- Email: [email protected]
Phase images of grooves in a perfectly conducting surface
15 September
1995
OPTICS COMMUNICATIONS Optics Communications
119 ( 1995) 485-490
Phase images of grooves in a perfectly conducting surface Juhani Huttunen a, Jari Turunen b a Department of Technical Physics, Helsinki University of Technology, FIN-02150 Espoo, Finland h Depurtment of Physics, University of Joensuu, PO. Bo.x 1 I I, FIN-80101 Joensuu, Finland Received 4 November
1994; revised version received 27 January
1995
Abstract Scattering of an arbitrary two-dimensional electromagnetic field by an isolated rectangular groove in a perfectly conducting substrate is modeled using a rigorous electromagnetic diffraction formalism. By calculating the phase of that part of the scattered field, which propagates in directions that fall within the numerical aperture of a microscope objective, we examine the resolution limits of a Linnik-type interference microscope with focused coherent illumination. It is shown that accurate groove width and depth determination by phase measurement should be possible, even when the groove width is somewhat smaller than the optical wavelength, if use is made of reference data obtained by rigorous computations. However, our results do not support some recent experimental results that suggest the possibility of achieving superresolution by interference microscopy.
1. Introduction Phase microscopy with instruments such as the Linnik-type interference microscope is an invaluable tool in the characterization of surface-relief structures, perhaps most notably in optical profilometry of ultrasmooth surfaces [ 11. The usefulness of this approach has also been demonstrated in metrology of sharp surface-relief structures [ 2,3], such as rectangular grooves, even when the groove width c is of the same order of magnitude as the optical wavelength A and the groove depth h is a sizable fraction of A. In fact, a proposition has been put forward that, with some a priort’ information about the surface-relief profile, superresolution could be achieved by observation of phase dislocations with the aid of a computerized, piezo-modulated Linnik-type microscope with focused coherent illumination [ 4,5], without the need for near-field detection or sophisticated computational procedures. 0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00418-l
It has been generally accepted for some time [ 6,7] that rigorous diffraction theory must be used in microscopy to accurately model the interaction between the illuminating field and the microstructure under inspection, in particular if the lateral dimensions of this microstructure are of the order of A and if the depth of the surface profile is a significant fraction of A. Compared to the solution of this scattering problem, the modeling of the optical system in a microscope is well understood. Recently the work initiated in Refs. [ 6,7] has been continued [g-lo]. As a result, we now have at our disposal a great deal of rigorously computed theoretical reference data on the scattering and imaging processes that take place in, e.g., bright-field, darkfield, confocal, and Nomarski microscopy. In this paper we apply rigorous diffraction theory to calculate the phase of the field scattered by a specific wavelength-scale surface-relief structure: an isolated groove in a perfectly conducting substrate illuminated by a coherent focused beam. Our research has been
motivated by the rather unexpected experimental results given in Refs. [ 3-51, which were backed up by an unconventional concept of wave dislocations that are generated at points of surface discontinuity and propagate to the image plane. Our aim is to simulate most of the experimental aspects of that work by means of rigorous diffraction theory and to examine the resolution limits of interferometric phase microscopy. We are aware of only one paper that deals with rigorous modeling of Linnik-type microscopy [ 21. Our work differs from that presented in Ref. [ 21 in several aspects. We are primarily interested in the fundamental resolution limits in the detection of subwavelengthscale features by phase measurement. Our theoretical model, which is a waveguide-mode method of the type considered in, e.g., Refs. [ 11,12], permits the rigorous treatment of non-peri~i~ structures under focused coherent illumination as is the case in the experiments reported in Refs. 13-51. 