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General spectral properties of Makyoh imaging of quasi-periodic surfaces
Optics & Laser Technology 43 (2011) 245–247
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Research Note
General spectral properties of Makyoh imaging of quasi-periodic surfaces Ferenc Riesz Hungarian Academy of Sciences, Research Institute for Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary
a r t i c l e in fo
abstract
Article history: Received 8 April 2010 Received in revised form 20 May 2010 Accepted 24 May 2010 Available online 8 June 2010
Imaging properties of the Makyoh imaging of periodic and quasi-periodic surfaces are analysed using a spectral approach based on a geometrical optical model. It is shown that in spite of the nonlinear nature of imaging, the Fourier representation can be used with some restrictions and limitations. Ray-tracing simulations are performed to illustrate the results. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Optical metrology Makyoh Geometrical optics
1. Introduction Makyoh (or magic-mirror) topography (MT) is a powerful topographic tool for qualitative visualization of the surface morphology of semiconductor wafers and other mirror-like surfaces [1–4]. MT gets its name from an ancient mirror of Japanese origin. Its operation is based on the following principle: the surface is illuminated by a uniform collimated light beam, and the reflected beam is intersected by a screen on which a reflected image is formed (Fig. 1). Irregularities of the surface act as convex or concave mirror regions and focus or defocus the reflected beam, and the sample surface morphology is then revealed as dark/bright contrast regions in the image. This technique has been applied mainly for the qualitative inspection of the surface quality of semiconductor wafers (see Ref. [4] for a review). MT has been successful due to its simplicity, low cost and real-time operation. The practical implementations usually employ additional optical elements and electronic cameras as imaging elements [2]. Although general imaging properties of MT are well understood through ray-tracing simulations [5] and geometrical optical [6] models, the interpretation of the obtained images is not always so. The imaging of isolated defects (typically, a hillock or a pit) is well understood [2,6]; however, less attention has been paid to periodic and quasi-periodic surfaces. Such surface features frequently appear in semiconductor technology as a macroscopic rough texture, swirl defects, sawing or polishing marks, etc. The aim of the present work is to provide a geometrical optical analysis of the spectral properties of the imaging of periodic and
E-mail address: [email protected] 0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.05.013
quasi-periodic surfaces to predict its general trends. First, the model of image formation is summarized, then, the imaging of a sine surface is analysed and then the general case is considered. Ray-tracing simulations are performed to illustrate the results. Since Makyoh is chiefly a qualitative tool, we restrict ourselves to a qualitative or semi-quantitative analysis. We restrict our analysis to a one-dimensional case.
2. Background The geometrical optical model of image formation has been detailed in Ref. [6]. The two basic imaging equations are fðrÞ ¼ r2L grad hðrÞ,
ð1Þ
and 1 IðfÞ ¼ ð12LCmin Þð12LCmax Þ :
ð2Þ
Eq. (1) represents a mapping of a surface point r to the image point f, while Eq. (2) gives the image points’ intensities I(f) as normalised to that of a flat surface. The reflectivity is assumed uniform. L is the screen-to-sample distance. Cmin and Cmax are the two principal curvatures at the point r, that is, the minimum and maximum, respectively, of the second derivatives of h(r), the height profile. These formulae tell us that increasing in 9L9 increases the separation of the point and its image and increases the contrast as well. Thus, the optimum setting is in that L range that produces high enough contrast for reliable observation while preserving the integrity of the surface topology. The illuminance is infinite (caustic limit) if L equals either (2Cmin)–1 or (2Cmax)–1; that is, when the screen is in focus of a surface area element.
246
F. Riesz / Optics & Laser Technology 43 (2011) 245–247
log A
image intensity
A/k2 = const. component with the highest intensity
screen
log 1/k Fig. 2. Fourier representation of the surface relief.
reflected beam
reflecting surface Fig. 1. Schematic representation of the Makyoh image formation.
It can be shown using a rigorous geometrical optical analysis [7] but it follows [8] also from Eqs. (1) and (2) that if the surface curvatures are negligible compared to L all over the surface, the image intensity can be approximated as IðrÞ ¼
1 2
12L r hðrÞ
1þ 2L r2 hðrÞ
ð3Þ
That is, the image intensity variation is proportional to the Laplacian of the surface relief.
