The effects of the axial field components of a focused beam under high NA on the second harmonic generation (SHG) in collagen was examined using a vectorial approach. We find that with high NA, the cross-component terms that are most likely to have an effect on SHG will be ExEx, ExEy, ExEz and EzEz as a result of tight focusing. By considering the tensor and the presence of the other electric field components the possibility of different polarization states of the generated second harmonic as a result of the nonlinear susceptibility tensor making it possible to generate radially polarized modes with linearly polarized beams.
©2006 Optical Society of America
There has been an increasing interest in the use of nonlinear optical processes for microscopy among which second harmonic generation (SHG) is one of them. Some advantages that SHG microscopy enjoys over other conventional microscopy include the use of intrinsic optical properties for imaging, the absence of photodamage, a highly localized interaction region and structural identification through the resolution of the tensor components of SHG emitting structures [1,2]. Currently, SHG microscopy has been applied to studying surfaces [3,4], microtubules , collagen [1,6], tumours , muscle  and in-vivo developmental biology .
As a nonlinear optical process, SHG is dependent on the optical properties of the material. This property is normally described by a nonlinear susceptibility tensor. For SHG, the tensor that describes its nonlinear optical properties is a second-order tensor with 27 separate elements, which results in an electric polarization that is dependent on the various components of the electric field . Currently, most papers dealing with SHG consider the effects of the focused beam have been approximated using an effectively scalar theory [11,12,13] or one that only focuses on the dominant component of the illuminating field [14,15]. In reality all three field components are present since one does not, in practice, encounter an infinite plane wave and also because microscopy utilizes high numerical aperture lenses to focus the beam. It is under such circumstances where a vectorial theory is appropriate . In this paper, we consider the effects of the various electric field components on the SGH from a hypothetical collagen fibril using the Richards and Wolf approach. We also find that the axial components of the electric field play a role in SHG in the case of collagen through the susceptibility tensor.
SHG is dependent on the nonlinear susceptibility tensor of the sample. This tensor is a tensor with 27 elements similar to the piezoelectric tensor. Under certain conditions the number of non-zero elements in this tensor reduces according to Kleinmann’s symmetry . In the case of collagen, a very good second harmonic generator, the tensor has been described as a C6 tensor [1,2,6]. When the axis of symmetry coincides with the laboratory z axis we can write:
Equation (1) represents the case when the axis of symmetry is parallel to the propagation of the excitation field. Alternatively, we may have the axis lying parallel to the polarization of the excitation field and assuming our field is polarized in the x direction, gives:
Previous values of dijk been found experimentally  and take the values dxzz=dxyy=1, dxxx:=0.09 and dyyx=dzzx=1.15. The equations indicate that SHG polarization can be induced through ‘cross-components’ such as EiEj (i ≠ j). For low NA focusing, the component of the electric field in the direction of polarization is strongest and applying that to Eq. (2), induces only |Px SHG| or |Py SHG| or both depending on the orientation of the polarization with respect to the collagen axis. When the collagen axis is along z the only SHG polarization induced will be |Pz SHG| The resultant SHG signal from such a configuration is similar to a dipole radiating along the z axis and will require a high NA collector for transmission type detection. This is perhaps why it has been reported that there is no apparent SHG signal when the orientation is such . One interesting aspect of this tensor is the fact that it is possible to elicit various modes of polarization different from the excitation beam. For example, the Pz SHG term in Eq. (2) implies a radially polarized mode obtainable with a mixing of the ExEz term. Likewise if we were to consider the case of PzSHG = dzxxExEx only we will have a radially polarized mode generated by a linearly polarized beam.
For high NA focusing, the electric field at the focus consists of transverse as well as axial components. Although weak, the transverse and axial components can have an effect when a tightly focused beam is used in nonlinear optical microscopy due to the tensorial nature of SHG. According to Richards and Wolf , the electric field components in the focal region are:
where r = (x 2 + y 2)1/2, α is the aperture half-angle and Jn( . ) is a Bessel function of the first type and order n. In the high NA case (here we assume a NA 1.4 objective) the ratios of the electric field components |Ex|:|Ey|:|Ez| are approximately 1:0.1:0.3 respectively. In Table 1 we find that the EiEj terms that are most likely to have an effect on SHG will be the cases where we have ExEx, ExEy, ExEz and EzEz. The other cross-component terms will have minimal effects since they are two orders of magnitude less than ExEx.
In order to solve for the far-field radiation of SHG, a Green’s function approach may be used  or a phased array approach [11,12]. We use the former approach and the interested reader is referred to the following references for further details [14,19]. We present only the end equation that is used for calculating the radiation pattern of SHG in the far-field (in spherical coordinates) :
where R̂ is the unit vector in the direction of the observer and k is now 2k due to frequency doubling, Θ is the polar angle of the observation point and Φ is the azimuthal angle of the observation point. For our calculations we took λ=1000nm, n=1.5, NA=1.4 and assumed that the refractive index of the lens and sample were the same.
In this work we assume, for simplicity, a hypothetical fibril of collagen that possesses the following properties. One is that the fibril is composed of many infinitesimally small subunits each possessing the C6 symmetry. These idealized subunits may then be arrayed in different orientations (with respect to their axis of symmetry) in ideal 1-dimensional lines or planes. Fig. 1 illustrates the geometry and orientation adopted in this theoretical study.
