other hand, the wave function in Eq. (10) may be derived from the standpoint of geometrical optics. Since the diffraction integral (9) is equivalent to the sum of a bundle of complex rays, the optical path length of each ray converging to (or diverging from) the mth-order focus may be written in a purely formal way as
87
where ˛ is the conic constant. The type of conic depends upon the value of the conic constant: ˛ < − 1 (hyperboloid), ˛ = − 1 (paraboloid), −1 < ˛ < 0 (ellipsoid with the major axis placed along the z-axis), ˛ = 0 (sphere), and ˛ > 0 (ellipsoid with the minor axis placed along the z-axis). Moreover, the distance from any point (x, y) in an opening of the zone plate to the observation point (x , y , z ) can be written in the fourth order approximation as r z +
(x − x)2 + (y − y)2 [(x − x)2 + (y − y)2 ]2 − . 2z 8z 3
Hence if we again eliminate x and y by inserting Eqs. (15)–(18) into Eq. (9), we find a ray-optical solution for the diffracted beam (x , y , z ) Cm Um
× exp
ikn
2(z
−
x2 + (y − ym − ibm u)2 − zm
− ibm )
[x2 + (y − ym − ibm u)2 ]2
8(z − z m − ibm )3
−ikW
(m)
(x , y , z )
,
(19)
is the factor independent of x and y , and the aberration of Cm where the Gaussian beam converging to (or diverging from) the mth-order focus is given explicitly by
W (m) (x , y , z ) =
1 (m) (z m + ibm )4 [x2 + (y − z u )2 ]2 S 8 I (z − z − ib )4 m
m
1 (m) (z m + ibm )3 + SII [x2 + (y − z u )2 ] 2 (z − z − ib )3 m
m
OPLm
x2 + (y − y − ibu)2 = n z − n(z + ib) − n 2(z + ib)
× (y − z u )(n u ) +
2
(13)
× (y − z u )2 (n u )2 +
By applying Fermat’s principle
∂OPLm = 0, ∂x
∂OPLm = 0, ∂y
x=
−z m − ibm x, (z − z m − ibm )
(14)
y=
−z m − ibm (y (z − z m − ibm )
− z u ),
(m)
SI
3. Fourth order correction to a paraxial Gaussian beam
(m)
SII
= =
(m)
(m)
SV Considering the terms of up to fourth order in aperture variables x and y, we can rewrite the wave function in Eq. (1) as C (x, y, z) z − z − ib
−
ikn z − z − ib +
[x2 + (y − y − ibu)2 ]2 8(z − z − ib)3
x2 + (y − y − ibu)2 2(z − z − ib)
,
(16)
and we may also characterize the phase of the zone plate by (x, y) =
n 2 n (x + y2 ) + 3 (1 + ˛)(x2 + y2 )2 , 2f 8f
× (y − z u )(n u )3 , n (z m
+ ibm )3
1
(z m + ibm )2
(17)
=
− −
n 3
(z + ib)
1 (m) (z m + ibm ) S 2 V (z − z m − ibm ) (20)
n m (1 + ˛), f3
(21)
, (z + ib)2
n (z m + ibm )
1 1 − 2. n2 n
+
1
1
(m)
SIII = SIV =
m
with the coefficients of (15)
where we have used the paraxial parameters in Eq. (11). If we now insert Eq. (15) into Eq. (9) to eliminate x and y, the complex amplitude in Eq. (10) is obtained directly, except for the factor independent of x and y .
× exp
1 (m) (z m + ibm )2 SIV 4 (z − z m − ibm )2
× [x2 + (y − z u )2 ](n u )2 +
we get the formal equations for tracing the path of the complex ray
1 (m) (z m + ibm )2 S 2 III (z − z − ib )2 m
(x − x)2 + (y − y)2 + y2 −n + n . 2z 2(f/m) x
(18)
−
1 , n(z + ib)
(21)
Each term in Eq. (20) corresponds to the Seidel-type wavefront aberrations for the spherical wave radiating to a complex image point (0, ym + ibm sin u , z m + ibm cos u ) [11]. The first term is the so-called spherical aberration that is independent of an inclination of the beam, while it depends upon the fourth power of the transverse coordinates (x , y ). The second term known as linear coma is proportional to the angle of inclination u and the cube of the transverse coordinates (x , y ). The third and fourth terms are linear astigmatism and field curvature, respectively. Both of them change according to the squares of the angle of inclination u and the transverse coordinates (x , y ). The last term called distortion is pro portional to the cube of the angle of inclination u . Hence if the axis of the incident beam is coincident with the z-axis (i.e., u = 0), the
S. Chang / Optik 124 (2013) 85–90
first term is responsible only for degrading the beam quality. However, the influence of off-axial aberrations becomes important as an inclination of the beam increases. The field distribution deformed by the first and fourth terms is rotationally symmetric with respect to the beam axis, while the second and third terms make the field distribution asymmetric. On the other hand, if b is large compared with both z and (f/m), we have from Eq. (11) f , m
zm
bm
n
f 2
m
bn
.
