ELSEVIER Ultramicroscopy 57 (1995) 153-159

ultramicroscopy

Analysis of the near field image formation of dielectric gratings

Yong Chen *, R.K. Kupka

Laboratoire de Microstructures et de Mic&lectronique (CNRS), 196 Avenue Henri Ravera, BP 107, F-92225 Bagneux, France

Received 20 October 1993; accepted 30 December 1993

Abstract

The electromagnetic field distribution near the surface of a dielectric grating is calculated using a modified beam propagation method under total reflection condition. Results of the image contrast, the polarization dependence and the angular spectrum are given and discussed in the context of near field optical microscopy.

1. Introduction

Near field optical microscopy is a powerful tool in the study of the electromagnetic field distribution near dielectric surfaces [ll. In many cases, the sample is illuminated by a light beam of large angle, incident from behind in the total reflection configuration. A dielectric tip (tapered optical fiber or small aperture) is placed near the surface to detect the local intensity [2-71. Contin- uous scanning of the tip or the sample over a two-dimensional sample area leads to an image of the morphological profile or the dielectric varia- tion of the sample surface.

Interpretation of a near field image is not easy due to the complexity of the excitation, of the near field formation and of the tip-sample cou- pling. A large amount of work has been done on the problem of diffraction of light through a subwavelength aperture with perfect conducting or dielectric screening [g-111. More recently ef- forts have been devoted to optical fiber detection [12-141. It has been shown that as the tip-sample

* Corresponding author.

separation increases, the image resolution drasti- cally decreases and this decrease of the image resolution cannot be described only by the expo- nential decay of the Fresnel’s evanescent wave, but is also related to the sample surface structure 1131. Moreover, the fields transmitted from the substrate can be partially reflected from the probe tip and again return to the sample surface where they interact. Thus, a self-consistent calculation is basically needed for a full interpretation of an imaged intensity profile [El. It is however helpful to first study the near field image formation with- out the tip interaction.

In the case of a dielectric grating structure, many physical effects can influence the near field formation. Under total reflection condition, for instance, evanescent waves are fundamental but the waveguide effect within the grating structure and the diffraction in the free space can play more significant roles to the near field formation. To numerically calculate such a wave propagation we investigate the feasibility of a beam propaga- tion method (BPM) based on fast Fourier trans- forms (FFT). We will show the efficiency and usefulness of this technique to determine the near field image formation. Dielectric gratings

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154 K Chen, R.K. Kupka / Ultrami~ros~opy 57 (1995) 1.53-159

are chosen as examples for the illustration, but the calculation algorithm can also be applied to much more complicated situations.

2. Computation algorithm

In this section we briefly outline the numerical implementation of the beam propagation method (BPM) in the case of a near field image calcula- tion.

The BPM is known as an iteration solution of the scalar Helmholtz equation. Initially proposed by Feit and Fleck [16], it is now widely used to calculate light propagation in optical fibers and in various waveguide geometries [17,18]. The BPM algorithm can also be applied to study the decay of the evanescent waves under total reflection condition. For this purpose, we define a field near the surface of a high index medium, as it is generated by a light beam with a large incident angle. We then evaluate the propagation of the wave step-by-step through the surface structure (grating or other geometries), using the BPM. The field obtained at the downstream side of the surface is then transferred to the observation plane by standard diffraction methods. The BPM algorithm inherently allows one- or two-dimen- sional calculations and various modifications have been proposed to meet special requirements. Here, let us consider the following scalar Helmholtz equation:

2

V:E+$+k$z2(x, y)E=O, (1)

where V: = a2/&x2 + a2/ay2, E is the transverse component of the electric field, k, = 27r/h, is the free space wave number, and n(x, y) the complex refraction index distribution. For the propagation along the z direction over a small enough distance, the formal solution of Eq. (1) can be written as

E(x, Y, 2)

=exp( +iz[V: +k$z’(x, y)]“‘)~(x, y, 0),

(2)

where the positive sign can be chosen for the forward propagation. The task is now to find an approximation for the exponential propagator which introduces least errors. In the case of oblique incidence, the square root in Eq. (2) can be developed as a sum of two operators,