3. Theory and discussion Consider now a one-dimensional sine surface with a period k and a peak-to-peak amplitude A. It is easy to show using Eq. (2) that the caustic limit is reached at Lcaustic ¼ k2 =ð4p2 AÞ
ð4Þ
Because of the geometric scaling of the problem, we can fix A and/or k and treat L values normalised to Lcaustic without losing generality.The extremes of intensities occur at reflections from the extremes of the sine surface. These maximum and minimum intensities Imax and Imin are given as 1/(17L/Lcaustic) based on the one-dimensional version of Eq. (2). The image contrast can be characterised by the modulation given by the definition M¼(Imax–Imin)/(Imax +Imin). It is easy to obtain that M¼L/Lcaustic. Detailed ray-tracing simulations for the sine surface have been presented in Ref. [6]. For small L (for about 9L/Lcaustic9 o0.1), the image intensity profile is closely a sine function, which can be obtained also analytically from the one-dimensional version of Eq. (3) as follows: 2 2 2p x 2p IðxÞ ¼ 1AL ¼ 12L sin 2p hðxÞ ð5Þ k k k With increase in L, the profiles become narrower, producing a sharp peak reaching infinite intensity at the caustic limit. For L beyond the caustic limit, the peaks split. The Makyoh imaging is non-linear in general, yet we can use the Fourier representation with some restrictions and amendments, at least for a qualitative analysis, as detailed in the
following. First, in the regime where Eq. (3) holds (all surface curvatures are negligible compared to L), the imaging is linear. For the general case, disregard temporarily Eq. (1), that is, assume f¼ r for the whole surface. In Eq. (2), intensity will then be a function of r rather than f. It follows then that it is 1/I(r), which will be linear in h(r). That is, although the superposition principle cannot be used to calculate the intensity map, the qualitative response can be predicted using the Fourier representation. Non-linearity means that higher intensities are ‘‘amplified’’ further. Now, let us take the mapping into account (Eq. (1)). A gradient varying over the surface means a varying phase shift in the image. The way this effect modifies the previously outlined scenario depends on the surface spectrum. For a quasi-periodic surface with non-systematic random phase distribution, the random phase shifts will cancel each other on a statistical basis. For a surface with a few distinct periodic components, the phase of higher-frequency components with smaller period will be modulated by the changing gradients of lower-frequency components. The intensity pertaining to these components will be the highest for those for which A/k2 is the largest. This can be represented in the log–log (or semi-log) Fourier space, where constant A/k2 values correspond to lines of the same gradient Fig. 2. It is important to note that these considerations do not apply for spatially limited surface features (i.e. isolated defects), because the correct phase is important in their Fourier representation. To illustrate our considerations, model surfaces were generated by superposing sine functions of different periods and amplitudes, and their Makyoh images were simulated with ray-tracing. A sine surface with A0 ¼1 and k0 ¼2p was taken as base and an additional sine profile with k1 ¼ 2p/10 period was added to it. Three amplitudes were selected for this additional component A1: 1/50, 1/100 and 1/200. The amplitude of A1 ¼1/100 just corresponds to the situation where the caustic limit of the additional component is the same as for the main one. The ray-tracing results are shown in Fig. 3. The depicted caustic limit Lcaustic is calculated for the main sine component. The results show that the additional component has a profound influence on the image; in spite of its small relative amplitude, all image profiles show a dominant component with its periodicity. At small L, the contribution of the main surface is hardly noticeable. At higher L, the differences are even more profound because of the non-linear amplification effect discussed before. For the A1 ¼1/200 case, the main component dominates at higher L values, while for the A1 ¼1/50 case, the contribution of the additional component dominates intermediate L. This surprisingly great impact on the images is caused by the k2-type dependence of the image intensities. The phase modulation of the higherfrequency component induced by the lower-frequency one is clearly seen as well at higher L values.