(m)
n m 2 (m + 1 + ˛), f3
(23)
(m)
=
1 1 − 2. n2 n
(23)
(m) SI
depends upon the cube of an inverse focal length,
We see that (m) SII (m) SIII
(m)
SIII = SIV =
m , n f
is proportional to the square of an inverse focal length, both (m)
(m)
and SIV are in inverse proportion to the focal length, and SV is a function of the refractive indices n and n for the case of the stop in contact with the zone plate. When n = ± n, the coefficient (m) of distortion always disappears (i.e., SV = 0). 4. Effects of aberrations on the beam quality As obvious from Eq. (19), the diffracted beam converging to (or diverging from) the mth-order focus departs from an ideal Gaussian beam, while fourth order corrections are made for the beam. We investigate the quality factor to quantify how much the corrected beam departs from a Gaussian [12]. The beam quality 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. (19) may be represented by a Fourier integral (fx , fy , z m ) Vm
dx dy Um (x , y , z m )
=
× exp[−i2(fx x + fy y )],
(24)
where fx (or fy ) denotes the component of spatial frequency in the x (or y) direction. If we define the lth moments of the irradiance distribution in space and in spatial frequency as
dx dy xl |Um (x , y , z m )|2
xl =
dx dy
|Um (x , y , z m )|2
df x df y fxl |Vm (fx , fy , z m )|2
fxl =
,
df x df y |Vm (fx , fy , z m )|2
respectively, the beam quality factor in the x-section (or sagittal section) of the mth-order focus is defined by Mx2 =
Wx x ,
(26)
where we have used the notation
Wx = 2
x2 − x 2 ,
x = 2
fx2 − fx 2 .
10
15
20
Fig. 3. Variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders m = 1 (solid line), 3 (dashed line), and 5 (dotted line). The incident beam radius w0 is taken as a variable, while other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = 0, and f = 900 mm.
Replacing x (or fx ) by y (or fy ) in Eqs. (25)–(27), we can also evaluate the beam quality factor in the y-section (or meridian section) of the mth-order focus (i.e., My2 ). It is important to note here that if fourth order terms in Eq. (19) are neglected, Mx2 (or My2 ) approaches unity. First we examine the effect of spherical aberration on the beam quality, where we let u = 0. In this case, there is no difference between Mx2 and My2 . Fig. 3 shows the variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders m = 1 (solid line), 3 (dashed line) and 5 (dotted line), where we take the incident beam radius w0 as a variable. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = 0 and f = 900 mm. The spherical aberration in Eq. (20) is independent of an inclination of the beam, while it depends upon the fourth power of the transverse coordinates (x , y ) . Therefore, if w0 gets close to zero, the spherical aberration disappears so that we have Mx2 = My2 1. However, Mx2 (or My2 ) changes to a certain value greater than 1 by increasing w0 . When ˛ = − 1 and b z, (f/m), the coefficient of spherical aberration in Eq. (23) depends upon the cube of the order m. Consequently, the dashed (or dotted) curve is shown to change more rapidly with w0 than the solid curve. Fig. 4 also shows the variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders (a) m = 1 and (b) m = 3, where we take the conic constant ˛ as a variable. The solid, dashed, and dotted lines are of f = 300 mm, 600 mm, and 900 mm, respectively. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, u = 0 and w0 = 10 mm. Since b( 5 × 105 mm) is much bigger than both z and (f/m), it follows from Eq. (23) that spherical aberration disappears at ˛ −(m2 + 1).
, (25)
5
Incident beam radius, w0 (mm)
m2 , f2
SV
m=1
1.0
0
(m)
m=3
1.2
0.8
(m)
SII
m=5
1.4
(22)
Inserting Eq. (22) into Eq. (21) leads us to SI
1.6
Beam quality factor
88
(27)
(28)
As shown in Fig. 4, therefore, Mx2 (or My2 ) for the diffracted beam converging to the focus of order m = 1 (or m = 3) approaches 1 at ˛ − 2 (or ˛ − 10). Next we look into the influence of off-axial aberrations on the / 0. If the axis of the incident (or beam quality, where we let u = diffracted) beam is slightly inclined with respect to the z-axis, Mx2 and My2 are not always equal. Fig. 5 shows the variations of Mx2 (solid line) and My2 (dashed line) for the diffracted beam converging to the focus of order m = 1, where we take the principal focal length f as a variable. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = −0.05 rad and w0 = 5 mm. When b (f/m) 1, the aberration coefficients in Eq. (23) disappear. Therefore, both Mx2 and My2 get close to 1 as f increases. The curve of Mx2 is affected by the first and fourth terms in Eq. (20) which correspond to spherical aberration and field curvature, respectively. The difference in the curves of Mx2 and My2 is caused by the second and third terms
S. Chang / Optik 124 (2013) 85–90
a
1.6
1.8
Bea am quality fac ctor
1.6
Beam quality factor
89
f=300 mm
1.4
1.2
f=600 mm 1.0
1.4
2
My
1.2
2
Mx
1.0
f=900 mm
0.00
0.8 -6
-4
-2
0
2
-0.02
-0.04
-0.06
-0.08
-0.10
Angle of inclination (radians)
Conic constant, Fig. 6. Curves of Mx2 (solid line) and My2 (dashed line) plotted as a function of the angle of inclination u, where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, w0 = 5 mm and f = 900 mm
Beam quality factor
1.6
f=300 mm
1.20 1.4
f=600 mm
1.15
1.2
1.0
f=900 mm
0.8 -14
-12
-10
-8
-6
Conic constant,
Beam quality factor
b
1.8
2
My 1.10
2
Mx
1.05
1.00
Fig. 4. Variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders (a) m = 1 and (b) m = 3. The solid, dashed, and dotted lines are of f = 300 mm, 600 mm, and 900 mm, respectively. The conic constant ˛ is taken as a variable, while other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, u = 0, and w0 = 10 mm.
