Abstract

Oblique plane imaging, using remote focusing with a tilted mirror, enables direct two-dimensional (2D) imaging of any inclined plane of interest in three-dimensional (3D) specimens. It can image real-time dynamics of a living sample that changes rapidly or evolves its structure along arbitrary orientations. It also allows direct observations of any tilted target plane in an object of which orientational information is inaccessible during sample preparation. In this work, we study the optical resolution of this innovative wide-field imaging method. Using the vectorial diffraction theory, we formulate the vectorial point spread function (PSF) of direct oblique plane imaging. The anisotropic lateral resolving power caused by light clipping from the tilted mirror is theoretically analyzed for all oblique angles. We show that the 2D PSF in oblique plane imaging is conceptually different from the inclined 2D slice of the 3D PSF in conventional lateral imaging. Vectorial optical transfer function (OTF) of oblique plane imaging is also calculated by the fast Fourier transform (FFT) method to study effects of oblique angles on frequency responses.

© 2014 Optical Society of America

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References

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    [CrossRef]

2012

F. Cutrale, E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. 75(11), 1461–1466 (2012).
[CrossRef] [PubMed]

2011

2008

E. J. Botcherby, R. Juskaitis, M. J. Booth, T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008).
[CrossRef]

C. Dunsby, “Optically sectioned imaging by oblique plane microscopy,” Opt. Express 16(25), 20306–20316 (2008).
[CrossRef] [PubMed]

2007

2005

2002

M. R. Arnison, C. J. R. Sheppard, “A 3D vectorial optical transfer function suitable for arbitrary pupil functions,” Opt. Commun. 211(1-6), 53–63 (2002).
[CrossRef]

1997

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

1994

1991

1987

1981

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

1977

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

1967

1959

B. Richards, E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

1939

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939).
[CrossRef]

Anselmi, F.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011).
[CrossRef] [PubMed]

Arnison, M. R.

M. R. Arnison, C. J. R. Sheppard, “A 3D vectorial optical transfer function suitable for arbitrary pupil functions,” Opt. Commun. 211(1-6), 53–63 (2002).
[CrossRef]

Bègue, A.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011).
[CrossRef] [PubMed]

Booth, M. J.

Botcherby, E. J.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939).
[CrossRef]

Cutrale, F.

F. Cutrale, E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. 75(11), 1461–1466 (2012).
[CrossRef] [PubMed]

Dunsby, C.

Emiliani, V.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011).
[CrossRef] [PubMed]

Frieden, B.

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Göbel, W.

W. Göbel, F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98(6), 3770–3779 (2007).
[CrossRef] [PubMed]

Gratton, E.

F. Cutrale, E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. 75(11), 1461–1466 (2012).
[CrossRef] [PubMed]

Gu, M.

Helmchen, F.

W. Göbel, F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98(6), 3770–3779 (2007).
[CrossRef] [PubMed]

Juskaitis, R.

E. J. Botcherby, R. Juskaitis, M. J. Booth, T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008).
[CrossRef]

E. J. Botcherby, R. Juskaitis, M. J. Booth, T. Wilson, “Aberration-free optical refocusing in high numerical aperture microscopy,” Opt. Lett. 32(14), 2007–2009 (2007).
[CrossRef] [PubMed]

Juškaitis, R.

Kawata, S.

Kawata, Y.

Kumar, S.

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

Li, Y.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

Li, Y. J.

Lyon, A. R.

MacLeod, K. T.

Matthews, H. J.

Ogden, D.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011).
[CrossRef] [PubMed]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Sheppard, C. J. R.

M. R. Arnison, C. J. R. Sheppard, “A 3D vectorial optical transfer function suitable for arbitrary pupil functions,” Opt. Commun. 211(1-6), 53–63 (2002).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11(2), 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4(8), 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Sikkel, M. B.

Smith, C. W.

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939).
[CrossRef]

Ventalon, C.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011).
[CrossRef] [PubMed]

Visser, T. D.

Wiersma, S. H.

Wilding, D.

Wilson, T.

Wolf, E.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

IEEE J. Microwave Opt. Acoust.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

J. Neurophysiol.

W. Göbel, F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98(6), 3770–3779 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microsc. Res. Tech.

F. Cutrale, E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. 75(11), 1461–1466 (2012).
[CrossRef] [PubMed]

Opt. Commun.

E. J. Botcherby, R. Juskaitis, M. J. Booth, T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281(4), 880–887 (2008).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

M. R. Arnison, C. J. R. Sheppard, “A 3D vectorial optical transfer function suitable for arbitrary pupil functions,” Opt. Commun. 211(1-6), 53–63 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttg.)

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

Phys. Rev.

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. 108(49), 19504–19509 (2011).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci.

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986).

M. Gu, Advanced Optical Imaging Theory (Springer, 2000).

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Figures (9)

Fig. 1
Fig. 1

Conceptual diagram of oblique plane imaging. OBJ: objective lens (EP: exit pupil; BFP: back focal plane); BS: beam splitter; L: lens; M: mirror. The beam path for an on-axis point object to the detector is shown in green, while the light clipping at the OBJ2 induced by the tilted mirror is illustrated in pink. Coordinates at (a) object space: an oblique plane ( x α y α ) inclined by α with respect to the focal plane (xy) of the OBJ1, (b) remote space: the x 1 y 1 image plane conjugate with the x α y α plane is rotated back to the x 1 ' y 1 ' plane (the focal plane of the OBJ2) by the α/2-tilted mirror, and (c) image space: the lateral detection plane ( x 2 y 2 ) is conjugate with the x α y α object plane. Rays (green, light blue) from two points on the oblique x α y α plane are focused on the x 2 y 2 plane.

