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Lateral resolution in the near field and far field phase images of π-phaseshifting structures
15 November 1994 OPTICS COMMUNICATIONS ELSEVIER
Optics Communications112 (1994) 189-200 F u l l length article
Lateral resolution in the near field and far field phase images of 7r-phaseshifting structures M i c h a e l Totzeck, M a r c o A. Krurnbtigel Optisches Institut der TechnischenUniversitlltBerlin, Hardenbergstrasse36, 10623 Berlin, Germany Received 19 April 1994;revisedmanuscriptreceived 3 August 1994
Abstract The resolution in the phase distribution of coherent images of strong phase objects is investigated using moment method near field calculations and simulated far field images. It is demonstrated that an apparent superresolution for ~r-phaseshifts corresponds to a threshold criterion and that the images depend strongly on polarization and refractive index.
1. Introduction Recently it was proposed by Tychinski et al. [ 1-3 ] that the far field "phase images", i.e. the phase of the coherent images of strong phaseshifting objects are not subject to the Rayleigh resolution limit of A/2: They reported a lateral resolution down to 10 nm (in determination of edge coordinates at A ffi 632 nm) by evaluating the phase of the microscopical image of topographical line structures in reflection. The best resolution was obtained for ~r-phaseshifts, i.e. structure heights of A/4 [4]. The surprising point of the method is the fact that the reported strong superresolution was achieved without using near field imaging or a subsequent numerical treatment of the images. As is well known, the maximum resolution obtainable with an optical system is restricted to 1/fmax with fmax the maximum transferred spatial frequency. The far field resolution limit of A/2 is a consequence of the fact that free space propagation acts as an optical low-pass filter. In this context, superresolution is equivalent to getting access to the f > 2/A-part of the spatial frequency spectrum, or in spatial terms: distin-
guishing structure elements with distances < A/2. Usually, superresolution (of stationary structures) requires a lot of experimental or numerical effort. Additionally, method and results depend strongly on the available a priori knowledge. Generally we may say that the more we know about the investigated structure, the more simple the applied superresolution methods may be, but the less informative are the images and the more susceptible to errors. The presented paper investigates the question of superresolution in the phase images of two dimensional dielectric structures with rectangular cross-section and strong phaseshifts. Several methods of superresolution imaging for these kind of objects are well known: (i) Scanning near field optical microscopy (SNOM), where the not propagating evanescent components of the frequency spectrum are measured by a sub-A probe that is moved by piezo-translators in the near field of the object [ 5 ]. This method is the most general one, since no a priori information about the object is required, although the image interpretation is sometimes rather difficult due to the complicated object-probe interaction in the near field.
0030-4018/94/~r/.00 (~) 1994 Elsevier ScienceB.V. All rights reserved SSDI 0030-4018 ( 94 ) 00451 -X
190
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200
(ii) Utilizing the analycity of the frequency spectrum by analytic continuation of the transferred part [6]. This method is mainly limited by noise (for a general discussion, see Ref. [7] ). (iii) Utilizing a priori information by finding a mathematical expression for the frequency spectrum with a set of free parameters and a subsequent numerical fit of these parameters to the transferred part of the spectrum. For dielectric bars, this procedure is possible for weak phase objects only, where the first Born approximation is valid [ 8 ]. It can not be applied here, because in order to obtain strong phaseshifts, objects of high refractive index are of main interest, where multiple scattered fields yield a strong contribution to the diffraction field, (iv) Utilizing a priori information in the form of threshold criteria, i.e. only the position of the edges of an object has to be determined. This least general method is widely used in critical dimension metrology [9,101. Another kind of superresolution that is closely related to phase imaging, is observed for two ~rphaseshifted coherent point objects (6-pulses): their images may be resolved for arbitrary small distances, since due to destructive interference of the corresponding Airy-discs there is a node between them [6]. However, this result can not be applied to arbitrary objects, because a third point object could not be ~'-phaseshifted with respect to both previous points. Considering a grating where neighboured grooves are ~r-phaseshifted with respect to each other, the resulting resolution improvement is limited to a factor of two, because a ~" phaseshift between the nearest neighbours results in identical phase of the next but one, between which the Rayleigh resolution limit holds. This is in agreement with the fact that utilizing ~'-phaseshifting masks in photolithography, a resolution improvement by a factor ~ 2 is possible [ 11,12].
2. Geometry of the imaging configuration For the investigation of superresolution in the images of strong phase objects the following imaginggeometry is considered: a dielectric strip of width b, thickness d, and refractive index n is illuminated by a normally incident plane wave of wavelength ho (Fig. 1). The strip is parallel to the y-axis. Some dis-
tance behind the object an imaging system of numerical aperture NA is situated. For calculation of the images the space is divided into two parts, linked by the plane z = z0 immediately behind the object. The optical field U(x, zo) in this plane is calculated by solving the near field diffraction problem of the particular phase object in part I. The diffraction near field is obtained from the sum of the incident field and the scattered field Us according to
U(x, zo) = exp(ikoz0) + Us(x, zo).
( 1)
Behind the object in part II this field propagates through free space to the imaging system. Two perpendicular linear polarizations are considered: in TM-polarization the electric field has only a y-component, i.e. U(x,z) = Ey(x,z), and in TEpolarization the magnetic field is U( x, z ) = Hy( x, z ). The imaging process is simulated by applying a spatial frequency filter to the near field U(x, zo), i.e.
[l(x) = . ~ { e - l ( f x , Az)M(fx).~{U(x)} },
(2)
where ~{} and ~ - l { } denote the Fourier transform and the inverse Fourier transform, respectively. The tilde marks the image. P - l ( f , , A z ) is the inverse propagation operator, which describes the focussing of the imaging system at the plane z = z0 - Az according to
P - l ( f x , A Z ) =exp[-i2zr(A-z + f~)t/2Az].
(3)
The phase image is ~ ( x ) = arg(U(x) )
(4)
and [U(x)[ denotes the amplitude image. The imaging system's optical transfer function
[ Aof x "~ M ( f , ) = rect~ 2--~-A} = 1,
Xofx < N^,
=½,
aofx = NA,
=0,
aofx > N ,
(5)
is assumed as real and symmetrical, i.e. the imaging system should be perfect in the sense that it acts as a pure bandpass filter without additional disturbances of the transmitted spatial frequencies. It should be emphasized, that while the imaging process of the real and imaginary part is linear (for a linear imaging system), the amplitude and phase imaging is nonlinear. Of particular interest for this paper is
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
focus plane
nearfleld plane
Incidentwave
/
= i
i >
phc obj,
,o;
TM'.®E i
Imaging system with numerical aperture N A
|
TE: ,I~,,H(, >
<
191
;!
d
\
>"Zoi< /N y
I
H
,Iriteracflon of incident I wave w th object I
free space propagation ]
Fig. 1. Geometry of image configuration.
the pronounced nonlinearity of the phase imaging for phaseshifts ~ It.
where H(x) is equivalent to the well known image of the perfectly absorbing halfplane (from ~'{H(x) } = 0.5(1 - i/zrfx)):
3. Far tidd images in geometrical optics approximation
H(x) = (1/~r)Si(koNAx) + 1.