F. Riesz / Optics & Laser Technology 43 (2011) 245–247
A1 = 1/100
A1 = 1/200
247
A1 = 1/50
4 L/Lcaustic = 1
2
0
4 L/Lcaustic = 0.5
Image intensity
2
0
4 L/Lcaustic = 0.2
2
0
4 L/Lcaustic = 0.1
2
0 0
π x
2π
0
π x
2π
0
π x
2π
Fig. 3. Ray-tracing simulations of the Makyoh image of a composite sine surface.
4. Summary
References
Imaging properties of Makyoh topography for periodic and quasi-periodic surfaces were analysed using a spectral approach. The main conclusions are as follows: (1) in spite of the non-linear laws of imaging, for small curvatures the imaging can be regarded to be linear; in the general case, with some restrictions and (2) the A/k2 type dependence of image intensity of the Fourier components causes a dominance of the higher-frequency components.
[1] Kugimiya K. Makyoh: the 2000 year old technology still alive. J. Cryst Growth 1990;103:420–422. [2] Blaustein P, Hahn S. Realtime inspection of wafer surfaces. Solid State Technol 1989;32:27–9. [3] Pei ZJ, Fisher GR, Bhagavat M, Kassir S. A grinding-based manufacturing method for silicon wafers: an experimental investigation. Int J Mach Tools Manuf 2005;45:1140–51. [4] Riesz F. Makyoh topography: a simple yet powerful optical method for flatness and defect characterisation of mirror-like surfaces. Proc SPIE 2004;5458: 86–100. [5] Koryta´r D, Hrivna´k M. Experimental and computer simulated Makyoh images of semiconductor wafers. Jpn J Appl Phys 1993;32:693–8. [6] Riesz F. Geometrical optical model of the image formation in Makyoh (magicmirror) topography. J Phys D 2000;33:3033–40. [7] Berry MV. Oriental magic mirrors and the Laplacian image. Eur J Phys 2006;27:109–18. [8] Riesz F. A note on Oriental magic mirrors and the Laplacian image. Eur J Phys 2006;27:N5–7.
Acknowledgements This work was supported, in part, by the Hungarian Scientific Research Fund (OTKA, Grant K 68534) and KPI (GVOP-3.2.1.-200404-0337/3.0).
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Research Note
General spectral properties of Makyoh imaging of quasi-periodic surfaces Ferenc Riesz Hungarian Academy of Sciences, Research Institute for Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary
a r t i c l e in fo
abstract
Article history: Received 8 April 2010 Received in revised form 20 May 2010 Accepted 24 May 2010 Available online 8 June 2010
Imaging properties of the Makyoh imaging of periodic and quasi-periodic surfaces are analysed using a spectral approach based on a geometrical optical model. It is shown that in spite of the nonlinear nature of imaging, the Fourier representation can be used with some restrictions and limitations. Ray-tracing simulations are performed to illustrate the results. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Optical metrology Makyoh Geometrical optics
1. Introduction Makyoh (or magic-mirror) topography (MT) is a powerful topographic tool for qualitative visualization of the surface morphology of semiconductor wafers and other mirror-like surfaces [1–4]. MT gets its name from an ancient mirror of Japanese origin. Its operation is based on the following principle: the surface is illuminated by a uniform collimated light beam, and the reflected beam is intersected by a screen on which a reflected image is formed (Fig. 1). Irregularities of the surface act as convex or concave mirror regions and focus or defocus the reflected beam, and the sample surface morphology is then revealed as dark/bright contrast regions in the image. This technique has been applied mainly for the qualitative inspection of the surface quality of semiconductor wafers (see Ref. [4] for a review). MT has been successful due to its simplicity, low cost and real-time operation. The practical implementations usually employ additional optical elements and electronic cameras as imaging elements [2]. Although general imaging properties of MT are well understood through ray-tracing simulations [5] and geometrical optical [6] models, the interpretation of the obtained images is not always so. The imaging of isolated defects (typically, a hillock or a pit) is well understood [2,6]; however, less attention has been paid to periodic and quasi-periodic surfaces. Such surface features frequently appear in semiconductor technology as a macroscopic rough texture, swirl defects, sawing or polishing marks, etc. The aim of the present work is to provide a geometrical optical analysis of the spectral properties of the imaging of periodic and
E-mail address: [email protected] 0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.05.013
quasi-periodic surfaces to predict its general trends. First, the model of image formation is summarized, then, the imaging of a sine surface is analysed and then the general case is considered. Ray-tracing simulations are performed to illustrate the results. Since Makyoh is chiefly a qualitative tool, we restrict ourselves to a qualitative or semi-quantitative analysis. We restrict our analysis to a one-dimensional case.