2. Theoretical model for the scattering problem We assume that the illumination wave is twodimensional such that it propagates in the xz-plane and TEGpolarized, with the tip of the electric field vector pointing in y-direction. The sole non-vanishing component of the electric field, E!, then satisfies a Helmholtz equation in the half-space z < 0 (refractive index nr) and also in the groove region 0 < z < 13,0 < x < c (refractive index nit), vanishing at the boundaries of the perfectly conducting substrate (Fig. 1). We denote the known angular spectrum of the incident field by A( a), and the unknown angular spectrum of the diffracted field by R(cr). Then the total field in region z < 0 may be written in the form cc
Ef,Lw) =
J
k = 27r/A, and we assume that A(o)
=.O if 1~~1>
k N.A., where N.A. denotes the numerical aperture of
a microscope objective. To obtain a field representation in the groove region II we separate the variables in the Helmholtz equation and demand that E, be zero at the perfectly conducting boundaries. This gives the solution in the form of a modal expansion
t?Fl
x (exptiy,,z)
-exp[iy,t2h
- z)l}a,,
(3)
where Xnt(x) = (2/~~“‘sin(~*~/~),
(4)
[(knn)’ - (mq/c)*] “’
if m < knlrc/7r
i [( m?r/c)* - (kntl)‘] “2 otherwise, (5) and a,, are as-yet undetermined parameters. The electromagnetic boundary conditions require the continuity of Ey and H, across the plane z = 0 in the interval 0 < x < c, and that Ey vanish at z = 0 if either x < 0 or x > c. The boundary condition for Ey gives, in view of Eqs. ( 1) and (3))
J
[A(n) + R(a) J exp(icux) da
--M
=
G Xm(xf[ Ilo
exp~i2~*~)]~~
if 0 < x < c
~~I%1
otherwise. (6)
Multiplication by exp( -i&x), integration from -co to cc), and use of the Fourier-integral definition of Dirac’s delta function then leads to
A(a) exp{i[crx -t r(a)zl}da
-ccl
cm
-t
J
R(a) exp{i[ax - r(a)z]}dcr.
(1)
-M
x [I - exp(i2y,h)]
a, - A(cu’),
(7)
where the integrals
Here r(a)
=
[ (kn1)2-
fy*p2 if lcrJ 5 krrr )*] ‘/’ otherwise,
i[a2 - (knt
(2)
&r(Cu)=
exp(-iax)X,(x) J
0
dx
(8)
.I. Huttunen, J. Turunen/Optics Communications 119 (1995) 485490
can be evaluated analytically. The boundary condition for H., gives 00 [A(a ,) - R(Q)] exp(icwn) dcu
r(a)
s -co
M _.
=
c
m&t tx) [l f exp(i2y,,h)]
an,
(9)
17t=l
when 0 < x < c. Multiplication by X, (x), integration over the interval 0 < x < c, and use of the orthonormality of X,,(x) leads to
=
2 ,n=
[I + expWy,,h)] Ispngh,
Yn1
( 10)
oc,
[I - exp(i2y,,h)]
!??=I +
In the truncation of the linear system ( 11) care has to be exercised to retain a sufficient number of waveguide modes for the modal representation of Etf (x, z ) to be sufficiently accurate. This can be checked by observing the convergence of the modal amplitudes a, and the angul~-sp~~rn components R(a) when the number of modes retained in the analysis is increased. For grooves with dimensions of the order of h, results accurate to several decimal places are obtained with 5 20 modes. The most time-consuming part of the computations is the numerical integration to obtain K,,. However, once KPn, are known, the diffraction problem can be solved from Eqs. ( 11) and (7) for grooves of different depths, and illu~nation waves of different form.
3. Numerical results
I
where the asterisk denotes complex conjugation and 8 represents the Kronecker delta symbol. Insertion of Eq. (7) into Eq. (10) yields a linear set of equations CC&,
487
Yls [ 1 +
exp(i2M )] S,jnr}anl
(23
=2 s --M
We apply the numericaI model presented above to some imaging problems pertinent to phase microscopy. We use the angular spectrum model customary in the study of optical images [ 13,141, since we are not concerned with near-field detection. Only those plane-wave components of the scattered field, which fall within the numerical aperture N.A. = nr sin B,,, of the optical system, then contribute to the image field (see Fig. 1) . Hence it is appropriate to consider
(11)
r(a) I,;@) A(a)da,
where we have denoted DC) s
r(a) &r(a) I;(&
da.
