## Vectorial point spread function and optical transfer function in oblique plane imaging |

Optics Express, Vol. 22, Issue 9, pp. 11140-11151 (2014)

http://dx.doi.org/10.1364/OE.22.011140

Acrobat PDF (1291 KB)

### 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

## 1. Introduction

1. F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and 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]

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

2. C. W. Smith, E. J. Botcherby, M. J. Booth, R. Juškaitis, and T. Wilson, “Agitation-free multiphoton microscopy of oblique planes,” Opt. Lett. **36**(5), 663–665 (2011). [CrossRef] [PubMed]

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

2. C. W. Smith, E. J. Botcherby, M. J. Booth, R. Juškaitis, and T. Wilson, “Agitation-free multiphoton microscopy of oblique planes,” Opt. Lett. **36**(5), 663–665 (2011). [CrossRef] [PubMed]

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

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

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

*et al.*[1

1. F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and 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]

*et al.*[2

2. C. W. Smith, E. J. Botcherby, M. J. Booth, R. Juškaitis, and T. Wilson, “Agitation-free multiphoton microscopy of oblique planes,” Opt. Lett. **36**(5), 663–665 (2011). [CrossRef] [PubMed]

3. C. W. Smith, E. J. Botcherby, and T. Wilson, “Resolution of oblique-plane images in sectioning microscopy,” Opt. Express **19**(3), 2662–2669 (2011). [CrossRef] [PubMed]

## 2. Schematic of the oblique plane imaging

1. F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and 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]

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

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

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

*α*/2-tilted mirror in the remote space. The pink beam in Fig. 1 shows how light is clipped at the OBJ2 induced by the tilted mirror. This light loss leads to a partial use of the OBJ2’s exit pupil (the blue arc) and this rotationally asymmetric pupil yields an anisotropic resolving power. The NA or half-cone angle of the OBJ2 should be chosen greater than the mirror tilt angle,

*α*/2, to prevent a complete loss of light from detection. For example, an axial plane imaging (

*α*= 90°) requires NA greater than 0.71 in air medium. In general, the use of as high NA as possible is desirable to minimize the clipping of the signal light.

## 3. Theory and formulation

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

12. B. Richards and 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]

### 3.1. Vectorial diffraction theory

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

*ω*is the temporal frequency of the field,

*G*is the Green function of a diverging spherical wave

*R*shown in Fig. 2,

*k*is the wave number in medium, and

*k*

*R*, Eq. (1) is reduced to

16. T. D. Visser and S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A **8**(9), 1404–1410 (1991). [CrossRef]

*O*compared with the distance

*R*in aplanatic systems, we can use the Debye approximation [17

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

*λ*is the wavelength in an immersed medium of which refractive index is

*n*. The Debye approximation makes intensity distribution axially symmetric along the optical axis (

*z*) and is valid if the Fresnel number

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

20. Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory,” J. Opt. Soc. Am. A **22**(1), 77–83 (2005). [CrossRef] [PubMed]

*f*is the focal length and

*λ*

_{0}is the vacuum wavelength. Most of the commercial objective lenses of any NA suffices

21. 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]

### 3.2. Pupil function in oblique plane imaging

*θ*: polar angle,

*ϕ*: azimuthal angle). As illustrated in Fig. 3(a), an overlap between the original circular pupil area of the objective lens and its reflected pupil area by the

*α*/2-tilted mirror forms the effective pupil. We divide this circularly asymmetric pupil in oblique plane imaging into Σ

_{1}(rotationally symmetric part) and Σ

_{2}(the rest area) for mathematical convenience of the integral calculation. Appendix A includes more details. Σ

_{1}disappears at high

*α*regime where

_{1}and Σ

_{2}can be described as where

*θ*,

_{C}*θ*

_{max},

*ϕ*

_{1}(

*θ*), and

*ϕ*

_{2}(

*θ*) are defined in Appendix A. The bounds of

*θ*,

*ϕ*are expressed as a function of

*α*, NA and

*n*. Figure 3(c) shows several pupil functions that change with oblique angles.

### 3.3. Point spread function

*ϕ*

_{0}with respect to the

*x*-axis, i.e.,

*α*is

12. B. Richards and 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]

### 3.4. Optical transfer function

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

23. B. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. **57**(1), 56–65 (1967). [CrossRef]

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

## 4. Numerical simulation results

*n*= 1.52) objective lens. We considered three different cases of NA: 1.30, 1.40 and 1.49. The L1 and the L2 were considered identical. The light source was assumed to be a self-luminous, unpolarized, quasi-monochromatic (

*λ*

_{0}= 519 nm), and isotropic point source. The Fresnel numbers of these high NA objective lenses are well above 1000, validating the Debye approximation.

