Interferometric optical microscopy of subwavelength grooves

Interferometric optical microscopy of subwavelength grooves

1 January 2001 Optics Communications 187 (2001) 29±38 www.elsevier.com/locate/optcom Interferometric optical microscopy of subwavelength grooves S...

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1 January 2001

Optics Communications 187 (2001) 29±38

www.elsevier.com/locate/optcom

Interferometric optical microscopy of subwavelength grooves S.P. Morgan *, E. Choi, M.G. Somekh, C.W. See School of Electrical and Electronic Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK Received 3 August 2000; received in revised form 16 October 2000; accepted 7 November 2000

Abstract The amplitude and phase response of a high numerical aperture interferometric microscope to subwavelength grooves are investigated both experimentally and theoretically. It is well known that for narrow and deep structures scalar di€raction theory is no longer valid and a rigorous vector di€raction model is required. Conical di€raction results presented demonstrate signi®cant di€erences in measurements taken in di€erent polarisation states. Signi®cant light coupling occurs when the polarisation state of light at the back focal plane of the microscope is aligned perpendicular to the groove (TE) whereas relatively poor coupling occurs when the polarisation is aligned along the groove (TM). The stronger coupling of TE incident light in the groove means that there is much greater contrast compared to TM. Under certain circumstances an inversion of the phase occurs in TE, which is intuitively explained in terms of interference between the top and bottom of the groove. The greater coupling that occurs in TE enables the depth of narrow grooves to be measured more accurately and over a wider range of depths than in TM. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Optical microscopy; Rigorous vector di€raction; Interferometry; Groove

1. Introduction Scalar di€raction models based on Fourier optics [1±3] have served the optical microscopy ®eld well for many years. However, as technology in the semiconductor industry develops the structures to be measured become laterally smaller, and very often deeper. When the lateral dimensions or heights are of the order of a wavelength, a scalar di€raction model is no longer valid and a rigorous vector di€raction model, based on MaxwellÕs equations, is required.

*

Corresponding author. Fax: +44-115-951-5616. E-mail address: [email protected] Morgan).

(S.P.

Early work applying vector di€raction theory to optical microscopy [4±14] restricted the illumination to a plane of incidence perpendicular to the grooves i.e. the non-conical di€raction problem, modelling line illumination with a cylindrical objective. These methods are less time consuming than modelling a spherical objective and are informative as they demonstrate trends in the response. However, in practical systems it is important to consider the response of a system imaging using a spherical objective i.e. the conical di€raction problem. Research by Sheridan and co-workers [7±10] applied rigorous di€raction theory with onedimensional illumination in a single polarisation state. The response of a range of microscopes including confocal [7], interferometric [8], near ®eld

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[9], bright and dark ®eld [10] was demonstrated. The amplitude and phase response of a Linnik microscope has also been demonstrated using onedimensional illumination [11]. Good agreement between theory and experiment was demonstrated by imaging a deep (k) di€raction grating at different defocuses. By comparison with scalar diffraction theory these authors have demonstrated the need to use a vector di€raction model. In contrast to our research, only subtle di€erences between TE and TM illumination polarisation states were reported. This is presumably as the lateral dimensions of the structures are fairly large compared to the wavelength (4k). The structures considered in this paper have smaller widths (
storage. For example Ref. [15] considers 0.5 duty cycle surface relief grating, numerical apertures (NAs) up to 0.6 and structure depths up to 0.35k. In this paper we are interested in high NA optical microscopy and have investigated images of a near isolated groove obtained with an interferometric microscope. Experimentally an NA of 0.95 is used to image silicon grooves of depth k=4. Using the model depths up to 1.5k are considered. The amplitude and phase response of grooves imaged in di€erent polarisation states becomes signi®cantly di€erent as the groove width decreases. This is explained in terms of the di€erence in coupling of modes into the groove and the interference of light scattered from the top and within the groove. The paper is structured as follows. The next section describes the rigorous di€raction model applied. Section 3 contains a brief description of the interferometric microscope used in the experiments. Section 4 presents theoretical and experimental measurements of a range of grooves decreasing in width. Discussions follow in Section 5 on the origins of the observed e€ects and how these may be used to characterise the groove. Finally, conclusions are made in Section 6. 2. Theory 2.1. Di€raction of a single plane wave The ®rst stage in the rigorous modelling of microscope systems is to calculate the light scattered by a structure from a single plane wave of arbitrary incident angle and polarisation. In this paper the model described by Li [18], which extends the work of Botten et al. [19±21] to conical di€raction, has been chosen. This method reduces conical di€raction to a set of non-conical di€raction problems resolved in the invariant z direction. Fig. 1 demonstrates the basic problem. A plane wave of arbitrary polarisation is incident on a lamellar di€raction grating (period d, groove width d1, height h) in a direction de®ned by angles h and /. Region I is a homogenous region, assumed to be air, which contains the incident plane wave and a set of re¯ected orders from the grating. The

