Simplifications of the bilinear transfer for microscopic binary objects

The deduction of features of scattering objects from the measurement of the emitted and scattered fields is an opti­ cal problem, the importance of which is growing in numerous fields such as metrology and biology.1 Experimentally, vari­ ous methods can be used: for instance, the direct analysis of the light field propagating in free space or the use of optical processors as imaging systems. In the latter case, optical microscopy in partially coherent light is still attractive, since it involves a simple instrument that is easy to use and that remains efficient for objects whose dimensions are of several wavelengths.2 Moreover, owing to partial coherence, a sca­ lar approach of the transfer is generally sufficient for de­ scribing the image formation accurately. Unfortunately, the partial coherent transfer is inherently nonlinear, and it is generally not possible, without restrictive hypotheses or pri­ or knowledge, to determine the object distribution from its magnified image. In order to determine with accuracy the parameters qf a given object, it is necessary to define, first, a unique relation between object and image distributions. The function connecting the object and image data is the bilinear t�ansfer function (BTF) (also called the cross-coef­ ficient function) in partially coherent light.3•4 Although the notion of bilinear transfer has simplified the mathematical formalism of the partially coherent imagery, its utility is not obvious in practice. The first approach to solving the inverse problem is to use a numerical procedure, which is today possible by means of the current numerical capacities. However, this approach presents some drawbacks involving the computing time, which cannot be neglected in certain cases requiring real­ time dimension. al control, involving signal processing; more­ over, a numerical approach requires a precise knowledge of the different parameters defining the system. · Several authors have tried to simplify the transfer by introducing the notion of an apparent transfer function (ATF),5 which depends on the class of objects that are con­ sidered (nature, shape, size, etc.). Despite this restriction, such. a function simply connects object and image spectra and thus permits the retrieval of the object features from the image irradiance. · A practical example of tr an sf er modeling was recently developed in a study of linewidth measurement technology.6 In that study, a simplified form of the partially coherent transfer equation was found experimentally. The simplifi­ cation, which is based on the decomposition of the overall transfer into two partial ones, permits the use of a fast numerical method for accurately retrieving the linewidth from the image spectrum data. This paper is intended to complete these experimental works with a theoretical approach by analyzing the transfer in partially coherent light for rectangular objects. It is shown that the modeling is well adapted for explaining the image behavior and, for example, the overshots that appear at the transition regions in the object. Finally, the modeling is tested by means of numerical simulations, and the role of each partial transfer on the image and its Fourier spectrum is pointed out.


INTRODUCTION
The deduction of features of scattering objects from the measurement of the emitted and scattered fields is an opti cal problem, the importance of which is growing in numerous fields such as metrology and biology. 1 Experimentally, vari ous methods can be used: for instance, the direct analysis of the light field propagating in free space or the use of optical processors as imaging systems.In the latter case, optical microscopy in partially coherent light is still attractive, since it involves a simple instrument that is easy to use and that remains efficient for objects whose dimensions are of several wavelengths.2Moreover, owing to partial coherence, a sca lar approach of the transfer is generally sufficient for de scribing the image formation accurately.Unfortunately, the partial coherent transfer is inherently nonlinear, and it is generally not possible, without restrictive hypotheses or pri or knowledge, to determine the object distribution from its magnified image.In order to determine with accuracy the parameters qf a given object, it is necessary to define, first, a unique relation between object and image distributions.The function connecting the object and image data is the bilinear t � ansfer function (BTF) (also called the cross-coef ficient function) in partially coherent light.3 •4 Although the notion of bilinear transfer has simplified the mathematical formalism of the partially coherent imagery, its utility is not obvious in practice.
The first approach to solving the inverse problem is to use a numerical procedure, which is today possible by means of the current numerical capacities.However, this approach presents some drawbacks involving the computing time, which cannot be neglected in certain cases requiring real time dimension .al control, involving signal processing; more over, a numerical approach requires a precise knowledge of the different parameters defining the system.
• Several authors have tried to simplify the transfer by introducing the notion of an apparent transfer function (ATF),5 which depends on the class of objects that are con sidered (nature, shape, size, etc.).Despite this restriction, such.a function simply connects object and image spectra and thus permits the retrieval of the object features from the image irradiance.A practical example of tr an sf er modeling was recently developed in a study of linewidth measurement technology.6 In that study, a simplified form of the partially coherent transfer equation was found experimentally.The simplifi cation, which is based on the decomposition of the overall transfer into two partial ones, permits the use of a fast numerical method for accurately retrieving the linewidth from the image spectrum data.
This paper is intended to complete these experimental works with a theoretical approach by analyzing the transfer in partially coherent light for rectangular objects.It is shown that the modeling is well adapted for explaining the image behavior and, for example, the overshots that appear at the transition regions in the object.
Finally, the modeling is tested by means of numerical simulations, and the role of each partial transfer on the image and its Fourier spectrum is pointed out.

