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Focusing a Gaussian laser beam without focal shift
15 July 1994 OPTICS COMMUNICATIONS ELSEVIER
Optics Communications 109 (1994) 368-374
Focusing a Gaussian laser beam without focal shift Akifumi Yoshida, Toshimitsu Asakura Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 060, Japan
Received 23 November 1993
Abstract
It is shown for the first time that the focal shift of a Gaussian laser beam can be offset with the use of the spherical aberration of a focusing lens. The diffraction integral, in the presence of aberrations, which is applicable to a low Fresnel number focusing system is derived. A new approximate formula is proposed for the normalized axial intensity in the presence of aberrations. This formula reduces to an exact formula for a nontruncated, aberration-free Gaussian beam. The expression for the coefficient of spherical aberration required to offset the focal shift is given as a function of the Fresnel number of the beam. The axial intensity distributions of Gaussian beams with the spherical aberration are presented, predicting the focusing without the focal shift.
1. Introduction
The propagation and focusing o f a Gaussian laser beam have been the subjects o f a great deal o f interest because o f the increasing use o f optical beams in a variety o f applications, such as laser communications, optical interconnections, and optical processings. Numerous research works have been done in the past three decades for a Gaussian laser beam which has a perfect spherical wavefront, i.e., for an aberration-free Gaussian beam [ 1-4 ]. When an ideal spherical Gaussian beam is passed through a focusing lens, its wavefront often deviates from its original perfect spherical shape. In other words, the wavefront o f the beam acquires aberrations by passing through an aberrating lens. The beam is then affected by the wavefront aberrations and the properties of propagation and focusing will deviate from those in an ideal spherical Gaussian beam. A number o f studies based mostly on the Debye approximation (the classical theory) have also been carried out on the propergation and focusing o f an aberrated Gaussian laser beam [ 5-11 ]. The classical
theory is fully justified when the Fresnel number o f a beam is much larger than unity. Here, the Fresnel n u m b e r of a Gaussian beam is defined as N = W2/2R ,
( 1)
where w is the beam radius at the aperture, which is defined as the distance from the beam center to the point where the intensity becomes equal to exp ( - 2) o f the value at the center, 2 is the wavelength o f light, and R is the radius o f curvature for the wavefront at the lens exit plane. In contrast to a large Fresnel number, it was found that, when the Fresnel number o f a beam is close to the order o f unity or less than unity, the accuracy o f classical theory is greatly degraded [3,12-14]. One of the interesting properties of an optical beam is a so-called focal shift which is described for a Gaussian beam as the shift o f a beam waist position from the geometrical focal point to a somewhat closer point to the aperture. It has been found that the focal shift becomes larger when the Fresnel n u m b e r N o f a beam becomes smaller [ 3,12-14 ]. On account o f this nature of the focal shift, the diffraction integral which
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)OO185-W
A. Yoshida, T. Asakura /Optics Communications 109 (1994) 368-3 74
is modified to be applicable to a low Fresnel number system with sufficient accuracy has been developed, and the focal shift phenomena have been investigated over the last decade on the basis of this modified theory [ 15-23 ]. Little investigation, however, has yet been performed by using the modified theory on an optical beam with the aberrated wavefront. Particularly, there have been no studies on the focal shift phenomena in the presence of aberrations. In Ref. [ 22 ], while the investigation based on the modified theory has been performed relating to aberrations, the intensity distribution along the optical axis in the presence of spherical aberration has not been evaluated, and this axial intensity distribution is the base of our present study. In this paper, the diffraction integral in the presense of aberrations, which is applicable to a low Fresnel number focusing system, is derived in Sect. 2 for the later discussion, since no studies have derived this integral in the presense of aberrations. In Sect. 3, a new approximate formula is proposed for the normalized intensity along the axis. In Sect. 4, this formula is proved to be reduced to an exact formula for a nontruncated and aberration-free Gaussian beam. The formula for an axial intensity ofa Gaussian beam in the presence of primary spherical aberration is derived on the basis of the above formula. In Sect. 5, it is shown that, with the positive primary spherical aberration, the focal shift of a Gaussian beam is possible to be offset. The amount of spherical aberration required for this purpose is evaluated. The axial intensity distributions are shown in the case where the spherical aberration offsetting the focal shift is present.
2. Derivation of the modified diffraction integral in the presence of aberrations
Consider, in Fig. 1, a Gaussian laser beam which passes through an aberrated lens, emerges from a circular aperture, and converges towards the axial point, z = R , on the z axis which is taken as the direction of propagation of the beam. Let (p, 0, z) be the cylindrical coordinates with an origin at the center O of the aperture. Let 0 and Q be the points, respectively, where a ray intersects the wavefront through O and the spherical reference sphere centered on a geomet-
369
z=R
~/
Spherical reference sphere centered on z = R
Wavefront
Fig. 1. Geometry and notation relating to the focusing of a converging spherical wave with aberrations.
