Three-dimensional image formation in fiber-optical second-harmonic-generation microscopy

Three-dimensional (3-D) image formation in fiber-optical second-harmonic-generation microscopy is revealed to be purely coherent and therefore can be described by a 3-D coherent transfer function (CTF) that exhibits the same spatial frequency passband as that of fiber-optical reflection-mode non-fluorescence microscopy. When the numerical aperture of the fiber is much larger than the angle of convergence of the illumination on the fiber aperture, the performance of fiber-optical second-harmonicgeneration microscopy behaves as confocal second-harmonic-generation microscopy. The dependence of axial resolution on fiber coupling parameters shows an improvement of approximately 7%, compared with that in fiber-optical two-photon fluorescence microscopy. © 2005 Optical Society of America OCIS codes: (180.6900) Three-dimensional microscopy; (110.2990) Image formation theory; (110.2350) Fiber optics imaging References and links 1. J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the optical scanning microscope,” Opt. Quantum Electron. 10, 435 (1978). 2. Y. Guo, P. P. Ho, H. Savage, D. Harris, P. Sacks, S. Schantz, F. Liu, N. Zhadin, and R. R. Alfano, “Optical harmonic generation from animal tissue by the use of picosecond and femtosecond laser pulses,” Opt. Lett. 22, 1323 (1997). 3. P. J. Campagnola and L. M. Loew, “Second harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms,” Nat. Biotech. 21, 1356 (2003). 4. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotech. 21, 1369 (2003). 5. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23, 1209 (1998). 6. W. Denk, J. H. Strickler, and W. W. Webb, “Two photon laser scanning fluorescence microscopy,” Science 248, 73 (1990). 7. D. Bird and M. Gu, “Compact two-photon fluorescence microscope based on a single-mode fiber coupler,” Opt. Lett. 27, 1031 (2002). 8. D. Bird and M. Gu, “Two-photon fluorescence endoscopy with a micro-optic scanning head,” Opt. Lett. 28, 1552 (2003). 9. D. Bird and M. Gu, “Fiber-optic two-photon scanning fluorescence microscopy,” J. Micros. 208, 35 (2002). 10. L. Fu, X. Gan, and M. Gu, “Use of single-mode fiber coupler for second-harmonic-generation microscopy,” Opt. Lett. 30, 385 (2005). 11. L. Fu, X. Gan, and M. Gu, “Nonlinear optical microscopy based on double-clad photonic crystal fibers,” Opt. Expess 13, 5528 (2005). 12. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996). 13. C. J. R. Sheppard and M. Gu, “The significance of 3D transfer functions in confocal scanning microscopy,” J. Micros. 165, 377 (1992). 14. M. Gu and D. Bird, “Three-dimensional optical-transfer-function analysis of fiber-optical two-photon fluorescence microscopy,” J. Opt. Soc. Am. A 20, 941 (2003). 15. M. Gu, C. J. R. Sheppard, and X. Gan, “Image formation in a fiber-optical confocal scanning microscope” J. Opt. Soc. Am. A 8, 1755 (1991). 16. S. Kimura and T. Wilson, “Confocal scanning optical microscope using single-mode fiber for signal detection,” Appl. Opt. 30, 2143 (1991). #9309 $15.00 USD Received 28 October 2005; revised 23 January 2006; accepted 23 January 2006 (C) 2006 OSA 6 February 2006 / Vol. 14, No. 3 / OPTICS EXPRESS 1175 17. M. Gu, X. Gan, and C. J. R. Sheppard, “Three-dimensional coherent transfer functions in fiber-optical confocal scanning microscopes,” J. Opt. Soc. Am. A 8, 1019 (1991). 18. S. Yazdanfar, L. H. Laiho, and P. T. C. So, “Interferometric second harmonic generation microscopy,” Opt. Express 12, 2739 (2004). 19. M. Gu and C. J. R. Sheppard, “Comparison of three-dimensional imaging properties between two-photon and single-photon fluorescence microscopy,” J. Micros. 177, 128 (1995).


Introduction
Second-harmonic-generation (SHG) microscopy was proposed by Gannaway and Sheppard in 1970s [1] and has been recently extensively explored for biological studies [2][3][4][5].Together with two-photon (2-p) excited fluorescence microscopy [6], SHG microscopy provides the cellular-level functionality and morphology information of a sample, an inherent sectioning ability for three-dimensional (3-D) imaging, and relatively deep optical penetration within biological tissue [2][3][4][5][6].In order to apply these new nonlinear optical imaging techniques into in vivo applications, fiber-optical components such as a fiber coupler have been adopted for miniaturization [7][8][9][10][11].Physically, 2-p excited fluorescence and SHG microscopy corresponds to incoherent and coherent imaging processes which can be understood by an optical transfer function and a coherent transfer function (CTF) [12,13], respectively.Therefore, a 3-D optical transfer function has been derived to describe the imaging performance in fiber-optical 2-p fluorescence microscopy [14].However, such a description is not applicable to fiberoptical SHG microscopy.
In the case of a non-fluorescent linear sample, fiber-optical scanning microscopy behaves fully coherently even for finite values of fiber spot size [15,16] and therefore can be described by a 3-D CTF that can be expressed by an analytical expression [17].The aim of this paper is to understand 3-D image formation in fiber-optical SHG microscopy using the concept of the 3-D CTF and to reveal the dependence of axial resolution on fiber coupling parameters.