0.95 0
5
10
15
20
Incident beam radius, w0 (mm) Fig. 7. Curves of Mx2 (solid line) and My2 (dashed line) plotted as a function of the incident beam radius w0 , where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, u = −0.01 rad and f = 900 mm.
2.0
Beam quality factor
1.8
different values greater than 1 as w0 increases. The difference in the changes of Mx2 and My2 results from the terms of linear coma and linear astigmatism in Eq. (20). In general, the fourth order theory of propagation of a slightly inclined Gaussian beam as described above may be applied to evaluate the degradation in the quality of the Gaussian beam focused by a zone plate.
1.6 1.4
2
My
1.2
2
Mx
1.0 0.8 400
600
800
1000
1200
Principal focal length, f (mm) Fig. 5. Curves of Mx2 (solid line) and My2 (dashed line) plotted as a function of the principal focal length f, where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, w0 = 5 mm and u = −0.05 rad.
in Eq. (20) which are called linear coma and linear astigmatism, respectively. In Fig. 6 we plot the variations of Mx2 (solid line) and My2 (dashed line) as a function of the angle of inclination u, where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, w0 = 5 mm and f = 900 mm. When the angle of inclination u is extremely small, we have Mx2 = My2 1. In this case the influence of spherical aberration is negligible. Therefore, the fourth term (i.e., field curvature) in Eq. (20) is responsible for the change of Mx2 , whereas the curve of My2 is under the influence of all off-axial aberrations in Eq. (20). Fig. 7 shows the variations of Mx2 (solid line) and My2 (dashed line) for the diffracted beam converging to the focus of order m = 1, where we take the incident beam radius w0 as a variable. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = −0.01 rad and f = 900 mm. If w0 goes to zero, all aberrations in Eq. (20) disappear so that we have Mx2 = My2 1. However, Mx2 and My2 change to the
5. Conclusions In this paper, we have discussed the Seidel-type wavefront aberrations which degrade the quality of the Gaussian beam focused by a zone plate. First we have examined the diffraction of a complexsource-point spherical wave (CSPSW) through the zone plate, while the terms of up to fourth order in aperture variables are taken into consideration. The CSPSW is known to be the sum of all the higher order corrections to the paraxial Gaussian beam. Then we have found a ray-optical solution for the diffracted beam that is represented explicitly in terms of Seidel-type wavefront aberrations. If the Rayleigh range of the incident beam is large compared with the principal focal length of the zone plate, spherical aberration (or linear coma) changes according to the cube (or square) of an inverse focal length, both linear astigmatism and field curvature are in inverse proportion to the focal length, and distortion is a function of the refractive indices of two media separated by the zone plate. Based on the derived formulas, we have also evaluated the effects of aberrations on the quality of the diffracted beam through the zone plate. If the axis of the incident beam is inclined slightly with respect to the optic axis, both spherical aberration and field curvature affect the beam quality in the sagittal section of the focal plane, whereas the beam quality in its meridian section is under the
90
S. Chang / Optik 124 (2013) 85–90
influence of all Seidel-type aberrations. The fourth order theory of propagation of a slightly inclined Gaussian beam as described here may be applied to evaluate the degradation in the quality of the Gaussian beam focused by a zone plate. References [1] G.A. Deschamps, Gaussian beam as a bundle of complex rays, Electron. Lett. 7 (1971) 684–685. [2] M. Couture, P.A. Belanger, From Gaussian beam to compex-source-point wave, Phys. Rev. A24 (1981) 355–359. [3] S. Chang, Evolution of a complex-source-point spherical wave through a symmetric system with spherical aberration, Optik 122 (2011) 2094–2100.
