104 OPTICS LETTERS / Vol. 33, No. 2 / January 15, 2008

First- and second-order statistics of partiallycoherent, high-numerical-aperture optical fields

Christer Rydberg*Micronic Laser Systems AB, P.O. Box 3141, SE-183 03 Täby, Sweden

*christer.rydberg@micronic.se

Received October 18, 2007; revised November 29, 2007; accepted November 29, 2007;posted December 3, 2007 (Doc. ID 88606); published January 4, 2008

The intensity probability distribution as well as the cross-spectral density of partially coherent optical fieldsgenerated through high-numerical-aperture illuminations are analyzed, and novel effects, not apparent inparaxial optical fields, are described. It is shown that the intensity probability distribution significantly dif-fers from what can be expected from a small-angle analysis, and the number of degrees of freedom for thedistribution is higher. It is further shown that the cross-spectral density of a high-angle optical field is afunction of the coordinate difference along the propagation direction of the field. © 2008 Optical Society ofAmerica

OCIS codes: 030.6600, 030.0030.

Modern microlithograpy uses partially coherent illu-mination and is increasingly tending toward highernumerical aperture (NA) of both the imaging and theillumination [1,2]. The resulting high angles warranta thorough treatment in the vector domain, takinginto account the nonscalar properties of the opticalfield. In this work we discuss changes of the first- andsecond-order statistics of the optical field apparentonly in the high-angle domain, changes that have animpact on microlithograpy.

First-order statistics, and in particular the inten-sity probability distribution of partially coherentfields, have been examined by many authors in theparaxial domain, where the first-order statistics is afunction of the polarization of the field. Lately, workhas also been done in the domain of how the degree ofpolarization changes for a highly focused field [3–5].

In microlithography the second-order statistics ofthe optical field is of great importance to the result ofthe imaging process. In the small-angle regime thereis a simple relationship—the van Cittert–Zerniketheorem—between the illumination and the correla-tion properties of the field. This work shows that thecorrelation, or more specifically the cross-spectraldensity, of an optical field resulting from a high-angleillumination, exhibits features that cannot accuratelybe accounted for in a paraxial analysis.

The second-order statistical properties of a generaloptical field can be expressed in space frequency bythe three-dimensional (3D) cross-spectral densitytensor, W�r1 ,r2 ,��, with the components

Wij�r1,r2,�� = �Ei*�r1,��Ej�r2,���, i,j = x,y,z, �1�

where � � is the ensemble average, Ei is the electricfield component in the direction indicated by the sub-script, � represents the complex conjugate, � is thefrequency of the optical field, and r1 and r2 are spa-tial coordinates. We assume a stationary optical field,and the variable substitution, �r=r2−r1, is war-ranted. The average intensity of the field can be ex-pressed as the trace of the cross-spectral density ten-

sor, �I����=Tr�W����, where the two coordinates are

0146-9592/08/020104-3/$15.00 ©

equal, �r=0. Under the assumption that the opticalfield components follow circular Gaussian statistics,as is the case for thermal, pseudothermal, and highlymultimode radiation (such as excimer lasers used inmicrolithography), the standard deviation of the in-stantaneous intensity can be expressed as the traceof the squared coherence matrix for �r=0, �I���=�Tr�W2���� [6]. One parameter describing the first-order statistics of the instantaneous intensity is thenumber of degrees of freedom, M, defined as

M =�I�2

�I2 =

Tr�W�2

Tr�W2�, �2�

where we have assumed that the field is cross spec-trally pure and omitted the frequency. This assump-tion is also made in the remainder of the text. Thenumber of degrees of freedom is important in microli-thography and laser ablation to determine the impactof what is sometimes called dynamic speckle, an un-certainty in the delivered dose due to a limited pulseduration and the finite bandwidth of the radiation[7–9].

It can be noted that the degree of polarization in3D space, P3, introduced in [3], is written as

P32 =

3Tr�W2�

2Tr�W�2 −1

2, �3�

and hence the number of degrees of freedom can berelated to the degree of polarization in 3D space as

M =3

2P32 + 1

. �4�

From this relation it can be deduced that the numberof degrees of freedom has the potential to be in therange [1, 3], with the extrema 1 and 3 correspondingto a fully polarized field and a completely unpolarized3D field, respectively.

In this work we consider the geometry illustratedin Fig. 1, with the aim of examining the properties of

the stationary optical field in the vicinity of the plane

2008 Optical Society of America

January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS 105

z=0, as generated by high-angle incoherent illumina-tion originating from directions indicated by thespherical-cap-shaped domain D.

Calculating the electric field, E, in one point r weneed to take into account the contributions from allillumination angles, D:

E�r� = �D

exp�ir · k� · U��,��dS, �5�

where U�� ,�� is the complex envelope representationof the electric field component of a plane wave arriv-ing from the direction �� ,�� with a wave vector k.

The intensity distribution of the plane waves,U�� ,��, can be expressed as a stochastic process thatis a function of the spatial coherence properties in theplane under study. Further, the polarization status ofthe illumination will determine the characteristics ofU�� ,��. The instantaneous intensity in r=0 can beformulated as

I = E2 = i=x,y,z

��D

Ui��,��dS�2

. �6�

It should be noted that the stochastic processes, Ux,Uy, and Uz, in the general case are not uncorrelatedwith each other, thus making further analyticalanalysis rather cumbersome.