Three-dimensional coherent transfer function in fiber-optical SHG microscope
In order to investigate the effect of optical fibers on illumination and collection separately, let us consider the schematic diagram of a fiber-optical SHG microscope as shown in Fig. 1.Two optical single-mode fibers F 1 and F 2 are used to deliver illumination at wavelength 0 2λ and collect SHG signal at wavelength 0 λ , respectively.When the illumination optical fiber F 1 and the collection optical fiber F 2 are identical, the performance of the system is equivalent to that using a fiber coupler [10].As SHG is a coherent process, the analysis of image formation in fiber-optical SHG microscopy is similar to that in fiber-optical non-fluorescence microscopy described elsewhere [17].Considering that the electrical field of the SHG emission from a sample is proportional to the square of the field of the illumination field on the sample, the image intensity from a scan point ( , , ) r = s s s s x y z in the fiber-optical SHG microscope can be expressed, if the optical axis is assumed along the z direction, as where the letters i r (i=0,1,2) represent vectors with components i x , i y , i z .The functions 1 ( , ) U x y and 2 ( , ) U x y are the amplitude mode profile on the output end of fiber F 1 and on the input end of fiber F 2 , respectively.* denotes the conjugate operation and the parameters 1 M and 2 M are diagonal matrices of the magnification factors of the illumination and collection lenses, respectively.The term in the first square brackets results from the quadratic dependence of SHG.
where 3 ⊗ denotes the 3-D convolution operation.eff h is the 3-D effective PSF for fiberoptical SHG microscopy and given by [ ] Here 2 ⊗ denotes the 2-D convolution operation.
It is necessary to point out that Eqs. ( 1) and ( 2) represent a superposition of the light amplitude from a sample and therefore implies that like fiber-optical non-fluorescence microscopy [15][16][17] fiber-optical SHG microscopy is purely coherent.This feature is of particular importance when one performs SHG interferometric microscopy/tomography [18].Therefore, fiber-optical SHG microscopy can be analyzed in terms of the 3-D CTF that is given by the 3-D Fourier transform of the effective PSF [12,17].The 3-D CTF, c(m), for fiber-optical SHG microscope can thus be described by and Here 3 F is the 3-D Fourier transform with respect to s r and m represents the spatial frequency vector with two transverse components m and n, and one axial component s.For a system using a circular lens, 1 ( ) m c is the 3-D CTF for a fiber-optical reflection-mode nonfluorescence microscope with wavelength 2λ 0 [17] and can be analytically expressed as: Similarly, Eq. ( 6) represents the 3-D CTF for a single circular lens with wavelength 0 λ and weighted by the Fourier transform of the fiber mode profile In Eqs. ( 7) and ( 8), α λ , respectively, where sinα is the numerical aperture of the objective of radius a.Here we have assumed that both illumination and collection fibers are single-mode fibers of mode spot radii a 1 and a 2 .It has been shown that j A is proportional to the square of the ratio of the numerical aperture of the objective in the illumination and collection paths to the numerical aperture of the fibers F 1 and F 2 [12].
According to Eq. ( 4), the 3-D CTF can be numerically evaluated, if the delta function in Eq. ( 8) is taken into account, by where σ represents the area overlapped by 2 It should be pointed out that 3-D CTF for fiber-optical SHG microscopy has a spatial frequency passband of 2 / 4 ( 0) 1 l s s < + < with axial and transverse cutoffs 1 and 2, respectively.Here is a constant axial spatial-frequency shift resulting from the reflection imaging geometry [12].This feature is the same as fiber-optical reflectionmode non-fluorescence microscopy [17].