[4] S. Chang, Fourth order correction to a paraxial Gaussian beam in a rotationallysymmetric system of non-spherical surfaces, Optik (2011), doi:10.1016/j.ijleo.2011.06.013. [5] S. Chang, Aberrations of a slightly inclined Gaussian beam in a symmetric optical system, Optik (2011), doi:10.1016/j.ijleo.2011.07.049. [6] F.A. Jenkins, H.E. White, Fundamentals of Optics, McGraw-Hill, London, 1981, pp. 44–77, 384–386. [7] A.V. Baez, Fresnel zone plate for optical image formation using extreme ultraviolet and soft x-radiation, J. Opt. Soc. Am. 51 (1961) 405–412. [8] R.E. Hopkins, M.J. Buzawa, Optics for laser scanning, Opt. Eng. 15 (1976) 90–94. [9] M. Young, Zone plates and their aberrations, J. Opt. Soc. Am. 62 (1972) 972–976. [10] M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford, 1980, pp. 378–383. [11] W.T. Welford, Aberrations of Optical Systems, Adam Hilger, Bristol, 1986, pp. 15–16, 130–140. [12] A.E. Siegman, New developments in laser resonators, Proc. SPIE 1224 (1990) 2–14.
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Aberrations of a Gaussian laser beam focused by a zone plate 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 16 June 2011 Accepted 13 November 2011
PACS: 42.30.Va Keywords: Gaussian beam Seidel aberrations Beam quality factor Zone plate
a b s t r a c t We discuss the Seidel-type wavefront aberrations which degrade the quality of the Gaussian beam focused by a zone plate. First we examine the diffraction of a complex-source-point spherical wave through the zone plate, while the terms of up to fourth order in aperture variables are under consideration. Then we find an explicit formula for the diffracted beam that is represented in terms of Seidel-type wavefront aberrations. We also evaluate the effects of aberrations on the quality of the diffracted beam through the zone plate. If the axis of the incident beam is inclined slightly with respect to the optic axis, both spherical aberration and field curvature affect the beam quality in the sagittal section of the focal plane, whereas the beam quality in its meridian section is under the influence of all Seidel-type aberrations. The fourth order theory of propagation of a slightly inclined Gaussian beam as described here may be applied to evaluate the degradation in the quality of the Gaussian beam focused by a zone plate. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction A complex-source-point spherical wave (CSPSW) is known to be the sum of all the higher order corrections to the paraxial Gaussian beam [1,2]. In earlier works [3–5], we examined the evolution of the CSPSW through a rotationally symmetric system composed of a set of refracting or reflecting surfaces. We derived an explicit formula for the fourth order-corrected Gaussian beam that was given in terms of Seidel-type wavefront aberrations. Such formula was applied to evaluate the degradation in the quality of the Gaussian beam focused by the refracting or reflecting surfaces. On the other hand, a zone plate as a diffractive element is also capable of focusing light at a point much the same as an ordinary lens or mirror [6]. The zone plate is even useful in spectral regions where conventional optics are unavailable and for special applications in the visible spectrum such as optics for laser scanning [7,8]. The defects in the image of a point object formed by a zone plate have been studied in terms of third-order and chromatic aberrations [9]. However, the homogeneous spherical wave, emanating from a point source, has different properties from an inhomogeneous laser beam. The effects of aberrations on the quality of the laser beam focused by a zone plate have not been investigated. In this paper, we discuss the Seidel-type wavefront aberrations which degrade the quality of the Gaussian beam focused by a zone plate. First we examine the CSPSW diffracted through the zone plate, while the terms of up to fourth order in aperture variables
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are taken into account. Then we find a ray-optical solution for the diffracted beam that is expressed explicitly in terms of Seidel-type wavefront aberrations. If the Rayleigh range of the incident beam is large compared with the principal focal length of the zone plate, spherical aberration (or linear coma) depends upon the cube (or the square) of an inverse focal length, both linear astigmatism and field curvature are in inverse proportion to the focal length, and distortion is a function of the refractive indices of two media separated by the zone plate. Based on the derived formulas, we also evaluate the effects of aberrations on the quality of the diffracted beam through the zone plate. If the axis of the incident beam is inclined slightly with respect to the optic axis, both spherical aberration and field curvature affect the beam quality in the sagittal section of the focal plane, whereas the beam quality in its meridian section is under the influence of all Seidel-type aberrations. The fourth order theory of propagation of a slightly inclined Gaussian beam as described here may be useful in evaluating the quality of the Gaussian beam focused by a zone plate. 2. Propagation of a paraxial Gaussian beam through a Fresnel zone plate Suppose that a Gaussian beam of vacuum wavelength is incident obliquely upon a zone plate separating two media of refractive indices n and n . The zone plate consists of a set of concentric annuli, alternately transparent and opaque. The Gaussian beam, after being diffracted through the zone plate, converges to (or appears to diverge from) several foci corresponding to a certain finite number of diffracted orders. Assuming that the aperture stop is in contact with the zone plate, we take a Cartesian reference
86
S. Chang / Optik 124 (2013) 85–90
Transmission coefficient, (x,y)
a
Fig. 1. A Gaussian beam is incident obliquely upon a zone plate separating two media of refractive indices n and n . The zone plate consists of a set of concentric annuli, alternately transparent and opaque. The Gaussian beam diffracted through the zone plate converges to (or appears to diverge from) several foci corresponding to a certain finite number of diffracted orders. The Cartesian coordinate system is referenced to the center of the zone plate normal to the z-axis. The axis of the incident (or diffracted) beam makes an angle of u (or u ) with the zaxis. The incident (or diffracted) beam of Rayleigh range b (or bm ) is equivalent paraxially to the spherical wave originating from (0, y + ib sin u, z + ib cos u) (or (0, ym + ibm sin u , z m + ibm cos u )).