[V: +k,2n2(x, y)]“‘=G +b, (3)

with

a^ = fV: +k,ZiZ2) i/2

,

&=k,[/n2(x, y) -nf sin2(i)

-/_I,

where i is the incident angle, ni is the bulk refraction index of the medium (k,ni sin(i) = ki, being the x-component of the wave number of incident wave), and Z is the weighted refraction index of the plane under consideration, e.g.,

n= ~~~

n(x, Y)~I.+, Y,O)I~ dx dy

\ l/2

\ /I IE(x, y,0)12 dx dy

I

(4)

The approximation defined by 6 was chosen to obtain the accurate solution for a single plane wave propagation in the case of oblique incidence [19]. Now, taking into account the commutative relation, Eq, (2) becomes

E( x, y, z) = eidr/2 ei5’ ei’*/*E(x, y, 0), (5) which represents the basis of the BPM algorithm and can be dealt with numerically. One may perceive that the split exponential factors of 6 describe a wave propagation in a homogeneous medium of refraction index ?i while the factor of 6 gives a local ‘phase’ modulation of the wave front.

The propagation problem in a homogeneous medium can be easily solved using the angular spectrum of plane waves 1201. Considering a dis- crete Fourier transform of the E field at z = 0 plane

Efx, Y, 0) = c A(k,, ky) ew[i(k,n + k,y)],

kx3k.v

(6)

Y, Chen, R.K. Kupka / Ultramicroscopy 57 (1995) 153-159 155

the E field in the plane at distance z/2 is given

by

E(x, Y, z/2)

= c 4L kJ k,.ky

xexp[(iz/2){k:]

xexp[i(k,x + k,y)]. (7)

Multiplying E(x, y, z/2) by the ‘phase’ cor- rection factor and then doing again the free space propagation from z/2 to z, one gets the solution E(x, Y, z>.

Repeated step-by-step application of this pro- cedure for the following z-subsequent planes al- lows to calculate the field distribution at the end surface of the sample. The propagation from this end surface to an observation plane further along can be easily calculated by using Eqs. (6) and (7) once again.

Note that Eqs. (6) and (7) are well known expressions of the angular spectrum description. For iT2ki < ki + k:, the square root in Eq. (7) becomes imaginary and the exponent negative. Thus, the field of this kind decays along z. The larger kz and/or kc, the smaller is its character- istic decay length. Under total reflection condi- tion, the field along z get rapidly damped be- cause most of the wave components are evane- scent, each one having a definitive characteristic decay length (see below).

Note also that with the BPM algorithm com- monly used, the operator V, is treated as a small correction within the paraxial approximation at k, y = 0. This is also the case for the oblique incidence if Eq. (3) is used where the paraxial approximation is applied along k,= k,ni sin(i). Clearly, for i = 0, Eq. (3) becomes b = k&(x, y) - ?I), which recovers the standard BPM approxi- mation [ 171. Higher-order approximations can also be made, but Eq. (3) seems to be the most effi- cient one which, at least, allows an accurate solu- tion for a single plane wave propagation.

Local variation of the field across the sample surface has important effects on both the abso- lute intensity and the image contrast. In the case of rectangular gratings, the field change at the

bottom and at the top interface can be easily calculated using macroscopic formulas. That is, neglecting multiple reflection and scattering, the change of the complex field amplitude (TE and TM polarizations) is given by

E; 2k: t,, = -y =-

EY k; + kl_ ’

s- 2k; t TM = H; - k: + (n’,‘n:)k; ’

(8)

(9)

where kk(ni) and ki(n,) are the z-components of the wave number (refraction index) before and after the transmission. Under total reflection con-

dition, k: is real but k: = kg/-~ may be imaginary.

It should be mentioned that even in the case of a rectangular grating, Eqs. (8) and (9) do not describe the exact change of the fields across the interfaces due to the effect of the scattering. In principle, the calculation should be performed for each wave number component at the wavefront. Nevertheless, given the wave numbers ki and k, then kk and k: represents the averaged propaga- tion which can be used as the first-order approxi- mation. A further going.investigation which de- composes the fields into the Fourier components and then uses Eqs. (8) and (9) for each of these components is currently carried out [19].

