Abstract

The Lagrange invariant is a well-known law for optical imaging systems formulated in the frame of ray optics. In this study, we reformulate this law in terms of wave optics and relate it to the resolution limits of various imaging systems. Furthermore, this modified Lagrange invariant is generalized for imaging along the z axis, resulting with the axial Lagrange invariant which can be used to analyze the axial resolution of various imaging systems. To demonstrate the effectiveness of the theory, analysis of the lateral and the axial imaging resolutions is provided for Fresnel incoherent correlation holography (FINCH) systems.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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2014 (4)

2013 (4)

2012 (4)

2011 (3)

2009 (1)

2007 (1)

2006 (1)

1997 (2)

1987 (1)

1985 (1)

1974 (1)

1971 (1)

1966 (3)

Bouchal, O.

O. Bouchal and Z. Bouchal, “Wide-field common-path incoherent correlation microscopy with a perfect overlapping of interfering beams,” J. Europ. Opt. Soc. Rap. Pub. 8, 13011 (2013).

Bouchal, P.

Bouchal, Z.

O. Bouchal and Z. Bouchal, “Wide-field common-path incoherent correlation microscopy with a perfect overlapping of interfering beams,” J. Europ. Opt. Soc. Rap. Pub. 8, 13011 (2013).

P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express 19(16), 15603–15620 (2011).
[Crossref] [PubMed]

Breckinridge, J. B.

Brooker, G.

Chmelík, R.

Cochran, G.

Faridian, A.

Fu, L.

Gao, B. Z.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

García, J.

García-Martínez, P.

Indebetouw, G.

Kapitán, J.

Katz, B.

Kelner, R.

Kim, E.-S.

Kim, M. K.

Kim, S.-G.

Lai, X.

Lee, B.

Li, H.

Li, Y.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

Lv, X.

Man, T.

Marathay, A. S.

Micó, V.

Naik, D. N.

Osten, W.

Pedrini, G.

Peng, X.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

Peters, P. J.

P. J. Peters, “Incoherent holograms with mercury light source,” Appl. Phys. Lett. 8(8), 209 (1966).
[Crossref]

Poon, T. C.

Psaltis, D.

Qin, W.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

Qu, X.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

Rosen, J.

R. Kelner, B. Katz, and J. Rosen, “Optical sectioning using a digital Fresnel incoherent-holography-based confocal imaging system,” Optica 1(2), 70–74 (2014).
[Crossref]

R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express 21(17), 20131–20144 (2013).
[Crossref] [PubMed]

B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
[Crossref] [PubMed]

N. Siegel, J. Rosen, and G. Brooker, “Reconstruction of objects above and below the objective focal plane with dimensional fidelity by FINCH fluorescence microscopy,” Opt. Express 20(18), 19822–19835 (2012).
[Crossref] [PubMed]

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref] [PubMed]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[PubMed]

N. T. Shaked, B. Katz, and J. Rosen, “Review of three-dimensional holographic imaging by multiple-viewpoint-projection based methods,” Appl. Opt. 48(34), H120–H136 (2009).
[Crossref] [PubMed]

J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
[Crossref] [PubMed]

Schilling, B. W.

Shaked, N. T.

Shinoda, K.

Siegel, N.

Sirat, G.

Storrie, B.

Suzuki, Y.

Wan, Y.

Wang, D.

Wang, V.

Worthington, H. R.

Wu, M. H.

Yang, X.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

Yao, H.

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

Young, M.

Yuan, J.

Zalevsky, Z.

Zeng, S.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

P. J. Peters, “Incoherent holograms with mercury light source,” Appl. Phys. Lett. 8(8), 209 (1966).
[Crossref]

J. Biomed. Opt. (1)

W. Qin, X. Yang, Y. Li, X. Peng, H. Yao, X. Qu, and B. Z. Gao, “Two-step phase-shifting fluorescence incoherent holographic microscopy,” J. Biomed. Opt. 19(6), 060503 (2014).
[Crossref] [PubMed]

J. Europ. Opt. Soc. Rap. Pub. (1)

O. Bouchal and Z. Bouchal, “Wide-field common-path incoherent correlation microscopy with a perfect overlapping of interfering beams,” J. Europ. Opt. Soc. Rap. Pub. 8, 13011 (2013).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Express (9)

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref] [PubMed]

P. Bouchal, J. Kapitán, R. Chmelík, and Z. Bouchal, “Point spread function and two-point resolution in Fresnel incoherent correlation holography,” Opt. Express 19(16), 15603–15620 (2011).
[Crossref] [PubMed]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[PubMed]