In order to investigate what kind of effects are to be expected for the imaging of strong phase objects, we assume propagation of the incident wave through the object according to geometrical optics (simulated images using rigorous electromagnetic near fields are discussed in Sect. 4). Consider a phaseshifting halfplane of transmittance
(9)
Si() denotes the sine-integral. For a lossless (Itol = 1) weak phaseshifting (~bo < 20 °) half plane the phase image ~(x) = ~b0H(x)
(10)
where H(x) denotes the Heaviside function
is proportional to the image of the perfectly absorbing halfplane, which is diffraction limited (Fig. 2b, NA = 1). For intermediate phaseshifts the phase distribution near the edge in the image becomes increasingly steeper with increasing phaseshifts. For Oo = ¢r the phase image is even discontinuous because the imaginary part of the complex image
H(x)=l,
t'(x) = 1 - 2H(x),
t(x) = 1 + [toexp(iOo) - 1] H(x),
(6)
x>O,
=½,
x =0,
--0,
x<0
(7)
and to is the amplitude transmittance. The phase image becomes ~ ( x ) --- arg[F(x) ] = arg{1 + [t0 exp(it~o) - 1] H ( x ) } ,
(8)
(11)
vanishes and the phase distribution indicates merely the change of sign of the real part, i.e. ~(x) = (~/2) {1 -t- sgn[1 - 2H(x) ]}.
(12)
Due to the nonlinearity of the phase imaging process, the steepness of an edge in the phase image of a strong phaseshifting object is no indicator for the bandwidth of the image (as for amplitude images of amplitude
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
192
a)
I?1
1.2
[a.u.]
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'-
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I
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i
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i
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-"
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-0.5
,
I
0
h
I
0.5
i
I
1
h
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1.5
,
2
Fig. 2. Amplitude (a) and normalized phase (b) of the calculated far field images (NA = 1) of a phaseshifting half plane with ~b0 = 10 ° (solid), $0 = 1500 (dashed) and $0 = 180° (dotted-dashed).
and weak phase objects). The corresponding amplitude distribution I~(x) l (Fig. 2a) is approximately constant for small phaseshifts but decreases strongly near the edge with increasing phaseshift down to It-] = 0 for ~b0 = ~r. It should be emphasized that the position of the phase discontinuity for ~b0 = ~" agrees with the true edge position x = 0, i.e. T(0) = 0, because H ( 0 ) = !2 is valid independent from the numerical aperture (cf. Eq. (11)). However, if the ,r-phaseshifting half plane is additionally absorbing (or reflecting) with to 4: 1, we obtain as an image the real function ~ x ) = 1 -- ( l + t 0 ) H ( x ) .
(13)
The resulting phase image ~ ( x ) = arg[t'(x)] (Fig. 3b) is still discontinuous but the positionn of the image edge is shifted due to t'(x) = 0 for H ( x ) = (1 + to) -1 (Fig. 3a): for the assumed perfect imaging system, the center of (odd) symmetry (x = O, ~= (1 - t0)/2) is conserved in the image, i.e. it denotes the true position of the edge. Only for to = 1 it agrees with the position of the phase discontinuity (~= 0). The deviation of the phase edge from the true edge position depends on the transmittance to and on the steepness of the edge image, i.e. on the numerical aperture NA. Fig. 4 shows a contour plot of the calculated deviations Ax0 (applying a root finding algorithm on t'(x)) depending on the amplitude transmittance to and the numerical aperture NA. For high numerical apertures the error remains even for high absorptances
M. Totzeck, M.A. Krurabfigel I Optics Communications 112 (1994) 189-200
193
o t(x) = 1 - (1 + to)H(x)
1
1 /
i(=)= I -
0 +
¢o1~(=1
.1
/ °--4
t.D
z [ [rad)
-I
..Q
>
4'(=)---
0~
(
~rnoglne~ot g
~ X o= I ~
d u e to
%I I
_~=)
>x
CM--}
Fig. 3. Lateral shift of the edge in the image of a ¢r-phaseshiftiag absorbing half plane of transmittance to: (a) real image, (b) phase image.
o
"--i i
-2 1.8-win////
-20
1.4
-"
/
~
0.2 0.1
~
O~ 5 ~ -~'~
0.3
I
2
a grey triangle.
o
0
1
Fig. 5. Contour plot of real image of lossless, ~r-phaseshiffing bars of increasing width b. The line 7"= 0 (thick) determines the width of the phase image. The true object widths are indicated by
t-
to 1.0
0.6 ~
0 x [Z.)
~
0.5 NA 0.7
0.9
Fig. 4. Contour plot of the lateral position error Ax0 depending on amplutude transmittance to and numerical aperture NA.
below 20%. For a decreasing numerical aperture the error increases strongly, to > 1 corresponds to a phaseshifting half plane of which the complementary part is absorbing. It results into errors with reversed sign. A ¢r-phaseshifting strip of finite width b yields also a purely real image
2(1
+tO)si(koNAb/2)
=l-(l+to) 1],
(14)
where the phase image indicates the lateral position of the changes of sign. Fig. 5 shows a contour plot of t'(x) (NA = 1,to = 1) depending on the strip width b (grey bar). The thick black line denotes t(x) -- 0. For b > 0.3A the phase image width follows the true width with errors up to 0.1h. For b < 0.22A it is always t'(x) > 0 and the phase image vanishes: The strip con-
>_ 1.
(15)
7/"
With the approximation Si(x) ~ x (first term of the polynomial series expansion [ 13 ] ) the linear relation is obtained (for x < 1) 2(1
+t°)NAb
h
t'(x) = 1 - (1 + to) ~'~t(x/2b)
× [ffl(x+b/2) +ffl(-x+b/2) -
sists basically of the superposed images of two phaseshifting half planes and t'(x) does not come down to the threshold-value t = 0 for too closely neighboured edges, i.e. too small strip widths. The condition for the observation of a phase image is t'(0) < 0, i.e. with Eq. (9) and x = 0 we obtain from Eq. (14)
_> 1,
(16)
For to = 1 and the numerical aperture NA = 1 we obtain from Eq. (16) a minimum strip width brain = 0.25A in good agreement with Fig. 5. If we consider a strip of constant width b but increasing phaseshift 4~0, the phase image steepens with increasing 4~o if b > brain and vanishes for b < brain (Fig. 6). The corresponding amplitude decreases to zero at the position of the phase discontinuity, but it remains well above zero for b < brain. The same effect could be obtained by decreasing the numerical aperture below a minimum value defined by Eq. (15). The extension of
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200
194
b=O.2Z,
?)
b=0.3~,
xc,i)
o.o
~
0.5
Fig. 6. Development of amplitude (a, c) and normalized phase (b, d) of the far field image (NA=I) of a strip with phaseshifl increasing from 5 ° to 180 °, left: width = 0.2& right: width = 0.3k. Object indicated by a bar.
this analysis to several phase strips is straight forward: Two neighboured edges at a distance Ax are resolved if NAAX > (NAb)rain. We conclude that the apparently increased resolution in the ~" phase edge corresponds to a threshold eriteflon for t" = 0. Compared with conventional threshold criteria, which refer to the maximum measured image amplitude, the t'= 0 threshold in the phase image refers to the true object transmission amplitude to, provided that to = 1. The consequence is an improved accuracy for very small objects (bn~n < b < b / N A ) . Due to reflection at the interfaces, however, the transmittance of dielectric objects obeys generally 1/01 < 1.