THEORY
Let us consider the space-invariant and inversion-invariant imaging system described in Fig. 1.The different optical elements are the thermal source S0, the field condenser lens L0, the condenser C, the object 0, and the objective Op.

A. Transfer Relation
The system can be described by the well-known transfer integraF Jc�,) = J J T ( u + u', u')O ( u :P u' ) o* (:� ) d 2 u'. (1)and u' are the position vectors u (u1, u2) and u'(u1', u2') in the frequency plane, l is the Fourier transform of the image irradiance, 6 is the Fourier transform of the object transmit tance (or reflectance), T is the BTF,}.. is the wavelength, and p and p ' are the distances from the object plane to the entrance pupil plane and from the exit pupil plane to the image plane, respectively.By putting w = u/'Ap, w' = u'/'Ap, p'/p = g and introducing the new functions T(w, w') = T(w • 'J\p, w ' • 'Ap), l(w) = l(w/g), Eq. (1) can be then rewritten as (3) f(w) = J J T(w + w', w')O(w + w')O*(w')d 2 w', (4) where T takes the form T(w, w') = J J S(a)P(w + a)P*(w' + a)d 2 a. (5) Sis the emittance in the image of the source projected in the pupil plane, and P is the pupil transmittance.
The type of object under test belongs to the category of two-dimensional objects (flat ones) and is represented phys ically in Fig. 2. Is and ¢s (IM and ¢M) are the intensity and the phase of a plane wave J.:s (J.:M) after reflection on the object.The latter can then be described mathematically by the one-dimensional functions where xis one of the coordinates of the position vector r(x, y) in the object plane, ¢0 is the phase difference ¢M -<Ps, and to is the quantity (!Mlls) 1 12• It was shown previously8 that the two-dimensional inte gral in the right-hand side of Eq. ( 4) reduces to a one dimensional one in the case of objects whose transmittance is constant along they direction: f(w) = J T(w + w', w')O(w + w')O*(w')dw'. (7)w) can be rewritten as a linear combination of two terms6: The term f(O), which corresponds to a constant irradiance in the object, has been omitted for the sake of clarity: and Parameters A and B, which depend on only the object parameters, are defined as follows: In order to continue the transfer modeling, it i.s necessary to simplify the integral term f'(w), for instance, by developing sin[7r(w + w')L] in sine and cosine functions and using the fractional expansion w'(w � w') = : c, -w � w' )- The Fourier transform of the image irradiance can then be rewritten in terms of a combination of three simpler terms: where The sine function is the ratio sin(7rwL)/7rwL./ 3(w) can be rewritten as by using the symmetry properties of the BTF4: T(w, w') = T*(w', -w), which is due to the reality of the image irradiance, and T(w, w') = T*(w, w'), which holds for inversion-invariant and space-invariant bi linear systems.
It is not possible to reduce the evaluation of the integrals further, except with approximations that are compatible with the experimental conditions.These approximations are studied in the Subsection 2.B.