C (x, y, z) = exp(iknr − iωt), r
(1)
2
2 1/2
r = [x2 + (y − y − ib sin u) + (z − z − ib cos u) ]
.
(2)
In the above the branch of r should be chosen such that its real part is equal to (z − z) when it is large, and the Rayleigh range of the incident beam is defined by b = nw02 / in terms of the waist radius w0 . When |u| 1 and x2 + (y − y)2 (z − z)2 + b2 , the Fresnel approximation allows the wave function in Eq. (1) to be rewritten as
C z − z − ib
ikn z − z − ib +
× exp
(x, y) =
x2 + (y − y − ibu)2 2(z − z − ib)
,
bm exp[−ikm(x, y)],
(4)
n 2 (x + y2 ), 2f
0
for m = ±1, otherwise,
or y/(2f /n')
1/2
or y/(2f /n')
2
3
2
3
1/2
1.0
0.5
0.0
-3
-2
-1
0
x/(2f /n')
1 1/2
Fig. 2. Transmission coefficient of a zone plate plotted as a function of position in the radial direction: (a) a sinusoidal (or Gabor) zone plate and (b) a binary (or Fresnel) zone plate.
for a sinusoidal (or Gabor) zone plate as illustrated in Fig. 2(a) or bm =
⎧ ⎨ 1/2 ⎩
for m = 0, (7)
sin(m/2)/m
for odd m,
0
for even m( = / 0),
for a binary (or Fresnel) zone plate as plotted in Fig. 2(b). If the distance from any point (x, y) in an opening of the zone plate to the observation point (x , y , z ) is in the quadric approximation written as r z +
(x − x)2 + (y − y)2 , 2z
1 (x , y , z ) iz
(8)
∞
× exp(ikn r ) =
dxdy
(x, y, 0)(x, y)
aperture bm Um (x , y , z ),
(9)
m=−∞
where the aperture size is assumed to be large enough to accept the (x , y , z ) stands for the mth-order paraxial Gaussian beam, and Um component of the diffracted beam. Solving the diffraction integral (9) analytically, we have
Cm
exp ikn
x
(5)
2
+ (y − ym − ibm u )2
2(z − z m − ibm )
,
(10)
is the factor independent of x and y , and the paraxial where Cm parameters are defined by
⎧ ⎨ 1/2 for m = 0, 1/4
1
1/2
b
(x , y , z ) Um
with a principal focal length of f, and the coefficients are of
⎩
0
-1
where the phase function becomes
bm =
-2
x/(2f /n')
(3)
m=−∞
(x, y) =
-3
the diffracted Gaussian beam through the zone plate may be evaluated in the Fresnel–Kirchhoff diffraction integral [10].
where we have dropped a time factor exp(− iωt). If the sign of n is positive (or negative), the wave function in Eq. (3) represents the Gaussian beam proceeding in the positive (or negative) z direction. The amplitude transmittance of a zone plate may also be expressed in general as follows: ∞
0.0
√
where C is the normalization constant, i(= −1) is the imaginary symbol, k(= 2/) is the magnitude of the wave vector in vacuum, ω is the angular frequency of the light, and
(x, y, z)
0.5
Transmission coefficient, (x,y)
system with origin at the center of the zone plate in which the z-axis corresponds to the optic axis (Fig. 1). The axis of the incident (or diffracted) beam placed at the yz plane makes an angle of u (or u ) with the z-axis. The angles of inclination u and u are positive when measured clockwise from the beam axis to the zaxis. If the center of the incident beam has coordinates (0, y, z) such that y = z tan u, the complex amplitude of the beam may be represented by the spherical wave originating from a complex source point (0, y + ib sin u, z + ib cos u) [1,2]
1.0
(6)
n
z m + ibm
n u nu,
n n , + (f/m) z + ib
(11)
S. Chang / Optik 124 (2013) 85–90
ym z m u .