### 4.1. Point spread function

*α*= 0° is equal to the conventional PSF of the circular aperture system of which resolution is isotropic in the lateral plane. The main lobe of the PSF stretches to the

*y*direction more apparently at higher oblique angle due to the reduced pupil area, which results in an anisotropic lateral resolving power. This numerical simulation shows that there is also a slight PSF stretch along the

_{α}*x*direction, which can be expected from the minor pupil loss along that direction as shown in Fig. 3(c). As a quantitative measure of these degradations, the FWHM was calculated in Fig. 5. The optical resolution decreases when the oblique angle increases from

_{α}*α*= 0° (conventional lateral imaging) to

*α*= 90° (axial plane imaging) due to the reduced effective NA. The FWHM ratio of such two extreme angles at the NA of 1.30 (1.40, 1.49) is 1.33 (1.16, 1.06) and 4.39 (2.92, 2.09) along the

*x*- and

_{α}*y*-axis, respectively.

_{α}*x*-axis to calculate a FWHM for each inclination α. While this rotation keeps the FWHM along the

*x*-axis unchanged (green dashed curve), it gives certain FWHM deterioration along the

*y*-axis (green line curve) originated from the well-known “ellipsoidal” PSF. Thus the rate of the increase in FWHM

_{y}calculated from the inclined PSF slows down near 90° and the FWHM converges to the FWHM

_{z}of the 3D PSF. This nature is quite different from the sharp increase near 90° in our simulation results together with the FWHM along the

*y*-axis at

_{α}*α*= 90° not limited to the FWHM

_{z}of the conventional PSF. It is clear that the tilt of the conventional PSF fails to predict both the minor

*x*-resolution loss and the

*y*-resolution trend over the oblique angles, both of which are attributed to the pupil area loss from the light clipping. This result tells that the inclined slice of the conventional 3D PSF is different from the light-clipped 2D PSF that we calculated.

### 4.2. Optical transfer function

*x*and

*y*directions led to a good agreement between numerical (FFT-based) and analytical (2D projected) OTFs: black vs. red curve in Fig. 6(b). The relative error in modulation transfer function (MTF) value was smaller than 0.013 over the entire spatial frequency. A numerical OTF from the PSF with 60-sidelobes almost perfectly overlapped with the analytical one, but this required about six times more computational time.

*x*and

_{α}*y*directions, followed by the 2D FFT operation. Figure 7 shows the 2D vectorial OTF of direct oblique plane imaging on the spatial frequency coordinates

_{α}*m*-

_{x}*m*(normalized by

_{y}*n*/

*λ*

_{0}), which correspond to the

*x*and

_{α}*y*directions in real space. As the oblique angle increases, the bandwidth (or cutoff frequency) along the

_{α}*m*-direction shrinks much faster than that along the

_{y}*m*-direction, which is qualitatively self-explanatory from the anisotropic PSFs in Fig. 4. Cross-sections of those OTFs were examined in Fig. 8. For

_{x}*α*= 0°, compared with the scalar Debye OTFs, the vectorial OTFs have lower modulation over the spatial frequency range. This MTF degradation is caused mainly by the depolarized light component along the optical axis that induces PSF broadening, which is neglected in the scalar theory. On the other hand, the cutoff frequency in the scalar Debye theory is

*2NA*/

*n*, i.e., 1.71 (1.84, 1.96) for the NA of 1.30 (1.40, 1.49), corresponding to 5.0 (5.4, 5.7) cycles/μm in physical coordinate. We determined the numerical cutoff frequency of the vectorial OTF at a threshold MTF of 0.01% to ignore the minor MTF oscillations (numerical artifacts) occurring near and above the cutoff frequency. The calculated vectorial cutoff frequencies for

*α*= 0° were consistent with the analytical scalar Debye cutoffs within 1% error.

*α*= 60, 90° in Fig. 8, we clearly see a downward trend in both MTF value and cutoff frequency with the oblique angle. This change is plotted in Fig. 9. The cutoff frequency along the

*m*axis drops by 21% (10%, 5%) for the NA of 1.30 (1.40, 1.49) as the oblique angles increases from 0° to 90°. Similarly, the

_{x}*m*cutoff frequency reduces up to 80% (71%, 60%). The cutoff frequency declines quantitatively less for lenses with higher NA.

