Abstract

For second harmonic generation (SHG) imaging, the specimen is often observed through an immersion medium and a cover glass whose refractive indices are usually different from that of the specimen. However, the currently used theoretical models are based on the assumption that the specimen is situated in a homogeneous medium. The limitation of these models is that they ignore the effects of the refractive index mismatches and the imaging depth. In this paper, we have demonstrated, for the first time to our knowledge, a rigorous model of SHG imaging through stratified media focused by radially polarized beams. Based on the proposed model, the detected SHG intensity patterns excited in a refractive index perfectly matched, aberration-free medium and in mismatched stratified media are compared. The effects of the imaging depth and effective numerical aperture (NA) on the performance of SHG imaging with oil immersion objectives are investigated by the stratified media model. It is found that the full width at half maximum (FWHM) in the axial direction at imaging depth of 80 µm is ~3.1 times as large as that of 10 µm imaging depth. While for the transverse FWHM, the increment is only about 23%. The quality of the SHG intensity distribution can be increased by reducing the NA appropriately at the expense of the detected signal strength. The proposed model is helpful to provide guidelines for the adaptive aberration correction in SHG imaging and can be used to optimize the experimental configuration.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Second harmonic generation (SHG), which is the conversion of the incident optical field at frequency ω to the second-harmonic field at frequency 2ω, plays an important role in basic biological research [1–6], clinical diagnostics [7] and nanoscale morphology detection of nano-objects [8–10]. As a virtual-transition-based nonlinear optical phenomenon, there is no energy absorption in SHG process. In SHG microscopy, no fluorescence labels are needed. The near-infrared excitation provides a deeper penetration depth. Another advantage of SHG microscopy lies in the added sensitivity toward local fundamental fields due to the nonlinear interaction. SHG only occurs at the focus region and the performance of SHG microscopy depends on the quality of the focal spot to a great extent.

Radially polarized beams have attracted increasing attention due to their unique focusing properties. When focused by an objective lens or an elliptical mirror with a high numerical aperture (NA), a strong longitudinal field component is generated in the vicinity of the focus with rotational symmetry about the optical axis [11,12]. For a radially polarized field distribution with annular aperture, the full width at half maximum (FWHM) of the focus spot is below the theoretical limit for linearly polarized light [13]. Based on these superiorities, radially polarized beams have been widely applied in many fields, such as particle acceleration [14], optical trapping [15–17], and high-resolution microscopy [9,18,19].

Hence, many studies of SHG imaging with radially polarized excitations have been carried out theoretically and experimentally. The SHG polarization induced in collagen by radially polarized beams has been calculated by vectorial approach [20]. Ohtsu et al. calculated the SHG wave patterns in the far field generated by radially polarized beams through a dielectric interface [21,22]. Sun et al. have calculated the far-field SHG angular radiation patterns of gold nanoparticles and it was found that for SHG microscopy with backward detection scheme, a tightly focused radially polarized beam could be used to improve the backward SHG signal [23]. In practical applications, SHG microscopy with radially polarized excitations has been conducted on rat-tail tendons [24], native corneal tissue [25], smooth metal and semiconductor surfaces [26], single CdSe nanowires [27], gold nanocones [9], and so on. It is worth emphasizing that, in SHG microscopy, the specimen is often observed through an immersion medium and a cover glass whose refractive indices are usually different from that of the specimen. When focusing beams through the stratified media, the aberration is introduced due to the mismatched refractive indices. The presence of the aberration will lead to a structural modification of the focused spot and then a degradation of the imaging resolution. However, the currently used mathematical models are inapplicable for the analysis of SHG imaging focused by radially polarized beams. These models are based on the assumption that the specimen is situated in a homogeneous medium of propagation or located at a dielectric interface. The effects of the refractive index mismatches and the imaging depth on SHG imaging with radially polarized beams have not yet been studied.

In this paper, we present a rigorous theoretical model for SHG microscopy with a radially polarized excitation. The SHG signal is collected in a backward detection scheme. The detected SHG intensity patterns obtained in a perfectly matched, aberration free medium and in a mismatched stratified media will be compared from several aspects. The effects of the imaging depth on the peak intensity, the shape and the size of the SHG emission spot will also be demonstrated. In addition, SHG response to an on-axis point object as a function of the effective NA will be studied. The results obtained in our study are helpful to provide guidelines for the adaptive aberration correction in SHG imaging.

2. Modeling of SHG imaging through stratified media

Figure 1 shows the configuration for excitation by radially polarized beams through a high NA objective and a three-layer medium, which corresponds to the most common geometry of a SHG microscope. The specimen is observed with an immersion medium and a cover glass. The first interface, perpendicular to the optical z axis, is located at z = −h1, the second interface at z = −h2. The wave numbers of the light beam in the immersion medium, cover glass and specimen are k1, k2 and k3 respectively. k1 = 2πn1/λ0, k2 = 2πn2/λ0, and k3 = 2πn3/λ0. n1, n2, and n3 are the refractive index of the immersion medium, cover glass and specimen respectively.

 figure: Fig. 1

Fig. 1 (a) Focusing of a near-infrared radially polarized beam through a three-layer stratified medium. The origin O of the (x, y, z) reference frame is at the unaberrated Gaussian focal point. (b) SHG radiation imaged through the same medium. The specimen is placed at the origin O’ of the (x’, y’, z’) reference frame. Superscripts 1, 2, 3, and d are for the three media and the detector region. OL: objective lens, DP: detector plane, DL: detector lens.

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According to the Richards-Wolf vectorial diffraction integral [11,28–30], we have derived the generalized formulae for the vectorial electric field in the focal region illuminated by radially polarized beams through a N-layer medium. Under the usual polar coordinate system notation, the radially polarized components and the longitudinal polarized components of the electric field in the focal region can be expressed as:

ENρ(ρ,z)=A0α1cos1/2θ1sinθ1l0(θ1)(Tp(N1)cosθN)J1(k1ρsinθ1)×exp(ik0Ψi)exp(ikNzcosθN)dθ1, (1-a)
ENz(ρ,z)=iA0α1cos1/2θ1sinθ1l0(θ1)(Tp(N1)sinθN)J0(k1ρsinθ1)×exp(ik0Ψi)exp(ikNzcosθN)dθ1, (1-b)
Ψi=hN1nNcosθNh1n1cosθ1,
where ρ, z are the radial coordinate and axial coordinate of an observation point near the focalregion respectively. α1 is the convergence semi-angle of the illumination, and is given as α1 = arcsin(NA/n1). l0(θ1) is the amplitude function in terms of θ1 and Jn(x) denotes a Bessel function of the first kind, of order n. Tp(N-1) is the transmission coefficient of the stratified medium describing the p-polarized light traversing N−1 media and computed as in [30]. Ψi denotes the aberration function.

