2978 OPTICS LETTERS / Vol. 31, No. 20 / October 15, 2006

Imaging theory of an aplanatic system with astratified medium based on the method for a vector

coherent transfer function

Hanming Guo, Songlin Zhuang, and Jiabi ChenShanghai Key Laboratory of Contemporary Optics System, College of Optics and Electronics, University of Shanghai

for Science and Technology, 516 Jungong Road, 200093 Shanghai, China

Zhongcheng LiangCollege of Optoelectronic Engineering, Nanjing University of Posts and Telecommunications, 66 Xin Mofan Road,

210003 Nanjing, China

Received June 7, 2006; revised July 27, 2006; accepted July 28, 2006;posted August 7, 2006 (Doc. ID 71768); published September 25, 2006

A simple formalism relating image fields to object fields, similar to that of the scalar and paraxial case, ispresented for an aplanatic system obeying the sine condition, which shows that the vector plane-wave spec-trum of image fields is equal to the product of the vector coherent transfer function due to the x- andy-polarized point electric field source and the scalar spectrum of the corresponding transverse object fields.Utilizing this formula and dyadic Green’s function, a rigorous imaging theory of an aplanatic system for thepoint electric current source through a stratified medium is readily developed. © 2006 Optical Society ofAmerica

OCIS codes: 110.2990, 260.1960, 050.1970.

A knowledge of precise structures of image fields isvery important to further improve the optics.1 In theparaxial and scalar cases, the scalar spectrum of im-age amplitudes is equal to the product of that of theGaussian image predicted by geometrical optics andthe coherent transfer function (CTF) of the opticalsystem.2 To consider effects of polarization and highaperture, vector diffraction theories must be used. Aclassical vector theory of an aplanatic system for alinearly polarized point source at infinity in the direc-tion of the axis was derived in 1959.3 Based on thistheory,3 Török and Wilson4 and Török et al.5 createdthe theory of microscopes by equivalently represent-ing a single lens system as a double lens system andtheir matrix formalism. Subsequently, thesetheories3–5 are further extended to different kinds ofmicroscope imaging for dipoles or small spherical ob-jects through stratified media.6–8 Recently, Guo et al.9

derived a vector diffraction model of an aplanatic sys-tem for a unit amplitude point electric field source(similar to Huygens’s secondary sources) located any-where at the optical axis by a vector plane-wavespectrum10 (VPWS) method.

If object fields are known, and there are arbitrarydistributions instead of a dipole, a small spherical ob-ject, or a point electric field source, the above vectortheories3–9 cannot be applied directly. In the paraxialand scalar cases, image amplitudes are found easilyas introduced above. Our aim, in this Letter, is to de-velop a simple formalism, similar to that of theparaxial and scalar cases, relating image fields to ob-ject fields of an aplanatic system obeying the sinecondition and create the imaging theory for an arbi-trarily oriented point electric current source througha stratified medium utilizing this formalism and dy-adic Green’s function, which is a typical example thatobject fields are complicated distributions in that a

0146-9592/06/202978-3/$15.00 ©

stratified medium results in spherical aberrations atthe object plane.

In this Letter, we assume that the time dependenceis exp�j�t�, and the imaging properties of any objectpoint within the view field are identical to those ofthe object point at the optical axis, except that thecenters of the image fields are located at their respec-tive Gaussian images. Hence the aplanatic systemobeying the sine condition considered can be viewedas a linear system. In a homogeneous medium, thetransform pairs of the VPWS of an electric field areexpressed as9

E�r� = − �−2� ��

E�s�exp�− jks · r�dsxdsy, �1�

E�s� =� ��

�xEx + yEy − zsz−1�sxEx

+ syEy��exp�jk�sxx + syy��dxdy, �2�

where E�s� denotes the VPWS of an electric field atz=0, s=xsx+ysy+zsz is a unit wave vector, and r is alocation vector. Replacing separately x and y by x−xo and y−yo in Eq. (1) yields the following shift theo-rem of the VPWS,

F�E�x − xo,y − yo�� = E�s�exp�jk�sxxo + syyo��, �3�

where the operator F� � means to have a Fouriertransform operation on an electric field.