B. Approximated Solutions for Integrals fi, fz, and k
It is supposed that the coherent cutoff frequency is several times as large as the frequency of the first zero in the spec trum of rectangular objects.This assumption means simply that the microscope works under good conditions, i.e., suffi ciently below the resolution limit.
With this hypothesis the integrals given in Subsection 2.A can be simplified.
Let us consider, first, the integral on the right-hand side of Eq. ( 13).It contains the product of the BTF T(w, w') with a sine function related to the object.With conventional op tics and in partially coherent illumination, the BTF has the aspects represented in Figs. as narrow as the Fourier spectrum of a rectangle function of the same width as that of the object.It is thus much more narrow than the BTF extension.In this case the value of the integral is different from zero only in the vicinity of w ' = 0, and Eq.(13) reduces to where T(w, O) is the value of the BTF along the axis w' = 0. Let us consider now the integral in Eq. (14).In microsco py, the cutoff frequency of the imaging system is always many times larger than the frequency of the first zero of the function sinc(7rwL) in order to form a well-resolved image in the output plane.The integrand on the right-hand side of Eq. ( 14) is thus the product of an extended slowly varying function, T(w + w', w'), and a rapidly oscillating term cos(27rw'L)/w'.Consequently, the value of the integral is sufficiently small, in comparison with the previous one, to be neglected.We shall use numerical simulations to demon strate more clearly the limits of the validity of these approxi mations.
The last term [Eq.(15)] is certainly the most interesting one, sfnce the integral does not contain information about the object.It is generally not negligible, and it plays a key role in the transfer process, as is shown in Section 3.
Finally, for large objects, taking into account the values of /1 and /3, the image spectrum can be written as the difference of two terms, where T A I= T(w, 0) is the first nonnormalized A TF and is the second nonnormalized ATF.We note that in the domain of validity of our hypotheses, the two ATF's do not depend on the width of the object under test.The partially coherent transfer is thus equiva lent to the juxtaposition of two transfers working in paralleL The first transfer, associated with the ATF TAi.acts as an incoherent transfer, where the modulation transfer function would be replaced by T(w, 0).The sine function is merely the Fourier transform of the object emittance.The shape of the function T(w, 0) can be deduced easily from the three dimensional representation of the BTF in Fig. 3.
The second transfer is not so classical.The cosine func tion can be explained as the contribution of the object edges.In our precise case of a binary object it is the Fourier trans form of the derivative of the object emittance.Its role is pointed out more clearly in the numerical simulation.

NUMERICAL SIMULATION
In the development in Section 2, the only condition on the system is that T( w, w') vary slowly related tow or w'.Our It was shown recently9 that, for uniform full sources and pupils, the BTF is modified slightly when the conventional two-dimensional functions are replaced in Eq. ( 5) by one dimensional equivalent functions.This point is illustrated in Fig. 3.In this case where a is one of the coordinates of the position vector a.
Since the source and the pupil are now constant, it is possible to introduce the ratio u= r/R, called the coherence coefficient.It is defined as the ratio of the radius of the image of the source projected in the exit pupil plane to the radius of the pupil aperture.Theoretical ly u varies from 0 (coherent illumination) to infinity (inco herent illumination), but often the highest value of u is about 1 (especially in reflection microscopy), corresponding to incoherent fields.
Note that it is obvious that the notion of large or small objects has no sense without a given reference, for instance, the coherent limit frequency of the system.The frequency wo of the first zero in the object spectrum is given by 1/L.For large objects [for example L = 4, w0 = 0.25, which is 4 times as small as the coherent limit frequency (we= 1)], the resulting image can be resolved well theoretically, whereas for small objects (for example, L = 1, wo = We = 1), the image is strongly affected by the limited resolution of the system.This approximation is justified effectively in a large range of metrological applications in partially coherent illumina tion even for submicrometer objects.
Integral 12 [Eq.( 14)] The case of integral 12 is not illustrated here.The integral can be neglected regardless of the coherence coefficient and the width of the object.
Integral 13 [Eq.( 16), Fig. 5] The term on the right-hand side of Eq. ( 16) corresponds to the ATF TA2• It does not depend on the object width L. The shape of the curve varies significantly when the coher ence of the illumination beam changes.It is well known that the sharp transitions of the transfer functions are responsi ble for oscillations in the images.It is interesting here that u = 0.67 corresponds to an ATF TA2 that varies slowly in the whole range of spatial frequencies.

B. Fourier Spectrum of the Image
It is now interesting to compare the spectrum of the image obtained directly from the transfer equation and that ob tained after modeling.Let us then introduce the physical parameters which are actual experiment conditions in linewidth mea surement microscopy.Figure 6 corresponds to the modulus of the image spec trum directly calculated from Eq. ( 7), i.e., without approxi mation (solid line) and after modeling (dotted line).We note the good similarity between the exact spectrum and the approximated one.