SHG intensity has a quadratic dependence on the optical field intensity of the input beam at the focus region. Because of this nonlinear relation, the resolution of SHG microscopy is below the diffraction limit. SHG also depends on the nonlinear susceptibility tensor of the specimen, χ(2), which is a third order tensor with 27 separate elements. When the frequency of the laser light source is far away from the resonant frequency of the specimen, the number of non-zero elements in this tensor reduces to 18 or less according to Kleinmann’s symmetry [20,31]. SHG polarization can be expressed as:

[PxPyPz]=[dxxxdxyydxzzdxyzdxxzdxxydyxxdyyydyzzdyyzdyxzdyxydzxxdzyydzzzdzyzdzxzdzxy][ENxENxENyENyENzENz2ENyENz2ENxENz2ENxENy].
The electric field E1 emitted by the SHG process traverses back the stratified medium and is written before the objective lens (in medium 1) as [29,32]:
E1x=12[Px(Ts+TpcosθNcosθ1)2PzTpsinθNcosθ1cosϕ1(TsTpcosθNcosθ1)(Pxcos2ϕ1+Pysin2ϕ1)], (4-a)
E1y=12[Py(Ts+TpcosθNcosθ1)2PzTpsinθNcosθ1sinϕ1(TsTpcosθNcosθ1)(Pxsin2ϕ1Pycos2ϕ1)], (4-b)
E1z=[PzTpsinθNsinθ1TpcosθNsinθ1(Pxcosϕ1+Pysinϕ1)]. (4-c)
(Px*,Py*,Pz*) denotes the Cartesian components of the complex conjugate of P. The transmission coefficients Ts and Tp for the stratified medium are computed as in [29,30], but for propagation from medium N towards medium 1. The electric vector through the detecting optical system is traced under the assumptions that the electric vector maintains its direction with respect to a meridional plane and the electric vector remains on the same side of a meridional plane on passing through the system.

The electric vector E being collimated by the lens (in the intermediate plane) is given by:

E=(cosθ1)1/2R1L11RE1,
R=[cosϕ1sinϕ10sinϕ1cosϕ10001],
L1=[cosθ10sinθ1010sinθ10cosθ1],
where the (cosθ1)−1/2 term derives from an inverse spherical-planar projection due to an aplanatic lens. The matrix R describes the coordinate transformation for rotation around the z axis and the matrix L1 describes the changes in the electric field as it traverses the lens [33]. According to Eq. (5), the Cartesian components of the electric vector field after the lens can be expressed as:
Ex=(cosθ1)1/2{Px(Tpcos2ϕ1cosθN+Tssin2ϕ1)+Py(TpcosθNTs)sinϕ1cosϕ1PzTpsinθNcosϕ1}, (8-a)
Ey=(cosθ1)1/2{Pxcosϕ1(TpcosθNTs)sinϕ1+Py(Tscos2ϕ1+Tpsin2ϕ1cosθN)PzTpsinϕ1sinθN}, (8-b)
Ez=0. (8-c)
The electric vector E2, behind the detector lens, is given by
E2=(cosθd)1/2R1L2RE,
L2=[cosθd0sinθd010sinθd0cosθd],
which yields:
E2x=(cosθd)1/2(cosθ1)1/2{Px[12Ts(1cos2ϕ1)+12TpcosθdcosθN(1+cos2ϕ1)]+Py[12TpcosθdcosθN12Ts]sin2ϕ1PzTpcosθdsinθNcosϕ1}, (11-a)
E2y=(cosθd)1/2(cosθ1)1/2{Px[12(Ts+TpcosθdcosθN)sin2ϕ1]+Py[12Ts(1+cos2ϕ1)+12TpcosθdcosθN(1cos2ϕ)]PzTpcosθdsinθNsinϕ1}, (11-b)
E2z=(cosθd)1/2(cosθ1)1/2{Px(TpsinθdcosθNcosϕ1)+Py(TpsinθdcosθNsinϕ1)+PzTpsinθdsinθN}. (11-c)
The solution for the electric field at the detector plane is hence given by using the integral formula of Richards and Wolf [28,34], as:
Edx=iAdPxId01+iAdPxId21cos2ϕd+iAdPyId21sin2ϕd2AdPzId11cosϕd, (12-a)
Edy=iAdPxId21sin2ϕdiAdPyId01iAdPyId21cos2ϕd2AdPzId11sinϕd, (12-b)
Edz=2AdPxId12cosϕd2AdPyId12sinϕd2iAdPzId02, (12-c)
where
Id01=0αd(cosθd)1/2(cosθ1)1/2(Tssinθd+TpsinθdcosθdcosθN)J0(kdρdsinθd)×exp(ikd0Ψdet)exp(ikdzdcosθd)dθd, (13-a)
Id02=0αd(cosθd)1/2(cosθ1)1/2(Tpsin2θdsinθN)J0(kdρdsinθd)×exp(ikd0Ψdet)exp(ikdzdcosθd)dθd, (13-b)
Id11=0αd(cosθd)1/2(cosθ1)1/2(TpcosθdsinθdsinθN)J1(kdρdsinθd)×exp(ikd0Ψdet)exp(ikdzdcosθd)dθd, (13-c)
Id12=0αd(cosθd)1/2(cosθ1)1/2(Tpsin2θdcosθN)J1(kdρdsinθd)exp(ikd0Ψdet)exp(ikdzdcosθd)dθd, (13-d)
Id21=0αd(cosθd)1/2(cosθ1)1/2(Tssinθd+TpsinθdcosθdcosθN)×J2(kdρdsinθd)exp(ikd0Ψdet)exp(ikdzdcosθd)dθd, (13-e)
with αd being the angular aperture of the detector lens, ρd, ϕd, and zd are the cylindrical coordinates of an observation point near the detection region and the azimuthal angle θd is related to the azimuthal angle θ1 by the relationship:
kd1sinα1kdsinαd=kd1sinθ1kdsinθd=M,
where M is the nominal magnification of the detector lens system. kd0, kd1 and kd are the wave numbers for the second-harmonic field in vacuum, the medium 1 (immersion medium), and the image space respectively, expressed as kd0 = 2π/λSHG, kd1 = 2πn1/λSHG, and kd = 2πnd/λSHG. λSHG denotes the wavelength of SHG. nd is the refractive index of the image space. The initial aberration function Ψdet is given by
Ψdet=n1h1cosθ1nNhN1cosθN.
The detected SHG intensity is then calculated according to:

ISHG=|Edx|2+|Edy|2+|Edz|2.