B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
[Crossref] [PubMed]

N. Siegel, J. Rosen, and G. Brooker, “Reconstruction of objects above and below the objective focal plane with dimensional fidelity by FINCH fluorescence microscopy,” Opt. Express 20(18), 19822–19835 (2012).
[Crossref] [PubMed]

D. N. Naik, G. Pedrini, and W. Osten, “Recording of incoherent-object hologram as complex spatial coherence function using Sagnac radial shearing interferometer and a Pockels cell,” Opt. Express 21(4), 3990–3995 (2013).
[Crossref] [PubMed]

R. Kelner, J. Rosen, and G. Brooker, “Enhanced resolution in Fourier incoherent single channel holography (FISCH) with reduced optical path difference,” Opt. Express 21(17), 20131–20144 (2013).
[Crossref] [PubMed]

Y. Wan, T. Man, and D. Wang, “Incoherent off-axis Fourier triangular color holography,” Opt. Express 22(7), 8565–8573 (2014).
[Crossref] [PubMed]

N. Siegel and G. Brooker, “Improved axial resolution of FINCH fluorescence microscopy when combined with spinning disk confocal microscopy,” Opt. Express 22(19), 22298–22307 (2014).
[Crossref] [PubMed]

Opt. Lett. (6)

Optica (1)

Other (3)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 4.4.5, P. 165, Chap. 8.6.2, P. 414, Chap. 8.8, P. 435.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company 2005), Chap. 6, P. 127, Chap. 5, P. 97, Chap. 9, P. 319.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1985) Chap. 15, P. 333.

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Figures (6)

Fig. 1
Fig. 1 Schematics of a single lens imaging system.
Fig. 2
Fig. 2 Schematics of FINCH: (a) Recording system. (b) Reconstruction system.
Fig. 3
Fig. 3 Lateral magnification ratio versus the axial object location for: (a) Four values of f1,2. (b) Four values of working distance zo with constant values of f2 = 140cm and f1 = 66.32cm.
Fig. 4
Fig. 4 Axial magnification ratio versus the axial object location for: (a) Four values of f1,2. (b) Four values of working distance zo with constant values of f2 = 140cm and f1 = 66.32cm.
Fig. 5
Fig. 5 Experimental results with RC1 (NBS 1963A) located at a fixed location of 30cm away from the objective lens and RC2 (1951 USAF) located at various zs locations of (a) 27cm to (g) 30cm. Left-hand column: two-lens imager with RC1 plane in focus; central column: FINCH reconstruction of RC1 plane of best focus; right-hand column: FINCH reconstruction of RC2 plane of best focus.
Fig. 6
Fig. 6 Experimental results with RC1 (NBS 1963A) located at a fixed location of 30cm away from the objective lens and RC2 (1951 USAF) located at various zs locations of (a) 30cm to (g) 33cm. Left-hand column: two-lens imager with RC1 plane in focus; central column: FINCH reconstruction of RC1 plane of best focus; right-hand column: FINCH reconstruction of RC2 plane of best focus.

Equations (42)

