Abstract

Besides the illumination wavelength also the numerical aperture (NA) of a microscope objective affects the fringe spacing in interference microscopy. Therefore, at high NA values an effective wavelength should be obtained by calibration. At step height structures both, the effective wavelength and the batwing effect strongly depend on the height-to-wavelength-ratio (HWR). Therefore, changes of the effective wavelength considering temporal and spatial coherence enable us to estimate the batwing effect in measurement results. For high NA systems and broadband illumination two different theoretical approaches for signal modeling are introduced to study the influence of the center wavelength, the temporal, and the spatial coherence of the illuminating light on measurement results of a rectangular grating. In both models diffraction is considered. While the first simulation model (Kirchhoff) is mostly analytical the second one (Richards-Wolf) is primarily numerical. Simulation results of both models show a good agreement with experimental measurement results.

© 2016 Optical Society of America

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References

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  1. E. Inglestam and L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
    [Crossref]
  2. G. Schulz and K. -E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
    [Crossref] [PubMed]
  3. J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
    [Crossref]
  4. C. J. R. Sheppard and K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
    [Crossref] [PubMed]
  5. H. Mykura and G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
    [Crossref]
  6. K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).
    [Crossref] [PubMed]
  7. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [Crossref] [PubMed]
  8. T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).
  9. P. de Groot and X. C. de Lega, “Signal modeling for low-coherence height-scanning interference microscopy,” Appl. Opt. 43, 4821–4830 (2004)
    [Crossref] [PubMed]
  10. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [Crossref]
  11. M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
    [Crossref]
  12. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).
  13. J. A. Ogilvy, Theory of Wave Scattering From Random Rough Surfaces (Taylor & Francis, 1991).
  14. D. F. McCammon and S. T. McDaniel, “On the convergence of a series solution to a modified helmholtz integral equation and validity of the kirchhoff approximation,” J. Ac. Soc. Am.,  79, 64–70, (1986).
    [Crossref]
  15. P. Lehmann, In-process Laser-Messmethoden auf der Grundlage der Fourieranalyse (Expert Verlag, 2003).
  16. P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
    [Crossref]
  17. P. de Groot and X. C. de Lega, “Determination of fringe order in white-light interference microscopy,” Appl. Opt. 41, 4571–4578 (2002).
    [Crossref] [PubMed]
  18. W. Xie, P. Lehmann, and J. Niehues, “Lateral resolution and transfer characteristics of vertical scanning white-light interferometers,” Appl. Opt. 51, 1795–1803 (2012).
    [Crossref] [PubMed]
  19. I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
    [Crossref]
  20. P. Lehmann, W. Xie, and J. Niehues, “Transfer characteristics of rectangular phase gratings in interference microscopy,” Opt. Lett. 37, 758–760 (2012).
    [Crossref] [PubMed]
  21. G. D. Durgin, “The practical behavior of various edge-diffraction formulas,” IEEE Antennas and Propagation Magazine,  51 (3), 24–35 (2009).
    [Crossref]

2012 (3)

2009 (1)

G. D. Durgin, “The practical behavior of various edge-diffraction formulas,” IEEE Antennas and Propagation Magazine,  51 (3), 24–35 (2009).
[Crossref]

2004 (1)

2002 (1)

1995 (1)

1991 (1)

1990 (1)

1989 (1)

1986 (1)

D. F. McCammon and S. T. McDaniel, “On the convergence of a series solution to a modified helmholtz integral equation and validity of the kirchhoff approximation,” J. Ac. Soc. Am.,  79, 64–70, (1986).
[Crossref]

1963 (1)

H. Mykura and G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

1958 (1)

E. Inglestam and L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[Crossref]

1956 (1)

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[Crossref]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[Crossref]

Abdulhalim, I.

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
[Crossref]

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
[Crossref]

Chim, S. S. C.

Corle, T. R.

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).