rical focal point, z = R . Here, the geometrical focal point indicates a point where the paraxial rays converge, as given by geometrical optics. The deformation of the actual wavefront from the ideal perfect spherical wavefront or the reference sphere is described by an aberration function ~R which is defined as the optical path difference between 0 and Q measured along the ray as in Ref. [ 5 ]. Here, the subscript R in ~bR indicates that the aberration function is defined with respect to the reference sphere centered at the point, z = R. By assuming a Gaussian amplitude distribution on the aperture plane and by including the wavefront aberration, the complex amplitude of the field emerging from the aperture at a point Q is expressed in the form u ( Q ) =A exp(--p2/w2) exp [ik(~R - - p 2 / 2 R ) ] ,
(2) where A is a constant amplitude factor, p the radial coordinate of point Q, w the beam radius, k--27t/2 the wave number, 2 being the wavelength of light, and R the radius of the reference sphere passing through O. In Eq. (2), the time-dependent factor exp ( - ito t) is omitted. Note that the spherical wavefront is expressed in Eq. (2) by the quadratic surface under the paraxial approximation, since the paraxial region is dealt with in this study. According to the Rayleigh-Sommerfeld diffraction formula which is a mathematical representation of the Huygens-Fresnel principle [24], the field amplitude at the observation point P that is not too close to the aperture plane is given by
A. Yoshida,T.Asakura/ OpticsCommunications109(1994)368-374
370
27t a
l ff exp(iks) cos a d S , u(P)= ~ u(Q) - -s
(3)
A
u(z)= ~zeXp(ikz) ~ ~ exp(-p2/w2) 0
where s is the distance QP, cos c~ is the inclination factor, and the integration extends over the portion of the spherical reference surface that approximately fills the aperture. Substitution of Eq. (2) into (3) yields
0
× exp{ik~R + 1 ( ~ -- 1)p2]} p dp d0.
(7,
In Eq. (7), the term ,
u(P) = ~
exp(-p2/w2)
exp[ik(~R-p2/2R) 1
exp(iKS) X cos o~dS. S
(4)
This equation can be simplified in the situation where the angle ol is very small. This is true when the distance z from the plane of observation to the aperture plane far exceeds the transverse dimensions of the aperture and the observation region of interest near the z axis. Under these conditions we can use the approximation cos ot_~ 1 and replace s in the denominator of Eq. (4) by z. The result is A
u(P)= ~zz ~f exp(-p2/w2) ×exp[ik(cI~-pZ/2R)
]exp (iks) dS.
(5)
Note that we have not replaced s by z in the exponent because the resulting errors are multiplied by a very large number k, and that, consequently, the difference between ks and kz could be comparable to or larger than 2m By applying the Fresnel approximation to the term exp(iks) in Eq. (5) and by letting the observation point P lie on the axis, the field amplitude at P along the z axis is given by
corresponds to the optical path length change caused by the change of the observation point z, or by the change of the radius R of the reference sphere when z is fixed. Note that, in this expression, the spherical surfaces of the wavefronts are replaced by the quadratic surfaces under the paraxial, Fresnel approximation.
3. Axial intensity formula Next, let us evaluate the axial intensity distribution when an aberrated Gaussian beam propagates along the optical axis. The wavefront deformation which is acquired by passing through an aberrating lens is expressed, as described in Sect. 2, by the aberration function, DR. It is convenient to normalize the axial intensity by the corresponding intensity that would be obtained at the geometrical focal point, z=R, in the absence of aberrations. The axial intensity expressed in this way is given, by using Eq. (7), in the form 2n
a
2
i(z)=(R)2 !!exp(-p2/w2, exp(ikF, pdpdO
27t a
u(z)=
A
~zzeXp(ikz) f 0
2rt a
f exp(--p2/w2)
--1
0
× exp [ i k ( ~ R - p 2 / Z R )
]
X exp(ikp2/2z) p dp dO,
where the phase factor F is defined as (6)
where a is the radius of the aperture. By rearranging the terms, Eq. (6) can be written as
F = qbR + I(--lz -- 1 ) p 2 .
(9)
The factor, (R/z) 2, in Eq. ( 8 ) is interpreted as being caused by the inverse square law dependence of the intensity on the distance from the aperture plane. The residual factor in Eq. (8) corresponds to the Strehl
A. Yoshida, T. Asakura / Optics Communications 109 (1994) 368-374
intensity or Strehl definition (SD) which is commonly defined as the ratio of the intensity at the best focus with aberrations present to the one at the geometrical focus without aberrations. When the phase factor Fis small, the exponential term, exp (ikF), can be approximated by the first three terms of a power expansion, leading to the known Marrchal formula for the Strehl definition, namely, SD~
1 -kE[F
2- (F)2]
,
(10)
where F n denotes the weighted average value of the nth power ofF, i.e., 2x a
0 0 2~t a
--1
o
The normalized axial intensity i (z) given by Eq. ( 8 ) can therefore be written as
371
when the beam center coincides with the optical axis, the only aberration which is usually serious is spherical aberrations. In this case, a primary spherical aberration which is defined with respect to the reference sphere centered at the geometrical focal point, z=R, can be expressed as t~R = (S1/w4)p 4 •
( 15 )
Here, for simplicity, the secondary and the higher order spherical aberrations are not taken into consideration in the present discussion. In Eq. (15) the spherical aberration coefficient $1 is defined in terms of the wavefront deformation at the location of the beam radius w rather than at the location of the aperture edge a. The definition of the coefficient in this way is suitable for the case where the beam radius w is much smaller than the aperture size a, since the power in the beam is concentrated near the center of the aperture and the power at the aperture edge is negligibly small. By defining the coefficient B as W2
i(z) = (R/z) 2 { 1 - k 2 [ p 2 _ (p)2 ] }.
( 12 )
Under the condition that the term, k 2 [ ~ 2 _ (~)2 ], is small as compared to unity, and by applying the following approximate formula,
and by using Eq. (9), the phase factor F is written in the form
l-x~l/(l+x)
(13)
F=~R+
(14)
We assume in this study, as mentioned above, that the aperture radius a is much larger than the beam radius w. Therefore, the Gaussian distribution of the optical power near the aperture edge is practically zero. Consequently, the upper integral limit a in Eq. ( 11 ) can be replaced by ~ . Then, Eq. ( 11 ) becomes
(]x]
Eq. (12) can be written as 1
i(z)=
I+k2[FE_(F)2 ] .
