To derive the dependence of SHG anisotropy on SHRE distribution within the focal volume, we start from the general description of the polarization $P$ in a medium: Display Formula
$P=\chi (1)E\u2192+\chi (2)E\u2192E\u2192+\cdots .$(2)
This equation is nothing but the bulk equivalent of Eq. (1). The $\chi (2)$ tensor describes the second-order susceptibility. In the general case, the susceptibility tensor $\chi ijk(2)(\omega 1,\omega 2)$ is a third-rank tensor with ($3\xd73\xd73$) elements. In the specific case of SHG, in which two fields with the same frequency ($\omega 1=\omega 2=\omega $) generate a third field with frequency $2\omega $, each component of the second-order polarization can be expressed as Display Formula$Pi(2)(2\omega )=\u2211j,k\chi ijk(2)(\omega ,\omega )Ej(\omega 1)Ek(\omega 2)=\u2211k,j\chi ikj(2)(\omega ,\omega )Ek(\omega 1)Ej(\omega 2)=\u2211j,k\chi ijk(2)(\omega ,\omega )Ej(\omega )Ek(\omega ).$(3)
Therefore, the susceptibility tensor has the following symmetry: Display Formula$\chi ijk(2)(\omega ,\omega )=\chi ikj(2)(\omega ,\omega ).$(4)
With these symmetries, the number of independent elements of the tensor decreases to 18. Hence, the second-order induced polarization can be written as a function of the components of the tensor $\chi ijk(2)$ and of the electric field $Ei$ as follows: Display Formula$(Px(2)Py(2)Pz(2))=(\chi xxx(2)\chi xyy(2)\chi xzz(2)\chi xyz(2)\chi xxz(2)\chi xxy(2)\chi yxx(2)\chi yyy(2)\chi yzz(2)\chi yyz(2)\chi yxz(2)\chi yxy(2)\chi zxx(2)\chi zyy(2)\chi zzz(2)\chi zyz(2)\chi zxz(2)\chi zxy(2))\u22c5(Ex2Ey2Ez22EyEz2ExEz2ExEy).$(5)
In a bulk sample made of a distribution of individual SHREs, the susceptibility tensor $\chi ijk(2)$ can be calculated, summing each individual hyperpolarizability term $\beta i\u2032j\u2032k\u2032$ (expressed in the molecule’s system of coordinates $x\u2032y\u2032z\u2032$): Display Formula$\chi ijk(2)=\u2211n\u2211i\u2032j\u2032k\u2032\u2009cos\u2009\phi ii\u2032\u2009cos\u2009\phi jj\u2032\u2009cos\u2009\phi kk\u2032\beta i\u2032k\u2032j\u2032.$(6)
Considering SHRE with a single dominant axis of hyperpolarizability and defining the molecular system of coordinates with the $y\u2032$ axis coinciding with the hyperpolarizability axis, the only nonzero component of $\beta $ is $\beta y\u2032y\u2032y\u2032$ (which, with abuse of notation, will be hereafter denoted simply as $\beta $). Then, Eq. (6) can be re-written as:Display Formula$\chi ijk(2)=\u2211n\u2009cos\u2009\phi iy\u2032\u2009cos\u2009\phi jy\u2032\u2009cos\u2009\phi ky\u2032\beta =N\beta \u2329cos\u2009\phi iy\u2032\u2009cos\u2009\phi jy\u2032\u2009cos\u2009\phi ky\u2032\u232a,$(7)
where $N$ is the number of the emitters. As noted above, biologically relevant SHG-emitting samples are characterized by a distribution of SHREs with cylindrical symmetry. We define the laboratory system of coordinates ($x$, $y$, $z$) with the $y$-axis along the axis of sample cylindrical symmetry [see Fig. 3(a)]. Under the assumption that, within the cylindrical symmetry, the emitters are oriented at a fixed polar angle $\u03d1$ with respect to the symmetry axis [see Fig. 3(b)], computation of the tensor elements using Eq. (7) produces the following nonzero components: Display Formula${\chi yyy(2)=N\beta \u2009cos3\u2009\u03d1\chi yxx(2)=\chi xxy(2)=\chi yzz(2)=\chi zyy(2)=N2\beta \u2009cos\u2009\u03d1\u2009sin2\u2009\u03d1.$(8)
The second-order susceptibility tensor, therefore, can be written as Display Formula$\chi (2)=(00000\chi xxy(2)\chi yxx(2)\chi yyy(2)\chi yzz(2)000000\chi zyz(2)00).$(9)
Considering an electric field propagating along the $z$-axis and linearly polarized at an angle $\psi $ with respect to the $y$-axis [Fig. 3(a)], Display Formula$E\u2192=E\u2009sin\u2009\psi e\u2227x+E\u2009cos\u2009\psi e\u2227y\u2009$(10)
and substituting Eqs. (9) and (10) into Eq. (5), the second-order polarization can be written as Display Formula$P\u2192(2)=2E2\u2009sin\u2009\psi \u2009cos\u2009\psi \chi yxx(2)e\u2227x+[E2\u2009sin2\u2009\psi \chi yxx(2)+E2\u2009cos2\u2009\psi \chi yyy(2)]e\u2227y.$(11)
The intensity of SHG ($ISHG$) is proportional to the square of the second-order polarization: Display Formula$ISHG\u221d[P\u2192(2)]2=E4[\chi yxx(2)]2{sin2\u20092\psi +[sin2\u2009\psi +\chi yyy(2)\chi yxx(2)\u2009cos2\u2009\psi ]2}.$(12)
The simple case illustrated in Fig. 1(a) (extended to $N$ molecules) can be described by setting the polar angle $\u03d1$ to zero so that Display Formula$ISHG\u221dE4N2\beta 2\u2009cos4\u2009\psi .$(13)
This equation provides a quantitative description of the coherent summation at the basis of SHG described in the previous section.