(12)
If the summations in Eqs. ( 11) are truncated and the integrals K,,, are evaluated numerically, the modal coefficients a,, and thus the field inside the groove region II are obtained from the linear system ( 11) . Then Eq. (7) gives the angular spectrum of the diffracted field in region I, and the diffraction problem for a groove is solved completely. In the numerical integrations the integrand of Kpm effectively vanishes for large enough values of LY, which enables one to use a finite range of integration, but it should be stressed that this range must be extended to the evanescent part of the field in region I.
Fig. 1. Geometry for diffraction of a two-di~nsional, linearly polarized, Gaussian electromagnetic field by a groove in a perfectly conducting screen. Here t& represents the maximum acceptance angle in the angular spectrum representation.
488
J. Huttunen, J. Turunen / Optics Communications
119 (1995) 485-490
the phase c$(X) = arg [ E, f x) 1 of the field distribution kntsinH,, E,(x) =
-
O(a) R( tu) exp( iax) da, s -km sin H,,,
(13)
c
where the aplanatic factor O(cu) is given by [2,141 O(CY)= coos, (I - sinZ*,M”)-“*~“*
8
(14)
and M denotes magni~cation. We choose M = 1 in all examples that follow but have established that values such as M = 50000 change the results only a little and therefore have no effect in the main conclusions regarding the possibility of achieving superresolution by phase microscopy. In the calculation of R(a), the illuminating Gaussian field is focused on top of the groove (plane z = 0 in Fig. 1) by the microscope objective with negligible truncation at all values of N.A. considered. Hence the illumination spot at z = 0 is Gaussian at a high accuracy. In Fig. 2 we show the amplitude and phase of E,.( x) for a groove of width c = 1.2h and depth h = h/4, with various values of the numerical aperture. The l/e half-width of the incident beam amplitude Ei(X, 0) is bv= 5h, and in all the examples considered nr = Ott = 1.0. One would expect, in the spirit of Kirchhoff’s boundary conditions, the amplitude of E,(x) to be the same as that of the incident field and its phase to reflect the object’s surface-relief profile such that in the groove region we would observe a relative shift of r radians. The rigarous analysis reveals significant departures from a Gaussian shape in the amplitude of E,.(x) for all N.A. considered, with minima in the vicinity of groove edges. The phase images are, as expected, easier to interpret. However, they do not give directly the information about the groove width and depth, even with the largest values of N.A. Fig. 3 represents a systematic numerical study of phase images with different groove widths and depths at N.A. = 0.9, and Table 1 illustrates more quantitatively the full-width at half-maximum (FWHM) of the phase profiles of Fig. 3 in comparison to the true groove width c/h. These theoretical results demonstrate the need to compare experimental phase images with rigorously computed reference data in precision groove-width determination even if c/A > 1. Such data enables the measurement of c also in the region
G
i.5
Ii1 .o 7 -z=K;G -
0.5 0.0 -9
-6
-3
0
3
6
3
3
6
9
(X-X,)/h
W
1.0
0.8
0.6
-9
-6
-3
0 (X-X,)/h
Fig. 2. (a) Amplitude and (b) phase images of a groove of width c = 1.2A and depth h = A/4 in a perfectly conducting substrate, with w = 5A and xc = c/2, evaluated for values N.A.=I.O (solid line), N.A.=0.75 (dotted line), and N.A.=OSO (dashed line) of the numerical aperture of the microscope objective.
0.5 5 c/A 5 1, although the signal-to-noise ratio in the experiments will become a more critical factor towards the lower end of this range. Information about groove widths below the limit c/A z 0.5 can not be achieved directly, because a free electromagnetic field (i.e., one that does not contain evanescent waves) can not exhibit su~~iently sharp phase variations. In groove-depth me~urement by phase microscopy, quantitative results similar to those given in Table 1 are difficult to provide because of the phase fluctuations. The fluctuations near the edges of the illumination spot could be moved further away from the phase image of the groove by using a larger value of w, but only at the expense of signal-to-noise ratio in phase me~urement. In view of the results given in Fig. 3, the groove depth can be determined from the phase image rather accurately if c/A > 1, and by comparison with rigorously calculated reference data also in the region
J. Huttunen, f. Turunen /Optics
1 .o
(a)
Communications
119 (1995) 485-490
489
.‘.. .__._._,,’
t
_. -9
-6
-3
0
3
6
-9
9
-6
-3
0
3
6
9
(X-XJ/h
(x-x,)/h 0.58
-9
-6
-3
0
3
6
9
-9
-6
-3
6
3
6
9
,
I
-9
-6
-3
0
3
6
9
-9
-6
-3
Cl
3
6
9
0.49
0.4
(X-X(.)/h
(x-x,)/A
Fig. 3. Phase images of grooves in a perfectly conducting substrate, with N.A. = 0.9, w = 5A and xC = c/Z, for groove depths h = A/8 (solid lines) and h = A/4 (dashed lines) when the groove width is (a) c = 4.8A, (b) c = 2,4A, (c) c = 1.2A,(d) c = 0.6A, (e) c = 0.3A, and (f) c = 0.15A. The horizontal dashed lines represent the phase baseline at # = VT.