3. Result

For l0(θ1) in Eq. (1), any prescribed formulation can be chosen by using an appropriate pupil filter in principle. The azimuthal Bessel-Gauss beam is more akin to the azimuthally polarized TE01 mode of the step-index optical fiber than to either the linear Bessel-Gauss or the Laguerre-Gaussian modes. In this paper, the Bessel-Gauss solution of Jordan and Hall is applied [11,35]:

l0(θ1)=exp[β02(sinθ1sinα1)2]J1(2β0sinθ1sinα1),
where α1 is the maximum of θ1. β0 is the ratio of the pupil radius and the beam waist, and is set to 1. Results for the unaberrated electric field distributions near the Gaussian focal point calculated via Eqs. (1), (2) and (17) agree well with the work of Youngworth et al. [11].

Myosin is a kind of endogeneous protein in living tissues and a very good second harmonic generator [36,37]. The tensor of myosin is described as C tensor [38,39]. When the axis of symmetry coincides with the laboratory x axis, SHG polarization can be written as Eq. (18) [40,41]. The relative values of dijk (relative to dzzx) of the myosin in adult xenopus have been measured experimentally [37], and dzzx = dyyx = 1, dxxx = 0.7, dxyy = dxzz = 1.12.

Px=dxzzENzENz+dxyyENyENy+dxxxENxENx, (18-a)
Py=2dyyxENyENx, (18-b)
Pz=2dzzxENzENx. (18-c)

Figure 2(a1) shows the SHG intensity patterns through focus (ϕd = 45° plane) for a refractive index perfectly matched, aberration free medium (n1 = n2 = n3), and Fig. 2(a2)–(a6) are for mismatched stratified media at different imaging depths. The corresponding SHG intensity patterns in the focal plane (x-y plane) are shown in Fig. 2(b1)–(b6). For comparison, the numerical simulations are computed at λ = 1060 nm and we consider an oil immersion (n1 = 1.518) objective of NA = 1.2. The nominal magnification of the detector lens system M is set to 50. For the mismatched stratified media, the specimen is located in a watery environment (n3 = 1.33) below a 120 µm cover glass (n2 = 1.525). The imaging depth (h2) is set to 10 µm, 20 µm, 40 µm, 60 µm and 80 µm respectively. In realistic experiments, the imaging depth can be adjusted just by moving the specimen stage in axial direction. As can be seen, the SHG intensity distribution for the aberration free medium is symmetric along z axis and has no sidelobe. The position of the peak intensity is located at the nominal focal point. By contrast, there is a focal shift in the presence of aberration for the mismatched stratified media and the SHG intensity distribution has a four-sidelobe pattern. The shift increases, and the stretching of the intensity distribution in z direction becomes more obvious with an increase of the imaging depth. The presence of the aberration breaks the symmetry of the SHG intensity distribution along z axis. The intensity ratio between the minor peaks and the focal point increases when the imaging depth increases.

 figure: Fig. 2

Fig. 2 The detected SHG intensity distributions at different depths below a 120 µm cover glass. SHG is obtained at λ = 1060 nm with a NA = 1.2 oil immersion objective, for a specimen located in a watery environment. (a1) The SHG intensity patterns through focus (ϕd = 45° plane) for aberration free medium (n1 = n2 = n3), (a2)-(a6) for mismatched stratified media at different imaging depths. The corresponding SHG intensity patterns in the focal plane (x-y plane) are shown in (b1)-(b6). All the intensity distributions are normalized by the respective maximum value.

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The corresponding axial and transverse intensity line profiles for different imaging depths are shown in Fig. 3. The scan positions of the transverse intensity line profiles are z = −1.52 µm, −3.70 µm, −6.94 µm, −10.1 µm and −13.18 µm for imaging depth of 10 µm, 20 µm, 40 µm, 60 µm and 80 µm respectively. As illustrated in Fig. 3(a), the axial FWHM values which are used to characterize the axial resolution of imaging are increased with the increment of the involved imaging depths. It is noticed in Fig. 3(b) that, for different imaging depths, the transverse FWHM values of the main lobe are almost identical. However, there are significant differences in the proportion of the amplitude of the side lobe relative to the main lobe (PSRM). The PSRM value is a gauge of the noise in SHG microscopy. A deeper imaging depth may lead to a larger imaging noise, and degrade the imaging resolution.

 figure: Fig. 3

Fig. 3 The normalized SHG intensity line profiles for different imaging depths. (a) The axial intensity line profiles. (b) The transverse intensity line profiles in the focal plane. The scan positions of the transverse intensity line profiles are z = −1.52 µm, −3.70 µm, −6.94 µm, −10.1 µm and −13.18 µm for imaging depth of 10 µm, 20 µm, 40 µm, 60 µm and 80 µm respectively.

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The performances of SHG imaging for different imaging depths are summarized in Table 1. The axial FWHM of the intensity distribution at imaging depth of 10 µm is calculated to be 1.03 μm. It increases to 3.20 μm at imaging depth of 80 µm and the value is ~3.1 times as large as that calculated for 10 µm imaging depth. The transverse FWHM varies from 0.237 μm to 0.292 μm, when the imaging depth varies from 10 μm to 80 μm. The increment is only about 23%. As can be seen, the refractive index mismatches have significant influences on the axial resolution.

Tables Icon

Table 1. The performances of SHG imaging for different imaging depths

The performances of SHG imaging for different excitation wavelengths are also studied. The imaging configuration described above is still considered. The excitation wavelength is set to 0.82 μm, 0.94 μm, 1.06 μm and 1.23 μm respectively. As seen from Fig. 4(a), the transverse FWHM value gets larger for nearly all the imaging depths when the excitation wavelength increases. The simulation results are consistent with the criterion of SHG imaging resolution. The minimum distance at which two points in SHG imaging can be clearly distinguished is 0.61λ/(2NA) [42]. It is important to note that the transverse FWHM does not increase monotonically with the increment of the imaging depth. The non-monotonicity can be explained by Eqs. (1) and (2). There is a cyclicality in the distribution of the electric field near the focal region when the aberration function or imaging depth increases. As shown in Fig. 4(b), the axial FWHM is not sensitive to the excitation wavelength and is mainly affected by the imaging depth. The optical section capability of SHG imaging may be degraded in a great imaging depth. Besides the FWHM of the SHG intensity distribution, the peak intensity of the detected SHG signal and the PSRM of the intensity distribution in the focal plane are two important criteria to evaluate the performance of SHG imaging. As shown in Fig. 4(c), the peak intensity of the detected SHG signal decreases monotonously for all the excitation wavelengths as the imaging depth increases. The observed result can be explained via the aberration induced by the refractive index mismatch at the interfaces. The refractive index mismatch becomes more aggravated with increasing the imaging depth. Due to the significant variance for different imaging depths, the peak intensity can be used as a merit function of the adaptive aberration correction in SHG imaging. As illustrated in Fig. 4(d), the values of PSRM exceed 0.2 when the imaging depth gets to 35 µm. It means that the noise in great penetration depth SHG imaging cannot be ignored and adaptive aberration correction should be applied.

 figure: Fig. 4

Fig. 4 (a) The transverse FWHM, (b) the axial FWHM, (c) the peak intensity of the detected SHG signal and (d) the PSRM of the SHG intensity distribution in the focal plane, as a function of the imaging depths for different excitation wavelengths. The excitation wavelength is set to 0.82 μm, 0.94 μm, 1.06 μm and 1.23 μm respectively. For each excitation wavelength, the peak intensity is normalized by the value of 10 μm imaging depth.