This is a formula proposed here to express the axial intensity distribution. Eq. (14) is an approximate formula applicable to a beam which is generally truncated by the aperture and has aberrations. It will be shown in the next section that Eq. (14) reduces to an exact expression for a Gaussian beam in the case where the beam is nontruncated and aberration-free.
1 z1-
2= ~ P
+ ~sP
"
(17)
F--~= f ~ exp(--p2/w2) F"p dp f ~ e x p ( - p E / w 2 ) p dp 2 ~
- w2 ~ exp(-pE/w 2) Fnpdp.
(18)
0
Substituting Eq. (17) into Eq. (18), we obtain
4. Axial intensity in the presence of spherical aberration
F=2S1 +B, and F 2 = 24S1 + 12SIB+2B 2. Therefore,
When an optical system is designed to be used with the laser beam propagating along the optical axis and
Substitution ofEq. (19) into Eq. (14) yields the normalized axial intensity for a nontruncated, aberrated
~_ F
(/~)2 =
20S 2 + 8S1B + B E .
( 19 )
A. Yoshida, T. Asakura / Optics Communications 109 (1994) 368-374
372
Gaussian beam given by
l+k2(20S~+8SIB+B2).
i(z)=
(20)
By using Eqs. (1) and (16), Eq. (20) can be rearranged into the form
duces to an exact formula for a nontruncated, aberration-free Gaussian beam in the limit where there is no truncation and aberration-free. By using Eqs. (22) and (24), the axial intensity distribution formula of Eq. (21 ) can be written as
i(z) =iM[ 1 + iM(Z/R-- zM/R ) 2
i(z) =RE[ (1 +~2N2-- 16Ir2NS+80~2S2)Z2 + ( 16IrERNS_2rcERN2)Z+IrER2N2] -1,
× ( 1 + ~zZN2 - 16Ir2NS+ 8 0 ~ 2 S
where S is the redefined primary spherical aberration coefficient which is measured in units of wavelengths as S=S~/2. When aberrations are present, the point of maximum intensity in the diffraction field is referred to as the so-called diffraction focus [ 5 ]. The diffraction focus defined in this way is obtained by equating to zero the derivative of the denominator of Eq. (21) with respect to z. Hence, the location ZM of the diffraction focus is obtained as (22)
For S = 0, i.e., in the absence of spherical aberrations, Eq. (22) reduces to Ir2N2
ZM=R 1 +ir2N2,
(S=0) .
(23)
This is a well-known formula for the beam waist position on axis for an aberration-free, nontruncated Gaussian beam [ 14,16 ]. Eq. (22) is derived from the axial intensity distribution formula of Eq. (14) which is introduced as an approximate formula for an aberrated beam. In the limit S ~ 0 , however, Eq. (22) turns out to be an exact formula for an aberrationfree Gaussian beam. Moreover, the maximum intensity iM at Z=ZM is given, by using Eq. (21), as 1+ iM =
16~z2NS+ 80ir2S 2 IRENE(I + 16ir2S 2)
Ir2N2 -
(24)
In the limit S-,0, Eq. (24) reduces also to a wellknown formula for an aberration-free, nontruncated Gaussian beam in the form 1 + Ir2N2
iM=
IraN z ,
(S=0) .
(26)
5. Focusing without focal shift With the positive primary spherical aberration, the focal shift of a Gaussian beam can be offset. This will be shown in the following. To accomplish this, the location ZM of maximum intensity has to be equal to R. By using Eq. (22), this condition becomes ZM
Ir2N 2-- 87c2NS ZM = R 1 + Ir2N2 -- 1 6 7 r 2 N S + 8 0 i r 2 S 2 "
2) ] - 1 .
(21)
(25)
From the results of Eqs. ( 21 ), (23 ) and ( 25 ), we can conclude that the proposed formula of Eq. (14) re-
IRENE-- 8IrENS
R - 1 + n 2 N 2 - 16n2NS+8On2S 2 = 1 .
(27)
For the special case of S = 0, Nhas to be much larger than unity, i.e., N>> 1, in order to satisfy Eq. (27). In other words, for an aberration-free Gaussian beam and for N>> 1, the beam waist of the focused beam is located at the geometrical focus. This point is mentioned in Sect. 1. That is, for N>> 1 the classical theory is applicable and the beam waist position coincides with the geometrical focal point. When S is not equal to zero and by using Eq. (27), we obtain the following quadratic equation for the value of the coefficient S that can offset the focal shift:
S z - (iV/10)S+ 1/80ir2
=
0.
(28)
The condition that Eq. (28) has the roots of real values is N > 5/ir= 0.71176.
(29)
Under this condition, Eq. (28) has the solutions in the form S = (1/20) (N+_x/N2-5/ir 2 ).
(30)
As seen above, under the condition of N>> 1, S has to be equal to zero in order to get the result ofzM = R . From this we see that, in Eq. (30), only the negative sign before the square root is acceptable. Thus, we obtain the required value for S as
A. Yoshida, T. Asakura / Optics Communications 109 (1994) 368-374
S = (1/20) ( N - ~ / N 2 - 5 / ] r 2) .