n = 1.6, widths 0.1k - k, thicknesses 0.01k - 1.5A) are calculated for linearly polarized, normally incident plane waves according to the pulse functions point matching version of the moment method [ 14,15]: For calculation of the electromagnetic field within the object, the cross-section of the dielectric bar is divided into single cells that are small enough (cross-section << k 2) to permit the approximation of the electric field inside them by a constant value. For N cells with center coordinates r~ = (x~, z~) a linear system of equations is obtained for each polarization. In TM-polaflzation it becomes with E ( r ~ ) = E~ N
= 1. . . . . N,
(17)
1-'=1
4. P h a s e i m a g e s c a l c u l a t e d f r o m r i g o r o u s n e a r fields
with =
In order to investigate the effects of polarization, refractive index, and object geometry on the phase images, rigorous two dimensional diffraction near fields of small dielectric bars (refractive indices n = 4.1 and
1 - ( n 2 - 1)
x [ 0 . 5 i ~ k o a H l ( k o a ) - 1], =
- K , koHo(kolrg -
r~l),
u g tz
(18)
195
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200
and in TE-polarization we obtain
and in TE-polarization N
+ oj:
( Z Hl ( kor ) EX Hy(ro) = HY(ro) + -iKsk° - - ~ " ~-~. __-~,\r
=
v--I N
(19)
- XH1 (kor)E~), r
(23)
/
v---1
wherex =x0-xv,z
where a~,~ =
1 - (n 2 - 1)[0.5i~rHl(koa)
x (koa-Jl(koa))-l],
v=l~,
= - ( K s / r 3)[z2korHo(kor) + (x 2 - z 2 ) H i ( k o r ) l , b~
=
=z0-z~,r
= V~+z
2 and
Z0 = (1£0/E0)1/2.
O,
v #~,
V=l~,
= -(Ks/r3)xz x [2Hl(kor) - korHo(kor)], dv~ = a ~ ,
v#l~,
v=/~,
= - ( K s / r 3) [x2korHo(ko r) + (Z 2 -- x2)Hl(kor)],
v Vtl~,
(20)
where x = x~ - x~, z = z~ - z~, r = x / ~ + z 2 and
Ks = (iera/2)Jl(koa)(n 2 - 1).
(21)
H0(), Hi () are the Hankel functions of first kind, zeroth order and first kind, first order, respectively. J1 () is the Bessel function of first order. The incident field is indicated by the index i. a is the radius of a circle having the same area as the Cell cross-section. After solving of the appropriate linear system of equations the diffraction field outside the cylinder is calculated at a point r0 in TM-polarization according to Ey(ro) = EY(ro) + 0.5i~rkoa(n 2 - 1) N
x Jl(koa) ~-~Ho(kor)E~ u=l
(22)
Subsequently, the far field image is calculated according to Eq. (2) with Az = zo +d/2, i.e. the filtered field is backpropagated to the center of the object corresponding to the assumed focus plane of the imaging system. To prove the applicability of the moment method for phase objects of refractive index n = 4.1, amplitude and phase of the computed near fields are compared with measurements of the near fields behind artificial dielectrics (Stycast 500, n = 4.1) obtained with a modified Mach-Zehnder intefferometer for 3 cm microwaves ( f = 9.0 GHz, A = 33.31 ram), which is described in Ref. [ 16]. It was applied in previous investigations to test the validity of the moment method for small (n = 1.02) and moderate (n = 1.6) refractive indices. The diffraction (near) field in a distance of z = 0.09A of a bar of 0.15A thickness and 0.76A width is shown in Fig. 7 (plotted are the deviations of amplitude AA and phase A~b in the diffraction field from the values o f t b e incident field Ai and ~i: AA = (A Ai )/Ai, A~ = ~ -- ~i ). The geometrical optics phaseshift, calculated via multiple beam interferences of a plate with thickness and refractive index of the object, is with - 1 7 1 ° near to 180 ° modulo 360 ° . Reflection reduces the amplitude of the transmitted field to 0.62 x incident field. The polarization dependence is reproduced accurately by the calculations, which yield a good agreement with the measurements. The calculated phase distribution in TE-polarization is discontinuous near the edge of the object. However, decreasing the distance by 0.02A yields a continuous curve in good agreement with the measurement. The reason for this effect is that the thickness of the object is near to the value where the phaseshift changes from 180 ° to - 180 °.
196
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
a)
[%]
o T M measurement •
TE
-60
- -
T M calculation - TE
- geom. opt o
o
20
-20
t
¼'
-60
-1002. 5
I
I
-2
-1.5
,
I
,
-I
I
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I
i*
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-0.5
I
1
115
2
2.5
b) 135
A~ [deg)
__, t
90 45 0 ih
-45
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-1
,
I
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-0.5
0
I
I
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1
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2
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Fig. 7. Measured (dots) and calculated (lines) near fields 0.09A behind a rectangular dielectric strip of 0.76A width, n = 4.1, and 0.15A thickness in TM- (eireles) and TE-polarization (full triangles).
4.1. Steep edges The moment method was applied to calculate the near fields in a distance z0 = 0.015A behind rectangular dielectric bars of constant width b and constant refractive index n, but increasing thickness d and thereby increasing phaseshift ~b. Fig. 8a shows as an example the phase distribution for TE-polarized illumination, n = 4.1, b = 0.63A, and d increasing from 0.01A to 0.15A. With increasing thickness, lateral oscillations due to diffraction become more pronounced. Fig. 8b shows the far field phase image calculated from Fig. 8a with NA = 1. Note the almost rectangular phase distribution for d ~ 0.15A. For a quantitative evaluation of the calculated phase images the half width bl/2, maximum phaseshift ~bmax,
and edge width Ab are calculated according to Fig. 9 and plotted as a function of the thickness d. The definition of the edge width Ab is somewhat arbitrary and should be understood only as a relative gauge. Prior to the calculations, 27r phaseshifts are removed from the phase distributions. Fig. 10 shows the evaluation of Fig. 8. The left axis displays widths and the right axis phaseshifts. The true object width b is indicated by a dashed line and the geometrical optics phaseshift calculated from multiple beam interferences is drawn as a dotted dashed line. While the maximum phaseshift (fight axis) follows the geometrical optics phaseshift for both the near field and the far field phase image (with an offset of about 30°), the half width and the edge width (left axis) show differences: Below thicknesses of d = 0.13A the
M. Totzeck, M.A. KrumbOgel / Optics Communications 112 (1994) 189--200
197
10% error, compared with the actual object width of 0.63,~, is in agreement with the error expected according to Fig. 4 for NA = 1, since [to[ ffi0.69 at d = 0.144. I[
4.2. Polarization dependence
•y, ,_,
Fig. 8. Phase distribution of the near field (a) and far field image (NA = 1) (b) of a bar of b = 0.63A width, refractive index n = 4.1, and thickness d increased from 0.01A to 0.15A. The true object width is indicated by a bar. TE-polafized illumination.