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We now look at the effects of varying effective NA when focusing to a certain depth in the specimens and consider an oil immersion objective with a nominal NA of 1.25. The variation of effective NA is realized by reducing the pupil size with the iris. The other specifications of the SHG imaging system are identical to the oil immersion objective system stated above. Figure 5 illustrates how the SHG response to an on-axis point object is affected by altering the pupil size. The distribution along each vertical section shows the axial distribution for a given NA. Each figure is normalized to the maximum intensity in that figure. Figure 5(a) shows the results when using this objective to focus into a perfectly matched, aberration free medium. The shape of the distribution is regular. The axial distributions when focusing to depths of 20 µm, 40 µm, 60 µm and 80 µm into the specimens with mismatch are shown in Figs. 5(b)–5(e). Focusing to a depth of 20 µm, the peak intensity corresponds to the NA of 1.16. The distribution is broader than in the aberration free case but there are no significant side lobes. The shape of the axial distribution degrades at higher NA and there is a reduction in the intensity when focusing to a depth of 40 µm. These effects are further exaggerated at an imaging depth of 60 µm. When the imaging depth gets to 80 µm, the axial distribution is heavily distorted due to the spherical aberration induced by the specimen. There are also significant side lobes present when the effective NA increases. It should be noticed that the quality of the SHG intensity distribution can be increased by reducing the effective NA appropriately although the SHG signal strength also reduces.

 figure: Fig. 5

Fig. 5 SHG response to an on-axis point object as a function of the effective NA for an oil immersion objective of maximum NA = 1.25. (a) Aberration free, (b) depth = 20 µm, (c) depth = 40 µm, (d) depth = 60 µm, (e) depth = 80 µm.

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4. Conclusion

To the best of our knowledge, this is the first demonstration about a rigorous model for SHG imaging through stratified media focused by radially polarized beams. The proposed model takes account of the refractive index mismatches and the imaging depth. Our study shows that, for myosin specimen, there will be a stretching of the SHG intensity distribution in z direction along with a greater focal shift when the imaging depth increases. The transverse FWHM is not sensitive to the aberration induced by the refractive index mismatch. The peak intensity of SHG distribution can be used as a merit function of the adaptive aberration correction in SHG imaging owing to the strict correspondence between the peak intensity and the specimen induced aberration. The quality of the SHG intensity distribution can be increased by reducing the NA appropriately at the expense of the detected signal strength. The proposed model provides a better prediction to the properties of SHG microscopy and can be used to optimize the experimental configuration.

Funding

National Natural Science Foundation of China (NSFC) (51775148); Science Foundation of Heilongjiang Province (QC2018079); China Postdoctoral Science Foundation (2017T100235).

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25. G. Latour, I. Gusachenko, L. Kowalczuk, I. Lamarre, and M. C. Schanne-Klein, “In vivo structural imaging of the cornea by polarization-resolved second harmonic microscopy,” Biomed. Opt. Express 3(1), 1–15 (2012). [CrossRef]   [PubMed]  

26. D. P. Biss and T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003). [CrossRef]   [PubMed]  

27. X. Wang, Z. Zeng, X. Zhuang, F. Wackenhut, A. Pan, and A. J. Meixner, “Second-harmonic generation in single CdSe nanowires by focused cylindrical vector beams,” Opt. Lett. 42(13), 2623–2626 (2017). [CrossRef]   [PubMed]  

28. 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]  

29. O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11(22), 2964–2969 (2003). [CrossRef]   [PubMed]  

30. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36(11), 2305–2312 (1997). [CrossRef]   [PubMed]  

31. D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962). [CrossRef]  

32. P. Török, “Propagation of electromagnetic dipole waves through dielectric interfaces,” Opt. Lett. 25(19), 1463–1465 (2000). [CrossRef]   [PubMed]  

33. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998). [CrossRef]  

34. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998). [CrossRef]  

35. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19(7), 427–429 (1994). [CrossRef]   [PubMed]  

36. S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006). [CrossRef]   [PubMed]  

37. F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15(19), 12286–12295 (2007). [CrossRef]   [PubMed]  

38. C. Odin, T. Guilbert, A. Alkilani, O. P. Boryskina, V. Fleury, and Y. Le Grand, “Collagen and myosin characterization by orientation field second harmonic microscopy,” Opt. Express 16(20), 16151–16165 (2008). [CrossRef]   [PubMed]  

39. M. Rivard, C. A. Couture, A. K. Miri, M. Laliberté, A. Bertrand-Grenier, L. Mongeau, and F. Légaré, “Imaging the bipolarity of myosin filaments with Interferometric Second Harmonic Generation microscopy,” Biomed. Opt. Express 4(10), 2078–2086 (2013). [CrossRef]   [PubMed]  

40. S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3(10), 2681–2693 (2012). [CrossRef]   [PubMed]  

41. C. Teulon, I. Gusachenko, G. Latour, and M. C. Schanne-Klein, “Theoretical, numerical and experimental study of geometrical parameters that affect anisotropy measurements in polarization-resolved SHG microscopy,” Opt. Express 23(7), 9313–9328 (2015). [CrossRef]   [PubMed]  

42. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23(15), 1209–1211 (1998). [CrossRef]   [PubMed]  

