Abstract

We develop a generalized model in order to calculate the point spread functions in both the focal and the detection planes for the electric field strengths. In these calculations, based on the generalized Jones matrices, we introduce all of the interdependent parameters that could influence the spatial resolution of a confocal optical microscope. Our proposed model is more nearly complete, since we make no ap proximations of the scattered electric fields. These results can be successfully applied to standard confocal optical techniques to get a better understanding for more quantitative interpretations of the probe.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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  24. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009

2008

2007

2006

2004

H. F. Arnoldus and J. T. Foley, “Transmission of dipole radiation through interfaces and the phenomenon of anti-critical angles,” J. Opt. Soc. Am. A 21, 1109-1117 (2004).
[CrossRef]

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy,” Opt. Commun. 235, 1-10 (2004).
[CrossRef] [PubMed]

2003

2000

1999

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127-141 (1999).
[CrossRef]

1998

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

1997

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127-135(1997).
[CrossRef]

P. Török and P. Varga, “Electromagnetic diffraction of light focussed through a stratified medium,” Appl. Opt. 36, 2305-2312 (1997).
[CrossRef]

1996

R. H. Webb, “Confocal optical microscopy,” Rep. Prog. Phys. 59, 427-471 (1996).
[CrossRef]

1995

1994

1987

Ammar, M.

Arnoldus, H. F.

Axelrod, D.

Booker, G. R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
[CrossRef]

Bourson, P.

R. Hammoum, M. D. Fontana, P. Bourson, and V. Ya. Shur, “Characterization of PPLN-microstructures by means of Raman spectroscopy,” Appl. Phys. A 91, 65-67 (2008).
[CrossRef]

Braat, J. J. M.

A. S. Van De Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

Chen, J.

Chung, E.

Foley, J. T.

Fontana, M. D.

R. Hammoum, M. D. Fontana, P. Bourson, and V. Ya. Shur, “Characterization of PPLN-microstructures by means of Raman spectroscopy,” Appl. Phys. A 91, 65-67 (2008).
[CrossRef]

Furukawa, H.

Guo, H.

Guo, S.

Haeberlé, O.

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy,” Opt. Commun. 235, 1-10 (2004).
[CrossRef] [PubMed]

O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “Point spread function of optical microscopes imaging through stratified media,” Opt. Express 11, 2964-2969 (2003).
[CrossRef] [PubMed]

O. Haeberlé, “Focussing of light through a stratified medium: a practical approach for computing microscope point spread function. Part I: Conventional microscopy,” Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Hammoum, R.

R. Hammoum, M. D. Fontana, P. Bourson, and V. Ya. Shur, “Characterization of PPLN-microstructures by means of Raman spectroscopy,” Appl. Phys. A 91, 65-67 (2008).
[CrossRef]

Hellen, E. H.

Higdon, P. D.

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127-141 (1999).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

Katagiri, T.

Kim, Y.-H.

Komachi, Y.

Kriezis, E. E.

Laczik, Z.

Liang, Z.

Maraval, P.

Masters, B. R.

Munro, P. R. T.

Pereira, S. F.

A. S. Van De Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

Sato, H.

Sheppard, C. J. R.

Shur, V. Ya.

R. Hammoum, M. D. Fontana, P. Bourson, and V. Ya. Shur, “Characterization of PPLN-microstructures by means of Raman spectroscopy,” Appl. Phys. A 91, 65-67 (2008).
[CrossRef]

So, P. T. C.

Sourisseau, C.

Tang, W. T.

Tashiro, H.

Tenjimbayashi, K.

Török, P.

P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express 16, 507-523 (2008).
[CrossRef] [PubMed]

O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “Point spread function of optical microscopes imaging through stratified media,” Opt. Express 11, 2964-2969 (2003).
[CrossRef] [PubMed]

P. Török, “Propagation of electromagnetic dipole waves through dielectric interfaces,” Opt. Lett. 25, 1463-1465(2000).
[CrossRef]

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127-141 (1999).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127-135(1997).
[CrossRef]

P. Török and P. Varga, “Electromagnetic diffraction of light focussed through a stratified medium,” Appl. Opt. 36, 2305-2312 (1997).
[CrossRef]

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325-332 (1995).

Van De Nes, A. S.

A. S. Van De Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

Varga, P.

Webb, R. H.

R. H. Webb, “Confocal optical microscopy,” Rep. Prog. Phys. 59, 427-471 (1996).
[CrossRef]

Wilson, T.

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127-141 (1999).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127-135(1997).
[CrossRef]

T. Wilson and B. R. Masters, “Confocal microscopy,” Appl. Opt. 33, 565-566 (1994).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
[CrossRef]

Zhuang, S.

Appl. Opt.

Appl. Phys. A

R. Hammoum, M. D. Fontana, P. Bourson, and V. Ya. Shur, “Characterization of PPLN-microstructures by means of Raman spectroscopy,” Appl. Phys. A 91, 65-67 (2008).
[CrossRef]

Appl. Spectrosc.