2. Background The geometrical optical model of image formation has been detailed in Ref. [6]. The two basic imaging equations are fðrÞ ¼ r2L grad hðrÞ,
ð1Þ
and 1 IðfÞ ¼ ð12LCmin Þð12LCmax Þ :
ð2Þ
Eq. (1) represents a mapping of a surface point r to the image point f, while Eq. (2) gives the image points’ intensities I(f) as normalised to that of a flat surface. The reflectivity is assumed uniform. L is the screen-to-sample distance. Cmin and Cmax are the two principal curvatures at the point r, that is, the minimum and maximum, respectively, of the second derivatives of h(r), the height profile. These formulae tell us that increasing in 9L9 increases the separation of the point and its image and increases the contrast as well. Thus, the optimum setting is in that L range that produces high enough contrast for reliable observation while preserving the integrity of the surface topology. The illuminance is infinite (caustic limit) if L equals either (2Cmin)–1 or (2Cmax)–1; that is, when the screen is in focus of a surface area element.
246
F. Riesz / Optics & Laser Technology 43 (2011) 245–247
log A
image intensity
A/k2 = const. component with the highest intensity
screen
log 1/k Fig. 2. Fourier representation of the surface relief.
reflected beam
reflecting surface Fig. 1. Schematic representation of the Makyoh image formation.
It can be shown using a rigorous geometrical optical analysis [7] but it follows [8] also from Eqs. (1) and (2) that if the surface curvatures are negligible compared to L all over the surface, the image intensity can be approximated as IðrÞ ¼
1 2
12L r hðrÞ
1þ 2L r2 hðrÞ
ð3Þ
That is, the image intensity variation is proportional to the Laplacian of the surface relief.
3. Theory and discussion Consider now a one-dimensional sine surface with a period k and a peak-to-peak amplitude A. It is easy to show using Eq. (2) that the caustic limit is reached at Lcaustic ¼ k2 =ð4p2 AÞ
ð4Þ
Because of the geometric scaling of the problem, we can fix A and/or k and treat L values normalised to Lcaustic without losing generality.The extremes of intensities occur at reflections from the extremes of the sine surface. These maximum and minimum intensities Imax and Imin are given as 1/(17L/Lcaustic) based on the one-dimensional version of Eq. (2). The image contrast can be characterised by the modulation given by the definition M¼(Imax–Imin)/(Imax +Imin). It is easy to obtain that M¼L/Lcaustic. Detailed ray-tracing simulations for the sine surface have been presented in Ref. [6]. For small L (for about 9L/Lcaustic9 o0.1), the image intensity profile is closely a sine function, which can be obtained also analytically from the one-dimensional version of Eq. (3) as follows: 2 2 2p x 2p IðxÞ ¼ 1AL ¼ 12L sin 2p hðxÞ ð5Þ k k k With increase in L, the profiles become narrower, producing a sharp peak reaching infinite intensity at the caustic limit. For L beyond the caustic limit, the peaks split. The Makyoh imaging is non-linear in general, yet we can use the Fourier representation with some restrictions and amendments, at least for a qualitative analysis, as detailed in the
following. First, in the regime where Eq. (3) holds (all surface curvatures are negligible compared to L), the imaging is linear. For the general case, disregard temporarily Eq. (1), that is, assume f¼ r for the whole surface. In Eq. (2), intensity will then be a function of r rather than f. It follows then that it is 1/I(r), which will be linear in h(r). That is, although the superposition principle cannot be used to calculate the intensity map, the qualitative response can be predicted using the Fourier representation. Non-linearity means that higher intensities are ‘‘amplified’’ further. Now, let us take the mapping into account (Eq. (1)). A gradient varying over the surface means a varying phase shift in the image. The way this effect modifies the previously outlined scenario depends on the surface spectrum. For a quasi-periodic surface with non-systematic random phase distribution, the random phase shifts will