Fig. 2
Fig. 2

Diffraction geometry. O: geometrical focus; X: an observation point; Y: a point on the exit pupil surface Σ where the incident field is refracted to E S . The distance YX ¯ is R| r r ' | .

Fig. 3
Fig. 3

(a) 2D and (b) 3D effective pupil geometry in the normalized object space. In (a), the circular pupil area of the objective lens and its mirror image by the α/2-tilted virtual mirror (red dashed line) are illustrated. Their overlapped area forms the effective pupil. (c) Oblique angle dependence of the effective pupil shapes (NA = 1.4, n = 1.52).

Fig. 4
Fig. 4

2D vectorial intensity PSF of the oblique plane imaging at different oblique angles (α) and NAs, for λ0 = 519 nm (unpolarized) and n = 1.52 (oil immersion). The axis unit is μm. The PSF elongates mainly along the vertical direction (ya) where the significant light clipping occurs in oblique plane imaging. Higher NA makes such an anisotropic resolving power less sensitive to the oblique angle.

Fig. 5
Fig. 5

FWHM of the vectorial PSFs at different NA values along the xα- and yα-axis over the oblique angle. For comparison, the FWHM from the inclined 3D vectorial PSF of the circular aperture system (the inset in the middle) is also plotted.

Fig. 6
Fig. 6

(a) Scalar Debye intensity PSF used to calculate an OTF by FFT (NA = 1.4, n = 1.52, λ0 = 519nm, α = 0°). The green, blue and red rectangles contain 5, 15 and 25 sidelobes respectively. (b) A comparison between FFT-based and analytical scalar Debye OTFs in the normalized spatial frequency by n/λ0. The PSF with enough sidelobes is necessary for an accurate OTF calculation. The inset shows details near the low frequency regimes.

Fig. 7
Fig. 7

2D vectorial OTF of the oblique plane imaging at different oblique angles (α) and NAs (λ0 = 519 nm, n = 1.52). The MTF cutoff contours are drawn in red. The lateral coordinate is normalized by n/λ0.

Fig. 8
Fig. 8

Vectorial OTF cross-sections along the horizontal (mx) and vertical (my) directions for α = 0, 60, and 90°. The scalar Debye OTF (both analytical and numerical) for α = 0° is also plotted for comparison. The spatial frequency is normalized by n/λ0 (λ0 = 519 nm, n = 1.52).

Fig. 9
Fig. 9

The variation in the cutoff frequency of the vectorial OTF along the horizontal (mx) and vertical (my) directions over the oblique angle.

Equations (7)

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E ( r )= 1 4π Σ [ jω( N ^ × B S )G+( N ^ × E S )× ' G+( N ^ E S ) ' G ]ds ,
E S = cos 1 2 θ( ( N ^ × k inc ) E inc | ( N ^ × k inc ) E inc | N ^ × k inc | N ^ × k inc | + [( N ^ × k inc )× k inc ] E inc | [( N ^ × k inc )× k inc ] E inc | ( N ^ × k inc )× N ^ | ( N ^ × k inc )× N ^ | )
P Σ 1 (θ,ϕ)={ 1, θ[0, θ C ], ϕ[0,2π], y C <0 0, otherwise
P Σ 2 (θ,ϕ)={ 1, θ[ θ C , θ max ], ϕ[ ϕ 1 (θ), ϕ 2 (θ)] 0, otherwise
E ϕ 0 ( x,y,z, )=( E x, ϕ 0 E y, ϕ 0 E z, ϕ 0 )= Σ ( sin(ϕ ϕ 0 )sinϕ+cos(ϕ ϕ 0 )cosθcosϕ sin(ϕ ϕ 0 )cosθ+cos(ϕ ϕ 0 )cosθsinϕ cos(ϕ ϕ 0 )sinθ ) ×( P Σ 1 (θ,ϕ)+ P Σ 2 (θ,ϕ) ) cos 1 2 θ × e ik( xsinθcosϕ+ysinθsinϕzcosθ ) sinθdϕdθ
I ϕ 0 ( x α , y α )=| ( I x, ϕ 0 I y, ϕ 0 I z, ϕ 0 ) |= | Σ ( sin(ϕ ϕ 0 )sinϕ+cos(ϕ ϕ 0 )cosθcosϕ sin(ϕ ϕ 0 )cosϕ+cos(ϕ ϕ 0 )cosθsinϕ cos(ϕ ϕ 0 )sinθ ) ×( P Σ 1 (θ,ϕ)+ P Σ 2 (θ,ϕ) ) cos 1 2 θ × e ik[ x α sinθcosϕ+ y α (cosαsinθsinϕsinαcosθ) ] sinθdϕdθ | 2
I( x α , y α )= 1 2π 0 2π E ϕ 0 * E ϕ 0 d ϕ 0 = 1 2 ( | I A | 2 + | I B | 2 ), I j = Σ 1 + Σ 2 E j e ik[ x α sinθcosϕ+ y α (cosαsinθsinϕsinαcosθ) ] sinθdϕdθ , E A = cos 1 2 θ( sin 2 ϕ+cosθ cos 2 ϕ sinϕcosϕ+cosθsinϕcosϕ cosϕsinθ ), E B = cos 1 2 θ( sinϕcosϕ+cosθsinϕcosϕ cos 2 ϕ+cosθ sin 2 ϕ sinϕsinθ )

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