References

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  6. G. Hall, K. B. Tilbury, K. R. Campbell, K. W. Eliceiri, and P. J. Campagnola, “Experimental and simulation study of the wavelength dependent second harmonic generation of collagen in scattering tissues,” Opt. Lett. 39(7), 1897–1900 (2014).
    [Crossref] [PubMed]
  7. J. M. Bueno, R. Palacios, A. Pennos, and P. Artal, “Second-harmonic generation microscopy of photocurable polymer intrastromal implants in ex-vivo corneas,” Biomed. Opt. Express 6(6), 2211–2219 (2015).
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  8. M. R. Tsai, S. Y. Chen, D. B. Shieh, P. J. Lou, and C. K. Sun, “In vivo optical virtual biopsy of human oral mucosa with harmonic generation microscopy,” Biomed. Opt. Express 2(8), 2317–2328 (2011).
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  10. G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
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  13. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  14. Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
    [Crossref]
  15. Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35(8), 1281–1283 (2010).
    [Crossref] [PubMed]
  16. M. Li, S. Yan, B. Yao, M. Lei, Y. Yang, J. Min, and D. Dan, “Trapping of rayleigh spheroidal particles by highly focused radially polarized beams,” J. Opt. Soc. Am. B 32(3), 468–472 (2015).
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  17. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [Crossref] [PubMed]
  18. M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
    [Crossref]
  19. G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
    [Crossref] [PubMed]
  20. E. Y. S. Yew and C. J. R. Sheppard, “Vectorial approach to studying second harmonic generation in collagen using linearly and radially polarized beams,” Proc. SPIE 6163, 61630L (2006).
    [Crossref]
  21. A. Ohtsu, “Second-harmonic wave induced by vortex beams with radial and azimuthal polarizations,” Opt. Commun. 283(20), 3831–3837 (2010).
    [Crossref]
  22. A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
    [Crossref]
  23. J. Sun, X. Wang, S. Chang, M. Zeng, S. Shen, and N. Zhang, “Far-field radiation patterns of second harmonic generation from gold nanoparticles under tightly focused illumination,” Opt. Express 24(7), 7477–7487 (2016).
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  24. E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275(2), 453–457 (2007).
    [Crossref]
  25. G. Latour, I. Gusachenko, L. Kowalczuk, I. Lamarre, and M. C. Schanne-Klein, “In vivo structural imaging of the cornea by polarization-resolved second harmonic microscopy,” Biomed. Opt. Express 3(1), 1–15 (2012).
    [Crossref] [PubMed]
  26. D. P. Biss and T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003).
    [Crossref] [PubMed]
  27. X. Wang, Z. Zeng, X. Zhuang, F. Wackenhut, A. Pan, and A. J. Meixner, “Second-harmonic generation in single CdSe nanowires by focused cylindrical vector beams,” Opt. Lett. 42(13), 2623–2626 (2017).
    [Crossref] [PubMed]
  28. 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]
  29. O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11(22), 2964–2969 (2003).
    [Crossref] [PubMed]
  30. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36(11), 2305–2312 (1997).
    [Crossref] [PubMed]
  31. D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
    [Crossref]
  32. P. Török, “Propagation of electromagnetic dipole waves through dielectric interfaces,” Opt. Lett. 25(19), 1463–1465 (2000).
    [Crossref] [PubMed]
  33. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998).
    [Crossref]
  34. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998).
    [Crossref]
  35. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19(7), 427–429 (1994).
    [Crossref] [PubMed]
  36. S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
    [Crossref] [PubMed]
  37. F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15(19), 12286–12295 (2007).
    [Crossref] [PubMed]
  38. C. Odin, T. Guilbert, A. Alkilani, O. P. Boryskina, V. Fleury, and Y. Le Grand, “Collagen and myosin characterization by orientation field second harmonic microscopy,” Opt. Express 16(20), 16151–16165 (2008).
    [Crossref] [PubMed]
  39. M. Rivard, C. A. Couture, A. K. Miri, M. Laliberté, A. Bertrand-Grenier, L. Mongeau, and F. Légaré, “Imaging the bipolarity of myosin filaments with Interferometric Second Harmonic Generation microscopy,” Biomed. Opt. Express 4(10), 2078–2086 (2013).
    [Crossref] [PubMed]
  40. S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3(10), 2681–2693 (2012).
    [Crossref] [PubMed]
  41. C. Teulon, I. Gusachenko, G. Latour, and M. C. Schanne-Klein, “Theoretical, numerical and experimental study of geometrical parameters that affect anisotropy measurements in polarization-resolved SHG microscopy,” Opt. Express 23(7), 9313–9328 (2015).
    [Crossref] [PubMed]
  42. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23(15), 1209–1211 (1998).
    [Crossref] [PubMed]

2017 (2)

2016 (1)

2015 (5)

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

M. Li, S. Yan, B. Yao, M. Lei, Y. Yang, J. Min, and D. Dan, “Trapping of rayleigh spheroidal particles by highly focused radially polarized beams,” J. Opt. Soc. Am. B 32(3), 468–472 (2015).
[Crossref]

J. M. Bueno, R. Palacios, A. Pennos, and P. Artal, “Second-harmonic generation microscopy of photocurable polymer intrastromal implants in ex-vivo corneas,” Biomed. Opt. Express 6(6), 2211–2219 (2015).
[Crossref] [PubMed]

C. Teulon, I. Gusachenko, G. Latour, and M. C. Schanne-Klein, “Theoretical, numerical and experimental study of geometrical parameters that affect anisotropy measurements in polarization-resolved SHG microscopy,” Opt. Express 23(7), 9313–9328 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (2)

2012 (3)

2011 (1)

2010 (4)

Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
[Crossref]

Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35(8), 1281–1283 (2010).
[Crossref] [PubMed]

A. Ohtsu, “Second-harmonic wave induced by vortex beams with radial and azimuthal polarizations,” Opt. Commun. 283(20), 3831–3837 (2010).
[Crossref]

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[Crossref]

2008 (2)

2007 (3)

2006 (2)

E. Y. S. Yew and C. J. R. Sheppard, “Vectorial approach to studying second harmonic generation in collagen using linearly and radially polarized beams,” Proc. SPIE 6163, 61630L (2006).
[Crossref]

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
[Crossref] [PubMed]

2004 (2)

2003 (3)

2000 (2)

1998 (3)

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 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(8), 1681–1698 (1998).
[Crossref]

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23(15), 1209–1211 (1998).
[Crossref] [PubMed]

1997 (2)

1994 (1)

1986 (1)

1962 (1)

D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
[Crossref]

1959 (1)

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]

Ai, M.

J. Liu, M. Ai, J. Tan, R. Wang, and X. Tan, “Focusing of cylindrical-vector beams in elliptical mirror based system with high numerical aperture,” Opt. Commun. 305, 71–75 (2013).
[Crossref]

Alfano, R. R.

Alkilani, A.

Amat-Roldan, I.

Ammar, M.

Araki, T.

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

Artal, P.

Artigas, D.

Ashida, K.

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

Bautista, G.

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

Beaurepaire, E.

Bertrand-Grenier, A.

Biss, D. P.

Boryskina, O. P.

Boulesteix, T.

Brown, T.

Brown, T. G.

Bueno, J. M.