S.P. Morgan et al. / Optics Communications 187 (2001) 29±38

Fig. 1. Plane wave incident on a di€raction grating at an arbitrary angle of incidence. The grating is assumed to be invariant in the z direction. Solutions for the re¯ected and transmitted amplitudes are obtained by ®nding a modal solution in the modulated region (II) and matching unknown electric and magnetic ®eld components at the boundaries.

electric ®eld in this region is represented as a Rayleigh expansion: Ez …x; y† ˆ Iz exp ‰ikx x ‡ iky yŠ ‡

‡1 X nˆÿ1

0 0 Rn exp‰ikxn x ‡ ikyn yŠ

…1†

where Ez , Iz and Rn are the components of the total, incident and re¯ected E-®elds in the invariant z direction respectively. kx , ky are the wave vectors of the input plane waves in the x and y directions 0 0 , kyn are the wave vectors of the respectively and kxn scattered plane waves. kx ˆ k sin h cos / ‡ 2np=d ky ˆ ÿk cosh

0 kxn ˆ k sin h0n cos/0n 0 kyn ˆ k cosh0n

where h0n /0n are the scattering angles of the nth order. The electric ®eld in the homogeneous substrate (region III) is represented as a set of di€racted transmitted orders: Ez …x; y† ˆ

‡1 X

0 0 Tn exp ‰ikxn x ‡ ikyn yŠ

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Similar equations are obtained for the magnetic ®eld in regions I and III. The unknowns are the complex amplitudes of the re¯ected and transmitted E and H ®elds. These are obtained by solving MaxwellÕs equations in the grating region (region II) and matching the unknown complex amplitudes of the modes to the unknown electric and magnetic ®elds at the boundaries with regions I and III. For materials with complex refractive index the modal solution involves applying a root ®nding algorithm in the complex plane [22] to obtain the necessary eigenvalues. In future work the e€ectiveness of other single plane wave solutions will be considered [13,23,24]. 2.2. Modelling of microscope systems The solution described in the previous section forms the basis of the microscope model as the focal plane distribution can be decomposed into a discrete set of polarised plane waves. The wave vector and polarisation state of each plane wave is de®ned by a point at the back focal plane (Fig. 2). For each incident plane wave, calculating the orders di€racted from the structure that are collected by the objective enables the microscope response to be obtained. 1 Once the scattering function has been calculated it is relatively straightforward to obtain the response of a range of microscopes at di€erent lateral and focal positions using appropriate vector summations 2 [2,3]. The ®eld at the back focal plane of the microscope at a particular object position (xpos ; ypos ) is given by U…kx0 ; kz0 † ˆ

Z

1

Z

1

s…kx ; ky ; kz ; kx0 ; ky0 ; kz0 †P …kx ; kz †P 0 …kx0 ; kz0 †

0 0 †xpos ‡ …ky ÿ kyn †ypos †† dkx dkz  exp…i……kx ÿ kxn

…3†

where s…kx ; ky ; kz ; kx0 ; ky0 ; kz0 † is the scattering function of the sample, P …kx ; kz †, P 0 …kx0 ; kz0 † are the pupil

…2†

nˆÿ1

The in®nite expansion of plane waves allows evanescent orders to be included which are essential to satisfy the boundary conditions. In practice, the series is truncated at a point where the results have converged (N ˆ 31 ensures accuracy 2 dps).

1 Relating a point on the back focal plane to a plane wave on the sample means that the back focal plane distribution needs to be sampled. It has been suggested [7] that four times the Littrow angle of the grating is adequate. In this paper we use a minimum of six times. 2 This paper deals with vector summations of the plane waves in contrast to the scalar summations in Refs. [2,3].