(31)
For N = 1 and 2, we obtain S=0.014879 wavelengths ( N = 1 ), and S=0.0065469 wavelengths ( N = 2 ) , respectively, as the required values. Fig. 2 shows the normalized axial intensity distributions along the z axis which are calculated from Eq. (26) for N = 1 and 2. In the figures the axial distance z is normalized by R. Therefore, the point z / R = 1 corresponds to the geometrical focal point. The intensity in each curve is normalized by the intensity at the geometrical focal point in the absence of aberrations. The curve A in each figure corresponds to the I
I
I
I
I
=
._~
1
U~
"1o
.~ 0.8
E o Z
0.6
case of an aberration-free Gaussian beam and, consequently, the normalized intensity at z / R = 1 is unity as defined. For curve A in each figure, the maxim u m point of intensity on axis, which is located at a somewhat closer point to the aperture side rather than at the geometrical focal point, corresponds to the beam waist position. The distance between the beam waist position and the geometrical focus in curve A is defined as the focal shift. It is seen that the focal shift of an aberration-free beam (curve A) is greater for the lower Fresnell number beam than for the higher Fresnel number beam. When the spherical aberration is present, the maximum intensity point or diffraction focus shifts from the original beam waist position of an aberration-free beam, and the magnitude of the intensity at the maximum point is degraded. In curve B of each figure, the diffraction focus coincides with the geometrical focus, since the amount of the spherical aberration coefficient S is chosen according to Eq. (31 ) so as to offset the focal shift. The magnitude of the normalized intensity at the geometrical focus is calculated by using Eq. (20) under the condition of z = R or B = 0. In this manner, the maximum intensity in curve B is calculated by
i ( z = R ) = 1/(1 + 80n2S 2) , '
'
018
'
'
'
'
'
'
112
z/R I
'
I
'
I
!
I
1
0.8 °
-
-
._N 0.6 o
z
0.4
0,2
I
,
I
,
I
,
0.8
I
1.2 z/R
Fig. 2. The normalized axial intensity distributions along the optical axis which are calculated from Eq. (26) for N = 1 and 2. The axial point z is normalized by the radius R of the reference sphere. Therefore, the location z / R = 1 corresponds to the geometrical focal point.
373
(32)
where, for curve B, the coefficient S is chosen so as to offset the focal shift as mentioned above. For curve B, we use Eq. (32) to give the maximum intensity. However, Eq. (32) is generally valid for small aberrations and gives a normalized intensity at the geometrical focal point irrespective of the position of a diffraction focus. The normalized intensity at the geometrical focal point is always smaller than unity as seen from Eq. (32). From this we can conclude that the intensity at the geometrical focus is a good measure for the amount of the spherical aberration wherever a diffraction focus locates. Moreover, as seen from Eq. (14), the intensity at the geometrical focus is a good measure for the amount of wavefront deformation irrespective of the types of aberrations wherever a diffraction focus locates.
6. Conclusion
The diffraction integral, in the presence of aberra-
374
A. Yoshida, T. Asakura /Optics Communications 109 (1994) 368-374
tions, which is applicable to a low Fresnel n u m b e r focusing system is d e r i v e d on the basis o f the Fresnel a p p r o x i m a t i o n . A new a p p r o x i m a t e f o r m u l a for the n o r m a l i z e d axial intensity in the presence o f aberrations is proposed. This f o r m u l a reduces to an exact formula for the case o f a nontruncated, aberrationfree G a u s s i a n beam. F o r the spherical aberration, an axial intensity dist r i b u t i o n f o r m u l a for a n o n t r u n c a t e d G a u s s i a n b e a m is derived. This formula is shown to be reduced to an exact expression for an aberration-free G a u s s i a n b e a m in the limit when the a b e r r a t i o n coefficient vanishes. It is shown for the first time that, with the positive p r i m a r y spherical aberration, the focusing without the focal shift is possible unless the Fresnel n u m b e r o f the b e a m is less than 0.71. The expression for the spherical a b e r r a t i o n coefficient required for the offsetting o f this focal shift is given as a function o f the Fresnel n u m b e r o f the beam. The axial intensity distributions for low Fresnel n u m b e r G a u s s i a n b e a m s with the spherical aberrations whose magnitudes are equal to those required for the offsetting o f this focal shift, are presented. These intensity distributions show the focusing without the focal shift. It is shown that the n o r m a l i z e d intensity at the geometrical focus is a good measure for the a m o u n t o f wavefront aberrations irrespective o f the position o f a diffraction focus.
Acknowledgements We would like to acknowledge the c o m p u t e r plotting by Y. Sakurada.