Campagnola, P. J.

G. Hall, K. B. Tilbury, K. R. Campbell, K. W. Eliceiri, and P. J. Campagnola, “Experimental and simulation study of the wavelength dependent second harmonic generation of collagen in scattering tissues,” Opt. Lett. 39(7), 1897–1900 (2014).
[Crossref] [PubMed]

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
[Crossref] [PubMed]

Campbell, K. R.

Chang, S.

Chen, S. Y.

Chen, Y.

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Couture, C. A.

Dan, D.

Deutsch, M.

Dhaka, V.

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Ding, B.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Eliceiri, K. W.

Fleury, V.

Freund, I.

Furukawa, H.

Gauderon, R.

Grasso, M.

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Guilbert, T.

Guo, Y.

Gusachenko, I.

Haeberlé, O.

Hall, D. G.

Hall, G.

Harris, D.

Hashimoto, M.

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 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(8), 1681–1698 (1998).
[Crossref]

Ho, P. P.

Huhtio, T.

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Huttunen, M. J.

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

Jiang, H.

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Jordan, R. H.

Kakko, J. P.

Karvonen, L.

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Kauppinen, E.

Kauranen, M.

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

Kleinmann, D. A.

D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
[Crossref]

Kontio, J. M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

Kowalczuk, L.

Kozawa, Y.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[Crossref]

Laliberté, M.

Lamarre, I.

Latour, G.

Le Grand, Y.

Légaré, F.

Lei, M.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Li, M.

Lipsanen, H.

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Liu, F.

Liu, J.

J. Liu, M. Ai, J. Tan, R. Wang, and X. Tan, “Focusing of cylindrical-vector beams in elliptical mirror based system with high numerical aperture,” Opt. Commun. 305, 71–75 (2013).
[Crossref]

Lou, P. J.

Loza-Alvarez, P.

Lukins, P. B.

Mäkitalo, J.

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

Meixner, A. J.

Millard, A. C.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
[Crossref] [PubMed]

Min, J.

Miri, A. K.

Mohler, W. A.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
[Crossref] [PubMed]

Mongeau, L.

Niioka, H.

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

Odin, C.

Ohtsu, A.

A. Ohtsu, “Second-harmonic wave induced by vortex beams with radial and azimuthal polarizations,” Opt. Commun. 283(20), 3831–3837 (2010).
[Crossref]

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[Crossref]

Olsen, B. R.

Palacios, R.

Pan, A.

Pennos, A.

Pfeffer, C. P.

Plotnikov, S. V.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
[Crossref] [PubMed]

Psilodimitrakopoulos, S.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Recher, G.

Richards, B.

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]

Rivard, M.

Rouède, D.

Sacks, P.

Salamin, Y. I.

Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
[Crossref]

Sato, S.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[Crossref]

Sauviat, M. P.

Savage, H.

Schanne-Klein, M. C.

Schantz, S.

Shen, S.

Sheppard, C. J. R.

E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275(2), 453–457 (2007).
[Crossref]

E. Y. S. Yew and C. J. R. Sheppard, “Vectorial approach to studying second harmonic generation in collagen using linearly and radially polarized beams,” Proc. SPIE 6163, 61630L (2006).
[Crossref]

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23(15), 1209–1211 (1998).
[Crossref] [PubMed]

Shieh, D. B.

Simonen, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

Sun, C. K.

Sun, J.

Suyama, T.

Tan, J.

J. Liu, M. Ai, J. Tan, R. Wang, and X. Tan, “Focusing of cylindrical-vector beams in elliptical mirror based system with high numerical aperture,” Opt. Commun. 305, 71–75 (2013).
[Crossref]

Tan, X.

J. Liu, M. Ai, J. Tan, R. Wang, and X. Tan, “Focusing of cylindrical-vector beams in elliptical mirror based system with high numerical aperture,” Opt. Commun. 305, 71–75 (2013).
[Crossref]

Tenjimbayashi, K.

Teulon, C.

Tiaho, F.

Tilbury, K. B.

Török, P.

Tsai, M. R.

Varga, P.

Wackenhut, F.

Wang, R.

J. Liu, M. Ai, J. Tan, R. Wang, and X. Tan, “Focusing of cylindrical-vector beams in elliptical mirror based system with high numerical aperture,” Opt. Commun. 305, 71–75 (2013).
[Crossref]

Wang, X.

Wilson, T.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 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(4-6), 300–315 (1998).
[Crossref]

Wolf, E.

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]

Yan, S.

Yang, Y.

Yao, B.

Yew, E. Y. S.

E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275(2), 453–457 (2007).
[Crossref]

E. Y. S. Yew and C. J. R. Sheppard, “Vectorial approach to studying second harmonic generation in collagen using linearly and radially polarized beams,” Proc. SPIE 6163, 61630L (2006).
[Crossref]

Yoshiki, K.

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

Youngworth, K.

Zang, X.

Zeng, M.

Zeng, Z.

Zhadin, N.

Zhan, Q.

Zhang, N.

Zhang, Y.

Zhuang, X.

Appl. Opt. (1)

Appl. Phys. B (1)

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98(4), 851–855 (2010).
[Crossref]

Appl. Phys. Express (1)

M. Hashimoto, H. Niioka, K. Ashida, K. Yoshiki, and T. Araki, “High-sensitivity and high-spatial-resolution imaging of self-assembled monolayer on platinum using radially polarized beam excited second-harmonic-generation microscopy,” Appl. Phys. Express 8(11), 112401 (2015).
[Crossref]

Biomed. Opt. Express (5)

Biophys. J. (1)

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90(2), 693–703 (2006).
[Crossref] [PubMed]

J. Mod. Opt. (1)

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998).
[Crossref]

J. Opt. Soc. Am. B (1)

Nano Lett. (2)

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12(6), 3207–3212 (2012).
[Crossref] [PubMed]

G. Bautista, J. Mäkitalo, Y. Chen, V. Dhaka, M. Grasso, L. Karvonen, H. Jiang, M. J. Huttunen, T. Huhtio, H. Lipsanen, and M. Kauranen, “Second-harmonic generation imaging of semiconductor nanowires with focused vector beams,” Nano Lett. 15(3), 1564–1569 (2015).
[Crossref] [PubMed]

Opt. Commun. (4)

A. Ohtsu, “Second-harmonic wave induced by vortex beams with radial and azimuthal polarizations,” Opt. Commun. 283(20), 3831–3837 (2010).
[Crossref]

E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275(2), 453–457 (2007).
[Crossref]

J. Liu, M. Ai, J. Tan, R. Wang, and X. Tan, “Focusing of cylindrical-vector beams in elliptical mirror based system with high numerical aperture,” Opt. Commun. 305, 71–75 (2013).
[Crossref]