Results and discussion
When 0 , which corresponds to the case when the numerical aperture of the fiber is A becomes infinity, ( , ) c l s becomes Eq. ( 8) or Eq.(7).No image is formed because ( , ) c l s is zero.For a SHG microscope based on a fiber coupler [10], we have 4A 1 = A 2 and the corresponding 3-D CTFs for A 1 = 0, 1, 5 are shown in Figs.2(a)-(c).Figure 2(a) represents the 3-D CTF for confocal SHG microscopy with a point source and a point detection.For a finite value of A 1 , the strength of the 3-D CTF is reduced in particular along the axial direction, as shown in Figs.2(b) and (c).Figure 2(d) also denotes the 3-D CTF when a point source is used (A 1 = 0) and a fiber is used for collection.In this case, the collection function of the objective becomes weak.Ultimately, when 2 A → ∞ , 2 ( , ) c l s approaches a delta function at l = 0 and thus the 3-D CTF for SHG microscopy is given by Eq. (7).
It is important to investigate the axial cross section, ( 0, ) c l s = of the 3-D CTF further as it gives the axial imaging performance.The solid curves in Fig. 3 represent the normalized axial cross section of the 3-D CTF for SHG microscopy using a fiber coupler ( 1 2 4A A = ), while the dashed line depicts the condition for 1 0 CTF increases linearly up to s = 1/3, which is contributed by the constant region in Eq. ( 7).
After the maximum value at s = 1/3, the CTF decreases and finally cuts off at s = 1.When 4A A = and 1 0 A ≠ , the strength of the CTF for s <1/3 is enhanced while that for s >1/3 is reduced.When eventually 1 2 4A A = →∞, the CTF approaches a delta function at s = 0, which means that there is no axial imaging ability.
To characterize axial resolution, one usually considers imaging of a perfect SHG reflector scanning through the focus of the objective.This axial response is a measure of axial resolution or the optical sectioning property [10,11] and can be calculated using the modulus squared of the Fourier transform of the axial cross section of the 3-D CTF at l = 0 [12,17].After mathematical manipulations, such an axial response can be expressed as  and is depicted as a dashed curve in the inset of Fig. 4, which confirms that SHG microscopy exhibits an inherent optical section property without necessarily using finite-sized detection.This feature also implies that SHG microscopy has an improvement of axial resolution by 35% compared with 2-p fluorescence microscopy without any pinhole [19].
In the case of SHG microscopy using a fiber coupler [10], the HWHM is approximately 4.72 for A is varied by using various values of numerical apertures of the objective to couple the SHG signal into the single-mode fiber coupler.It is shown that experimental results further confirm the dependence of the axial resolution on the normalized fiber spot size parameters.The deviations between the theory and the experimental data might be due to the presence of spherical aberration and the finite thickness of the SHG layer.Compared with the HWHM in fiber-optical 2-p fluorescence microscopy [14], the axial resolution in fiber-optical SHG microscopy is increased approximately by 7%.Fig. 4. Half width at half maximum of the axial response, Δu 1/2 , as a function of the normalized fiber spot size parameter when A 2 =4A 1 (bottom axis) and when A 1 = 0 (up axis).The squares are experimental results for A 1 =2.0, 4.2, 6.4, 7.3, 8.4, respectively.Inset: Normalized axial response of a perfect SHG reflector in fiber-optical SHG microscopy using a fiber coupler (A 2 =4A 1 ) for different values of the normalized optical spot size parameter A 1 .The dashed curve represents the case for A 1 = 0 and A 2 → ∞.

Conclusions
In conclusion, the 3-D CTF has been reported to analyze the three-dimensional image formation in fiber-optical SHG microscopy.Its spatial frequency passband is identical to that for fiber-optical reflection-model non-fluorescence microscopy though the strength of the 3-D CTF is reduced due to the finite-sized fiber aperture.For a system based on a fiber coupler and a given illumination wavelength, the axial resolution in SHG microscopy is increased approximately by 7%, compared with 2-p fluorescence microscopy.
amplitude point spread functions (PSFs) for the objective in illumination and collection paths, respectively [17].ε(r) is the object function representing the SHG strength of the object.Using the 3-D convolution relation, one can simplify Eq. (1) as

2 )
is the normalized fiber spot size for illumination and collection fibers.d j is the distance between the fiber ends and the objective.The variables l (

Fig. 3 .
Fig.3.Axial cross section of the 3-D CTF for fiber-optical SHG microscopy using a fiber coupler (A 2 =4A 1 ) for different values of the normalized optical spot size parameter A 1 .The dashed curve represents the case for A 1 = 0 and A 2 → ∞.
axial response and the half width at half maximum (HWHM, Δu 1/2 ) as a function of the normalized fiber spot size parameters j A are shown in Fig.4.It shows that the SHG axial resolution approaches 5.57 for 1 0 A = and 2 A → ∞ .In this case, Eq.
The HWHM as a function of A 1 exhibits a linear dependence when 15 A > .Under the experimental conditions[10] of A 1 = 2.0, 4.2, 6.4, 7.3, 8.4, the measured values of Δu 1/2 are shown as square spots in Fig.4.Δu 1/2 is derived from the axial response of the fiber-optic SHG microscope to a thin layer of AF-50 dye.Parameter 1