So far the BPM algorithm is presented for a transverse component of the electric field. From the Maxwell relations, the only non-zero field components are (E,, H,, Hz) for the TE polar- ization and (H,, E,, E,) for the TM polarization. For the TE polarization E is E, and the two magnetic components can be obtained from

1 aD VxH=cl.

To extend the modeling to TM polarization, one needs to propagate the H, field (E,, + H,, t,, -+ t,,), and then using the Faraday relation,

1 aB VxE= -cat

to calculate the two E field components.

156 Y. Chen, R.K. Kupka / Ultramicroscopy 57 (1995) 153-159

3. Numerical examples

In a near field optical microscope, the de- tected optical signal is related to the energy flow collected by the probing tip. Under total reflec- tion condition, the tip can also be considered as the frustration element to frustrate the evane- scent fields. In principle, without tip or any other frustration mechanism the z-component of the Poynting vector outside a plane surface is zero and, therefore, there will be no energy flow nor- mal to the sample surface. If the tip is now brought near the surface, one suddenly detects energy which indicates that fields with finite z- component Poynting vector are present. Thus, the tip itself acts as a scattering center and alters the field distribution! Before presenting our nu- merical BPM results of the near field intensity profile, we show in Fig. 1 the transmission coeffi- cients (TE and TM) versus the separation be- tween two media of infinite and flat surfaces (n = 1.5) [21] and compare them with the expo- nential decay of the evanescent field intensity (i = 45”). Although there is a conceptional differ- ence between the two representations (the trans- mission or tunneling curves take multiple reflec- tion into account whereas the exponential decay of the evanescent wave simply describes the for- ward ‘propagation’), the similar decreasing be-

z W-4 Fig. 1. Transmission coefficients (TE and TM) versus the

separation between two media of flat and infinite surfaces

(ni = 1.5), in comparison with the exponential decay of the

Fresnel’s evanescent waves (i = 45”).

Incident plane wave Totally reflected wave

t .z -- Evanescent wave

Fig. 2. Geometry of a rectangular grating and model for the

near field image generation under total reflection condition.

havior can be observed. In fact, if the diameter of the tip head is infinitely small no multiple reflec- tion should occur which corresponds to a simple exponential decay. In return, if the tip head sur- face is large and flat, the field intensity variation is close to the tunneling curves TE and TM.

A surface grating structure can be obtained by selective etching or compositional doping, both corresponding to a refraction index variation. For illustration we consider an etched rectangular structure profile, as shown in Fig. 2, where h is the etching depth, 1 and s are the line and space width. Gratings with arbitrary profiles can also be considered, which may need a slight modification of the transmission algorithm given by (8) and (9). Under total reflection condition a plane wave is incident from the top at an angle i with respect to the vertical axis z. For the sake of simplicity, all the following numerical calculations have been done with fixed values: ni = 1.5 for the material refraction index, i = 45” for the incident angle, A, = 633 nm for the excitation wavelength, and 1 = s = 100 nm and h = 50 nm for the line and space width and the depth of the grating struc- ture.

Fig. 3 shows an intensity profile I E,(x) I * of the TE polarization near the grating surface at a distance of 30 nm. Clearly, this image intensity profile does not describe all the details of the grating topology, but it still shows a high intensity contrast variation with a subwavelength resolu- tion. In Fig. 3 the maximum and minimum of the intensity profile correspond, respectively, to the bottom and top grating areas because of the

Y. Chen, R.K. Kupka / Ultramicroscopy 57 (1995) 153-159 157

4. I I ’ ’ ’

h=Wnm, z=3Onm

Fig. 3. TE field intensity near the grating surface at a distance of 30 nm.