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148(4-6), 300–315 (1998).
[Crossref]

Opt. Express (10)

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15(19), 12286–12295 (2007).
[Crossref] [PubMed]

C. Odin, T. Guilbert, A. Alkilani, O. P. Boryskina, V. Fleury, and Y. Le Grand, “Collagen and myosin characterization by orientation field second harmonic microscopy,” Opt. Express 16(20), 16151–16165 (2008).
[Crossref] [PubMed]

C. Teulon, I. Gusachenko, G. Latour, and M. C. Schanne-Klein, “Theoretical, numerical and experimental study of geometrical parameters that affect anisotropy measurements in polarization-resolved SHG microscopy,” Opt. Express 23(7), 9313–9328 (2015).
[Crossref] [PubMed]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[Crossref] [PubMed]

G. Bautista, J. P. Kakko, V. Dhaka, X. Zang, L. Karvonen, H. Jiang, E. Kauppinen, H. Lipsanen, and M. Kauranen, “Nonlinear microscopy using cylindrical vector beams: Applications to three-dimensional imaging of nanostructures,” Opt. Express 25(11), 12463–12468 (2017).
[Crossref] [PubMed]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

C. Odin, T. Guilbert, A. Alkilani, O. P. Boryskina, V. Fleury, and Y. Le Grand, “Collagen and myosin characterization by orientation field second harmonic microscopy,” Opt. Express 16(20), 16151–16165 (2008).
[Crossref] [PubMed]

C. P. Pfeffer, B. R. Olsen, and F. Légaré, “Second harmonic generation imaging of fascia within thick tissue block,” Opt. Express 15(12), 7296–7302 (2007).
[Crossref] [PubMed]

O. Haeberlé, M. Ammar, H. Furukawa, K. Tenjimbayashi, and P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11(22), 2964–2969 (2003).
[Crossref] [PubMed]

J. Sun, X. Wang, S. Chang, M. Zeng, S. Shen, and N. Zhang, “Far-field radiation patterns of second harmonic generation from gold nanoparticles under tightly focused illumination,” Opt. Express 24(7), 7477–7487 (2016).
[Crossref] [PubMed]

Opt. Lett. (10)

D. P. Biss and T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28(11), 923–925 (2003).
[Crossref] [PubMed]

X. Wang, Z. Zeng, X. Zhuang, F. Wackenhut, A. Pan, and A. J. Meixner, “Second-harmonic generation in single CdSe nanowires by focused cylindrical vector beams,” Opt. Lett. 42(13), 2623–2626 (2017).
[Crossref] [PubMed]

P. Török, “Propagation of electromagnetic dipole waves through dielectric interfaces,” Opt. Lett. 25(19), 1463–1465 (2000).
[Crossref] [PubMed]

R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19(7), 427–429 (1994).
[Crossref] [PubMed]

Y. Guo, P. P. Ho, H. Savage, D. Harris, P. Sacks, S. Schantz, F. Liu, N. Zhadin, and R. R. Alfano, “Second-harmonic tomography of tissues,” Opt. Lett. 22(17), 1323–1325 (1997).
[Crossref] [PubMed]

G. Hall, K. B. Tilbury, K. R. Campbell, K. W. Eliceiri, and P. J. Campagnola, “Experimental and simulation study of the wavelength dependent second harmonic generation of collagen in scattering tissues,” Opt. Lett. 39(7), 1897–1900 (2014).
[Crossref] [PubMed]

Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35(8), 1281–1283 (2010).
[Crossref] [PubMed]

I. Freund and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11(2), 94–96 (1986).
[Crossref] [PubMed]

T. Boulesteix, E. Beaurepaire, M. P. Sauviat, and M. C. Schanne-Klein, “Second-harmonic microscopy of unstained living cardiac myocytes: measurements of sarcomere length with 20-nm accuracy,” Opt. Lett. 29(17), 2031–2033 (2004).
[Crossref] [PubMed]

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23(15), 1209–1211 (1998).
[Crossref] [PubMed]

Phys. Rev. (1)

D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
[Crossref]

Phys. Rev. A (1)

Y. I. Salamin, “Direct particle acceleration by two identical crossed radially polarized laser beams,” Phys. Rev. A 82(1), 013823 (2010).
[Crossref]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

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]

Proc. SPIE (1)

E. Y. S. Yew and C. J. R. Sheppard, “Vectorial approach to studying second harmonic generation in collagen using linearly and radially polarized beams,” Proc. SPIE 6163, 61630L (2006).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Focusing of a near-infrared radially polarized beam through a three-layer stratified medium. The origin O of the (x, y, z) reference frame is at the unaberrated Gaussian focal point. (b) SHG radiation imaged through the same medium. The specimen is placed at the origin O’ of the (x’, y’, z’) reference frame. Superscripts 1, 2, 3, and d are for the three media and the detector region. OL: objective lens, DP: detector plane, DL: detector lens.
Fig. 2
Fig. 2 The detected SHG intensity distributions at different depths below a 120 µm cover glass. SHG is obtained at λ = 1060 nm with a NA = 1.2 oil immersion objective, for a specimen located in a watery environment. (a1) The SHG intensity patterns through focus (ϕd = 45° plane) for aberration free medium (n1 = n2 = n3), (a2)-(a6) for mismatched stratified media at different imaging depths. The corresponding SHG intensity patterns in the focal plane (x-y plane) are shown in (b1)-(b6). All the intensity distributions are normalized by the respective maximum value.
Fig. 3
Fig. 3 The normalized SHG intensity line profiles for different imaging depths. (a) The axial intensity line profiles. (b) The transverse intensity line profiles in the focal plane. The scan positions of the transverse intensity line profiles are z = −1.52 µm, −3.70 µm, −6.94 µm, −10.1 µm and −13.18 µm for imaging depth of 10 µm, 20 µm, 40 µm, 60 µm and 80 µm respectively.
Fig. 4
Fig. 4 (a) The transverse FWHM, (b) the axial FWHM, (c) the peak intensity of the detected SHG signal and (d) the PSRM of the SHG intensity distribution in the focal plane, as a function of the imaging depths for different excitation wavelengths. The excitation wavelength is set to 0.82 μm, 0.94 μm, 1.06 μm and 1.23 μm respectively. For each excitation wavelength, the peak intensity is normalized by the value of 10 μm imaging depth.
Fig. 5
Fig. 5 SHG response to an on-axis point object as a function of the effective NA for an oil immersion objective of maximum NA = 1.25. (a) Aberration free, (b) depth = 20 µm, (c) depth = 40 µm, (d) depth = 60 µm, (e) depth = 80 µm.