Particularly for small object widths, the polarization dependence of the phase images is pronounced. Fig. 11 shows the evaluated phase images for NA = 1, n -- 4.1, and an object width of 0.27A for TE- (a) and TM-polarization (b). In TE-polarization (Fig. l l a ) smooth curves are obtained where the maximum phaseshift remains well below the geometrical optics phaseshift and the half width is considerably larger than the object width, except the object thicknesses where Omax yields a maximum. There is a distinct correlation between ~bmax,bl/2 and Ab: With increasing phaseshift tl~maxboth Ab, and bl/2 decrease. The minimum edge width at d = 0.24A occurs at ~brnax = -185 °. The corresponding half width of 0.42A differs by 55% from the actual object width. This value is caused by a near field distribution that is already too wide, compared with the object. In TM-polarization the development of the values with increasing thickness is rather erratic for d < 0.15~. The maximum phaseshift oscillates around ~bgeo with deviations up to 150 °. Here, the measured half width bl/2 agrees with the actual one for ~bmax < 50 °. The corresponding edge width Ab = 0.1 - 0.2A is well above zero. The polarization dependence increases strongly with decreasing object width. Generally, TM-polarized illumination yields larger phaseshifts and more intensely modulated fields than TE-polarization. This effect is caused by a stronger scattered field in TMpolarization [ 17].
Fig. 9. Determination of half width
4.3. Influence of refractive index
near field edge width varies around 0.1A, while the values in the far field image are twice as large. Both far and near field images show decreasing edge widths with increasing phaseshifts. Immediately before the minimum value a sharp local increase is observed. The far field image yields a minimum Ab for 0max = 180°. The corresponding half width is 0.70A. The obtained
For a refractive index of n = 1.6, the obtained far field phase images (Fig, l l c ) are diffraction limited even for maximum phaseshifts around 180 °. Although the edge width decreases with increasing phaseshift, the half width increases strongly. A possible reason for this effect is that at ~b ~, 180 ° the thickness d is comparable to the wavelength and large compared to the width, i.e. waveguiding effects become dominant.
hi~2, edge width Ab, and maximum phase.shift Om~, from the calculated phase distributions.
198
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200 near field
......-
0.8
200
4~rnc= \
O.O
_....J
~
'geo
o
IO0
~
ol e
b 1/2
0.4
Ah
0.2
--
Oi~
,
,
I
0.03
--
,
41.
,
,ll.
I
0.00
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200
0.8 100 . . . . . .
. . . . . . . .
0.6 \
0]
oi/2
D=O.63X
0.4
e
Ab
-100
0.2
0.03
0.06
d D,]
0.0g
0.12
0.1
300
Fig. 10. Evaluation of the calculated phase images of Fig. 8, bl/2 = half width, Ab = edge width, ~brnax = m a x i m u m phaseshift, ~bgeo = phaseshift according to geometrical optics, d = object thickness. Dashed horizontal line = true object width.
4.4. Image of very small objects
5. Conclusions
For very small objects of b = 0.1A we observed a strongly increased width of the phase distribution in the near field and consequently a broadened far field image (for sufficiently high NA). Fig. 12 shows the "vanishing" of the phase image (cf. Fig. 6) with decreasing numerical aperture: there is a pronounced transition from a strong to a weak phase image between NA = 0.6 and NA = 0.5. Using Eq. (16) with the amplitude transmittance to = 0.5 and b = 0.6A we obtain the condition NA > 0.56 for a visible phase image.
It was demonstrated in geometrical optics approximation for the images of 7r-phaseshifting, dielectric objects that an apparent superresolution of the edges corresponds to a threshold criterion in the real part of the complex image (7(x) = 0) with the consequences that, (i) absorbing or reflecting objects yield nevertheless discontinuous edges but at a wrong position, where the error increases with decreasing amplitude transmission and numerical aperture and (ii) 7r-phaseshifting strips must have a minimum width (depending on amplitude transmission and numerical aperture) in order to obtain a phase image. These effects were essentially confirmed using
M. Totzeck, M.A. Krumbtigel I Optics Communications 112 (1994) 189-200
a) 1
199
200
TE-polarizatlon
I.__ ~
n=4.1
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n=4.1 100 o
0.6 r-'l
.0.
0.4
-100 0.2
%
. . . . . . . . 0.05
*
0.1
. . . .
|
d [),]
0.15
....
*
0.2
•
,
i i I.~00 O.
1
c) 0.8
TM-polarization n=l.6 . ~
~
200
,_0.5 ¢..,
J
0.4 |
.~.o
0.2
00
,
,
I
0.3
,
,
I
0.6
,
,
,I 0.0
,
,
I
1.2
,
,
-~00
1.5'
d [),]
Fig. 11. Evaluation of the calculated phase images for dielectric bars of 0.27A width. (a) TE-polarization, n = 4.1, (b) TM-polarization, n = 4.1, (c) TM-polarization, n = 1.6. Note the changed scaling of the x-axis in (c). Symbols: s¢¢ Fig. 10.
200
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
Summing up: phase imaging is a nonlinear process and the interpretation of the images should be performed cautiously.
100
a)
c%] NA= I
60
2O
Acknowledgements -20
We are indebted to S. B~iumer,Prof. J. Kross, W. Mirand~ and Dr. K.-E Schr0der for valuable discussions.
-~o -100
-1.6
-1.2
-0.8
-'0.4
X
0
IX]
0.4
0.8
1.2
1.6
References
20
-40 [deg] ~0 n i i
im m mm mm
i i i
-100
~ i
mm
i' i
-120
, i t
m l I
-80
mm
n
,
i f
",I /
-160
-1,6
-1.2
"O.S
"0.4
0
X
[),3
0.4
O.S
1.2
1.6
2
Fig. 12. Amplitude (a) and phase (b) of the images of a dielectric bar of 0.1A width and 0.16A thickness for different numerical apertures NA. The geometrical optics transmittance is indicated by a grey bar.
simulated far field images calculated from rigorous diffraction near fields behind dielectric strips of increasing thickness. Additionally, it was shown that edge width and half width depend strongly on polarization and refractive index. A phase image that is comparable to the image of geometrical optics fields is better possible for objects with a high refractive index and TE-polarized illumination.
[1] V.R Tychinski, Optics Comm. 74 (1989) 41. [2] V.E "l~chinski, Optics Comm. 81 (1991) 131. [3] V.E Tychinski, A.V. Tarrov, Sov. J. Quantum Electron. 20 (1990) 206. [4] C.H.E Velzei, R. Masselink, A. Tavrov, V. Tychinski, Int. Conf. on Optical Metrology and Nanotechnology, EOS Topical Meetings Digest Series 4 (1994) 22. [5 ] D.W. Pohl, Advances in Optical and Electron Microscopy, 12 (1991) 243. [6] J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968). [7] PJ. Sementilli, B.R. Hunt, M.S. Nadar, J. Opt. Soc. Am. A 10 (1993) 2265. [8] M.A. Krumb(igel, M. Totzeck, Optics Comm. 98 (1993) 47. [9] D. Nyyssonen, J. Opt. Soc, Am. 72 (1982) 1425. [10] D. Nyyssonen, Opt. Eng. 26 (1987) 81. [11] M.D. Levenson, N.S. Viswanathan, R.A. Simpson, Trans. Electron Devices 29 (1982) 1828. [ 12] M.D. Levenson, Physics Today, July (1993) 28. [13] M. Abramowitz, I.A. Stegan, Handbook of Mathematical Functions (Dover FubI., New York) 232. [14] J.H. Richmond, II~.l~.P. Trans. Antennas Propag. AP-13 (1965) 334. [15] J.H. Richmond, IEEE Trans. Antennas Propag. AP-14 (1966) 460. [16] G. Koppelmann, M. Totzeck, J. Opt. Soc. Am. A 8 (1991) 554. [17] M. Totzeck, Appl. Optics 32 (1993) 1901
Optics Communications112 (1994) 189-200 F u l l length article
Lateral resolution in the near field and far field phase images of 7r-phaseshifting structures M i c h a e l Totzeck, M a r c o A. Krurnbtigel Optisches Institut der TechnischenUniversitlltBerlin, Hardenbergstrasse36, 10623 Berlin, Germany Received 19 April 1994;revisedmanuscriptreceived 3 August 1994
Abstract The resolution in the phase distribution of coherent images of strong phase objects is investigated using moment method near field calculations and simulated far field images. It is demonstrated that an apparent superresolution for ~r-phaseshifts corresponds to a threshold criterion and that the images depend strongly on polarization and refractive index.