Fig. 4. Averaged field intensity (I) (a), maximum (I,,,,) and minimum (IminI intensity difference (b), and image contrast (c) versus the distance to the grating surface.

exponential decay at different local distances. We think that the exponential decay at different local distances is important and might roughly prompt the field intensity contrast. However, in addition to this, the waveguide effect inside the grating transient layer and the diffraction in free space also play a very significant role in the near field image construction, otherwise the graph in Fig. 3 should exactly follow the grating topographic pro- file. Calculations show that the intensity profile strongly depends on grating parameters and the excitation conditions. In Fig. 4 we plot TE mode variations versus the distance to the grating sur- face of (a) the averaged intensity (I), (b) the intensity difference taken from the maximum (I,,) and minimum C&j, and (c) the image contrast defined by

CI = (LaX -rmin>/(~max +rmin)* (10)

In Figs. 3 and 4, values larger than unity are physical meaningful which stem from the fact that the transmission coefficients (Eqs. (8) and (9)) can become larger than one. Note that the aver- aged intensity (a) does not give an useful signal for the image profile but it prompts the distance to the grating surface. It is shown that the maxi- mum and minimum intensity different (b) de- creases much faster than (al, indicating advan- tages for short distance detection. Concerning the contrast curve (c), although it shows a quite smooth variation, the intensity profile at short

distances should contain much more information of the higher spatial frequencies. This will be discussed below using an angular spectrum de- scription.

One of the most important features in near field optics is its drastic dependence on the polar- ization. Fig. 5 shows the intensity profile 1 E,(x) I ’ calculated using the same grating but with a TM polarization. Higher image contrast is obtained and apparently much more information can be resolved because of the fundamental differences between the TE and TM wave propagation.

h=50nm, z-30nm

6.

ot% 0. 600.

x Pm) Fig. 5. TM field intensity near the grating surface at a dis- tance of 30 nm.

158 Y Chen, R.K. Kupka / Ultramicroscopy 57 (1995) 153-159

0. 20. 40. 60. 80.

= (nm) Fig. 6. Intensity of the first eight diffracted orders versus the

distance to the grating surface.

In Fig. 6 we show the intensity versus the distance from the surface of the first eight diffracted orders I A[k, = k,n, sin(i) + m21r/(l + s)] ( 2, numerically calculated respectively for the TE (top) and TM (bottom) polarizations. Sim- ilar curves can be obtained using analytical ex- pressions based on the grating diffraction. From grating theory it is known that a periodic struc- ture generates an infinite number of diffracted orders, centered around the incident beam direc- tion. If the wave is evanescent, the grating formu- las still hold and we generate evanescent waves with higher and lower diffraction orders. Ignoring details of the grating profile, the diffracted wave of mth order exponentially decays along z with a characteristic decay length given by 1131

I, = A0

27r/[ni sin(i) +~~~h,/(l+s)]~- 1 T (11)

where m = 0, i 1, rt2,. . . and ni is the bulk re- fraction index. For m = 0, formula (11) is exactly the decay length of the Fresnel’s evanescent wave near a flat surface medium. As can be seen from Eq. (11) and Fig. 6, higher-order diffracted waves have shorter characteristic decay lengths and pos- sibly, some negative diffracted orders result in true propagating waves. Remarkable polarization dependences can be also observed in Fig. 6, but for both TE and TM polarizations, only waves of the first three diffracted orders can significantly contribute to the near field image fo~ation at a distance of 20 nm.

4. Conclusion

In conclusion we have developed a novel algo- rithm of the beam propagation method under total reflection condition for the study of near field image formation. Comparing with the other computation techniques (Ref. [13]), the BMP ap- proach requires much less CPU time since no perturbative development has been involved. It is also qualitatively supported because of the slow variation of the structure along the evanescent wave decay direction. As an example the near field image formation with a periodic grating st~cture has been analyzed for both TE and TM polarizations. It has been shown that in both TE and TM cases the wavefront near the grating surface does not follow the grating topographic profile because of the ‘waveguide’ effect and the diffraction in the free space. High spatial fre- quencies decay rapidly as the distance increases but the first diffracted order can have a relatively large decay distance. A strong polarization effect has also been demonstrated through the near field image calculation and the variation of the first diffracted orders which indicate possible ad- vantages for the TM mode detection.

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