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In this case we calculate the one-dimensional scan response, using two-dimensional illumination, of an interferometric microscope given by [3] Z Z i…xpos † ˆ 2 Re…U …kx0 ; kz0 †R …kx0 ; kz0 †† dkx0 dkz0 P 0 …kx0 ;kz0 †

Z Z ˆ

P 0 …X ;Z†

2 Re…U …X ; Z†R …X ; Z†† dX dZ …4†

where R…kx0 ; kz0 † is the reference beam ®eld and  denotes the complex conjugate. In the experiments and simulations contained in this paper, the polarisation state of the reference beam is aligned with the input beam polarisation. 3 The amplitude and phase calculation is performed at the fringe carrier frequency. As the model is based on scattering from a di€raction grating an isolated groove can only be approximated and the results will be in¯uenced from scattering from other grooves. However, the spacing between grooves is ®xed at 1.7 lm (2 focal spot widths at NA ˆ 0:95), thus reducing the contribution of scattering from other grooves in the grating.

3. Ultrastable interferometric microscope con®guration Fig. 2. The focal distribution can be decomposed into a set of plane waves de®ned at the back focal (Fourier) plane of the objective. Using di€erent summations of the scattered waves enables a range of microscopes and scanning to be simulated.

functions of the objective for the illuminating and scattered ®elds respectively and the ®eld U …kx0 ; kz0 † maps to position U …X ; Z† on the back focal plane. The x and y terms in the exponent in Eq. (3) account for lateral scanning and defocus respectively. A uniform spectral distribution in k space is assumed at the back focal plane. Sampling in k space at the back focal plane accounts for the obliquity factor that occurs in high NA systems [25], as the contribution from large h values is less. In all results presented the focus is at the top of the groove structure.

The scanning interferometric microscope (Fig. 3) has been described in detail elsewhere [26] so will only be described brie¯y. Its principal advantage is the use of a computer generated hologram as a beamsplitter to project both reference and focused sample beams onto the sample. With this con®guration, piston microphonics are common to both beams, providing noise cancellation and a highly stable phase measurement. The reference is tilted with respect to the sample beam to obtain fringes. These are projected onto a CCD camera, 3 It is also interesting to consider the case of a crossed polariser, which might potentially be more sensitive to light that su€ers multiple scattering. However in the cases considered here the signal levels were low in this channel.

S.P. Morgan et al. / Optics Communications 187 (2001) 29±38

Fig. 3. System diagram of the interferometric microscope. The principal element is the computer generated hologram which projects both reference and focused sample beams onto the sample, thus eliminating microphonics.

where the amplitude and phase of the fringe carrier frequency are extracted. The sample under investigation is a set of isolated grooves in silicon (refractive index ˆ 3:796 ‡ 0:013i at 0:688 lm [27]). Groove widths range from 10 to 0.34 lm and are 0.16 lm deep. A line scan is obtained by scanning the sample a distance of 12 lm in steps of 0.1 lm. Images were obtained with an Olympus MSPlan 100 objective with a quoted NA of 0.95. The illumination wavelength was 0.688 lm and images in two linear polarisation states were taken. Light polarised parallel to the grooves at the back focal plane of the objective is de®ned as TM, and light polarised perpendicular to the grooves as TE.

4. Results The experimental results (Fig. 4) show the amplitude and phase response for a series of grooves with nominal widths from 4.99 to 0.34 lm (depth ˆ 0:16 lm). Line scans of ®ve grooves are shown here to illustrate the results. The left column shows the amplitude response for TE (solid line) and TM (dashed line) illumination. The right column shows the corresponding phase response. For d1 ˆ 4:99 lm grooves there is relatively little difference between two scans apart from greater

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scattering at the edges in TE. However, as the groove width decreases signi®cant di€erences are observed. Both the amplitude contrast and phase in TM illumination decrease with decreasing groove width. The amplitude contrast in TE illumination shows an initial increase and then also decreases. In addition, the corresponding phase initially increases, inverts at a width between 0.55 and 0.44 lm and decreases. The amplitude contrast is always much greater in the TE scans. Fig. 5 demonstrates modelled line scans for a set of grooves …d1 ˆ 0:3; 0:25; 0:2 lm† in the region of the phase inversion. Again the left (right) column demonstrates the amplitude (phase) response of the TE (solid) and TM (dashed) cases. Scanning is shown over a 2 lm range and similar trends to the experiments are observed. Fig. 6 shows modelled results of the amplitude contrast and phase response over a wider range of groove widths. In TM the phase and amplitude contrast decrease with decreasing groove width. In TE we see an initial increase and then decrease in amplitude contrast with a corresponding increase, phase inversion and decrease in phase. It should be noted that the phase inversion is not caused by phase wrapping at the maximum phase values but occurs at all points in the line scan. The inversion occurs at a width of 0.26 lm, which is smaller than the width it occurs in the experiment. The reasons for this di€erence are discussed in the next section. An intuitive picture of these e€ects can be described in terms of interference between light scattered from the groove and that scattered from the surface of the structure (Fig. 7a). For narrow widths the focal spot overlaps the groove and the surface 4 and as the spot is scanned the response is a weighted vector summation of these contributions. In TM illumination there is relatively little coupling into the groove, which results in the small