(11) zm
In the above denotes the distance from the zone plate to the mth-order focus measured in the z direction, ym is the height of the beam axis at the mth-order focus, and bm is the Rayleigh range of the mth-order component of the diffracted beam, related to the waist radius wm 2 n wm .
bm =
(12)
The wave function in Eq. (10) describes the Gaussian beam converging to (or diverging from) the mth-order focus if m = / 0, and the incident beam if m = 0. This function is equivalent paraxially to the spherical wave radiating to a complex image point (0, ym + ibm sin u , z m + ibm cos u ) as shown in Fig. 1. The sign of n is positive (or negative) when the diffracted beam proceeds in the positive (or negative) z direction. Therefore, if the zone plate is a reflecting surface, we have to put n = − n. It also follows from Eqs. (6) to (7) that a sinusoidal (or Gabor) zone plate produces the foci of orders m = ± 1, while a binary (or Fresnel) zone plate generates the foci of odd orders m = ± 1, ±3, ±5, etc. Further if both b and bm are equal to zero, the first of Eq. (11) reduces to the Gaussian formula for a thin lens of focal length (f/m) in geometrical optics [6]. In this case, the function (10) represents the spherical wave centered at the image point (0, ym , z m ) which is conjugate to the object point (0, y, z). On the other hand, the wave function in Eq. (10) may be derived from the standpoint of geometrical optics. Since the diffraction integral (9) is equivalent to the sum of a bundle of complex rays, the optical path length of each ray converging to (or diverging from) the mth-order focus may be written in a purely formal way as
87
where ˛ is the conic constant. The type of conic depends upon the value of the conic constant: ˛ < − 1 (hyperboloid), ˛ = − 1 (paraboloid), −1 < ˛ < 0 (ellipsoid with the major axis placed along the z-axis), ˛ = 0 (sphere), and ˛ > 0 (ellipsoid with the minor axis placed along the z-axis). Moreover, the distance from any point (x, y) in an opening of the zone plate to the observation point (x , y , z ) can be written in the fourth order approximation as r z +
(x − x)2 + (y − y)2 [(x − x)2 + (y − y)2 ]2 − . 2z 8z 3
Hence if we again eliminate x and y by inserting Eqs. (15)–(18) into Eq. (9), we find a ray-optical solution for the diffracted beam (x , y , z ) Cm Um
× exp
ikn
2(z
−
x2 + (y − ym − ibm u)2 − zm
− ibm )
[x2 + (y − ym − ibm u)2 ]2
8(z − z m − ibm )3
−ikW
(m)
(x , y , z )
,
(19)
is the factor independent of x and y , and the aberration of Cm where the Gaussian beam converging to (or diverging from) the mth-order focus is given explicitly by
W (m) (x , y , z ) =
1 (m) (z m + ibm )4 [x2 + (y − z u )2 ]2 S 8 I (z − z − ib )4 m
m
1 (m) (z m + ibm )3 + SII [x2 + (y − z u )2 ] 2 (z − z − ib )3 m
m
OPLm
x2 + (y − y − ibu)2 = n z − n(z + ib) − n 2(z + ib)
× (y − z u )(n u ) +
2
(13)
× (y − z u )2 (n u )2 +
By applying Fermat’s principle
∂OPLm = 0, ∂x
∂OPLm = 0, ∂y
x=
−z m − ibm x, (z − z m − ibm )
(14)
y=
−z m − ibm (y (z − z m − ibm )
− z u ),
(m)
SI
3. Fourth order correction to a paraxial Gaussian beam
(m)
SII
= =
(m)
(m)
SV Considering the terms of up to fourth order in aperture variables x and y, we can rewrite the wave function in Eq. (1) as C (x, y, z) z − z − ib
−
ikn z − z − ib +
[x2 + (y − y − ibu)2 ]2 8(z − z − ib)3
x2 + (y − y − ibu)2 2(z − z − ib)
,
(16)
and we may also characterize the phase of the zone plate by (x, y) =
n 2 n (x + y2 ) + 3 (1 + ˛)(x2 + y2 )2 , 2f 8f
× (y − z u )(n u )3 , n (z m
+ ibm )3
1
(z m + ibm )2
(17)
=
− −
n 3
(z + ib)
1 (m) (z m + ibm ) S 2 V (z − z m − ibm ) (20)
n m (1 + ˛), f3
(21)
, (z + ib)2
n (z m + ibm )
1 1 − 2. n2 n
+
1
1
(m)
SIII = SIV =
m
with the coefficients of (15)
where we have used the paraxial parameters in Eq. (11). If we now insert Eq. (15) into Eq. (9) to eliminate x and y, the complex amplitude in Eq. (10) is obtained directly, except for the factor independent of x and y .