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n o y o tan θ o = n i y i tan θ i ,
n o y o R λ z o = n i y i R λ z i .
M T M W = 1.
y i W i = y o W o y o n o R λ z o .
| M T M W | > 1 ,
| M ¯ T | = z h z s ,
z r = ( z h z 1 ) ( z 2 z h ) z 2 z 1 .
z h = 2 f 1 f 2 f 1 + f 2 .
R H = | R ( z k z h ) z k | , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) .
tan θ r = R H z r = | R ( z k z h ) / z k ( z h z 1 ) ( z 2 z h ) / ( z 2 z 1 ) | = | R ( z 2 z 1 ) z k ( z 3 k z h ) | , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) .
| M ¯ W | = | W i W o | = | tan θ o tan θ r | = | R / z s R ( z 2 z 1 ) / z k ( z 3 k z h ) | = | z k ( z 3 k z h ) z s ( z 2 z 1 ) | , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) .
| M ¯ T M ¯ W | = | z h / z s z k ( z 3 k z h ) / z s ( z 2 z 1 ) | = | z h ( z 2 z 1 ) z k ( z 3 k z h ) | , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) .
| M ¯ T M ¯ W | = 2.
z k = z o z s f k z s f k + z o z s z o f k , k = 1 , 2
| M ¯ T M ¯ W | = 2 α z o ( f 2 f 1 ) 2 f 2 f 1 | 1 α | + α z o ( f 2 f 1 ) .
| α 1 | α = z o ( f 2 f 1 ) 2 f 2 f 1 .
M Δ = Δ i Δ o = n o z i 2 n i z o 2 = n i n o M W 2 = n i n o M T 2 ,
M A = | d z i d z o | = n i n o M T 2 .
M A M Δ = 1.
M ¯ A = | d z r d z s | = | z h 3 ( f 1 + f 2 ) z o 3 ( f 2 f 1 ) ( 1 α ) α 3 | .
M ¯ Δ = | Δ i Δ o | = ( z h [ α z o ( f 2 f 1 ) + 2 f 2 f 1 | α 1 | ] 2 α 2 z o 2 ( f 2 f 1 ) ) 2 .
| M ¯ A M ¯ Δ | = | 8 f 1 f 2 ( 1 α ) α z o ( f 2 f 1 ) [ α z o ( f 2 f 1 ) + 2 f 1 f 2 | 1 α | ] 2 | .
| M ¯ A M ¯ Δ | | 8 f 1 f 2 ( 1 α ) z o ( f 2 f 1 ) | .
H = | C 1 L [ z 1 x s z s ( z h z 1 ) ] Q [ 1 z h z 1 ] e i θ + C 2 L [ z 2 x s z s ( z 2 z h ) ] Q [ 1 z 2 z h ] | 2 ,
H F = L [ x s z 1 z s ( z h z 1 ) + z 2 x s z s ( z 2 z h ) ] Q [ 1 z h z 1 + 1 z 2 z h ] = L [ x s ( z 2 z 1 ) z h z s ( z h z 1 ) ( z 2 z h ) ] Q [ ( z 2 z 1 ) ( z h z 1 ) ( z 2 z h ) ] = L [ x s z h z s z r ] Q [ 1 z r ] .
| M T | = z h z s ,
z r = ( z h z 1 ) ( z 2 z h ) z 2 z 1 .
| M T | = z h z o ,
z r = ( z h f 1 ) ( f 2 z h ) f 2 f 1 .
z r = f 2 f 1 ( f 2 f 1 ) ( f 2 + f 1 ) 2 = z h ( f 2 f 1 ) 2 ( f 2 + f 1 ) .
H = | Q [ 1 z s ] Q [ 1 z o ] ( Q [ 1 f 1 ] e i θ + Q [ 1 f 2 ] ) Q [ 1 z h ] | 2 = | Q [ 1 z a + z h ] e i θ + Q [ 1 z b + z h ] | 2 ,
z a = f 1 z e f 1 z e , z b = f 2 z e f 2 z e , z e = z o z s z o z s = α z o 1 α .
H F = Q [ 1 z a + z h 1 z b + z h ] .
z r = ( z a + z h ) ( z b + z h ) z b z a .
z r = ( z h f 1 ) ( f 2 z h ) f 2 f 1 z h 2 f 1 f 2 z e 2 ( f 2 f 1 ) = z h ( f 2 f 1 ) 2 ( f 2 + f 1 ) z h 2 f 1 f 2 ( 1 α ) 2 α 2 z o 2 ( f 2 f 1 ) .
M ¯ A = d z r d z s = d z r d z e d z e d z s = 2 z h 2 f 1 f 2 ( f 2 f 1 ) z e 3 d z e d z s = 2 z h 2 f 1 f 2 ( f 2 f 1 ) z e 3 d d z s ( z o z s z o z s ) = 2 z h 2 f 1 f 2 ( f 2 f 1 ) z o 2 α 2 z e = z h 3 ( f 1 + f 2 ) z o 3 ( f 2 f 1 ) ( 1 α ) α 3 .
M ¯ Δ = | Δ i Δ o | = | R 2 z r 2 R H 2 z s 2 | .
R H = | R ( z k z h ) z k | , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) ,
z k = z o z s f k z s f k + z o z s z o f k , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) .
R H = R [ α z o f k z h ( α z o + ( α 1 ) f k ) ] α z o f k , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) .
M ¯ Δ = R 2 z r 2 R H 2 z s 2 = ( z r f k α z o f k z h [ α z o + | α 1 | f k ] ) 2 , k = 1 i f z h 2 z 1 z 2 / ( z 1 + z 2 ) k = 2 i f z h > 2 z 1 z 2 / ( z 1 + z 2 ) ,
M ¯ Δ = ( z h [ α z o ( f 2 f 1 ) + 2 f 2 f 1 | α 1 | ] 2 α 2 z o 2 ( f 2 f 1 ) ) 2 .

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