4 It should be noted that at high NA the focal spot is elliptical [25] so the weighting between the groove and surface contributions will be di€erent for TE and TM. However the next section demonstrates that the groove scattering in di€erent polarisation states is the dominant e€ect.

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Fig. 4. Experimental line scans for di€erent groove widths (depth ˆ 0:16 lm, NA ˆ 0:95, wavelength ˆ 0:688 lm). The left-hand column shows the amplitude response for TE (±±) and TM (± ± ±) illumination. The right-hand column shows the corresponding phase response. The inversion in the phase response using TE illumination occurs at a width between 0.55 and 0.44 lm.

phase di€erence and amplitude of the groove component (Fig. 7b). As the spot is scanned there is therefore a small phase shift and little amplitude contrast caused by the weighted summation of these components. For smaller groove widths the contribution from the groove decreases, thus further reducing the amplitude contrast and phase. In TE illumination there is signi®cant coupling into the groove. For the depths considered in these experiments (k=4) the relative phase di€erence between groove and surface is close to p (Fig. 7c and d). The destructive interference of these components cause the large amplitude contrast and phase shifts observed in the TE line scans. The

phase inversion with decreasing width occurs when the light coupling causes the groove contribution to change from groove vector (i) in Fig. 7c to groove vector (ii) in Fig. 7d. The phase response determined by the weighted summation of the surface vector with groove vector (i) is the inverse of the phase response due to the summation of the surface vector and groove vector (ii). Inversion is most likely to occur when the groove depth introduces a relative phase di€erence between the surface and groove close to p. Any small variation in the phase due to di€erences in light coupling, such as the coupling and propagation of bound surface waves, will cause the change from groove

S.P. Morgan et al. / Optics Communications 187 (2001) 29±38

Fig. 5. Modelled line scans for groove widths around the region of the phase inversion widths (depth ˆ 0:16 lm, NA ˆ 0:95, wavelength ˆ 0:688 lm). The left-hand column shows the amplitude response for TE (±±) and TM (± ± ±) illumination. The right-hand column shows the corresponding phase response.

Fig. 6. Modelled results showing the e€ect of decreasing the groove width on the amplitude and phase response for TE (±±) and TM (± ± ±) illumination. The phase inversion occurs in TE polarisation at a groove width of 0.26 lm.

vector (i) to (ii) and the corresponding phase inversion. The most important observations are that TE and TM imaging provide signi®cantly di€erent responses for narrow grooves that can be predicted from a vector di€raction model. In addition, the coupling in TE is much greater which makes the

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Fig. 7. For narrow grooves the focal spot overlaps both the surface and the groove (a). The response is therefore a vector summation of the two components. For TM (b) the coupling into the groove is small which causes relatively small phase shifts and amplitude contrast. For TE (c) the coupling is greater. The phase inversion is observed when the scattered component from the groove changes from vector (i) to vector (ii) in (c).

response more sensitive to variations in the groove depth. This property suggests that TE polarised light could potentially be used to measure the depth of narrow grooves more e€ectively than TM. This con®rms the observations in Ref. [13] but here we are imaging dielectric grooves where the intuitive argument presented in Ref. [13] does not apply so readily. Moreover, we consider groove depths from 10ÿ4 to 1 lm (NA ˆ 0:95) for di€erent groove widths. Fig. 8 shows the TE, TM and ÔcorrectÕ 5 phase shifts (unwrapped) against height for a groove width (a) d1 ˆ 0:4 lm, (b) d1 ˆ 0:3 lm (c) d1 ˆ 0:2 lm. The correct phase shift includes an obliquity factor [25] to account for the mean cosine of the incident radiation differing substantially from 1 when imaging with high NA. Fig. 8a demonstrates that for small depths TE provides an accurate measure of the depth whereas

5 The ÔcorrectÕ phase shift is calculated using 4p  depth  obl:factor=k.

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Fig. 8. The stronger coupling of TE illumination into the grooves enables the deeper structures to be probed (a) d1 ˆ 0:4 lm (b) d1 ˆ 0:3 lm (c) d1 ˆ 0:2 lm.