0.5 5 c/A 5 1. Examination of field distributions inside subwavelength grooves has shown that the effcctive penetration depth of the electric field inside such a groove is only a fraction of A, which explains the independence of the phase delay on h if c/h < 0.5. The grooves considered in the examples of Fig. 3 are relatively shallow in units of the optical wavelength (h 2 A/4). However, scattering by deeper grooves has also been considered. For example, if h = 0.75h and c > A, the width of the phase profile is approxi-
mately the same as in the case h = 0.25& but the peak value of qS(X) is slightly reduced. If, on the other hand, c < h/2, the groove depth only has a small effect in the form of Cp(x), which agrees with the results obtained for shallow grooves. In this region the peak value of #J(X) decreases with c/A in a rather uniform fashion. Hence one could envisage a possibility to obtain supe~esolution (with LZpriori info~ation about the object), provided that we measure the phase-delay amplitude instead of the FWHM of the diffracted phase
490
J. Huttunen, J. Turunen / Optics Communications
Table I FWHM of the phase profiles in Fig. 3, compared to the true groove width c/A. c/A
FWHMJA (h = h/8)
FWHM/A (h = 114)
4.80 2.40 1.20 0.60 0.30 0.15
4.64 2.24 0.96 0.73 0.68 0.67
4.58 2.12 0.96 0.76 0.68 0.67
profile. Observing the scales in Figs. 3e and 3f, this would require a good experimental signal-to-noise ratio and an ultrasmooth surface outside the groove.
119 (1995) 485-M
While this paper was being reviewed, a paper peared in which phase images of dielectric objects analyzed rigorously [ 151. The geometries analyzed Ref. [ 151 are different from ours, but the results in agreement with our conclusion.
apare in are
Acknowledgements The work of J. Huttunen was funded by the Academy of Finland. J. Turunen acknowledges a grant from Emil Aaltonen Foundation. We are grateful to C.H.F. Velzel for bringing the topic considered here to our attention, and appreciate discussions with A.T. Friberg.
References
4. Conclusions
[II J. van Wingerden, The results of our numerical study confirm the possibility of achieving subwavelength transverse resolution in phase microscopy provided that some a priori information about the object is available. Clearly, however, an extensive rigorous electromagnetic diffraction analysis is necessary before quantitative information about the dimensions of the object can be retrieved from the experimental results. To this end, the formalism presented above should be extended to finitely conducting materials. However, our rigorous calculations do not predict the possibility of achieving superresolution (beyond A/2) by phase microscopy without an exceedingly high signal-to-noise ratio in phase measurement. Hence our conclusion is in contrast to the experimental observations in Refs. [ 3-51, but in agreement with conventional expectations.
.