References [ 1] K. Kogelnik, Bell Syst. Tech. J. 44 (1964) 455. [2] L.D. Dickson, Appl. Optics 9 (1970) 1854. [3] D.A. Holmes, J.E. Korka and P.V. Avizonis, Appl. Optics 11 (1972) 565. [4] A. Yariv, Optical Electronics, 3d Ed. (Holt, Rinehart and Winston, New York, 1985 ) Chap. 2. [5 ] M. Born and E. Wolf, Principles of optics (Pergamon, New York, 1970) Chap. 9. [6] J.J. Stamnes, Waves in focal regions (Hilger, Bristol, 1986), Sec. 12.6. [7] D.D. Lowenthal, Appl. Optics 13 (1974) 2126. [ 8 ] J.T. Hunt, P.A. Renard and R.G. Nelson, Appl. Optics 15 (1976) 1458. [9] A. Yoshida and T. Asakura, Optics Comm. 14 ( 1975 ) 211. [ 10] A. Yoshida and T. Asakura, Optics Comm. 25 (1978) 133. [ 11 ] A. Yoshida, Appl. Optics 21 (1982) 1812. [ 12 ] Y. Li and E. Wolf, Optics Comm. 39 ( 1981 ) 211. [ 13 ] Y. Li and E. Wolf, Optics Comm. 42 ( 1982 ) 151. [ 14 ] J.J. Stamnes. Waves in Focal Regions (Hilger, Bristol, 1986 ) Sec. 12.2. [ 15 ] M.P. Givens, Optics Comm. 41 (1982) 145. [ 16] W.H. Carter, Appl. Optics 21 (1982) 1989. [ 17] Y. Li and H. Platzer, Optics Acta 30 ( 1983) 1621. [ 18] G.D. Sucha and W.H. Carter, Appl. Optics 23 (1984) 4345. [ 19] V.N. Mahajan, Appl. Optics 22 (1983) 3042. [20] S. Szapiel, Optics Lett. 8 (1983) 327. [21 ] R. Herloski, J. Opt. Soc. Am. A 2 ( 1985 ) 1027. [22] V.N. Mahajan, J. Opt. Soc. Am. A 3 (1986) 470. [23] Y. Li, Opt. Eng. 32 (1993) 774. [24] J.W. Goodman, Introduction to Fourier optics (McGrawHill, New York, 1968) Sec. 3.4.
Optics Communications 109 (1994) 368-374
Focusing a Gaussian laser beam without focal shift Akifumi Yoshida, Toshimitsu Asakura Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 060, Japan
Received 23 November 1993
Abstract
It is shown for the first time that the focal shift of a Gaussian laser beam can be offset with the use of the spherical aberration of a focusing lens. The diffraction integral, in the presence of aberrations, which is applicable to a low Fresnel number focusing system is derived. A new approximate formula is proposed for the normalized axial intensity in the presence of aberrations. This formula reduces to an exact formula for a nontruncated, aberration-free Gaussian beam. The expression for the coefficient of spherical aberration required to offset the focal shift is given as a function of the Fresnel number of the beam. The axial intensity distributions of Gaussian beams with the spherical aberration are presented, predicting the focusing without the focal shift.
1. Introduction
The propagation and focusing o f a Gaussian laser beam have been the subjects o f a great deal o f interest because o f the increasing use o f optical beams in a variety o f applications, such as laser communications, optical interconnections, and optical processings. Numerous research works have been done in the past three decades for a Gaussian laser beam which has a perfect spherical wavefront, i.e., for an aberration-free Gaussian beam [ 1-4 ]. When an ideal spherical Gaussian beam is passed through a focusing lens, its wavefront often deviates from its original perfect spherical shape. In other words, the wavefront o f the beam acquires aberrations by passing through an aberrating lens. The beam is then affected by the wavefront aberrations and the properties of propagation and focusing will deviate from those in an ideal spherical Gaussian beam. A number o f studies based mostly on the Debye approximation (the classical theory) have also been carried out on the propergation and focusing o f an aberrated Gaussian laser beam [ 5-11 ]. The classical
theory is fully justified when the Fresnel number o f a beam is much larger than unity. Here, the Fresnel n u m b e r of a Gaussian beam is defined as N = W2/2R ,
( 1)
where w is the beam radius at the aperture, which is defined as the distance from the beam center to the point where the intensity becomes equal to exp ( - 2) o f the value at the center, 2 is the wavelength o f light, and R is the radius o f curvature for the wavefront at the lens exit plane. In contrast to a large Fresnel number, it was found that, when the Fresnel number o f a beam is close to the order o f unity or less than unity, the accuracy o f classical theory is greatly degraded [3,12-14]. One of the interesting properties of an optical beam is a so-called focal shift which is described for a Gaussian beam as the shift o f a beam waist position from the geometrical focal point to a somewhat closer point to the aperture. It has been found that the focal shift becomes larger when the Fresnel n u m b e r N o f a beam becomes smaller [ 3,12-14 ]. On account o f this nature of the focal shift, the diffraction integral which
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)OO185-W
A. Yoshida, T. Asakura /Optics Communications 109 (1994) 368-3 74
is modified to be applicable to a low Fresnel number system with sufficient accuracy has been developed, and the focal shift phenomena have been investigated over the last decade on the basis of this modified theory [ 15-23 ]. Little investigation, however, has yet been performed by using the modified theory on an optical beam with the aberrated wavefront. Particularly, there have been no studies on the focal shift phenomena in the presence of aberrations. In Ref. [ 22 ], while the investigation based on the modified theory has been performed relating to aberrations, the intensity distribution along the optical axis in the presence of spherical aberration has not been evaluated, and this axial intensity distribution is the base of our present study. In this paper, the diffraction integral in the presense of aberrations, which is applicable to a low Fresnel number focusing system, is derived in Sect. 2 for the later discussion, since no studies have derived this integral in the presense of aberrations. In Sect. 3, a new approximate formula is proposed for the normalized intensity along the axis. In Sect. 4, this formula is proved to be reduced to an exact formula for a nontruncated and aberration-free Gaussian beam. The formula for an axial intensity ofa Gaussian beam in the presence of primary spherical aberration is derived on the basis of the above formula. In Sect. 5, it is shown that, with the positive primary spherical aberration, the focal shift of a Gaussian beam is possible to be offset. The amount of spherical aberration required for this purpose is evaluated. The axial intensity distributions are shown in the case where the spherical aberration offsetting the focal shift is present.