× exp
1 (m) (z m + ibm )2 SIV 4 (z − z m − ibm )2
× [x2 + (y − z u )2 ](n u )2 +
we get the formal equations for tracing the path of the complex ray
1 (m) (z m + ibm )2 S 2 III (z − z − ib )2 m
(x − x)2 + (y − y)2 + y2 −n + n . 2z 2(f/m) x
(18)
−
1 , n(z + ib)
(21)
Each term in Eq. (20) corresponds to the Seidel-type wavefront aberrations for the spherical wave radiating to a complex image point (0, ym + ibm sin u , z m + ibm cos u ) [11]. The first term is the so-called spherical aberration that is independent of an inclination of the beam, while it depends upon the fourth power of the transverse coordinates (x , y ). The second term known as linear coma is proportional to the angle of inclination u and the cube of the transverse coordinates (x , y ). The third and fourth terms are linear astigmatism and field curvature, respectively. Both of them change according to the squares of the angle of inclination u and the transverse coordinates (x , y ). The last term called distortion is pro portional to the cube of the angle of inclination u . Hence if the axis of the incident beam is coincident with the z-axis (i.e., u = 0), the
S. Chang / Optik 124 (2013) 85–90
first term is responsible only for degrading the beam quality. However, the influence of off-axial aberrations becomes important as an inclination of the beam increases. The field distribution deformed by the first and fourth terms is rotationally symmetric with respect to the beam axis, while the second and third terms make the field distribution asymmetric. On the other hand, if b is large compared with both z and (f/m), we have from Eq. (11) f , m
zm
bm
n
f 2
m
bn
.
(m)
n m 2 (m + 1 + ˛), f3
(23)
(m)
=
1 1 − 2. n2 n
(23)
(m) SI
depends upon the cube of an inverse focal length,
We see that (m) SII (m) SIII
(m)
SIII = SIV =
m , n f
is proportional to the square of an inverse focal length, both (m)
(m)
and SIV are in inverse proportion to the focal length, and SV is a function of the refractive indices n and n for the case of the stop in contact with the zone plate. When n = ± n, the coefficient (m) of distortion always disappears (i.e., SV = 0). 4. Effects of aberrations on the beam quality As obvious from Eq. (19), the diffracted beam converging to (or diverging from) the mth-order focus departs from an ideal Gaussian beam, while fourth order corrections are made for the beam. We investigate the quality factor to quantify how much the corrected beam departs from a Gaussian [12]. The beam quality 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. (19) may be represented by a Fourier integral (fx , fy , z m ) Vm
dx dy Um (x , y , z m )
=
× exp[−i2(fx x + fy y )],
(24)
where fx (or fy ) denotes the component of spatial frequency in the x (or y) direction. If we define the lth moments of the irradiance distribution in space and in spatial frequency as
dx dy xl |Um (x , y , z m )|2
xl =
dx dy
|Um (x , y , z m )|2
df x df y fxl |Vm (fx , fy , z m )|2
fxl =
,
df x df y |Vm (fx , fy , z m )|2
respectively, the beam quality factor in the x-section (or sagittal section) of the mth-order focus is defined by Mx2 =
Wx x ,
(26)
where we have used the notation
Wx = 2
x2 − x 2 ,
x = 2
fx2 − fx 2 .
10
15
20
Fig. 3. Variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders m = 1 (solid line), 3 (dashed line), and 5 (dotted line). The incident beam radius w0 is taken as a variable, while other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = 0, and f = 900 mm.
Replacing x (or fx ) by y (or fy ) in Eqs. (25)–(27), we can also evaluate the beam quality factor in the y-section (or meridian section) of the mth-order focus (i.e., My2 ). It is important to note here that if fourth order terms in Eq. (19) are neglected, Mx2 (or My2 ) approaches unity. First we examine the effect of spherical aberration on the beam quality, where we let u = 0. In this case, there is no difference between Mx2 and My2 . Fig. 3 shows the variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders m = 1 (solid line), 3 (dashed line) and 5 (dotted line), where we take the incident beam radius w0 as a variable. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = 0 and f = 900 mm. The spherical aberration in Eq. (20) is independent of an inclination of the beam, while it depends upon the fourth power of the transverse coordinates (x , y ) . Therefore, if w0 gets close to zero, the spherical aberration disappears so that we have Mx2 = My2 1. However, Mx2 (or My2 ) changes to a certain value greater than 1 by increasing w0 . When ˛ = − 1 and b z, (f/m), the coefficient of spherical aberration in Eq. (23) depends upon the cube of the order m. Consequently, the dashed (or dotted) curve is shown to change more rapidly with w0 than the solid curve. Fig. 4 also shows the variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders (a) m = 1 and (b) m = 3, where we take the conic constant ˛ as a variable. The solid, dashed, and dotted lines are of f = 300 mm, 600 mm, and 900 mm, respectively. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, u = 0 and w0 = 10 mm. Since b( 5 × 105 mm) is much bigger than both z and (f/m), it follows from Eq. (23) that spherical aberration disappears at ˛ −(m2 + 1).