121 131 [4]
[ 51 161 171 [ 81 191 [ 101
[III 1121 [ 131 I141 [ 151
H.J. Frankena, B.A. van der Swan, Opt. Eng. 31 ( 1992) 2450, and references cited therein. D.M. Gale, M.I. Pether, EC. Reavell, Proc. SPIE 811 (1987) 40. VP Tychinsky, I.N. Masalov, V.L. Pankov, D.V. Ublinsky, Optics Comm. 74 ( 1989) 37. VP Tychinsky, Optics Comm. 74 (1989) 41. V. Tychinsky, Optics Comm. 81 ( 199 I ) I3 I. D. Nyyssonen, J. Opt. Sot. Am 72 ( 1982) 1425. D. Nyyssonen, C.P Kirk, J. Opt. Sot. Am. A 5 ( 1988) 1270. C.-M. Yuan, A.J. Strojwas, J. Opt. Sot. Am. A 8 (1991) 778. J.T. Sheridan, C.J.R. Sheppard, Optics Comm. 105 ( 1994) 367. J.F. Aguilar, E.R. Mendez, J. Opt. Sot. Am. A 11 ( 1994) 155. O.M. Mendez, M. Cadilhac, R. Petit, J. Opt. Sot. Am. 73 (1983) 328. A. Roberts, J. Opt. Sot. Am. A 4 (1987) 1970. A. Walther, Am. J. Phys. 35 (1967) 808. E. Wolf, Proc. Roy. Sot. (London) A253 (1959) 349. M. Totzeck, M.A. Krumbhgel, Optics Comm. 112 (1994) 189.
1995
OPTICS COMMUNICATIONS Optics Communications
119 ( 1995) 485-490
Phase images of grooves in a perfectly conducting surface Juhani Huttunen a, Jari Turunen b a Department of Technical Physics, Helsinki University of Technology, FIN-02150 Espoo, Finland h Depurtment of Physics, University of Joensuu, PO. Bo.x 1 I I, FIN-80101 Joensuu, Finland Received 4 November
1994; revised version received 27 January
1995
Abstract Scattering of an arbitrary two-dimensional electromagnetic field by an isolated rectangular groove in a perfectly conducting substrate is modeled using a rigorous electromagnetic diffraction formalism. By calculating the phase of that part of the scattered field, which propagates in directions that fall within the numerical aperture of a microscope objective, we examine the resolution limits of a Linnik-type interference microscope with focused coherent illumination. It is shown that accurate groove width and depth determination by phase measurement should be possible, even when the groove width is somewhat smaller than the optical wavelength, if use is made of reference data obtained by rigorous computations. However, our results do not support some recent experimental results that suggest the possibility of achieving superresolution by interference microscopy.
1. Introduction Phase microscopy with instruments such as the Linnik-type interference microscope is an invaluable tool in the characterization of surface-relief structures, perhaps most notably in optical profilometry of ultrasmooth surfaces [ 11. The usefulness of this approach has also been demonstrated in metrology of sharp surface-relief structures [ 2,3], such as rectangular grooves, even when the groove width c is of the same order of magnitude as the optical wavelength A and the groove depth h is a sizable fraction of A. In fact, a proposition has been put forward that, with some a priort’ information about the surface-relief profile, superresolution could be achieved by observation of phase dislocations with the aid of a computerized, piezo-modulated Linnik-type microscope with focused coherent illumination [ 4,5], without the need for near-field detection or sophisticated computational procedures. 0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00418-l
It has been generally accepted for some time [ 6,7] that rigorous diffraction theory must be used in microscopy to accurately model the interaction between the illuminating field and the microstructure under inspection, in particular if the lateral dimensions of this microstructure are of the order of A and if the depth of the surface profile is a significant fraction of A. Compared to the solution of this scattering problem, the modeling of the optical system in a microscope is well understood. Recently the work initiated in Refs. [ 6,7] has been continued [g-lo]. As a result, we now have at our disposal a great deal of rigorously computed theoretical reference data on the scattering and imaging processes that take place in, e.g., bright-field, darkfield, confocal, and Nomarski microscopy. In this paper we apply rigorous diffraction theory to calculate the phase of the field scattered by a specific wavelength-scale surface-relief structure: an isolated groove in a perfectly conducting substrate illuminated by a coherent focused beam. Our research has been