2. Derivation of the modified diffraction integral in the presence of aberrations
Consider, in Fig. 1, a Gaussian laser beam which passes through an aberrated lens, emerges from a circular aperture, and converges towards the axial point, z = R , on the z axis which is taken as the direction of propagation of the beam. Let (p, 0, z) be the cylindrical coordinates with an origin at the center O of the aperture. Let 0 and Q be the points, respectively, where a ray intersects the wavefront through O and the spherical reference sphere centered on a geomet-
369
z=R
~/
Spherical reference sphere centered on z = R
Wavefront
Fig. 1. Geometry and notation relating to the focusing of a converging spherical wave with aberrations.
rical focal point, z = R . Here, the geometrical focal point indicates a point where the paraxial rays converge, as given by geometrical optics. The deformation of the actual wavefront from the ideal perfect spherical wavefront or the reference sphere is described by an aberration function ~R which is defined as the optical path difference between 0 and Q measured along the ray as in Ref. [ 5 ]. Here, the subscript R in ~bR indicates that the aberration function is defined with respect to the reference sphere centered at the point, z = R. By assuming a Gaussian amplitude distribution on the aperture plane and by including the wavefront aberration, the complex amplitude of the field emerging from the aperture at a point Q is expressed in the form u ( Q ) =A exp(--p2/w2) exp [ik(~R - - p 2 / 2 R ) ] ,
(2) where A is a constant amplitude factor, p the radial coordinate of point Q, w the beam radius, k--27t/2 the wave number, 2 being the wavelength of light, and R the radius of the reference sphere passing through O. In Eq. (2), the time-dependent factor exp ( - ito t) is omitted. Note that the spherical wavefront is expressed in Eq. (2) by the quadratic surface under the paraxial approximation, since the paraxial region is dealt with in this study. According to the Rayleigh-Sommerfeld diffraction formula which is a mathematical representation of the Huygens-Fresnel principle [24], the field amplitude at the observation point P that is not too close to the aperture plane is given by
A. Yoshida,T.Asakura/ OpticsCommunications109(1994)368-374
370
27t a
l ff exp(iks) cos a d S , u(P)= ~ u(Q) - -s
(3)
A
u(z)= ~zeXp(ikz) ~ ~ exp(-p2/w2) 0
where s is the distance QP, cos c~ is the inclination factor, and the integration extends over the portion of the spherical reference surface that approximately fills the aperture. Substitution of Eq. (2) into (3) yields
0
× exp{ik~R + 1 ( ~ -- 1)p2]} p dp d0.
(7,
In Eq. (7), the term ,
u(P) = ~
exp(-p2/w2)
exp[ik(~R-p2/2R) 1
exp(iKS) X cos o~dS. S
(4)
This equation can be simplified in the situation where the angle ol is very small. This is true when the distance z from the plane of observation to the aperture plane far exceeds the transverse dimensions of the aperture and the observation region of interest near the z axis. Under these conditions we can use the approximation cos ot_~ 1 and replace s in the denominator of Eq. (4) by z. The result is A
u(P)= ~zz ~f exp(-p2/w2) ×exp[ik(cI~-pZ/2R)
]exp (iks) dS.
(5)
Note that we have not replaced s by z in the exponent because the resulting errors are multiplied by a very large number k, and that, consequently, the difference between ks and kz could be comparable to or larger than 2m By applying the Fresnel approximation to the term exp(iks) in Eq. (5) and by letting the observation point P lie on the axis, the field amplitude at P along the z axis is given by
corresponds to the optical path length change caused by the change of the observation point z, or by the change of the radius R of the reference sphere when z is fixed. Note that, in this expression, the spherical surfaces of the wavefronts are replaced by the quadratic surfaces under the paraxial, Fresnel approximation.
3. Axial intensity formula Next, let us evaluate the axial intensity distribution when an aberrated Gaussian beam propagates along the optical axis. The wavefront deformation which is acquired by passing through an aberrating lens is expressed, as described in Sect. 2, by the aberration function, DR. It is convenient to normalize the axial intensity by the corresponding intensity that would be obtained at the geometrical focal point, z=R, in the absence of aberrations. The axial intensity expressed in this way is given, by using Eq. (7), in the form 2n
a
2
i(z)=(R)2 !!exp(-p2/w2, exp(ikF, pdpdO
27t a
u(z)=
A
~zzeXp(ikz) f 0
2rt a
f exp(--p2/w2)
--1
0
× exp [ i k ( ~ R - p 2 / Z R )
]
X exp(ikp2/2z) p dp dO,
where the phase factor F is defined as (6)
where a is the radius of the aperture. By rearranging the terms, Eq. (6) can be written as
F = qbR + I(--lz -- 1 ) p 2 .
(9)
The factor, (R/z) 2, in Eq. ( 8 ) is interpreted as being caused by the inverse square law dependence of the intensity on the distance from the aperture plane. The residual factor in Eq. (8) corresponds to the Strehl
A. Yoshida, T. Asakura / Optics Communications 109 (1994) 368-374
intensity or Strehl definition (SD) which is commonly defined as the ratio of the intensity at the best focus with aberrations present to the one at the geometrical focus without aberrations. When the phase factor Fis small, the exponential term, exp (ikF), can be approximated by the first three terms of a power expansion, leading to the known Marrchal formula for the Strehl definition, namely, SD~
1 -kE[F
2- (F)2]
,
(10)
where F n denotes the weighted average value of the nth power ofF, i.e., 2x a
0 0 2~t a
--1
o
The normalized axial intensity i (z) given by Eq. ( 8 ) can therefore be written as
371
when the beam center coincides with the optical axis, the only aberration which is usually serious is spherical aberrations. In this case, a primary spherical aberration which is defined with respect to the reference sphere centered at the geometrical focal point, z=R, can be expressed as t~R = (S1/w4)p 4 •
( 15 )
Here, for simplicity, the secondary and the higher order spherical aberrations are not taken into consideration in the present discussion. In Eq. (15) the spherical aberration coefficient $1 is defined in terms of the wavefront deformation at the location of the beam radius w rather than at the location of the aperture edge a. The definition of the coefficient in this way is suitable for the case where the beam radius w is much smaller than the aperture size a, since the power in the beam is concentrated near the center of the aperture and the power at the aperture edge is negligibly small. By defining the coefficient B as W2
i(z) = (R/z) 2 { 1 - k 2 [ p 2 _ (p)2 ] }.