We calculated the induced SHG polarization by assuming that the C6 structure was valid in the focal plane and applying Eqs. (1)–(4). Different orientations of the fibre axis result in different induced SHG polarization as can be seen from Fig. 2, which can be visualized as having the layout of Fig. 1(a) except that the linear array of subunits has been extended over the focal plane.
Although the field in the x direction is strongest for a focused beam linearly polarized along x, we find that it is not the |Px SHG| that is dominant but rather the |Pz SHG|. The |Px SHG| term is also significant as it is half as strong as the |Pz SHG| term when the axis is parallel to z axis. This |Px SHG| component is generated by the product of the x and z components of the incident field. The |Px SHG| component is nearly 5 times weaker than the |Pz SHG| term when the axis of the fibre is parallel to the x axis as seen in Fig. 3. This is explained by looking at Eqs. (1) and (2) and noting the values of the associated d coefficients. The dominant |Pz SHG| component is now generated by the product of the x and z components of the incident field.
Given that the induced SHG polarization is different from the applied field and exhibits different orientations, we calculated the SHG radiation patterns from a hypothetical collagen fibril of zero thickness (Fig. 1). We thus had four cases viz. orientation of the axis of symmetry in the x direction and “extending” in the x or z directions and orientation of the axis of symmetry in the z and ‘extending’ in the x or z directions. We used Eq. (5) to calculate the final radiation pattern in the Θ direction.
3.1 Extension in the x direction
When the axis of symmetry is parallel to z and the extension is in x we would expect short lengths of the fibril to radiate in a manner similar to that of a dipole along the z axis. This is because for ranges very close to zero the only induced SHG polarization is in the axial direction (Fig. 2(a) and Fig. 2(c)). A very short fibril length will thus approximate a dipole along the z axis and will radiate as shown in Fig. 4(a). Similarly for the case when the axis of symmetry is parallel to x, Fig. 3 indicates that a very short length of such a layout will result in a radiation pattern similar to a dipole along the x axis as seen in Fig. 5(a). Fig. 5(a) is, however, not rotationally symmetric. This is due to the fact that in Eq. (5), the Pz SHG term is dependent only on Θ and not on Φ.
We expect the radiation pattern to change as the length varies simply by looking at the relative strengths of the induced SHG polarization in Fig. 2(a) and Fig. 2(c). Figures 4 and Fig. 5 give an idea of how the SHG radiation pattern ‘evolves’ as the fibril gets longer in the x direction. The need for phase matching also creates a radiation pattern that has lobes radiating away from the z axis [12,14]. Since the fibril is only along the x axis, it is possible for a b-SHG (backward propagating SHG) as seen in Fig. 4 and Fig. 5 (where dotted lines denote the z = 0 level) illustrate the relative strengths of the b-SHG to the f-SHG.
3.2 Extension in the z direction
For the case when the extension of the collagen is in the z direction the radiation pattern for a short length is determined by the induced SHG polarization that is non-zero near the origin as previously. More interestingly is the lack of b-SHG as seen in Fig. 6 and Fig. 7. This is primarily due to coherence effects. In the axial direction, the phase changes and a Gouy phase shift occurs. Each SHG-producing element along the axial direction thus radiates equally backwards and forwards. The relative phase shift between each element implies that the b-SHG will interfere destructively and thus bulk objects typically have a strong f-SHG signal only. Figure 7 has also been calculated by calculated previously based on inhomgenous scatterers in membranes  and hemispherical interfaces .
Thus there exists a certain amount of observable SHG in reflection for a hypothetical line of SHG emitting elements along the transverse axes. The presence of a ‘thickness’ or extension in the axial direction means that the SHG radiates in a forward direction due to coherence effects. Phase matching considerations also indicate that SHG will rarely propagate along the z axis but will, instead, propagate at an angle away from the z axis. In Fig. 4, the angle of propagation away from the z axis is approximately 20°, 29° for figure 5, 47° for Fig. 6 and 40° for Fig. 7. A high NA condenser is thus necessary to collect transmitted SHG light more efficiently.
We have examined the far-field SHG radiation patterns for a variety of cases using a full vectorial approach to describe a linearly polarized excitation field. We find that for high NA focusing, the axial component of the electric field plays a role in the generation of SHG through the susceptibility tensor in the case of collagen. We find that the EiEj terms that are most likely to have an effect on SHG will be the cases where we have ExEx, ExEy, ExEz and EzEz. A further point of interest is the possibility of different polarization states of the generated second harmonic as a result of the nonlinear susceptibility tensor. This possibility of generating radially polarized modes with a linearly polarized beams is of interest. There have also been several reports on aperturing on harmonic generation [20,21]. Effects of aperturing could be made independent of the order of the harmonic as well as the gas species and that the harmonic efficiencies in terms of aperture size was peaked indicating a certain aperture size which is optimum . It was also demonstrated that higher NA objectives improved the resolution and varied the intensity of the SHG . The reason assigned to this was that in certain cases, the orientation of the crystal was such that it favoured SHG with the excitation beam parallel to the axis of propagation. With an increasing NA, the spectrum of waves was spread out over a wider angle and hence less energy was available for SHG in the direction optimal for phase matching. Similarly, cases where the intensity was observed to increase with NA could be explained in a similar fashion. It is clear that as the NA influences the strengths of the various components of the electric field, further study of the effects of NA on harmonic generation incorporating a vectorial theory of the electric field at the focus is advantageous for nonlinear microscopy techniques. Applications for such techniques will include identifying the three-dimensional orientation of molecules and tracking changes in the susceptibility tensor which will have potential interest in cancer and various diseases caused by modifications to the tissue/cellular structure.
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