Because an optical system is corrected only for afixed object plane and its image plane so as to obeythe sine condition in practical cases, we will restrictour attention to the relation among electric fields atthese object and image planes. From now on, unless

explicitly stated, the object and image planes sepa-

2006 Optical Society of America

October 15, 2006 / Vol. 31, No. 20 / OPTICS LETTERS 2979

rately refer to that fixed object plane and its imageplane. Let Eo�x ,y� and Ei�x ,y� separately be objectand image fields. For a linear system, Eo may be de-noted as

Eo�x,y� =� ��

Eo�xo,yo���x − xo,y − yo�dxodyo.

Following Ref. 2, the operator L� � of a linear system,which we imagine to operate on object fields to pro-duce image fields, may be brought within the integralso as to yield the image fields

Ei�x,y� =� ��

�Eo�xo,yo��h�x,y;xo,yo�dxodyo, �4�

where h�x ,y ;xo ,yo�=L�eo�xo ,yo���x−xo ,y−yo�� iscalled the point spread function (PSF) for the objectpoint �xo ,yo� of the aplanatic system, eo is the unitelectric vector of Eo�xo ,yo�, and eo�xo ,yo���x−xo ,y−yo� represents the unit amplitude point electric fieldsource called the object point �xo ,yo�. Having a Fou-rier transform operation in Eq. (4) at the imageplane, one may obtain the VPWS of Ei�x ,y�:

Ei�six,siy� =� ��

�Eo�xo,yo��h�six,siy;xo,yo�dxodyo, �5�

where six and siy are transverse Cartesian compo-nents of the unit wave vector si in the image space.Reference 9 has given the VPWS of the PSF for theobject point located at the optical axis (i.e., xo=yo=0)[Eq. (19) of Ref. 9], from which, and the formula (3),we may obtain the VPWS of the PSF for the objectpoint �xo ,yo�, namely,

h�six,siy;xo,yo� = �Hc1�six,siy�eox�xo,yo�

+ Hc2�six,siy�eoy�xo,yo��exp�jkiM�sixxo

+ siyyo��, �6�

with �=M cos−�1/2� �o cos−�1/2� �i and

Hc1x�six,siy� = − j��cos �o sin2 � + cos �i cos2 ��,

Hc1y�six,siy� = − j��cos �i − cos �o�sin � cos �,

Hc1z�six,siy� = − j� sin �i cos �, �7a�

Hc2x�six,siy� = Hc1y�six,siy�,

Hc2y�six,siy� = − j��cos �o cos2 � + cos �i sin2 ��,

Hc2z�six,siy� = − j� sin �i sin �, �7b�

where the spherical polar coordinates with origin atOi have been used with 0��i�i and 0���2 sosi= �−sin �i cos � ,−sin �i sin � , cos �i�. i is the angu-lar semiaperture on the image side. �o is the anglethat the corresponding incident ray so=xsox+ysoy

+zsoz with the origin at Oo in the object space makes

with the positive z axis and cos �o= �1− �Mni sin �i /no�2�1/2 where M=−�nodi� / �nido� is thenominal magnification of the aplanatic system. Formore details, the reader is referred to Ref. 9. As thereare identities niMsix=nosox and niMsiy=nosoy,

9 substi-tuting Eq. (6) with eo= �Eo�−1Eo into Eq. (5) and thencomparing the result with formula (2), we can obtainthe VPWS of the image fields,