( 12 )
Under the condition that the term, k 2 [ ~ 2 _ (~)2 ], is small as compared to unity, and by applying the following approximate formula,
and by using Eq. (9), the phase factor F is written in the form
l-x~l/(l+x)
(13)
F=~R+
(14)
We assume in this study, as mentioned above, that the aperture radius a is much larger than the beam radius w. Therefore, the Gaussian distribution of the optical power near the aperture edge is practically zero. Consequently, the upper integral limit a in Eq. ( 11 ) can be replaced by ~ . Then, Eq. ( 11 ) becomes
(]x]
Eq. (12) can be written as 1
i(z)=
I+k2[FE_(F)2 ] .
This is a formula proposed here to express the axial intensity distribution. Eq. (14) is an approximate formula applicable to a beam which is generally truncated by the aperture and has aberrations. It will be shown in the next section that Eq. (14) reduces to an exact expression for a Gaussian beam in the case where the beam is nontruncated and aberration-free.
1 z1-
2= ~ P
+ ~sP
"
(17)
F--~= f ~ exp(--p2/w2) F"p dp f ~ e x p ( - p E / w 2 ) p dp 2 ~
- w2 ~ exp(-pE/w 2) Fnpdp.
(18)
0
Substituting Eq. (17) into Eq. (18), we obtain
4. Axial intensity in the presence of spherical aberration
F=2S1 +B, and F 2 = 24S1 + 12SIB+2B 2. Therefore,
When an optical system is designed to be used with the laser beam propagating along the optical axis and
Substitution ofEq. (19) into Eq. (14) yields the normalized axial intensity for a nontruncated, aberrated
~_ F
(/~)2 =
20S 2 + 8S1B + B E .
( 19 )
A. Yoshida, T. Asakura / Optics Communications 109 (1994) 368-374
372
Gaussian beam given by
l+k2(20S~+8SIB+B2).
i(z)=
(20)
By using Eqs. (1) and (16), Eq. (20) can be rearranged into the form
duces to an exact formula for a nontruncated, aberration-free Gaussian beam in the limit where there is no truncation and aberration-free. By using Eqs. (22) and (24), the axial intensity distribution formula of Eq. (21 ) can be written as
i(z) =iM[ 1 + iM(Z/R-- zM/R ) 2
i(z) =RE[ (1 +~2N2-- 16Ir2NS+80~2S2)Z2 + ( 16IrERNS_2rcERN2)Z+IrER2N2] -1,
× ( 1 + ~zZN2 - 16Ir2NS+ 8 0 ~ 2 S
where S is the redefined primary spherical aberration coefficient which is measured in units of wavelengths as S=S~/2. When aberrations are present, the point of maximum intensity in the diffraction field is referred to as the so-called diffraction focus [ 5 ]. The diffraction focus defined in this way is obtained by equating to zero the derivative of the denominator of Eq. (21) with respect to z. Hence, the location ZM of the diffraction focus is obtained as (22)
For S = 0, i.e., in the absence of spherical aberrations, Eq. (22) reduces to Ir2N2
ZM=R 1 +ir2N2,
(S=0) .
(23)
This is a well-known formula for the beam waist position on axis for an aberration-free, nontruncated Gaussian beam [ 14,16 ]. Eq. (22) is derived from the axial intensity distribution formula of Eq. (14) which is introduced as an approximate formula for an aberrated beam. In the limit S ~ 0 , however, Eq. (22) turns out to be an exact formula for an aberrationfree Gaussian beam. Moreover, the maximum intensity iM at Z=ZM is given, by using Eq. (21), as 1+ iM =
16~z2NS+ 80ir2S 2 IRENE(I + 16ir2S 2)
Ir2N2 -
(24)
In the limit S-,0, Eq. (24) reduces also to a wellknown formula for an aberration-free, nontruncated Gaussian beam in the form 1 + Ir2N2
iM=
IraN z ,
(S=0) .
(26)
5. Focusing without focal shift With the positive primary spherical aberration, the focal shift of a Gaussian beam can be offset. This will be shown in the following. To accomplish this, the location ZM of maximum intensity has to be equal to R. By using Eq. (22), this condition becomes ZM
Ir2N 2-- 87c2NS ZM = R 1 + Ir2N2 -- 1 6 7 r 2 N S + 8 0 i r 2 S 2 "
2) ] - 1 .
(21)
(25)
From the results of Eqs. ( 21 ), (23 ) and ( 25 ), we can conclude that the proposed formula of Eq. (14) re-
IRENE-- 8IrENS
R - 1 + n 2 N 2 - 16n2NS+8On2S 2 = 1 .