Since the conventional polarization states in theparaxial domain can be ambiguous in the high-angledomain, we define three different polarization cases:linear polarization, unpolarized radiation, and circu-lar polarization. The high-angle linear polarized fieldis defined so that plane waves emanating in the di-rection �� ,�� with the wave vector k are expressed as

Ulin��,�� = �− k · ey�U��,��, �7�

where we in the following assume that U�� ,�� is ascalar random noise process, i.e., that there is no cor-relation between U�� ,�� and U��+�� ,�+��� for anynonzero values of �� and ��. This assumption is accu-rate for widespread high-NA and spatial stationaryfields, where all points spaced near the plane z=0 areequivalent, and the instantaneous intensity forms arapidly varying high-NA speckle pattern. Obviously,for highly focused conventional beams, which requirea high degree of coherence between different angular

Fig. 1. Geometry of the coordinate system. The opticalfield in the plane z=0 is created by a number of planewaves arriving from the direction � in the spherical-cap do-main D.

components U�� ,��, this analysis is not valid. Turn-

ing to an unpolarized field, the electric field compo-nent of a plane wave emanating in the direction �� ,��can be written as

Uunp��,�� = U1��,��e� + U2��,��e�, �8�

where again U1�� ,�� and U2�� ,�� are two uncorre-lated scalar random noise processes. Finally, a circu-larly polarized field can be expressed in the same wayas the unpolarized one in Eq. (8), with the additionsthat the two processes U1 and U2 are not uncorre-lated but rather the same process phase shifted � /2radians:

Ucirc��,�� = U1��,��e� + iU1��,��e�. �9�

To evaluate both the probability distribution of theintensity and the elements in the cross-spectral den-sity tensor of a spherical-cap-shaped illuminationwith polarized, unpolarized, and circularly polarizedillumination, we resort to numerical simulations. Thesimulations are performed by evaluating the instan-taneous intensity using Eqs. (6) and (1) with the ran-dom processes, U1 and U2, while the elements in thecross-spectral density tensor can be obtained by us-

Fig. 2. (Color online) Numerical simulation of the numberof degrees of freedom parameter for the intensity probabil-ity distribution of three different polarization states for acircular cap as a function of �max. Also highlighted in theillustration is the number of degrees of freedom of the in-tensity probability distribution for �max=� /2.

Fig. 3. (Color online) Cross-spectral density for of an opti-cal field generated by a spherical-cap illumination of circu-lar polarization with �max=� /2 along �x (or �y due to sym-metry). The cross-spectral density is different whenconsidering the electric field along the x axis and y axis(Wxx and Wyy are equal due to symmetry) or along the zaxis �Wzz�. As a reference the cross-spectral density of a

scalar field is also indicated in the illustration.

106 OPTICS LETTERS / Vol. 33, No. 2 / January 15, 2008

ing the definition, Eq. (1), in combination with Eq.(5). The statistics are then estimated by repeatingthe calculations a sufficient number of times.

To visualize the change in character of the inten-sity probability density Fig. 2 illustrates the numberof degrees of freedom as a function of the angular ex-tent of the cap. The simulations indicate that theprobability distribution of the intensity does changewith the extent of the spherical cap, an effect notpresent in a paraxial analysis.

In Fig. 3 the cross-spectral density is numericallysimulated for a half-sphere illumination of circularpolarization. It is shown that the cross-spectral den-sity over �x is different for the electric field compo-nent along the x axis �Wxx� or the z axis �Wzz�. Alsoindicated in Fig. 3 is the cross-spectral density in thehigh-angle scalar approximation. In Fig. 4 the cross-spectral density components (Wxx and Wzz) of an op-tical field resulting from a half-sphere illumination ofcircular polarization is again numerically simulated,

Fig. 4. (Color online) Cross-spectral density of an opticalfield generated by a spherical-cap illumination of circularpolarization with �max=� /2 along �z. Again the cross-spectral density is different when considering the electricfield along the x axis and y axis (Wxx and Wyy are equal dueto symmetry) or along the z axis �Wzz�. In the scalar ap-proximation the cross-spectral density along �z is coherent,and hence Wxx��z�, Wyy��z�, and Wzz��z� are all constant.

now as a function of �z. A striking difference between

the cross-spectral density of a low-angle field and ahigh-angle optical field is apparent, namely, thecross-spectral density of a high-NA optical field willbe a function of �z. In the low-angle domain coher-ence properties along the propagation direction havesolely been attributed to temporal coherence due tothe bandwidth of the radiation. Here we point to thefact that a full treatment of the coherence propertiesof a high-NA field must include the cross-spectraldensity as function of coordinate differences alongthe propagation direction. This effect is of practicalconsequence for depth-of-focus calculations ofhigh-NA illuminations in microlithographic applica-tions.

In conclusion, we have shown that both first-orderstatistics, in the form of the intensity probability dis-tribution, and the second-order statistics, in the formof the cross-spectral density, differ substantially in aparaxial compared with a vector treatment. Thenumber of degrees of freedom of the intensity prob-ability distribution is higher in a full-vector treat-ment. Further, it has been shown that the cross-spectral density is a function of coordinate differencealong the propagation direction of the optical field.

References

1. J. H. Bruning, Proc. SPIE 6520, 652004 (2007).2. N. Lindlein, S. Quabis, U. Peschel, and G. Leuchs, Opt.

Express 15, 5827 (2007).3. T. Setälä, A. Shevchenko, M. Kaivola, and A. T.

Friberg, Phys. Rev. E 66, 016615 (2002).4. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, Opt.

Commun. 248, 333 (2005).5. K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, J.

Opt. Soc. Am. A 22, 561 (2005).6. J. W. Goodman, Statistical Optics (Wiley, 2000).7. P. E. Dyer, C. D. Walton, and K. A. Akeel, Opt. Lett. 30,

1336 (2005).8. C. Rydberg, J. Bengtsson, and T. Sandström, J.

Microlithography, Microfabrication and Microsystems5, 33004-1 (2006).

9. J. W. Goodman, Speckle Phenomena in Optics: Theory

and Applications (Roberts and Company, 2006).