Ei�six,siy� = Hc1�six,siy�Eox�sox,soy�

+ Hc2�six,siy�Eoy�sox,soy�. �8�

We should emphasize that formula (8) is valid onlyfor the electric fields at the object and image planesin that the optical system must obey the sine condi-tion. Once the known electric fields are not at the ob-ject plane, we must first derive object fields by theVPWS method before the application of formula (9),which is different from the paraxial and scalar case.It is easily seen from Eqs. (7a) that Hc1 and Hc2 aredetermined by the nominal magnification and theexit pupil of the aplanatic system. Eox and Eoy sepa-rately denote scalar spectrums of the transverse ob-ject fields. As the definition of the CTF in the scalarcase, Hc1 and Hc2 are defined as the vector CTF dueto the x- and y-polarized point electric field source, re-spectively. Hence formula (9) shows that, for a vectortreatment and high aperture, the VPWS of imagefields is equal to the product of the vector CTF owingto the x- and y-polarized point electric field sourceand scalar spectra of the corresponding transverseobject fields, which is our major conclusion in thisLetter.

Now, take the electric current density J as an ar-bitrarily oriented point electric current source ofstrength pe at r=0 (i.e., the point Oo�) in the medium(1), i.e., J�r�=pe��r� (see Fig. 1). The first dielectricinterface is placed perpendicular to the z axis at z=z1. Subsequent interfaces are placed at z=z2 , . . . ,zm−1. By the multilayered electric dyadicGreen’s function (MEDGF) GIm,1�r ,0�, the electricfields immediate to the z=zm−1 interface at the mthlayer due to J may be viewed as11

Em�r,0� = − j��1GIm,1�r,0� · pe. �9�

In terms of formula (44) for the MEDGF of Ref. 11and the assumption that the medium �m� is semi-

Fig. 1. Geometry of imaging of an aplanatic system for the

point electric current source through a stratified medium.

2980 OPTICS LETTERS / Vol. 31, No. 20 / October 15, 2006

infinite homogeneous (i.e., the TE and TM effectiverefection coefficients at the mth layer are zero), forpropagation in a positive z direction, there is

GIm,1�r,0� =− j

42 � ��

dkxdky

exp�− j�1z1�

2�1�n�n� m,1��

+ nm� n1� m,1�� �km�1/k1�m��exp�− j�kxx

+ kyy��, �10�

where km=xkx+yky+z�m is the wave vector in themedium �m�. n� and nm� separately denote the TEand TM polarizations of the electric field vector. m,1��

and m,1�� separately refer to the TE and TM effectivetransmission coefficients. The presentation of the ac-tual form of the variables in Eq. (10) is omitted forbrevity. For details, the reader is referred to Ref. 11.

As kx=kmsmx and ky=kmsmy, dkxdky=km2 dsmxdsmy.

Substituting Eq. (10) into Eq. (9) and then comparingthe result with formula (1), we can obtain the VPWSEm�smx ,smy� of the electric field Em�x ,y� across aplane immediately behind the z=zm−1 interface. Asformula (8) is valid only for object fields, we assumethat the object plane of the aplanatic system L is atz=zo� (see Fig. 1), which is a virtual plane at the mthlayer. In terms of formula (1), it is easy to see that theVPWS of object fields is determined by the expressionEo�smx ,smy�=Em�smx ,smy�exp�j�m�zm−1−zo���. So,

Eox = A�k1smyA m,1�� + smxs1zB m,1�� �, �11a�

Eoy = − A�k1smxA m,1�� − smys1zB m,1�� �, �11b�

with A=smypex−smxpey, B=k1smxs1zpex+k1smys1zpey−kmsmt

2 pez, and A= ���1 /2k12smt

2 s1z�exp�jkmsmz�zm−1−zo���exp�−jk1s1zz1�, where smt=xsmx+ysmy.