TM underestimates the correct phase response. Beyond a depth of 0.01 lm TM is relatively insensitive to changes in the depth. TE polarisation, on the other hand, follows the correct phase to a depth of 0.05 lm and then overestimates the response. Beyond this depth TE shows a strong monotonic variation to depths of 0.9 lm thus allowing for calibration of the response. Similar trends are observed in Fig. 8b and c demonstrating that TE is more e€ective than TM at measuring the depth of grooves of di€erent width. At a groove width of d1 ˆ 0:3 lm (Fig. 8b) the TE

phase response becomes insensitive at a depth of 0.5 lm whereas for a width of d1 ˆ 0:2 lm (Fig. 8c) the response is insensitive at a depth of 0.1 lm. 5. Discussion Experimental and modelled results have both shown that there are signi®cant di€erences in images of narrow grooves taken with TE and TM polarisation states. The experimental results demonstrate relatively little di€erence (apart from

S.P. Morgan et al. / Optics Communications 187 (2001) 29±38

scattering at the edges) for wide grooves. As the groove width decreases the amplitude contrast and phase measured using TM illumination decreases. The TE amplitude response shows an initial increase in contrast and then a decrease with decreasing groove width. The corresponding phase response initially increases, then inverts and decreases. These e€ects have also been predicted using a vector di€raction model, demonstrating that it is essential under these circumstances to use a vector approach as scalar theory does not incorporate polarisation e€ects. In previous work [11] the essential physical feature that is brought out by the use of a vector di€raction model is that a scalar model does not account for shadowing. In addition to this point we demonstrate the di€erent coupling mechanisms of TE and TM modes. It should be noted that these e€ects are not simply due to the elliptical focal spot obtained in di€erent polarisation states when imaging at high NA [28]. If this were the case then TM would be more e€ective in measuring the height as in this case the smallest axis of the ellipse scans across the groove. Instead there is a dependence on the light coupling in the groove structure. The strong coupling of TE into the groove indicates that when probing deeper structures TE provides a better estimate of the depth over a wider range of depths. It is interesting to note that TE cannot be used to probe all depths and widths e€ectively and that as the groove width decreases, the maximum depth that can be probed decreases. This is in contrast to one-dimensional illumination results [13] where only a single width has been considered no maximum depth observed. The model is able to predict the signi®cant di€erences in the response in both polarisation states and for di€erent widths, giving con®dence that the model is valid. However, there are some di€erences in the absolute values between model and experiment. The potential sources of error are in specifying only nominal values of groove width, depth and NA. Also the groove shape is assumed to be rectangular and the pupil function uniform. In addition a periodic groove structure, separated by two focal spot widths, is modelled rather than a truly isolated groove. A future model will consider di€erent groove shapes and more isolated grooves,

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which will enable the di€erences in the absolute values to be investigated. Having demonstrated that vector di€raction provides an accurate model of the forward problem, one of the main questions that must be posed is whether this information can be used to solve the inverse problem. Huttenen and Turunen [12] have suggested that, by making a priori assumptions about the type of structure and using a rigorous model to provide a look up table, the width and height of subwavelength structures can be obtained. In such a case it is likely that data obtained in TE illumination will be better conditioned than TM, which may nevertheless provide valuable additional information.

6. Conclusions The amplitude and phase responses of an interferometric microscope to approximately isolated grooves in silicon have been demonstrated experimentally. A vector di€raction model for well separated periodic grooves predicts the salient features of the measured performance. The results show a signi®cant di€erence in the response under illumination with di€erent polarisation states as the groove width becomes less than a wavelength. These di€erent responses have been explained in terms of interference between the component of the focal spot that interacts with the groove and the component re¯ected from the surface. Di€erent responses occur in di€erent polarisation states as the coupling of modes into the groove is polarisation dependent. TE shows stronger coupling into the groove and this has been demonstrated to measure depth more accurately and over a wider range of depths than in TM.

Acknowledgements We are grateful to the Engineering and Physical Sciences Research Council (UK) for supporting this work and to the National Physical Laboratory (Teddington, UK) for providing the sample.

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