Creath, K.

de Groot, P.

de Lega, X. C.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[Crossref]

Durgin, G. D.

G. D. Durgin, “The practical behavior of various edge-diffraction formulas,” IEEE Antennas and Propagation Magazine,  51 (3), 24–35 (2009).
[Crossref]

Elssner, K. -E.

Gates, J. W.

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[Crossref]

Inglestam, E.

E. Inglestam and L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[Crossref]

Johansson, L. P.

E. Inglestam and L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[Crossref]

Kino, G. S.

G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
[Crossref] [PubMed]

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).

Larkin, K. G.

Lehmann, P.

McCammon, D. F.

D. F. McCammon and S. T. McDaniel, “On the convergence of a series solution to a modified helmholtz integral equation and validity of the kirchhoff approximation,” J. Ac. Soc. Am.,  79, 64–70, (1986).
[Crossref]

McDaniel, S. T.

D. F. McCammon and S. T. McDaniel, “On the convergence of a series solution to a modified helmholtz integral equation and validity of the kirchhoff approximation,” J. Ac. Soc. Am.,  79, 64–70, (1986).
[Crossref]

Mykura, H.

H. Mykura and G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[Crossref]

Niehues, J.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering From Random Rough Surfaces (Taylor & Francis, 1991).

Rhead, G. E.

H. Mykura and G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Schulz, G.

Sheppard, C. J. R.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
[Crossref]

Xie, W.

Ann. Phys. (2)

P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[Crossref]

I. Abdulhalim, “Spatial and temporal coherence effects in interference microscopy and full-field optical coherence tomography,” Ann. Phys. 524, 787–804 (2012).
[Crossref]

Appl. Opt. (7)

IEEE Antennas and Propagation Magazine (1)

G. D. Durgin, “The practical behavior of various edge-diffraction formulas,” IEEE Antennas and Propagation Magazine,  51 (3), 24–35 (2009).
[Crossref]

J. Ac. Soc. Am. (1)

D. F. McCammon and S. T. McDaniel, “On the convergence of a series solution to a modified helmholtz integral equation and validity of the kirchhoff approximation,” J. Ac. Soc. Am.,  79, 64–70, (1986).
[Crossref]

J. Sci. Instrum. (3)

E. Inglestam and L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[Crossref]

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
[Crossref]

H. Mykura and G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[Crossref]

Opt. Lett. (1)

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
[Crossref]

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

J. A. Ogilvy, Theory of Wave Scattering From Random Rough Surfaces (Taylor & Francis, 1991).

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic Press, 1996).

P. Lehmann, In-process Laser-Messmethoden auf der Grundlage der Fourieranalyse (Expert Verlag, 2003).

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Figures (14)