motivated by the rather unexpected experimental results given in Refs. [ 3-51, which were backed up by an unconventional concept of wave dislocations that are generated at points of surface discontinuity and propagate to the image plane. Our aim is to simulate most of the experimental aspects of that work by means of rigorous diffraction theory and to examine the resolution limits of interferometric phase microscopy. We are aware of only one paper that deals with rigorous modeling of Linnik-type microscopy [ 21. Our work differs from that presented in Ref. [ 21 in several aspects. We are primarily interested in the fundamental resolution limits in the detection of subwavelengthscale features by phase measurement. Our theoretical model, which is a waveguide-mode method of the type considered in, e.g., Refs. [ 11,12], permits the rigorous treatment of non-peri~i~ structures under focused coherent illumination as is the case in the experiments reported in Refs. 13-51. 2. Theoretical model for the scattering problem We assume that the illumination wave is twodimensional such that it propagates in the xz-plane and TEGpolarized, with the tip of the electric field vector pointing in y-direction. The sole non-vanishing component of the electric field, E!, then satisfies a Helmholtz equation in the half-space z < 0 (refractive index nr) and also in the groove region 0 < z < 13,0 < x < c (refractive index nit), vanishing at the boundaries of the perfectly conducting substrate (Fig. 1). We denote the known angular spectrum of the incident field by A( a), and the unknown angular spectrum of the diffracted field by R(cr). Then the total field in region z < 0 may be written in the form cc
Ef,Lw) =
J
k = 27r/A, and we assume that A(o)
=.O if 1~~1>
k N.A., where N.A. denotes the numerical aperture of
a microscope objective. To obtain a field representation in the groove region II we separate the variables in the Helmholtz equation and demand that E, be zero at the perfectly conducting boundaries. This gives the solution in the form of a modal expansion
t?Fl
x (exptiy,,z)
-exp[iy,t2h
- z)l}a,,
(3)
where Xnt(x) = (2/~~“‘sin(~*~/~),
(4)
[(knn)’ - (mq/c)*] “’
if m < knlrc/7r
i [( m?r/c)* - (kntl)‘] “2 otherwise, (5) and a,, are as-yet undetermined parameters. The electromagnetic boundary conditions require the continuity of Ey and H, across the plane z = 0 in the interval 0 < x < c, and that Ey vanish at z = 0 if either x < 0 or x > c. The boundary condition for Ey gives, in view of Eqs. ( 1) and (3))
J
[A(n) + R(a) J exp(icux) da
--M
=
G Xm(xf[ Ilo
exp~i2~*~)]~~
if 0 < x < c
~~I%1
otherwise. (6)
Multiplication by exp( -i&x), integration from -co to cc), and use of the Fourier-integral definition of Dirac’s delta function then leads to
A(a) exp{i[crx -t r(a)zl}da
-ccl
cm
-t
J
R(a) exp{i[ax - r(a)z]}dcr.
(1)
-M
x [I - exp(i2y,h)]
a, - A(cu’),
(7)
where the integrals
Here r(a)
=
[ (kn1)2-
fy*p2 if lcrJ 5 krrr )*] ‘/’ otherwise,
i[a2 - (knt
(2)
&r(Cu)=
exp(-iax)X,(x) J
0
dx
(8)
.I. Huttunen, J. Turunen/Optics Communications 119 (1995) 485490
can be evaluated analytically. The boundary condition for H., gives 00 [A(a ,) - R(Q)] exp(icwn) dcu
r(a)
s -co
M _.
=
c
m&t tx) [l f exp(i2y,,h)]
an,
(9)
17t=l
when 0 < x < c. Multiplication by X, (x), integration over the interval 0 < x < c, and use of the orthonormality of X,,(x) leads to
=
2 ,n=
[I + expWy,,h)] Ispngh,
Yn1
( 10)
oc,
[I - exp(i2y,,h)]
!??=I +
In the truncation of the linear system ( 11) care has to be exercised to retain a sufficient number of waveguide modes for the modal representation of Etf (x, z ) to be sufficiently accurate. This can be checked by observing the convergence of the modal amplitudes a, and the angul~-sp~~rn components R(a) when the number of modes retained in the analysis is increased. For grooves with dimensions of the order of h, results accurate to several decimal places are obtained with 5 20 modes. The most time-consuming part of the computations is the numerical integration to obtain K,,. However, once KPn, are known, the diffraction problem can be solved from Eqs. ( 11) and (7) for grooves of different depths, and illu~nation waves of different form.