C. Image Reconstruction
To illustrate the effect of the simplifications on the image profile, the latter is reconstructed from the spectrum with out approximation in Fig. 7(a) and from the approximated solution in Fig. 7(b).The validity of the modeling is con firmed even for small objects (L = 2.5, i.e., LE = 1.35 µm).This operation is performed by using a numerical inverse fast Fourier transform.
Moreover, it was predicted previously that the transfer depending on TA1 could be considered as describing the classical noncoherent transfer, the second term being con nected to a derivative effect.This point is confirmed by the curves in Fig. 8, in which the contribution of each term of the spectrum is exhibited in the image plane.
It is clear from relation (18) that the relative weight of each elementary image depends on the object parameters to and ¢0• For example, if then the image is described only by Fig. 8(b).This result is well known; the image of a purely phase object exhibits variations of intensity only in the region of object transition.The proposed modeling permits us to associate with this phase-contrast effect a perfectly determined transfer func tion (TA2) that can be calculated for any coherence degree.
It is also interesting that the transfer T A2 cannot be can celed for purely amplitude objects (i.e., <Po = 0) because In other words, the overshots in the image are not due only .Fig. 7. Image profile obtained (a) without approximation (Fourier transform of the spectrum plotted in Fig. 6) and (b) after modeling.
!----.   to the phase difference ¢0• To cancel the second transfer, it would be necessary to verify the equation that is, which is never verified if to � 1.
For reducing the overall transfer to only the first term, it is thus necessary to suppose low-contrast amplitude objects, that is, t0 = 1 ± €, <Po= 0.
In this case, We again find the conditions of linearization of the bilin ear transfer,4 which can be met easily in practice.
All these cases are effectively encountered practically, as shown in Fig. 9.

D. Generalization of this Modeling
It is clear that perfectly rectangular objects are rarely met in practice even in integrated electronics.The usual sample object is one with slightly smoothed edges.Results of some experiments made on this type of object show that the pro posed approach of the transfer remains valid.6It can be explained as follows: an object with smoothed edges can be described as a perfect rectangle in convolution with a SII!Oothing function.The effect of the latter can be taken into account in the transfer, consequently introducing a small modification in the shapes of both TAI and T A2 •

CONCLUSION
For rectangular two-dimensional objects (flat objects) the bilinear transfer can be simplified by introducing two partic ular ATF's.The first one acts on the Fourier spectrum of the intensity reemitted by the object.It thus has certain similarities with an incoherent transfer.The second ATF acts on a c ' osine function that is the Fourier transform of the object derivative.In the more general case of an object with slightly smoothed edges, it describes the edge transfer.Such a transfer modeling can be then considered a two channel transmission, with the relative weight of each chan nel depending on the object parameters (¢0 and to).The interest of such a modeling is, on the one hand, a simple interpretation of the image profile in partially coherent light and, on the other hand, the possibility of carrying out fast procedures to determine such critical parameters as the linewidth of wafers by solving relation (18) numerically.Images of microscopic objects of width from 0.8 µm to a few micrometers can be measured accurately.6 •1 0 This modeling will be developed and extended in future studies to the more physical case of binary three-dimension al objects.

Fig. 1 .
Fig. 1.Microscope in Kohler's illumination.The image Ss of the thermal source (S0) is projected in the pupil plane (P) of the objec tive (Ob).

Fig. 2 .
Fig. 2. (a) Schematic representation of the object; ns and nM are the indices of the two media constituting the object.Plane waves before (io) and after (is and �M) reflection over the object are shown.(b), (c) Variation of To and ¢0 over the object.
Fig. 3. Three-dimensional representation of the BTF in partially coherent light (a) for a circular objective aperture and (b) for a one dimensional aperture (coherence coefficient, u = 0.67).

A.
Disc u ssion of the C u rves Integral 11 [Eq.(13), Fig. 4] Let us recall that we have assumed that 11 � LT (w, O) in our working conditions.

Fig. 5 .
Fig. 5. Integral I 3 for the same coherence coefficients as in Fig.4.

Fig. 8 .
Fig. 8. (a) Partial image profile due to the first apparent transfer.(b) Partial image profile due to the second apparent transfer (deriv ative effect).