1. Introduction Recently it was proposed by Tychinski et al. [ 1-3 ] that the far field "phase images", i.e. the phase of the coherent images of strong phaseshifting objects are not subject to the Rayleigh resolution limit of A/2: They reported a lateral resolution down to 10 nm (in determination of edge coordinates at A ffi 632 nm) by evaluating the phase of the microscopical image of topographical line structures in reflection. The best resolution was obtained for ~r-phaseshifts, i.e. structure heights of A/4 [4]. The surprising point of the method is the fact that the reported strong superresolution was achieved without using near field imaging or a subsequent numerical treatment of the images. As is well known, the maximum resolution obtainable with an optical system is restricted to 1/fmax with fmax the maximum transferred spatial frequency. The far field resolution limit of A/2 is a consequence of the fact that free space propagation acts as an optical low-pass filter. In this context, superresolution is equivalent to getting access to the f > 2/A-part of the spatial frequency spectrum, or in spatial terms: distin-
guishing structure elements with distances < A/2. Usually, superresolution (of stationary structures) requires a lot of experimental or numerical effort. Additionally, method and results depend strongly on the available a priori knowledge. Generally we may say that the more we know about the investigated structure, the more simple the applied superresolution methods may be, but the less informative are the images and the more susceptible to errors. The presented paper investigates the question of superresolution in the phase images of two dimensional dielectric structures with rectangular cross-section and strong phaseshifts. Several methods of superresolution imaging for these kind of objects are well known: (i) Scanning near field optical microscopy (SNOM), where the not propagating evanescent components of the frequency spectrum are measured by a sub-A probe that is moved by piezo-translators in the near field of the object [ 5 ]. This method is the most general one, since no a priori information about the object is required, although the image interpretation is sometimes rather difficult due to the complicated object-probe interaction in the near field.
0030-4018/94/~r/.00 (~) 1994 Elsevier ScienceB.V. All rights reserved SSDI 0030-4018 ( 94 ) 00451 -X
190
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200
(ii) Utilizing the analycity of the frequency spectrum by analytic continuation of the transferred part [6]. This method is mainly limited by noise (for a general discussion, see Ref. [7] ). (iii) Utilizing a priori information by finding a mathematical expression for the frequency spectrum with a set of free parameters and a subsequent numerical fit of these parameters to the transferred part of the spectrum. For dielectric bars, this procedure is possible for weak phase objects only, where the first Born approximation is valid [ 8 ]. It can not be applied here, because in order to obtain strong phaseshifts, objects of high refractive index are of main interest, where multiple scattered fields yield a strong contribution to the diffraction field, (iv) Utilizing a priori information in the form of threshold criteria, i.e. only the position of the edges of an object has to be determined. This least general method is widely used in critical dimension metrology [9,101. Another kind of superresolution that is closely related to phase imaging, is observed for two ~rphaseshifted coherent point objects (6-pulses): their images may be resolved for arbitrary small distances, since due to destructive interference of the corresponding Airy-discs there is a node between them [6]. However, this result can not be applied to arbitrary objects, because a third point object could not be ~'-phaseshifted with respect to both previous points. Considering a grating where neighboured grooves are ~r-phaseshifted with respect to each other, the resulting resolution improvement is limited to a factor of two, because a ~" phaseshift between the nearest neighbours results in identical phase of the next but one, between which the Rayleigh resolution limit holds. This is in agreement with the fact that utilizing ~'-phaseshifting masks in photolithography, a resolution improvement by a factor ~ 2 is possible [ 11,12].
2. Geometry of the imaging configuration For the investigation of superresolution in the images of strong phase objects the following imaginggeometry is considered: a dielectric strip of width b, thickness d, and refractive index n is illuminated by a normally incident plane wave of wavelength ho (Fig. 1). The strip is parallel to the y-axis. Some dis-
tance behind the object an imaging system of numerical aperture NA is situated. For calculation of the images the space is divided into two parts, linked by the plane z = z0 immediately behind the object. The optical field U(x, zo) in this plane is calculated by solving the near field diffraction problem of the particular phase object in part I. The diffraction near field is obtained from the sum of the incident field and the scattered field Us according to
U(x, zo) = exp(ikoz0) + Us(x, zo).
( 1)
Behind the object in part II this field propagates through free space to the imaging system. Two perpendicular linear polarizations are considered: in TM-polarization the electric field has only a y-component, i.e. U(x,z) = Ey(x,z), and in TEpolarization the magnetic field is U( x, z ) = Hy( x, z ). The imaging process is simulated by applying a spatial frequency filter to the near field U(x, zo), i.e.
[l(x) = . ~ { e - l ( f x , Az)M(fx).~{U(x)} },
(2)
where ~{} and ~ - l { } denote the Fourier transform and the inverse Fourier transform, respectively. The tilde marks the image. P - l ( f , , A z ) is the inverse propagation operator, which describes the focussing of the imaging system at the plane z = z0 - Az according to
P - l ( f x , A Z ) =exp[-i2zr(A-z + f~)t/2Az].
(3)
The phase image is ~ ( x ) = arg(U(x) )
(4)
and [U(x)[ denotes the amplitude image. The imaging system's optical transfer function
[ Aof x "~ M ( f , ) = rect~ 2--~-A} = 1,
Xofx < N^,
=½,
aofx = NA,
=0,
aofx > N ,
(5)
is assumed as real and symmetrical, i.e. the imaging system should be perfect in the sense that it acts as a pure bandpass filter without additional disturbances of the transmitted spatial frequencies. It should be emphasized, that while the imaging process of the real and imaginary part is linear (for a linear imaging system), the amplitude and phase imaging is nonlinear. Of particular interest for this paper is
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
focus plane
nearfleld plane
Incidentwave
/
= i
i >
phc obj,
,o;
TM'.®E i
Imaging system with numerical aperture N A
|
TE: ,I~,,H(, >
<
191
;!
d
\
>"Zoi< /N y
I
H
,Iriteracflon of incident I wave w th object I
free space propagation ]
Fig. 1. Geometry of image configuration.
the pronounced nonlinearity of the phase imaging for phaseshifts ~ It.
where H(x) is equivalent to the well known image of the perfectly absorbing halfplane (from ~'{H(x) } = 0.5(1 - i/zrfx)):
3. Far tidd images in geometrical optics approximation
H(x) = (1/~r)Si(koNAx) + 1.