J. Microsc.

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127-141 (1999).
[CrossRef]

J. Mod. Opt.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

O. Haeberlé, “Focussing of light through a stratified medium: a practical approach for computing microscope point spread function. Part I: Conventional microscopy,” Opt. Commun. 216, 55-63 (2003).
[CrossRef]

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part II: confocal and multiphoton microscopy,” Opt. Commun. 235, 1-10 (2004).
[CrossRef] [PubMed]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127-135(1997).
[CrossRef]

Opt. Express

Opt. Lett.

Rep. Prog. Phys.

A. S. Van De Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323-2363 (2006).
[CrossRef]

R. H. Webb, “Confocal optical microscopy,” Rep. Prog. Phys. 59, 427-471 (1996).
[CrossRef]

Other

J.B.Pawley, ed., Handbook of Biological Confocal Microscopy (Springer, 2006).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Simplified schematic of the confocal microscope. (a) Schematic of the imaging geometry of the pinhole detector. (b) Labeling scheme for the scattered waves for a dipole embedded in a sample. (c) Labeling the s- and p unit vectors relative to the Cartesian coordinate system.

Fig. 2
Fig. 2

Illumination PSF. (a) Calculated time-averaged electric energy density distribution. (b) Illumination PSF for four discrete displacements d of the interface for two wavelengths. (c) Resolution versus probe depth illumination. Here we used an objective lens with NA = 0.9 and a 100 × magnification system. The black lines and the gray lines refer to the 633 and the 514.5 nm laser wavelengths, respectively. The fluctuations present in (c) are due to the relatively small discrimination steps for our calculus.

Fig. 3
Fig. 3

Detection point spread function (PSF). (a)–(c) Detection PSF components, | E x | 2 , | E y | 2 , and | E z | 2 , for electric dipoles oriented perpendicular to the interface. (d)–(f) Detection PSF, | E x | 2 , | E y | 2 , and | E z | 2 , components for electric dipoles embedded in the x y plane, and oriented along the x axis. The dipoles are assumed to exist only in the whole focal plan for a focus in the air, d = 0 .

Fig. 4
Fig. 4

Effective numerical aperture of a dry lens assuming a focal length f = 250 μm . This is a function of the probing depth into the sample. Comparison is made by assuming other values for the refractive index n 2 .

Fig. 5
Fig. 5

Detection intensities for dipoles oriented perpendicular to the interface assuming parallel and crossed polarizers for an objective lens of (a)  NA = 0.5 and magnification 50 × , (b)  NA = 0.9 and magnification 100 × .

Equations (35)