, (25)
5
Incident beam radius, w0 (mm)
m2 , f2
SV
m=1
1.0
0
(m)
m=3
1.2
0.8
(m)
SII
m=5
1.4
(22)
Inserting Eq. (22) into Eq. (21) leads us to SI
1.6
Beam quality factor
88
(27)
(28)
As shown in Fig. 4, therefore, Mx2 (or My2 ) for the diffracted beam converging to the focus of order m = 1 (or m = 3) approaches 1 at ˛ − 2 (or ˛ − 10). Next we look into the influence of off-axial aberrations on the / 0. If the axis of the incident (or beam quality, where we let u = diffracted) beam is slightly inclined with respect to the z-axis, Mx2 and My2 are not always equal. Fig. 5 shows the variations of Mx2 (solid line) and My2 (dashed line) for the diffracted beam converging to the focus of order m = 1, where we take the principal focal length f as a variable. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = −0.05 rad and w0 = 5 mm. When b (f/m) 1, the aberration coefficients in Eq. (23) disappear. Therefore, both Mx2 and My2 get close to 1 as f increases. The curve of Mx2 is affected by the first and fourth terms in Eq. (20) which correspond to spherical aberration and field curvature, respectively. The difference in the curves of Mx2 and My2 is caused by the second and third terms
S. Chang / Optik 124 (2013) 85–90
a
1.6
1.8
Bea am quality fac ctor
1.6
Beam quality factor
89
f=300 mm
1.4
1.2
f=600 mm 1.0
1.4
2
My
1.2
2
Mx
1.0
f=900 mm
0.00
0.8 -6
-4
-2
0
2
-0.02
-0.04
-0.06
-0.08
-0.10
Angle of inclination (radians)
Conic constant, Fig. 6. Curves of Mx2 (solid line) and My2 (dashed line) plotted as a function of the angle of inclination u, where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, w0 = 5 mm and f = 900 mm
Beam quality factor
1.6
f=300 mm
1.20 1.4
f=600 mm
1.15
1.2
1.0
f=900 mm
0.8 -14
-12
-10
-8
-6
Conic constant,
Beam quality factor
b
1.8
2
My 1.10
2
Mx
1.05
1.00
Fig. 4. Variations of Mx2 (or My2 ) for the diffracted beams converging to the foci of orders (a) m = 1 and (b) m = 3. The solid, dashed, and dotted lines are of f = 300 mm, 600 mm, and 900 mm, respectively. The conic constant ˛ is taken as a variable, while other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, u = 0, and w0 = 10 mm.
0.95 0
5
10
15
20
Incident beam radius, w0 (mm) Fig. 7. Curves of Mx2 (solid line) and My2 (dashed line) plotted as a function of the incident beam radius w0 , where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, u = −0.01 rad and f = 900 mm.
2.0
Beam quality factor
1.8
different values greater than 1 as w0 increases. The difference in the changes of Mx2 and My2 results from the terms of linear coma and linear astigmatism in Eq. (20). In general, the fourth order theory of propagation of a slightly inclined Gaussian beam as described above may be applied to evaluate the degradation in the quality of the Gaussian beam focused by a zone plate.
1.6 1.4
2
My
1.2
2
Mx
1.0 0.8 400
600
800
1000
1200
Principal focal length, f (mm) Fig. 5. Curves of Mx2 (solid line) and My2 (dashed line) plotted as a function of the principal focal length f, where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, w0 = 5 mm and u = −0.05 rad.
in Eq. (20) which are called linear coma and linear astigmatism, respectively. In Fig. 6 we plot the variations of Mx2 (solid line) and My2 (dashed line) as a function of the angle of inclination u, where we take the parameters of z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, m = 1, w0 = 5 mm and f = 900 mm. When the angle of inclination u is extremely small, we have Mx2 = My2 1. In this case the influence of spherical aberration is negligible. Therefore, the fourth term (i.e., field curvature) in Eq. (20) is responsible for the change of Mx2 , whereas the curve of My2 is under the influence of all off-axial aberrations in Eq. (20). Fig. 7 shows the variations of Mx2 (solid line) and My2 (dashed line) for the diffracted beam converging to the focus of order m = 1, where we take the incident beam radius w0 as a variable. Other parameters are fixed at z = 0, = 632.8 nm, n = n = 1, ˛ = − 1, u = −0.01 rad and f = 900 mm. If w0 goes to zero, all aberrations in Eq. (20) disappear so that we have Mx2 = My2 1. However, Mx2 and My2 change to the
5. Conclusions In this paper, we have discussed the Seidel-type wavefront aberrations which degrade the quality of the Gaussian beam focused by a zone plate. First we have examined the diffraction of a complexsource-point spherical wave (CSPSW) through the zone plate, while the terms of up to fourth order in aperture variables are taken into consideration. The CSPSW is known to be the sum of all the higher order corrections to the paraxial Gaussian beam. Then we have found a ray-optical solution for the diffracted beam that is represented explicitly in terms of Seidel-type wavefront aberrations. If the Rayleigh range of the incident beam is large compared with the principal focal length of the zone plate, spherical aberration (or linear coma) changes according to the cube (or square) of an inverse focal length, both linear astigmatism and field curvature are in inverse proportion to the focal length, and distortion is a function of the refractive indices of two media separated by the zone plate. Based on the derived formulas, we have also evaluated the effects of aberrations on the quality of the diffracted beam through the zone plate. If the axis of the incident beam is inclined slightly with respect to the optic axis, both spherical aberration and field curvature affect the beam quality in the sagittal section of the focal plane, whereas the beam quality in its meridian section is under the
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