3. Numerical results
I
where the asterisk denotes complex conjugation and 8 represents the Kronecker delta symbol. Insertion of Eq. (7) into Eq. (10) yields a linear set of equations CC&,
487
Yls [ 1 +
exp(i2M )] S,jnr}anl
(23
=2 s --M
We apply the numericaI model presented above to some imaging problems pertinent to phase microscopy. We use the angular spectrum model customary in the study of optical images [ 13,141, since we are not concerned with near-field detection. Only those plane-wave components of the scattered field, which fall within the numerical aperture N.A. = nr sin B,,, of the optical system, then contribute to the image field (see Fig. 1) . Hence it is appropriate to consider
(11)
r(a) I,;@) A(a)da,
where we have denoted DC) s
r(a) &r(a) I;(&
da.
(12)
If the summations in Eqs. ( 11) are truncated and the integrals K,,, are evaluated numerically, the modal coefficients a,, and thus the field inside the groove region II are obtained from the linear system ( 11) . Then Eq. (7) gives the angular spectrum of the diffracted field in region I, and the diffraction problem for a groove is solved completely. In the numerical integrations the integrand of Kpm effectively vanishes for large enough values of LY, which enables one to use a finite range of integration, but it should be stressed that this range must be extended to the evanescent part of the field in region I.
Fig. 1. Geometry for diffraction of a two-di~nsional, linearly polarized, Gaussian electromagnetic field by a groove in a perfectly conducting screen. Here t& represents the maximum acceptance angle in the angular spectrum representation.
488
J. Huttunen, J. Turunen / Optics Communications
119 (1995) 485-490
the phase c$(X) = arg [ E, f x) 1 of the field distribution kntsinH,, E,(x) =
-
O(a) R( tu) exp( iax) da, s -km sin H,,,
(13)
c
where the aplanatic factor O(cu) is given by [2,141 O(CY)= coos, (I - sinZ*,M”)-“*~“*
8
(14)
and M denotes magni~cation. We choose M = 1 in all examples that follow but have established that values such as M = 50000 change the results only a little and therefore have no effect in the main conclusions regarding the possibility of achieving superresolution by phase microscopy. In the calculation of R(a), the illuminating Gaussian field is focused on top of the groove (plane z = 0 in Fig. 1) by the microscope objective with negligible truncation at all values of N.A. considered. Hence the illumination spot at z = 0 is Gaussian at a high accuracy. In Fig. 2 we show the amplitude and phase of E,.( x) for a groove of width c = 1.2h and depth h = h/4, with various values of the numerical aperture. The l/e half-width of the incident beam amplitude Ei(X, 0) is bv= 5h, and in all the examples considered nr = Ott = 1.0. One would expect, in the spirit of Kirchhoff’s boundary conditions, the amplitude of E,(x) to be the same as that of the incident field and its phase to reflect the object’s surface-relief profile such that in the groove region we would observe a relative shift of r radians. The rigarous analysis reveals significant departures from a Gaussian shape in the amplitude of E,.(x) for all N.A. considered, with minima in the vicinity of groove edges. The phase images are, as expected, easier to interpret. However, they do not give directly the information about the groove width and depth, even with the largest values of N.A. Fig. 3 represents a systematic numerical study of phase images with different groove widths and depths at N.A. = 0.9, and Table 1 illustrates more quantitatively the full-width at half-maximum (FWHM) of the phase profiles of Fig. 3 in comparison to the true groove width c/h. These theoretical results demonstrate the need to compare experimental phase images with rigorously computed reference data in precision groove-width determination even if c/A > 1. Such data enables the measurement of c also in the region
G
i.5
Ii1 .o 7 -z=K;G -
0.5 0.0 -9
-6
-3
0
3
6
3
3
6
9
(X-X,)/h
W
1.0
0.8
0.6
-9
-6
-3
0 (X-X,)/h
Fig. 2. (a) Amplitude and (b) phase images of a groove of width c = 1.2A and depth h = A/4 in a perfectly conducting substrate, with w = 5A and xc = c/2, evaluated for values N.A.=I.O (solid line), N.A.=0.75 (dotted line), and N.A.=OSO (dashed line) of the numerical aperture of the microscope objective.