Tables (1)

Tables Icon

Table 1 The performances of SHG imaging for different imaging depths

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

E Nρ (ρ,z)=A 0 α 1 cos 1/2 θ 1 sin θ 1 l 0 ( θ 1 ) ( T p (N1) cos θ N ) J 1 ( k 1 ρsin θ 1 ) ×exp(i k 0 Ψ i )exp(i k N zcos θ N )d θ 1 ,
E Nz (ρ,z)=iA 0 α 1 cos 1/2 θ 1 sin θ 1 l 0 ( θ 1 ) ( T p (N1) sin θ N ) J 0 ( k 1 ρsin θ 1 ) ×exp(i k 0 Ψ i )exp(i k N zcos θ N )d θ 1 ,
Ψ i = h N1 n N cos θ N h 1 n 1 cos θ 1 ,
[ P x P y P z ]=[ d xxx d xyy d xzz d xyz d xxz d xxy d yxx d yyy d yzz d yyz d yxz d yxy d zxx d zyy d zzz d zyz d zxz d zxy ][ E Nx E Nx E Ny E Ny E Nz E Nz 2 E Ny E Nz 2 E Nx E Nz 2 E Nx E Ny ].
E 1 x = 1 2 [ P x ( T s + T p cos θ N cos θ 1 )2 P z T p sin θ N cos θ 1 cos ϕ 1 ( T s T p cos θ N cos θ 1 )( P x cos2 ϕ 1 + P y sin2 ϕ 1 )],
E 1 y = 1 2 [ P y ( T s + T p cos θ N cos θ 1 )2 P z T p sin θ N cos θ 1 sin ϕ 1 ( T s T p cos θ N cos θ 1 )( P x sin2 ϕ 1 P y cos2 ϕ 1 )],
E 1 z =[ P z T p sin θ N sin θ 1 T p cos θ N sin θ 1 ( P x cos ϕ 1 + P y sin ϕ 1 )].
E= (cos θ 1 ) 1/2 R 1 L 1 1 R E 1 ,
R=[ cos ϕ 1 sin ϕ 1 0 sin ϕ 1 cos ϕ 1 0 0 0 1 ],
L 1 =[ cos θ 1 0 sin θ 1 0 1 0 sin θ 1 0 cos θ 1 ],
E x = (cos θ 1 ) 1/2 { P x ( T p cos 2 ϕ 1 cos θ N + T s sin 2 ϕ 1 ) + P y ( T p cos θ N T s )sin ϕ 1 cos ϕ 1 P z T p sin θ N cos ϕ 1 },
E y = (cos θ 1 ) 1/2 { P x cos ϕ 1 ( T p cos θ N T s )sin ϕ 1 + P y ( T s cos 2 ϕ 1 + T p sin 2 ϕ 1 cos θ N ) P z T p sin ϕ 1 sin θ N },
E z =0.
E 2 = (cos θ d ) 1/2 R 1 L 2 RE,
L 2 =[ cos θ d 0 sin θ d 0 1 0 sin θ d 0 cos θ d ],
E 2x = (cos θ d ) 1/2 (cos θ 1 ) 1/2 { P x [ 1 2 T s (1cos2 ϕ 1 )+ 1 2 T p cos θ d cos θ N (1+cos2 ϕ 1 )] + P y [ 1 2 T p cos θ d cos θ N 1 2 T s ]sin2 ϕ 1 P z T p cos θ d sin θ N cos ϕ 1 },
E 2y = (cos θ d ) 1/2 (cos θ 1 ) 1/2 { P x [ 1 2 ( T s + T p cos θ d cos θ N )sin2 ϕ 1 ] + P y [ 1 2 T s (1+cos2 ϕ 1 )+ 1 2 T p cos θ d cos θ N (1cos2ϕ)] P z T p cos θ d sin θ N sin ϕ 1 },
E 2z = (cos θ d ) 1/2 (cos θ 1 ) 1/2 { P x ( T p sin θ d cos θ N cos ϕ 1 ) + P y ( T p sin θ d cos θ N sin ϕ 1 )+ P z T p sin θ d sin θ N }.
E dx =i A d P x I d01 +i A d P x I d21 cos2 ϕ d +i A d P y I d21 sin2 ϕ d 2 A d P z I d11 cos ϕ d ,
E dy =i A d P x I d21 sin2 ϕ d i A d P y I d01 i A d P y I d21 cos2 ϕ d 2 A d P z I d11 sin ϕ d ,
E dz =2 A d P x I d12 cos ϕ d 2 A d P y I d12 sin ϕ d 2i A d P z I d02 ,
I d01 = 0 α d (cos θ d ) 1/2 (cos θ 1 ) 1/2 ( T s sin θ d + T p sin θ d cos θ d cos θ N ) J 0 ( k d ρ d sin θ d ) ×exp(i k d0 Ψ det )exp(i k d z d cos θ d )d θ d ,
I d02 = 0 α d (cos θ d ) 1/2 (cos θ 1 ) 1/2 ( T p sin 2 θ d sin θ N ) J 0 ( k d ρ d sin θ d ) ×exp(i k d0 Ψ det )exp(i k d z d cos θ d )d θ d ,
I d11 = 0 α d (cos θ d ) 1/2 (cos θ 1 ) 1/2 ( T p cos θ d sin θ d sin θ N ) J 1 ( k d ρ d sin θ d ) ×exp(i k d0 Ψ det )exp(i k d z d cos θ d )d θ d ,
I d12 = 0 α d (cos θ d ) 1/2 (cos θ 1 ) 1/2 ( T p sin 2 θ d cos θ N ) J 1 ( k d ρ d sin θ d ) exp(i k d0 Ψ det )exp(i k d z d cos θ d )d θ d ,
I d21 = 0 α d (cos θ d ) 1/2 (cos θ 1 ) 1/2 ( T s sin θ d + T p sin θ d cos θ d cos θ N ) × J 2 ( k d ρ d sin θ d )exp(i k d0 Ψ det )exp(i k d z d cos θ d )d θ d ,
k d1 sin α 1 k d sin α d = k d1 sin θ 1 k d sin θ d =M,
Ψ det = n 1 h 1 cos θ 1 n N h N1 cos θ N .
I SHG = | E dx | 2 + | E dy | 2 + | E dz | 2 .
l 0 ( θ 1 )=exp[ β 0 2 ( sin θ 1 sin α 1 ) 2 ]J1( 2β0 sin θ 1 sin α 1 ),
P x = d xzz E Nz E Nz + d xyy E Ny E Ny + d xxx E Nx E Nx ,
P y =2 d yyx E Ny E Nx ,
P z =2 d zzx E Nz E Nx .

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