(27)
For the special case of S = 0, Nhas to be much larger than unity, i.e., N>> 1, in order to satisfy Eq. (27). In other words, for an aberration-free Gaussian beam and for N>> 1, the beam waist of the focused beam is located at the geometrical focus. This point is mentioned in Sect. 1. That is, for N>> 1 the classical theory is applicable and the beam waist position coincides with the geometrical focal point. When S is not equal to zero and by using Eq. (27), we obtain the following quadratic equation for the value of the coefficient S that can offset the focal shift:
S z - (iV/10)S+ 1/80ir2
=
0.
(28)
The condition that Eq. (28) has the roots of real values is N > 5/ir= 0.71176.
(29)
Under this condition, Eq. (28) has the solutions in the form S = (1/20) (N+_x/N2-5/ir 2 ).
(30)
As seen above, under the condition of N>> 1, S has to be equal to zero in order to get the result ofzM = R . From this we see that, in Eq. (30), only the negative sign before the square root is acceptable. Thus, we obtain the required value for S as
A. Yoshida, T. Asakura / Optics Communications 109 (1994) 368-374
S = (1/20) ( N - ~ / N 2 - 5 / ] r 2) .
(31)
For N = 1 and 2, we obtain S=0.014879 wavelengths ( N = 1 ), and S=0.0065469 wavelengths ( N = 2 ) , respectively, as the required values. Fig. 2 shows the normalized axial intensity distributions along the z axis which are calculated from Eq. (26) for N = 1 and 2. In the figures the axial distance z is normalized by R. Therefore, the point z / R = 1 corresponds to the geometrical focal point. The intensity in each curve is normalized by the intensity at the geometrical focal point in the absence of aberrations. The curve A in each figure corresponds to the I
I
I
I
I
=
._~
1
U~
"1o
.~ 0.8
E o Z
0.6
case of an aberration-free Gaussian beam and, consequently, the normalized intensity at z / R = 1 is unity as defined. For curve A in each figure, the maxim u m point of intensity on axis, which is located at a somewhat closer point to the aperture side rather than at the geometrical focal point, corresponds to the beam waist position. The distance between the beam waist position and the geometrical focus in curve A is defined as the focal shift. It is seen that the focal shift of an aberration-free beam (curve A) is greater for the lower Fresnell number beam than for the higher Fresnel number beam. When the spherical aberration is present, the maximum intensity point or diffraction focus shifts from the original beam waist position of an aberration-free beam, and the magnitude of the intensity at the maximum point is degraded. In curve B of each figure, the diffraction focus coincides with the geometrical focus, since the amount of the spherical aberration coefficient S is chosen according to Eq. (31 ) so as to offset the focal shift. The magnitude of the normalized intensity at the geometrical focus is calculated by using Eq. (20) under the condition of z = R or B = 0. In this manner, the maximum intensity in curve B is calculated by
i ( z = R ) = 1/(1 + 80n2S 2) , '
'
018
'
'
'
'
'
'
112
z/R I
'
I
'
I
!
I
1
0.8 °
-
-
._N 0.6 o
z
0.4
0,2
I
,
I
,
I
,
0.8
I
1.2 z/R
Fig. 2. The normalized axial intensity distributions along the optical axis which are calculated from Eq. (26) for N = 1 and 2. The axial point z is normalized by the radius R of the reference sphere. Therefore, the location z / R = 1 corresponds to the geometrical focal point.
373
(32)
where, for curve B, the coefficient S is chosen so as to offset the focal shift as mentioned above. For curve B, we use Eq. (32) to give the maximum intensity. However, Eq. (32) is generally valid for small aberrations and gives a normalized intensity at the geometrical focal point irrespective of the position of a diffraction focus. The normalized intensity at the geometrical focal point is always smaller than unity as seen from Eq. (32). From this we can conclude that the intensity at the geometrical focus is a good measure for the amount of the spherical aberration wherever a diffraction focus locates. Moreover, as seen from Eq. (14), the intensity at the geometrical focus is a good measure for the amount of wavefront deformation irrespective of the types of aberrations wherever a diffraction focus locates.
6. Conclusion
The diffraction integral, in the presence of aberra-
374
A. Yoshida, T. Asakura /Optics Communications 109 (1994) 368-374
tions, which is applicable to a low Fresnel n u m b e r focusing system is d e r i v e d on the basis o f the Fresnel a p p r o x i m a t i o n . A new a p p r o x i m a t e f o r m u l a for the n o r m a l i z e d axial intensity in the presence o f aberrations is proposed. This f o r m u l a reduces to an exact formula for the case o f a nontruncated, aberrationfree G a u s s i a n beam. F o r the spherical aberration, an axial intensity dist r i b u t i o n f o r m u l a for a n o n t r u n c a t e d G a u s s i a n b e a m is derived. This formula is shown to be reduced to an exact expression for an aberration-free G a u s s i a n b e a m in the limit when the a b e r r a t i o n coefficient vanishes. It is shown for the first time that, with the positive p r i m a r y spherical aberration, the focusing without the focal shift is possible unless the Fresnel n u m b e r o f the b e a m is less than 0.71. The expression for the spherical a b e r r a t i o n coefficient required for the offsetting o f this focal shift is given as a function o f the Fresnel n u m b e r o f the beam. The axial intensity distributions for low Fresnel n u m b e r G a u s s i a n b e a m s with the spherical aberrations whose magnitudes are equal to those required for the offsetting o f this focal shift, are presented. These intensity distributions show the focusing without the focal shift. It is shown that the n o r m a l i z e d intensity at the geometrical focus is a good measure for the a m o u n t o f wavefront aberrations irrespective o f the position o f a diffraction focus.
Acknowledgements We would like to acknowledge the c o m p u t e r plotting by Y. Sakurada.
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