Fig. 1
Fig. 1 Schematic illustration of imaging from the object plane to the image plane.
Fig. 2
Fig. 2 Schematic illustration of the optical setup regarding Richards-Wolf-modeling.
Fig. 3
Fig. 3 Local coordinate system of defocused imaging regarding Richards-Wolf-modeling.
Fig. 4
Fig. 4 Interference correlograms with different properties of temporal and spatial coherence, assuming P1(θe) = P2(θe) = 1 and Rs(k, θe) = 1. The black dashed line represents the correlogram from the upper plateau and the green dashed line the one from the lower plateau. The solid lines show the correlogram at the upper and lower edge position of the rectangular grating. (a) Correlogram assuming perfect temporal and spatial coherence: θe → 0, h0 = λ0/4. (b) Correlogram assuming partial temporal coherence (Gaussian envelope) and full spatial coherence: θe → 0, h0 = λ0/4, where λ0 is the center wavelength of the Gaussian spectrum. (c) Correlograms assuming finite spatial coherence and full temporal coherence: NA = 0.9, h0 = 0.2 μm and λ0 = 0.6 μm. λeff is the effective wavelength resulting from the NA effect and λeff = 0.8 μm. (d) Correlograms assuming finite temporal and spatial coherence: θ ∈ [−arcsin(NA), arcsin(NA)], h0 = 0.2 μm, λ0 = 0.6 μm. λeff = 0.8 μm results.
Fig. 5
Fig. 5 Simulated interference correlograms and their spectra. The blue curve in the spectrum represents the nominal spectrum of the LED, the red curve is the part of the spectrum applied to calculate the effective wavelength according to a centroid method. (a) Correlograms and their spectra using a royal blue LED. (b) Correlograms and their spectra using a red LED.
Fig. 6
Fig. 6 Upper figures: Profiles of a rectangular grating resulting from envelope evaluation (blue curve) and phase evaluation (red curve); Bottom figures: corresponding course of the effective wavelength. (a) Topography and effective wavelength using royal blue LED. (b) Topography and effective wavelength using red LED.
Fig. 7
Fig. 7 Graphical illustration of simulation results of different LED illumination, assuming NA = 0.9.
Fig. 8
Fig. 8 Fringe images of maximum contrast captured by the CMOS camera using red. (a) and royal blue. (b) LED illumination.
Fig. 9
Fig. 9 Measurement results using royal blue and red LED illumination. (a) Measured correlograms and their spectra using royal blue LED. The blue curve in upper figure represents the unfiltered correlogram, while the red curve shows the correlogram filtered by a Gaussian filter. The blue curve in the bottom figure represents the nominal spectrum of the LED, the red curve is the part of the spectrum applied to calculate the effective wavelength according to a centroid method. (b) Measured correlograms and their spectra using red LED. (c) Measured topography and effective wavelength using royal blue LED. Upper figure: Profiles of a rectangular grating resulting from envelope evaluation (blue curve) and phase evaluation (red curve); Bottom figure: corresponding course of the effective wavelength. (d) Measured topography and effective wavelength using red LED.
Fig. 10
Fig. 10 Normalized spectral density of LEDs of different colors
Fig. 11
Fig. 11 Simulation results according to Richards-Wolf-modelling corresponding to Fig. 4 using the same parameters. The black dashed line represents the correlogram from the upper plateau and the green dashed line the one from the lower plateau. The solid lines show the correlogram at the upper and lower edge position of the rectangular grating. (a) Correlograms for temporally and spatially coherent illumination. (b) Correlograms for partial temporal coherence (Gaussian envelope) and full spatial coherence. (c) Correlogram for finite spatial coherence and full temporal coherence. (d) Correlogram for partial temporal coherence and finite spatial coherence.
Fig. 12
Fig. 12 Comparison of interferograms using Richards-Wolf-modelling compared to the analytic result from [19].The red and black curves are results of the analytic solution, which are the interferogram and its envelope respectively. The blue and green curves result from Richards-Wolf-modelling.
Fig. 13
Fig. 13 Schematic representation of the Linnik interferometer
Fig. 14
Fig. 14 Comparison of measurement and simulation results using amber LED illumination. The blue curve at the bottom diagrams of Fig. 14(a) and Fig. 14(b) represents the nominal spectrum of the LED, the red curve is the part of the spectrum applied to calculate the effective wavelength according to a centroid method. The upper figures in Fig. 14(c) and Fig. 14(d) show profiles of a rectangular grating resulting from envelope evaluation (blue curve) and phase evaluation (red curve).