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E 02 ( x , y , z ) = A ( ϕ 1 ) 2 [ A + ( τ p cos ϕ 2 + τ s ) + ( τ p cos ϕ 2 τ s ) ( A + cos 2 θ + i B sin 2 θ ) i B ( τ p cos ϕ 2 + τ s ) + ( τ p cos ϕ 2 τ s ) ( i B cos 2 θ + A + sin 2 θ ) 2 τ p sin ϕ 2 ( A + cos θ + i B sin θ ) ] ,
l 0 ( ϕ 1 , NA ) = ( F n NA ) exp ( 2 sin 2 ϕ i / NA 2 ) = l 01 ( NA ) × D ( ϕ 1 , NA ) .
E 2 x = i K [ I 0 A + + I 2 ( A + cos 2 θ Q + i B sin 2 θ Q ) ] ,
E 2 y = K [ I 0 B I 2 ( B cos 2 θ Q + i A + sin 2 θ Q ) ] ,
E 2 z = 2 K I 1 ( A + cos θ Q + i B sin θ Q ) ,
I 0 = 0 α D ( ϕ 1 ) ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp ( i k 0 Ψ ( ϕ 1 , ϕ 2 , d ) ) ( τ s + τ p cos ϕ 2 ) × J 0 ( k 1 r Q sin ϕ Q sin ϕ 1 ) exp ( i k 2 r Q cos ϕ Q cos ϕ 2 ) d ϕ 1 ,
I 1 = 0 α D ( ϕ 1 ) ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp ( i k 0 Ψ ( ϕ 1 , ϕ 2 , d ) ) τ p sin ϕ 2 × J 1 ( k 1 r Q sin ϕ Q sin ϕ 1 ) exp ( i k 2 r Q cos ϕ Q cos ϕ 2 ) d ϕ 1 ,
I 2 = 0 α D ( ϕ 1 ) ( cos ϕ 1 ) 1 / 2 ( sin ϕ 1 ) exp ( i k 0 Ψ ( ϕ 1 , ϕ 2 , d ) ) × ( τ s τ p cos ϕ 2 ) × J 2 ( k 1 r Q sin ϕ Q sin ϕ 1 ) exp ( i k 2 r Q cos ϕ Q cos ϕ 2 ) d ϕ 1 ,
K = k 1 2 f l 01 2 k 2 = π n 1 2 f l 01 λ n 2 ,
E s ( r ) = i 2 π n 2 2 + d 2 k ρ 1 k 0 ν 2 [ k 0 2 n 1 2 p ( p · k ) k ] exp [ i k · ( r H · e ^ z ) ] for     z H ,
e ^ s = 1 k ρ k ρ e ^ z , e ^ p , t = 1 n 1 [ ν 1 ( k ρ k ρ ) + η · e ^ z ] .
E 01 = [ E 01 , ρ E 01 , s E 01 , z ] = [ ( ν 1 / n 1 ) τ p p · e ^ p τ s p · e ^ s ( η / n 1 ) τ p p · e ^ p ] = [ ( ν 1 / n 1 n 2 ) τ p ( ν 2 p · e ^ ρ + η p · e ^ z ) τ s p · e ^ s ( η / / n 1 n 2 ) τ p ( ν 2 p · e ^ ρ + η p e ^ z ) ] ,
i Φ = i d ( k 2 s 2 z k 1 s 1 z ) = i d k 0 ( ν 2 ν 1 ) .
E 00 ( x , y , z ) ( γ ) = C ( ϕ 1 , ϕ 0 ) · N · R b · L 0 b · L 1 b · E 01 ( ρ , s , z ) ,
C ( ϕ 1 , ϕ 0 ) = ( cos ϕ 0 cos ϕ 1 ) 1 / 2 .
L 1 b = [ cos ϕ 1 0 sin ϕ 1 0 1 0 sin ϕ 1 0 cos ϕ 1 ] , L 0 b = [ cos ϕ 0 0 sin ϕ 0 0 1 0 sin ϕ 0 0 cos ϕ 0 ] .
R b = [ cos θ sin θ 0 sin θ cos θ 0 0 0 1 ] , N = [ cos 2 γ sin γ cos γ 0 sin γ cos γ sin 2 γ 0 0 0 1 ] ,
E 0 ( x , y , z ) = i k 0 2 π 0 σ 0 2 π sin ϕ 0 · E 00 ( x , y , z ) exp [ i k 0 ( r d sin ϕ 0 sin ϕ d · cos ( θ θ d ) z d cos ϕ 0 ) ] exp ( i Φ ) d θ d ϕ 0 ,
E 0 x = i k 0 2 [ p x K 0 I + 2 i p z K 1 I cos θ d + K 2 I ( p x cos 2 θ d + p y sin 2 θ d ) ] ,
E 0 y = i k 0 2 [ p y K 0 I + 2 i p z K 1 I sin θ d + K 2 I ( p x sin 2 θ d p y cos 2 θ d ) ] ,
E 0 z = k 0 [ K 1 I I · ( p x cos θ d + p y sin θ d ) i p z K 0 I I ]
K 0 I = 0 σ ( cos ϕ 0 cos ϕ 1 ) 1 / 2 ( τ s + τ p cos ϕ 2 cos ϕ 0 ) sin ϕ 0 · J 0 ( k 0 ρ     sin ϕ 0 ) exp [ i ( k 0 z d cos ϕ 0 Φ ) ] d ϕ 0 ,
K 1 I = 0 σ ( cos ϕ 0 cos ϕ 1 ) 1 / 2 τ p sin ϕ 2 cos ϕ 0 sin ϕ 0 · J 1 ( k 0 ρ sin ϕ 0 ) exp [ i ( k 0 z d cos ϕ 0 Φ ) ] d ϕ 0 ,
K 2 I = 0 σ ( cos ϕ 0 cos ϕ 1 ) 1 / 2 ( τ s τ p cos ϕ 2 cos ϕ 0 ) sin ϕ 0 · J 2 ( k 0 ρ sin ϕ 0 ) exp [ i ( k 0 z d cos ϕ 0 Φ ) ] d ϕ 0 ,
K 1 I I = 0 σ ( cos ϕ 0 cos ϕ 1 ) 1 / 2 τ p 2 cos ϕ 2 sin 2 ϕ 1 sin 2 ϕ 0 · J 1 ( k 0 ρ sin ϕ 0 ) exp [ i ( k 0 z d cos ϕ 0 Φ ) ] d ϕ 0 ,
K 0 I I = 0 σ ( cos ϕ 0 cos ϕ 1 ) 1 / 2 τ p 2 sin ϕ 2 sin 2 ϕ 1 sin 2 ϕ 0 · J 0 ( k 0 ρ sin ϕ 0 ) exp [ i ( k 0 z d cos ϕ 0 Φ ) ] d ϕ 0 .
e 0 x = cos 2 γ E 0 x + sin γ cos γ E 0 y ,
e 0 y = sin γ cos γ E 0 x + sin 2 γ E 0 y ,
e 0 z = E 0 z .
k 1 sin ϕ 1 k 0 sin ϕ 0 = sin ϕ 1 sin ϕ 0 = M , r d = r Q · M ,
z d = z Q M · M ( z Q ) ,
M ( z Q ) = M 1 + ( z Q / t ) × M × ( 1 + M ) ,
I ( | E 0 x | 2 + | E 0 y | 2 + | E 0 z | 2 ) / P T ,
I d = D | E 0 | 2 S D d D = 0 R 0 2 π | E 0 | 2 f ( ρ d ) ρ d d ρ d d θ p ,
NA eff = r eff r eff 2 + f 2 NA eff 2 = ( f · NA ) 2 ( f · NA ) 2 + [ d n 2 2 NA 2 + ( f d ) 1 NA 2 ] 2 ,

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