Note that there are sox=smx, soy=smy, and k1s1t=k2s2t= ¯ =kmsmt. Substitution of Eqs. (7), (8) and(11) into formula (1), for an arbitrary point P�� ,� ,z�in the image space, a well-known procedure de-scribed in many prior publications3,9 results in thefollowing expressions for the Cartesian coordinatecomponents of the electric fields of the point P,

Eix�r� = jC�k1pexA0a + j2Mkipez cos �A1

a + k1�pex cos 2�

+ pey sin 2��A2a�,

Eiy�r� = jC�k1peyA0a + j2Mkipez sin �A1

a + k1�pex sin 2�

− pey cos 2��A2a�,

Eiz�r� = 2C�jMkipezA0b − k1�pex cos � + pey sin ��A1

b�,

�12�

with C= ��� Mk2� / �8k2� and

1 i 1

Ana,b = �

0

i

cos−�1/2� �m cos1/2 �i sin �iBna,b exp�− j��

� Jn�ki� sin �i�exp�− jki cos �iz�d�i,

with �=k1 cos �1z1−km cos �m�zm−1−zo��, B0a=cos−1 �1

��cos �m m,1�� +cos2 �1 cos �i m,1�� �, B1a=sin �i cos �i m,1�� ,

B2a=cos−1 �1�cos �m m,1�� −cos2 �1 cos �i m,1�� �, B0

b

=sin2 �i m,1�� , B1b=sin �i cos �1 m,1�� , and Jn�·� being the

Bessel function of the first kind and order n.For an electric dipole source of moment

exp�j�t�pe��r�, the current density is J�r�= j� exp�j�t�pe��r�.12 Utilizing this relation, ourmathematical models [formulas (12)] may be used todescribe image formations in optical microscopy withthe help of dipole waves. However, our mathematicalmodels do not require that the dipole must be manywavelengths from the first interface, which is an ap-proximation made in Ref. 13. When the object spaceis a homogeneous medium, there are m,1�� = m,1��

=exp�jko cos �o�z1−zm−1�� in terms of their actualform [Eq. (A12) of Ref. 11] and �1=�m=�o. It is obvi-ous that the phase term of Bn

a,b exp�−j�� isexp�−jko cos �ozo��, which denotes the defocus term. Inaddition, the relation between the angles �o and �ishould be determined by the nominal magnificationof the optical system and independent of the defocusterm (Refs. 4 and 5 believe they are related) in thatthe sine condition is satisfied only for a fixed objectplane and its image plane in practical cases.

This research was supported by the National BasicResearch Program of China (2005CB724304), theShanghai Leading Academic Discipline Project(T0501), the National Natural Science Foundation ofChina (60478045 and 60377006), and the ShanghaiFoundation for Development of Science and Technol-ogy (04dz05110). Corresponding author S. Zhuang’se-mail address is slzhuangx@yahoo.com.cn. H. Guo’se-mail address is ghanming@vip.sina.com.References

1. J. T. Foley and E. Wolf, Opt. Lett. 30, 1312 (2005).2. J. W. Goodman, Introduction to Fourier Optics

(McGraw-Hill, 1968).3. B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A

253, 358 (1959).4. P. Török and T. Wilson, Opt. Commun. 137, 127 (1997).5. P. Török, P. D. Higdon, and T. Wilson, J. Mod. Opt. 45,

1681 (1998).6. O. Haeberlé, Opt. Commun. 216, 55 (2003).7. O. Haeberlé, M. Ammar, H. Furukawa, K.

Tenjimbayashi, and P. Török, Opt. Express 11, 2964(2003).

8. O. Haeberlé, Opt. Commun. 235, 1 (2004).9. H. Guo, J. Chen, and S. Zhuang, “Resolution of

aplanatic systems with various semiapertures, viewedfrom the two sides of the diffracting aperture,” J. Opt.Soc. Am. A (to be published).

10. H. Guo, J. Chen, and S. Zhuang, Opt. Express 14, 2095(2006).

11. S. Barkeshli and P. H. Pathak, IEEE Trans. MicrowaveTheory Tech. 40, 128 (1992).

12. R. L. Hartman, J. Opt. Soc. Am. A 17, 1067 (2000).13. P. Török, Opt. Lett. 25, 1463 (2000).