Tables (2)

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Table 1 Overview of simulation results at different LED illumination

Tables Icon

Table 2 Basic components

Equations (22)

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h ( x 0 , y 0 ) = h ( x 0 ) = h 0 2 cos ( 2 π / Λ x 0 ) | cos ( 2 π / Λ x 0 ) | .
U s ; k , θ e ( θ s ) = j U 0 e j k r 4 π r S R s ( k , θ e ) e j ( k e k s ) r 0 ( k e k s ) n ^ d S ,
k e = k ( sin θ e 0 cos θ e ) , k s = k ( sin θ s 0 cos θ s ) ,
r 0 = ( x 0 0 h ( x 0 ) ) , n ^ = ( 0 0 1 ) .
q = k s k e = ( q x q y q z ) = k ( sin θ s sin θ e 0 cos θ s + cos θ e ) .
U s ; k , θ e ( θ s ) = j U 0 e j k r 4 π r A P 1 ( θ e ) R s ( k , θ e ) q z exp { j q z h ( x 0 ) } exp ( j q x x 0 ) d x 0 .
U r ; k , θ e ( θ s ) = P 1 ( θ e ) R r ( k , θ e ) q z exp ( j q x x 0 ) d x 0 .
exp { j q z h ( x 0 ) } = n = C n exp ( j 2 n π Λ x 0 ) ,
C n = { cos ( q z h 0 / 2 ) , n = 0 0 , n is even ( 1 ) 0.5 ( n + 1 ) 2 j / n π sin ( q z h 0 / 2 ) . n is odd
θ s , n = arcsin ( n λ Λ + sin θ e ) .
arcsin ( NA ) θ s , n arcsin ( NA ) .
n min n n max , and n ,
n min = ceil ( ( NA sin θ e ) Λ λ ) , n max = floor ( ( NA sin θ e ) Λ λ ) ,
U s ; k , θ e ( θ s ) = P 1 ( θ e ) R s ( k , θ e ) n min n max C n q z exp ( j 2 n π Λ x 0 ) exp ( j q x x 0 ) d x 0 = P 1 ( θ e ) R s ( k , θ e ) n min n max C n q z exp ( j 2 n π Λ x 0 ) exp ( j q x x 0 ) d x 0 .
U s ; k , θ e ( θ s ) = P 1 ( θ e ) R s ( k , θ e ) n min n max C n q z δ ( q x n 2 π Λ ) .
U r ; k , θ e ( θ s ) = P 1 ( θ e ) R r ( k , θ e ) q z , n = 0 δ ( q x ) ,
I ( x , Δ z ; k ) = θ e , max θ e , max ( P 1 ( θ e ) P 2 ( θ e ) sin θ e { n min n max C n q z R s ( k , θ e ) exp ( j 2 n π Λ x ) exp ( 2 k cos θ e Δ z ) + 2 k cos θ e R r ( k , θ e ) } ) 2 d θ e .
I ( x , Δ z ) = 0 I ( x , Δ z ; k ) F ( k ) S ( k ) d k ,
I ( x , x c ; k , θ e ) = | E 0 2 | + 2 | E 1 2 | + | E 2 2 | ,
E 0 ( x , x c ; k , θ e ) = P 1 ( θ e ) P 2 ( θ e ) cos θ e sin θ e ( 1 + cos θ e ) J 0 ( k r P sin θ e sin θ P ) [ R s ( k , θ e ) exp ( j 2 k Δ z cos θ e ) + R r ( k , θ e ) ] , E 1 ( x , x c ; k , θ e ) = P 1 ( θ e ) P 2 ( θ e ) cos θ e sin 2 θ e J 1 ( k r P sin θ e sin θ P ) [ R s ( k , θ e ) exp ( j 2 k Δ z cos θ e ) + R r ( k , θ e ) ] , E 2 ( x , x c ; k , θ e ) = P 1 ( θ e ) P 2 ( θ e ) cos θ e sin θ e ( 1 cos θ e ) J 2 ( k r P sin θ e sin θ P ) [ R s ( k , θ e ) exp ( j 2 k Δ z cos θ e ) + R r ( k , θ e ) ] .
I ( x , Δ z ) = 0 0 θ e , max x c = δ x x c = δ x { I ( x , x c ; k , θ e ) } d θ e F ( k ) S ( k ) d k ,
P 1 ( θ e ) = P 2 ( θ e ) = cos ς θ e ,

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