Abstract

We describe an optical sectioning microscopy system with no moving parts based on a micro-structured stripe-array light emitting diode (LED). By projecting arbitrary line or grid patterns onto the object, we are able to implement a variety of optical sectioning microscopy techniques such as grid-projection structured illumination and line scanning confocal microscopy, switching from one imaging technique to another without modifying the microscope setup. The micro-structured LED and driver are detailed and depth discrimination capabilities are measured and calculated. ©2007 Optical Society of America

© 2007 Optical Society of America

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References

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  1. T. Wilson and Barry R. Masters, Confocal microscopy, 1990Academic Press, San Diego
  2. W. B. Amos, J. G. White, and J. B. Pawley (Plenum, New York, 1995), pp. 403–415
  3. C. J. R. Sheppard and X. Q. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 25, 1169–1185 (1988)
    [CrossRef]
  4. M.A.A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905–1907 (1997)
    [CrossRef]
  5. OptiGrid, QiOptiq Imaging Solutions, http://www.qioptiqimaging.com
  6. ApoTome, Carl Zeiss MicroImaging GmbH, http://www.zeiss.com
  7. P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
    [CrossRef] [PubMed]
  8. C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135–140 (2006)
    [CrossRef] [PubMed]
  9. O. Bormuth, J. Howard, and E. Schaffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. 226, 1–5 (2007)
    [CrossRef] [PubMed]
  10. S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaN/GaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236–3238
  11. H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
    [CrossRef]
  12. C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
    [CrossRef]
  13. H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1975
  15. M.A.A. Neil, R. Juskaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference illumination,” Opt. Commun. 153, 1–4 (1998)
    [CrossRef]
  16. L. H. Schaeffer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004)
    [CrossRef]
  17. P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)
  18. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [CrossRef]

2007 (1)

O. Bormuth, J. Howard, and E. Schaffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. 226, 1–5 (2007)
[CrossRef] [PubMed]

2006 (2)

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135–140 (2006)
[CrossRef] [PubMed]

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

2004 (2)

L. H. Schaeffer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004)
[CrossRef]

C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
[CrossRef]

2003 (1)

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

2001 (1)

P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
[CrossRef] [PubMed]

1998 (1)

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference illumination,” Opt. Commun. 153, 1–4 (1998)
[CrossRef]

1997 (1)

1995 (1)

P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)

1988 (1)

C. J. R. Sheppard and X. Q. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 25, 1169–1185 (1988)
[CrossRef]

1969 (1)

Amos, W. B.

W. B. Amos, J. G. White, and J. B. Pawley (Plenum, New York, 1995), pp. 403–415

Benedetti, P.A.

P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)

Bormuth, O.

O. Bormuth, J. Howard, and E. Schaffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. 226, 1–5 (2007)
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1975

Choi, H.W.

C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
[CrossRef]

Choi,, H.W.

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

Dawson, M.D.

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
[CrossRef]

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

Edwards, P.R.

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

Evangelista, V.

P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)

French, P.W.M.

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

Gong, Z.

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

Gu, E.

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
[CrossRef]

Guidarini, D.

P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)

Herman, P.

P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
[CrossRef] [PubMed]

Howard, J.

O. Bormuth, J. Howard, and E. Schaffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. 226, 1–5 (2007)
[CrossRef] [PubMed]

Jeon, C.W

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

Jeon, C.W.

C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
[CrossRef]

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

Jiang, H. X.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaN/GaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236–3238

Jin, S. X.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaN/GaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236–3238

Juskaitis, R.

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference illumination,” Opt. Commun. 153, 1–4 (1998)
[CrossRef]

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905–1907 (1997)
[CrossRef]

Klimant, I.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135–140 (2006)
[CrossRef] [PubMed]

Lakowicz, J. R.

P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
[CrossRef] [PubMed]

Li, J.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaN/GaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236–3238

Lin, H.-J.

P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
[CrossRef] [PubMed]

Lin, J. Y.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaN/GaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236–3238

Maliwal, B. P.

P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
[CrossRef] [PubMed]

Mao, X. Q.

C. J. R. Sheppard and X. Q. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 25, 1169–1185 (1988)
[CrossRef]

Martin, R.W.

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

Masters, Barry R.

T. Wilson and Barry R. Masters, Confocal microscopy, 1990Academic Press, San Diego

Mayr, T.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135–140 (2006)
[CrossRef] [PubMed]

Moser, C.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135–140 (2006)
[CrossRef] [PubMed]

Neil, M.A.A.

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference illumination,” Opt. Commun. 153, 1–4 (1998)
[CrossRef]

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905–1907 (1997)
[CrossRef]

Pawley, J. B.

W. B. Amos, J. G. White, and J. B. Pawley (Plenum, New York, 1995), pp. 403–415

Schaeffer, L. H.

L. H. Schaeffer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004)
[CrossRef]

Schaffer, E.

O. Bormuth, J. Howard, and E. Schaffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. 226, 1–5 (2007)
[CrossRef] [PubMed]

Schaffer, J.

L. H. Schaeffer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004)
[CrossRef]

Schuster, D.

L. H. Schaeffer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004)
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and X. Q. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 25, 1169–1185 (1988)
[CrossRef]

Stokseth, P. A.

Vestri, S.

P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)

White, J. G.

W. B. Amos, J. G. White, and J. B. Pawley (Plenum, New York, 1995), pp. 403–415

Wilson, T.

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference illumination,” Opt. Commun. 153, 1–4 (1998)
[CrossRef]

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905–1907 (1997)
[CrossRef]

T. Wilson and Barry R. Masters, Confocal microscopy, 1990Academic Press, San Diego

Wolf, E.