_{y}## 5. Conclusion and discussion

## Appendix A. Derivation of the point *C* coordinate and the bounds of the pupil function

*x*

^{2}+

*y*

^{2}+

*z*

^{2}= 1 with

*x*= 0) gives the coordinate values of the point

*C*as

## Appendix B. The modification of Eq. (9) for faster numerical calculation

_{1}, the double integral can be reduced to a single integral as shown in [12

12. B. Richards and 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]

_{2}, the integration area can be reduced by half due to the even symmetry of the bounds of

*ϕ*. These considerations lead to

## Acknowledgments

## References and links

1. | F. Anselmi, C. Ventalon, A. Bègue, D. Ogden, and V. Emiliani, “Three-dimensional imaging and photostimulation by remote-focusing and holographic light patterning,” Proc. Natl. Acad. Sci. U.S.A. |

2. | C. W. Smith, E. J. Botcherby, M. J. Booth, R. Juškaitis, and T. Wilson, “Agitation-free multiphoton microscopy of oblique planes,” Opt. Lett. |

3. | C. W. Smith, E. J. Botcherby, and T. Wilson, “Resolution of oblique-plane images in sectioning microscopy,” Opt. Express |

4. | W. Göbel and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. |

5. | C. Dunsby, “Optically sectioned imaging by oblique plane microscopy,” Opt. Express |

6. | S. Kumar, D. Wilding, M. B. Sikkel, A. R. Lyon, K. T. MacLeod, and C. Dunsby, “High-speed 2D and 3D fluorescence microscopy of cardiac myocytes,” Opt. Express |

7. | F. Cutrale and E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. |

8. | E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “Aberration-free optical refocusing in high numerical aperture microscopy,” Opt. Lett. |

9. | E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. |

10. | C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A |

11. | M. Gu, |

12. | B. Richards and 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. |

13. | J. D. Jackson, |

14. | J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. |

15. | J. J. Stamnes, |

16. | T. D. Visser and S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A |

17. | C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEEE J. Microwave Opt. Acoust. |

18. | E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. |

19. | Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A |

20. | Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory,” J. Opt. Soc. Am. A |

21. | E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

22. | C. J. R. Sheppard, M. Gu, Y. Kawata, and S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A |

23. | B. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. |

24. | C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) |

25. | M. R. Arnison and C. J. R. Sheppard, “A 3D vectorial optical transfer function suitable for arbitrary pupil functions,” Opt. Commun. |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.4850) Imaging systems : Optical transfer functions

(180.0180) Microscopy : Microscopy

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: February 25, 2014

Revised Manuscript: April 8, 2014

Manuscript Accepted: April 8, 2014

Published: May 1, 2014

**Virtual Issues**

Vol. 9, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Jeongmin Kim, Tongcang Li, Yuan Wang, and Xiang Zhang, "Vectorial point spread function and optical transfer function in oblique plane imaging," Opt. Express **22**, 11140-11151 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-11140

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### References

- 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]
- C. W. Smith, E. J. Botcherby, M. J. Booth, R. Juškaitis, T. Wilson, “Agitation-free multiphoton microscopy of oblique planes,” Opt. Lett. 36(5), 663–665 (2011). [CrossRef] [PubMed]
- C. W. Smith, E. J. Botcherby, T. Wilson, “Resolution of oblique-plane images in sectioning microscopy,” Opt. Express 19(3), 2662–2669 (2011). [CrossRef] [PubMed]
- W. Göbel, F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98(6), 3770–3779 (2007). [CrossRef] [PubMed]
- C. Dunsby, “Optically sectioned imaging by oblique plane microscopy,” Opt. Express 16(25), 20306–20316 (2008). [CrossRef] [PubMed]
- S. Kumar, D. Wilding, M. B. Sikkel, A. R. Lyon, K. T. MacLeod, C. Dunsby, “High-speed 2D and 3D fluorescence microscopy of cardiac myocytes,” Opt. Express 19(15), 13839–13847 (2011). [CrossRef] [PubMed]
- F. Cutrale, E. Gratton, “Inclined selective plane illumination microscopy adaptor for conventional microscopes,” Microsc. Res. Tech. 75(11), 1461–1466 (2012). [CrossRef] [PubMed]
- 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]
- 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. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4(8), 1354–1360 (1987). [CrossRef]
- M. Gu, Advanced Optical Imaging Theory (Springer, 2000).
- 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]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939). [CrossRef]
- J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986).
- T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8(9), 1404–1410 (1991). [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]
- E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981). [CrossRef]
- Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22(1), 68–76 (2005). [CrossRef] [PubMed]
- Y. J. Li, “Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory,” J. Opt. Soc. Am. A 22(1), 77–83 (2005). [CrossRef] [PubMed]
- 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]
- 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]
- B. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. 57(1), 56–65 (1967). [CrossRef]
- C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).
- 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]

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