0.5 5 c/A 5 1, although the signal-to-noise ratio in the experiments will become a more critical factor towards the lower end of this range. Information about groove widths below the limit c/A z 0.5 can not be achieved directly, because a free electromagnetic field (i.e., one that does not contain evanescent waves) can not exhibit su~~iently sharp phase variations. In groove-depth me~urement by phase microscopy, quantitative results similar to those given in Table 1 are difficult to provide because of the phase fluctuations. The fluctuations near the edges of the illumination spot could be moved further away from the phase image of the groove by using a larger value of w, but only at the expense of signal-to-noise ratio in phase me~urement. In view of the results given in Fig. 3, the groove depth can be determined from the phase image rather accurately if c/A > 1, and by comparison with rigorously calculated reference data also in the region
J. Huttunen, f. Turunen /Optics
1 .o
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Communications
119 (1995) 485-490
489
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Fig. 3. Phase images of grooves in a perfectly conducting substrate, with N.A. = 0.9, w = 5A and xC = c/Z, for groove depths h = A/8 (solid lines) and h = A/4 (dashed lines) when the groove width is (a) c = 4.8A, (b) c = 2,4A, (c) c = 1.2A,(d) c = 0.6A, (e) c = 0.3A, and (f) c = 0.15A. The horizontal dashed lines represent the phase baseline at # = VT.
0.5 5 c/A 5 1. Examination of field distributions inside subwavelength grooves has shown that the effcctive penetration depth of the electric field inside such a groove is only a fraction of A, which explains the independence of the phase delay on h if c/h < 0.5. The grooves considered in the examples of Fig. 3 are relatively shallow in units of the optical wavelength (h 2 A/4). However, scattering by deeper grooves has also been considered. For example, if h = 0.75h and c > A, the width of the phase profile is approxi-
mately the same as in the case h = 0.25& but the peak value of qS(X) is slightly reduced. If, on the other hand, c < h/2, the groove depth only has a small effect in the form of Cp(x), which agrees with the results obtained for shallow grooves. In this region the peak value of #J(X) decreases with c/A in a rather uniform fashion. Hence one could envisage a possibility to obtain supe~esolution (with LZpriori info~ation about the object), provided that we measure the phase-delay amplitude instead of the FWHM of the diffracted phase
490
J. Huttunen, J. Turunen / Optics Communications
Table I FWHM of the phase profiles in Fig. 3, compared to the true groove width c/A. c/A
FWHMJA (h = h/8)
FWHM/A (h = 114)
4.80 2.40 1.20 0.60 0.30 0.15
4.64 2.24 0.96 0.73 0.68 0.67
4.58 2.12 0.96 0.76 0.68 0.67
profile. Observing the scales in Figs. 3e and 3f, this would require a good experimental signal-to-noise ratio and an ultrasmooth surface outside the groove.
119 (1995) 485-M
While this paper was being reviewed, a paper peared in which phase images of dielectric objects analyzed rigorously [ 151. The geometries analyzed Ref. [ 151 are different from ours, but the results in agreement with our conclusion.
apare in are
Acknowledgements The work of J. Huttunen was funded by the Academy of Finland. J. Turunen acknowledges a grant from Emil Aaltonen Foundation. We are grateful to C.H.F. Velzel for bringing the topic considered here to our attention, and appreciate discussions with A.T. Friberg.
References
4. Conclusions
[II J. van Wingerden, The results of our numerical study confirm the possibility of achieving subwavelength transverse resolution in phase microscopy provided that some a priori information about the object is available. Clearly, however, an extensive rigorous electromagnetic diffraction analysis is necessary before quantitative information about the dimensions of the object can be retrieved from the experimental results. To this end, the formalism presented above should be extended to finitely conducting materials. However, our rigorous calculations do not predict the possibility of achieving superresolution (beyond A/2) by phase microscopy without an exceedingly high signal-to-noise ratio in phase measurement. Hence our conclusion is in contrast to the experimental observations in Refs. [ 3-51, but in agreement with conventional expectations.
.
121 131 [4]
[ 51 161 171 [ 81 191 [ 101
[III 1121 [ 131 I141 [ 151
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