M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1975

Zhang, H.X.

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

Appl. Phys. Lett. (1)

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaN/GaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236–3238

IEE Photon. Technol. Lett. (1)

H.X. Zhang, E. Gu, C.W Jeon, Z. Gong, M.D. Dawson, M.A.A. Neil, and P.W.M. French, “Microstripe-Array InGaN Light-Emitting Diodes With Individually Addressable Elements,” IEE Photon. Technol. Lett. 18, 1681– 1683, (2006)
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H.W. Choi,, C.W. Jeon, M.D. Dawson, P.R. Edwards, and R.W. Martin, “Fabrication and performance of paralleladdressed InGaN micro-LED arrays,” IEEE Photon. Technol. Lett. 15, 510–512 (2003)
[CrossRef]

J. Microsc. (4)

P. Herman, B. P. Maliwal, H.-J. Lin, and J. R. Lakowicz, “Frequency-domain fluorescence microscopy with the LED as a light source,” J. Microsc. 203, 176–181 (2001)
[CrossRef] [PubMed]

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135–140 (2006)
[CrossRef] [PubMed]

O. Bormuth, J. Howard, and E. Schaffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. 226, 1–5 (2007)
[CrossRef] [PubMed]

L. H. Schaeffer, D. Schuster, and J. Schaffer, “Structured illumination microscopy: artefact analysis and reduction utilizing a parameter optimization approach,” J. Microsc. 216, 165–174 (2004)
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard and X. Q. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 25, 1169–1185 (1988)
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

M.A.A. Neil, R. Juskaitis, and T. Wilson, “Real time 3D fluorescence microscopy by two beam interference illumination,” Opt. Commun. 153, 1–4 (1998)
[CrossRef]

Opt. Lett. (1)

Photon. Technol. Lett. (1)

C.W. Jeon, H.W. Choi, E. Gu, and M.D. Dawson, “High-density matrix-addressable AlInGaN-based 368-nm microarray light-emitting diodes,” Photon. Technol. Lett. 16, 2421–2423 (2004)
[CrossRef]

Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE (1)

P.A. Benedetti, V. Evangelista, D. Guidarini, and S. Vestri, “Electronic multiconfocal points microscopy,” Three dimensional microscopy: Image acquisition and processing II, Proc. SPIE 2412, 56–62 (1995)

Other (5)

T. Wilson and Barry R. Masters, Confocal microscopy, 1990Academic Press, San Diego

W. B. Amos, J. G. White, and J. B. Pawley (Plenum, New York, 1995), pp. 403–415

M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1975

OptiGrid, QiOptiq Imaging Solutions, http://www.qioptiqimaging.com

ApoTome, Carl Zeiss MicroImaging GmbH, http://www.zeiss.com

Supplementary Material (3)

» Media 1: MOV (10 KB)     
» Media 2: MOV (1187 KB)     
» Media 3: MOV (791 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Picture of the micro-stripe LED

Fig. 2.
Fig. 2.

Driver board

Fig. 3.
Fig. 3.

Microscope setup

Fig. 4.
Fig. 4.

scanning scheme used for grid-projection

Fig. 5.
Fig. 5.

20x images of stained pollen grains acquired with grid-projection structured illumination. (a) Modulated raw image [Media 1], (b) Sectioned image, (c) Conventional image

Fig. 6.
Fig. 6.

20x magnification images of stained pollen grains (a) conventional image, (b) maximum projection confocal image and (c) automatic slit confocal image. Movies of (a) and (b) show how the images evolve as the line is scanned across the sample [Media 2] [Media 3]

Fig. 7.
Fig. 7.

Theoretical and experimental axial responses of the structured illumination system

Fig. 8.
Fig. 8.

Theoretical and experimental sectioning strengths of a slit scanning confocal microscope.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

I sec = 2 3 ( I 1 I 2 ) 2 + ( I 2 I 3 ) 2 + ( I 3 I 1 ) 2
I conv = I 1 + I 2 + I 3 3
I sec = i = 1 N Mask i · I i
Mask i ( x ) x [ Nx , Ny ] = { 1 if MAXPOS z ( x ) = i 0 else
I sec ( x ) x [ Nx , Ny ] = MAX i = 1 N [ I i ( x ) ]
I WF ( x ) = i = 1 N I i ( x ) N
I sec ( u ) = MTF i ( u , ν ) × MTF d ( u , ν ) MTF ( u , ν ) 2
u = ( 8 π n λ ) z sin 2 ( α 2 ) ν = ( 2 π Λ ) ( n sin ( α ) λ )
MTF ( u , ν ) = { g ( ν ) 2 J 1 [ u ν ( 1 ν 2 ) ] u ν ( 1 ν 2 ) if 0 < ν < 2 0 otherwise
g ( ν ) = 1 0.69 ν + 0.0076 ν 3 + 0.043 ν 3
I ˜ sec ( u ) = 2 J 1 [ u ν ( 1 ν 2 ) ] u ν ( 1 ν 2 ) 2
I ( u ) + OTF ( u , ν x , 0 ) 2 sinc ( ν x d i 2 π ) sinc ( ν x d d 2 π ) d ν x
d i , d = 2 π λ D i , d sin α sinc ( x ) = sin ( x ) x
OTF ( u , ν x , 0 ) = { g ( ν x ) 2 J 1 [ u ν x ( 1 ν x 2 ) ] u ν x ( 1 ν x 2 ) if 2 < ν x < 2 0 otherwise

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