In order to investigate what kind of effects are to be expected for the imaging of strong phase objects, we assume propagation of the incident wave through the object according to geometrical optics (simulated images using rigorous electromagnetic near fields are discussed in Sect. 4). Consider a phaseshifting halfplane of transmittance
(9)
Si() denotes the sine-integral. For a lossless (Itol = 1) weak phaseshifting (~bo < 20 °) half plane the phase image ~(x) = ~b0H(x)
(10)
where H(x) denotes the Heaviside function
is proportional to the image of the perfectly absorbing halfplane, which is diffraction limited (Fig. 2b, NA = 1). For intermediate phaseshifts the phase distribution near the edge in the image becomes increasingly steeper with increasing phaseshifts. For Oo = ¢r the phase image is even discontinuous because the imaginary part of the complex image
H(x)=l,
t'(x) = 1 - 2H(x),
t(x) = 1 + [toexp(iOo) - 1] H(x),
(6)
x>O,
=½,
x =0,
--0,
x<0
(7)
and to is the amplitude transmittance. The phase image becomes ~ ( x ) --- arg[F(x) ] = arg{1 + [t0 exp(it~o) - 1] H ( x ) } ,
(8)
(11)
vanishes and the phase distribution indicates merely the change of sign of the real part, i.e. ~(x) = (~/2) {1 -t- sgn[1 - 2H(x) ]}.
(12)
Due to the nonlinearity of the phase imaging process, the steepness of an edge in the phase image of a strong phaseshifting object is no indicator for the bandwidth of the image (as for amplitude images of amplitude
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
192
a)
I?1
1.2
[a.u.]
~bo--10°
• -
,.,._../
\
o.g
/
""-""
/
\
0.6
'-
i
#
0.3
~._ #~o= 180 ° i
I
,
I
-1.5
"'2
,
-1
I
i
-0.5
'7
i
0
I
i
0.5
I
,
1
x (x) I
i
1.5
b) 0.8x
qb / dpo
%= 150 °
U:;
-"
0.6
0,
\
~o-- 10°
0.2
"L:L*e18oo
\
o -0.2 2 -
x (Z.) '
I
-1.5
i
I
-1
J
I
-0.5
,
I
0
h
I
0.5
i
I
1
h
I
1.5
,
2
Fig. 2. Amplitude (a) and normalized phase (b) of the calculated far field images (NA = 1) of a phaseshifting half plane with ~b0 = 10 ° (solid), $0 = 1500 (dashed) and $0 = 180° (dotted-dashed).
and weak phase objects). The corresponding amplitude distribution I~(x) l (Fig. 2a) is approximately constant for small phaseshifts but decreases strongly near the edge with increasing phaseshift down to It-] = 0 for ~b0 = ~r. It should be emphasized that the position of the phase discontinuity for ~b0 = ~" agrees with the true edge position x = 0, i.e. T(0) = 0, because H ( 0 ) = !2 is valid independent from the numerical aperture (cf. Eq. (11)). However, if the ,r-phaseshifting half plane is additionally absorbing (or reflecting) with to 4: 1, we obtain as an image the real function ~ x ) = 1 -- ( l + t 0 ) H ( x ) .
(13)
The resulting phase image ~ ( x ) = arg[t'(x)] (Fig. 3b) is still discontinuous but the positionn of the image edge is shifted due to t'(x) = 0 for H ( x ) = (1 + to) -1 (Fig. 3a): for the assumed perfect imaging system, the center of (odd) symmetry (x = O, ~= (1 - t0)/2) is conserved in the image, i.e. it denotes the true position of the edge. Only for to = 1 it agrees with the position of the phase discontinuity (~= 0). The deviation of the phase edge from the true edge position depends on the transmittance to and on the steepness of the edge image, i.e. on the numerical aperture NA. Fig. 4 shows a contour plot of the calculated deviations Ax0 (applying a root finding algorithm on t'(x)) depending on the amplitude transmittance to and the numerical aperture NA. For high numerical apertures the error remains even for high absorptances
M. Totzeck, M.A. Krurabfigel I Optics Communications 112 (1994) 189-200
193
o t(x) = 1 - (1 + to)H(x)
1
1 /
i(=)= I -
0 +
¢o1~(=1
.1
/ °--4
t.D
z [ [rad)
-I
..Q
>
4'(=)---
0~
(
~rnoglne~ot g
~ X o= I ~
d u e to
%I I
_~=)
>x
CM--}
Fig. 3. Lateral shift of the edge in the image of a ¢r-phaseshiftiag absorbing half plane of transmittance to: (a) real image, (b) phase image.
o
"--i i
-2 1.8-win////
-20
1.4
-"
/
~
0.2 0.1
~
O~ 5 ~ -~'~
0.3
I
2
a grey triangle.
o
0
1
Fig. 5. Contour plot of real image of lossless, ~r-phaseshiffing bars of increasing width b. The line 7"= 0 (thick) determines the width of the phase image. The true object widths are indicated by
t-
to 1.0
0.6 ~
0 x [Z.)
~
0.5 NA 0.7
0.9
Fig. 4. Contour plot of the lateral position error Ax0 depending on amplutude transmittance to and numerical aperture NA.
below 20%. For a decreasing numerical aperture the error increases strongly, to > 1 corresponds to a phaseshifting half plane of which the complementary part is absorbing. It results into errors with reversed sign. A ¢r-phaseshifting strip of finite width b yields also a purely real image
2(1
+tO)si(koNAb/2)
=l-(l+to) 1],
(14)
where the phase image indicates the lateral position of the changes of sign. Fig. 5 shows a contour plot of t'(x) (NA = 1,to = 1) depending on the strip width b (grey bar). The thick black line denotes t(x) -- 0. For b > 0.3A the phase image width follows the true width with errors up to 0.1h. For b < 0.22A it is always t'(x) > 0 and the phase image vanishes: The strip con-
>_ 1.
(15)
7/"
With the approximation Si(x) ~ x (first term of the polynomial series expansion [ 13 ] ) the linear relation is obtained (for x < 1) 2(1
+t°)NAb
h
t'(x) = 1 - (1 + to) ~'~t(x/2b)
× [ffl(x+b/2) +ffl(-x+b/2) -
sists basically of the superposed images of two phaseshifting half planes and t'(x) does not come down to the threshold-value t = 0 for too closely neighboured edges, i.e. too small strip widths. The condition for the observation of a phase image is t'(0) < 0, i.e. with Eq. (9) and x = 0 we obtain from Eq. (14)
_> 1,
(16)
For to = 1 and the numerical aperture NA = 1 we obtain from Eq. (16) a minimum strip width brain = 0.25A in good agreement with Fig. 5. If we consider a strip of constant width b but increasing phaseshift 4~0, the phase image steepens with increasing 4~o if b > brain and vanishes for b < brain (Fig. 6). The corresponding amplitude decreases to zero at the position of the phase discontinuity, but it remains well above zero for b < brain. The same effect could be obtained by decreasing the numerical aperture below a minimum value defined by Eq. (15). The extension of
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200
194
b=O.2Z,
?)
b=0.3~,
xc,i)
o.o
~
0.5
Fig. 6. Development of amplitude (a, c) and normalized phase (b, d) of the far field image (NA=I) of a strip with phaseshifl increasing from 5 ° to 180 °, left: width = 0.2& right: width = 0.3k. Object indicated by a bar.