cancel each other on a statistical basis. For a surface with a few distinct periodic components, the phase of higher-frequency components with smaller period will be modulated by the changing gradients of lower-frequency components. The intensity pertaining to these components will be the highest for those for which A/k2 is the largest. This can be represented in the log–log (or semi-log) Fourier space, where constant A/k2 values correspond to lines of the same gradient Fig. 2. It is important to note that these considerations do not apply for spatially limited surface features (i.e. isolated defects), because the correct phase is important in their Fourier representation. To illustrate our considerations, model surfaces were generated by superposing sine functions of different periods and amplitudes, and their Makyoh images were simulated with ray-tracing. A sine surface with A0 ¼1 and k0 ¼2p was taken as base and an additional sine profile with k1 ¼ 2p/10 period was added to it. Three amplitudes were selected for this additional component A1: 1/50, 1/100 and 1/200. The amplitude of A1 ¼1/100 just corresponds to the situation where the caustic limit of the additional component is the same as for the main one. The ray-tracing results are shown in Fig. 3. The depicted caustic limit Lcaustic is calculated for the main sine component. The results show that the additional component has a profound influence on the image; in spite of its small relative amplitude, all image profiles show a dominant component with its periodicity. At small L, the contribution of the main surface is hardly noticeable. At higher L, the differences are even more profound because of the non-linear amplification effect discussed before. For the A1 ¼1/200 case, the main component dominates at higher L values, while for the A1 ¼1/50 case, the contribution of the additional component dominates intermediate L. This surprisingly great impact on the images is caused by the k2-type dependence of the image intensities. The phase modulation of the higherfrequency component induced by the lower-frequency one is clearly seen as well at higher L values.
F. Riesz / Optics & Laser Technology 43 (2011) 245–247
A1 = 1/100
A1 = 1/200
247
A1 = 1/50
4 L/Lcaustic = 1
2
0
4 L/Lcaustic = 0.5
Image intensity
2
0
4 L/Lcaustic = 0.2
2
0
4 L/Lcaustic = 0.1
2
0 0
π x
2π
0
π x
2π
0
π x
2π
Fig. 3. Ray-tracing simulations of the Makyoh image of a composite sine surface.
4. Summary
References
Imaging properties of Makyoh topography for periodic and quasi-periodic surfaces were analysed using a spectral approach. The main conclusions are as follows: (1) in spite of the non-linear laws of imaging, for small curvatures the imaging can be regarded to be linear; in the general case, with some restrictions and (2) the A/k2 type dependence of image intensity of the Fourier components causes a dominance of the higher-frequency components.
[1] Kugimiya K. Makyoh: the 2000 year old technology still alive. J. Cryst Growth 1990;103:420–422. [2] Blaustein P, Hahn S. Realtime inspection of wafer surfaces. Solid State Technol 1989;32:27–9. [3] Pei ZJ, Fisher GR, Bhagavat M, Kassir S. A grinding-based manufacturing method for silicon wafers: an experimental investigation. Int J Mach Tools Manuf 2005;45:1140–51. [4] Riesz F. Makyoh topography: a simple yet powerful optical method for flatness and defect characterisation of mirror-like surfaces. Proc SPIE 2004;5458: 86–100. [5] Koryta´r D, Hrivna´k M. Experimental and computer simulated Makyoh images of semiconductor wafers. Jpn J Appl Phys 1993;32:693–8. [6] Riesz F. Geometrical optical model of the image formation in Makyoh (magicmirror) topography. J Phys D 2000;33:3033–40. [7] Berry MV. Oriental magic mirrors and the Laplacian image. Eur J Phys 2006;27:109–18. [8] Riesz F. A note on Oriental magic mirrors and the Laplacian image. Eur J Phys 2006;27:N5–7.
Acknowledgements This work was supported, in part, by the Hungarian Scientific Research Fund (OTKA, Grant K 68534) and KPI (GVOP-3.2.1.-200404-0337/3.0).