this analysis to several phase strips is straight forward: Two neighboured edges at a distance Ax are resolved if NAAX > (NAb)rain. We conclude that the apparently increased resolution in the ~" phase edge corresponds to a threshold eriteflon for t" = 0. Compared with conventional threshold criteria, which refer to the maximum measured image amplitude, the t'= 0 threshold in the phase image refers to the true object transmission amplitude to, provided that to = 1. The consequence is an improved accuracy for very small objects (bn~n < b < b / N A ) . Due to reflection at the interfaces, however, the transmittance of dielectric objects obeys generally 1/01 < 1.
n = 1.6, widths 0.1k - k, thicknesses 0.01k - 1.5A) are calculated for linearly polarized, normally incident plane waves according to the pulse functions point matching version of the moment method [ 14,15]: For calculation of the electromagnetic field within the object, the cross-section of the dielectric bar is divided into single cells that are small enough (cross-section << k 2) to permit the approximation of the electric field inside them by a constant value. For N cells with center coordinates r~ = (x~, z~) a linear system of equations is obtained for each polarization. In TM-polaflzation it becomes with E ( r ~ ) = E~ N
= 1. . . . . N,
(17)
1-'=1
4. P h a s e i m a g e s c a l c u l a t e d f r o m r i g o r o u s n e a r fields
with =
In order to investigate the effects of polarization, refractive index, and object geometry on the phase images, rigorous two dimensional diffraction near fields of small dielectric bars (refractive indices n = 4.1 and
1 - ( n 2 - 1)
x [ 0 . 5 i ~ k o a H l ( k o a ) - 1], =
- K , koHo(kolrg -
r~l),
u g tz
(18)
195
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200
and in TE-polarization we obtain
and in TE-polarization N
+ oj:
( Z Hl ( kor ) EX Hy(ro) = HY(ro) + -iKsk° - - ~ " ~-~. __-~,\r
=
v--I N
(19)
- XH1 (kor)E~), r
(23)
/
v---1
wherex =x0-xv,z
where a~,~ =
1 - (n 2 - 1)[0.5i~rHl(koa)
x (koa-Jl(koa))-l],
v=l~,
= - ( K s / r 3)[z2korHo(kor) + (x 2 - z 2 ) H i ( k o r ) l , b~
=
=z0-z~,r
= V~+z
2 and
Z0 = (1£0/E0)1/2.
O,
v #~,
V=l~,
= -(Ks/r3)xz x [2Hl(kor) - korHo(kor)], dv~ = a ~ ,
v#l~,
v=/~,
= - ( K s / r 3) [x2korHo(ko r) + (Z 2 -- x2)Hl(kor)],
v Vtl~,
(20)
where x = x~ - x~, z = z~ - z~, r = x / ~ + z 2 and
Ks = (iera/2)Jl(koa)(n 2 - 1).
(21)
H0(), Hi () are the Hankel functions of first kind, zeroth order and first kind, first order, respectively. J1 () is the Bessel function of first order. The incident field is indicated by the index i. a is the radius of a circle having the same area as the Cell cross-section. After solving of the appropriate linear system of equations the diffraction field outside the cylinder is calculated at a point r0 in TM-polarization according to Ey(ro) = EY(ro) + 0.5i~rkoa(n 2 - 1) N
x Jl(koa) ~-~Ho(kor)E~ u=l
(22)
Subsequently, the far field image is calculated according to Eq. (2) with Az = zo +d/2, i.e. the filtered field is backpropagated to the center of the object corresponding to the assumed focus plane of the imaging system. To prove the applicability of the moment method for phase objects of refractive index n = 4.1, amplitude and phase of the computed near fields are compared with measurements of the near fields behind artificial dielectrics (Stycast 500, n = 4.1) obtained with a modified Mach-Zehnder intefferometer for 3 cm microwaves ( f = 9.0 GHz, A = 33.31 ram), which is described in Ref. [ 16]. It was applied in previous investigations to test the validity of the moment method for small (n = 1.02) and moderate (n = 1.6) refractive indices. The diffraction (near) field in a distance of z = 0.09A of a bar of 0.15A thickness and 0.76A width is shown in Fig. 7 (plotted are the deviations of amplitude AA and phase A~b in the diffraction field from the values o f t b e incident field Ai and ~i: AA = (A Ai )/Ai, A~ = ~ -- ~i ). The geometrical optics phaseshift, calculated via multiple beam interferences of a plate with thickness and refractive index of the object, is with - 1 7 1 ° near to 180 ° modulo 360 ° . Reflection reduces the amplitude of the transmitted field to 0.62 x incident field. The polarization dependence is reproduced accurately by the calculations, which yield a good agreement with the measurements. The calculated phase distribution in TE-polarization is discontinuous near the edge of the object. However, decreasing the distance by 0.02A yields a continuous curve in good agreement with the measurement. The reason for this effect is that the thickness of the object is near to the value where the phaseshift changes from 180 ° to - 180 °.
196
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
a)
[%]
o T M measurement •
TE
-60
- -
T M calculation - TE
- geom. opt o
o
20
-20
t
¼'
-60
-1002. 5
I
I
-2
-1.5
,
I
,
-I
I
I
I
i*
0.5
-0.5
I
1
115
2
2.5
b) 135
A~ [deg)
__, t
90 45 0 ih
-45
-90 -135 -1802.5
I
I
I
-2
-1.5
-1
,
I
" - ' ~ " "
-0.5
0
I
I
0.5
1
,
I
I
1.5
2
2.5
Fig. 7. Measured (dots) and calculated (lines) near fields 0.09A behind a rectangular dielectric strip of 0.76A width, n = 4.1, and 0.15A thickness in TM- (eireles) and TE-polarization (full triangles).
4.1. Steep edges The moment method was applied to calculate the near fields in a distance z0 = 0.015A behind rectangular dielectric bars of constant width b and constant refractive index n, but increasing thickness d and thereby increasing phaseshift ~b. Fig. 8a shows as an example the phase distribution for TE-polarized illumination, n = 4.1, b = 0.63A, and d increasing from 0.01A to 0.15A. With increasing thickness, lateral oscillations due to diffraction become more pronounced. Fig. 8b shows the far field phase image calculated from Fig. 8a with NA = 1. Note the almost rectangular phase distribution for d ~ 0.15A. For a quantitative evaluation of the calculated phase images the half width bl/2, maximum phaseshift ~bmax,
and edge width Ab are calculated according to Fig. 9 and plotted as a function of the thickness d. The definition of the edge width Ab is somewhat arbitrary and should be understood only as a relative gauge. Prior to the calculations, 27r phaseshifts are removed from the phase distributions. Fig. 10 shows the evaluation of Fig. 8. The left axis displays widths and the right axis phaseshifts. The true object width b is indicated by a dashed line and the geometrical optics phaseshift calculated from multiple beam interferences is drawn as a dotted dashed line. While the maximum phaseshift (fight axis) follows the geometrical optics phaseshift for both the near field and the far field phase image (with an offset of about 30°), the half width and the edge width (left axis) show differences: Below thicknesses of d = 0.13A the
M. Totzeck, M.A. KrumbOgel / Optics Communications 112 (1994) 189--200
197
10% error, compared with the actual object width of 0.63,~, is in agreement with the error expected according to Fig. 4 for NA = 1, since [to[ ffi0.69 at d = 0.144. I[
4.2. Polarization dependence
•y, ,_,
Fig. 8. Phase distribution of the near field (a) and far field image (NA = 1) (b) of a bar of b = 0.63A width, refractive index n = 4.1, and thickness d increased from 0.01A to 0.15A. The true object width is indicated by a bar. TE-polafized illumination.
Particularly for small object widths, the polarization dependence of the phase images is pronounced. Fig. 11 shows the evaluated phase images for NA = 1, n -- 4.1, and an object width of 0.27A for TE- (a) and TM-polarization (b). In TE-polarization (Fig. l l a ) smooth curves are obtained where the maximum phaseshift remains well below the geometrical optics phaseshift and the half width is considerably larger than the object width, except the object thicknesses where Omax yields a maximum. There is a distinct correlation between ~bmax,bl/2 and Ab: With increasing phaseshift tl~maxboth Ab, and bl/2 decrease. The minimum edge width at d = 0.24A occurs at ~brnax = -185 °. The corresponding half width of 0.42A differs by 55% from the actual object width. This value is caused by a near field distribution that is already too wide, compared with the object. In TM-polarization the development of the values with increasing thickness is rather erratic for d < 0.15~. The maximum phaseshift oscillates around ~bgeo with deviations up to 150 °. Here, the measured half width bl/2 agrees with the actual one for ~bmax < 50 °. The corresponding edge width Ab = 0.1 - 0.2A is well above zero. The polarization dependence increases strongly with decreasing object width. Generally, TM-polarized illumination yields larger phaseshifts and more intensely modulated fields than TE-polarization. This effect is caused by a stronger scattered field in TMpolarization [ 17].
Fig. 9. Determination of half width
4.3. Influence of refractive index
near field edge width varies around 0.1A, while the values in the far field image are twice as large. Both far and near field images show decreasing edge widths with increasing phaseshifts. Immediately before the minimum value a sharp local increase is observed. The far field image yields a minimum Ab for 0max = 180°. The corresponding half width is 0.70A. The obtained
For a refractive index of n = 1.6, the obtained far field phase images (Fig, l l c ) are diffraction limited even for maximum phaseshifts around 180 °. Although the edge width decreases with increasing phaseshift, the half width increases strongly. A possible reason for this effect is that at ~b ~, 180 ° the thickness d is comparable to the wavelength and large compared to the width, i.e. waveguiding effects become dominant.
hi~2, edge width Ab, and maximum phase.shift Om~, from the calculated phase distributions.
198
M. Totzeck, M.A. Krumbiigel / Optics Communications 112 (1994) 189-200 near field
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Fig. 10. Evaluation of the calculated phase images of Fig. 8, bl/2 = half width, Ab = edge width, ~brnax = m a x i m u m phaseshift, ~bgeo = phaseshift according to geometrical optics, d = object thickness. Dashed horizontal line = true object width.
4.4. Image of very small objects
5. Conclusions
For very small objects of b = 0.1A we observed a strongly increased width of the phase distribution in the near field and consequently a broadened far field image (for sufficiently high NA). Fig. 12 shows the "vanishing" of the phase image (cf. Fig. 6) with decreasing numerical aperture: there is a pronounced transition from a strong to a weak phase image between NA = 0.6 and NA = 0.5. Using Eq. (16) with the amplitude transmittance to = 0.5 and b = 0.6A we obtain the condition NA > 0.56 for a visible phase image.
It was demonstrated in geometrical optics approximation for the images of 7r-phaseshifting, dielectric objects that an apparent superresolution of the edges corresponds to a threshold criterion in the real part of the complex image (7(x) = 0) with the consequences that, (i) absorbing or reflecting objects yield nevertheless discontinuous edges but at a wrong position, where the error increases with decreasing amplitude transmission and numerical aperture and (ii) 7r-phaseshifting strips must have a minimum width (depending on amplitude transmission and numerical aperture) in order to obtain a phase image. These effects were essentially confirmed using
M. Totzeck, M.A. Krumbtigel I Optics Communications 112 (1994) 189-200
a) 1
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Fig. 11. Evaluation of the calculated phase images for dielectric bars of 0.27A width. (a) TE-polarization, n = 4.1, (b) TM-polarization, n = 4.1, (c) TM-polarization, n = 1.6. Note the changed scaling of the x-axis in (c). Symbols: s¢¢ Fig. 10.
200
M. Totzeck, M.A. Krumbiigel I Optics Communications 112 (1994) 189-200
Summing up: phase imaging is a nonlinear process and the interpretation of the images should be performed cautiously.
100
a)
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Acknowledgements -20
We are indebted to S. B~iumer,Prof. J. Kross, W. Mirand~ and Dr. K.-E Schr0der for valuable discussions.
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References
20
-40 [deg] ~0 n i i
im m mm mm
i i i
-100
~ i
mm
i' i
-120
, i t
m l I
-80
mm
n
,
i f
",I /
-160
-1,6
-1.2
"O.S
"0.4
0
X
[),3
0.4
O.S
1.2
1.6
2
Fig. 12. Amplitude (a) and phase (b) of the images of a dielectric bar of 0.1A width and 0.16A thickness for different numerical apertures NA. The geometrical optics transmittance is indicated by a grey bar.
simulated far field images calculated from rigorous diffraction near fields behind dielectric strips of increasing thickness. Additionally, it was shown that edge width and half width depend strongly on polarization and refractive index. A phase image that is comparable to the image of geometrical optics fields is better possible for objects with a high refractive index and TE-polarized illumination.
[1] V.R Tychinski, Optics Comm. 74 (1989) 41. [2] V.E "l~chinski, Optics Comm. 81 (1991) 131. [3] V.E Tychinski, A.V. Tarrov, Sov. J. Quantum Electron. 20 (1990) 206. [4] C.H.E Velzei, R. Masselink, A. Tavrov, V. Tychinski, Int. Conf. on Optical Metrology and Nanotechnology, EOS Topical Meetings Digest Series 4 (1994) 22. [5 ] D.W. Pohl, Advances in Optical and Electron Microscopy, 12 (1991) 243. [6] J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968). [7] PJ. Sementilli, B.R. Hunt, M.S. Nadar, J. Opt. Soc. Am. A 10 (1993) 2265. [8] M.A. Krumb(igel, M. Totzeck, Optics Comm. 98 (1993) 47. [9] D. Nyyssonen, J. Opt. Soc, Am. 72 (1982) 1425. [10] D. Nyyssonen, Opt. Eng. 26 (1987) 81. [11] M.D. Levenson, N.S. Viswanathan, R.A. Simpson, Trans. Electron Devices 29 (1982) 1828. [ 12] M.D. Levenson, Physics Today, July (1993) 28. [13] M. Abramowitz, I.A. Stegan, Handbook of Mathematical Functions (Dover FubI., New York) 232. [14] J.H. Richmond, II~.l~.P. Trans. Antennas Propag. AP-13 (1965) 334. [15] J.H. Richmond, IEEE Trans. Antennas Propag. AP-14 (1966) 460. [16] G. Koppelmann, M. Totzeck, J. Opt. Soc. Am. A 8 (1991) 554. [17] M. Totzeck, Appl. Optics 32 (1993) 1901