Theoretical simulation study of linearly polarized light on microscopic second-harmonic generation in collagen type I

Ying Chang; Changshui Chen; Jianxin Chen; Ying Jin; Xiao-Yuan Deng

doi:10.1117/1.3174427

1 July 2009 Theoretical simulation study of linearly polarized light on microscopic second-harmonic generation in collagen type I

Ying Chang, Changshui Chen, Jianxin Chen, Ying Jin, Xiao-Yuan Deng

Author Affiliations +

Journal of Biomedical Optics, Vol. 14, Issue 4, 044016 (July 2009). https://doi.org/10.1117/1.3174427

Abstract

A theoretical model is established for dealing with second-harmonic generation (SHG) in type I collagen excited by linearly polarized light focused by a microscope. With this model, the effects of the polarization angle α, numerical aperture (NA), as well as the ratio of hyperpolarizability ρ=β xxx/β xyy on SHG emission have been investigated. Simulation results reveal that SHG emission power changes periodically as α. The use of lower NA leads to weaker SHG emission but is more concentrated in two closer lobes, whereas more distributed emission in two detached lobes appear at higher NA. As the introduction of polarization direction, which is not along with the fiber axis (alpha ≠ 0 deg), one more element β xyy is valid in our case than β xxx alone, while their ratio ρ plays a very important role for collagen features characterization. SHG emission with ρ shows complicated modality that SHG emission is different at different α and not symmetric at ± ρ except at α=0 deg, suggesting the important impact of polarization working on ρ for SHG emission. Our theoretical simulation results provide useful clues for experimental study of microscopic SHG emission in collagen excited by linearly polarized beam.

1. Introduction

Second-harmonic generation (SHG) microscopy, based on radiative SHG under a focused beam has attracted more and more interest because it has been proved to be a powerful tool for visualization of cell and tissue structures and functions with microresolution.^{1, 2}

Collagen has the particular relevance of microscopic SHG applications onto biological imaging.^{3, 4} Collagen is the most abundant intrinsic structural protein in tissues. Collagen has a highly crystalline triple-helix structure, which is not centrosymmetric,^{5, 6} and the molecules are organized on the scale of the wavelength of light, which suggests that collagen is a promising biomedical material capable of SHG. Meanwhile, collagen possesses first hyperpolarizabilities $β$ large enough for practical SHG.⁷

SHG in collagen fiber has been widely investigated, experimentally and theoretically. However, although many experiments on SHG in collagen have been accomplished under the microscope, theoretically, SHG in collagen under a focused beam has far from thorough investigation, such as the polarization effect. Thus far, polarization effect on SHG under focused beam has been investigated only on uniformly distributed dye dipoles on a planar cell membrane.⁸ In collagen, such studies have been concentrated on the excitation by collimated light rather than focused light.^{9, 10} Under a focused beam, only in a very special condition, such as the excitation beam, which polarizes parallel to the fibril axis, has been dealt with.¹¹ SHG is a nonlinear optical phenomenon involving coherent photon scattering. Because of the coherent nature of SHG, the radiation of SHG is usually highly directional and depends critically on the emission source, such as its spatial extent, polarization of the excitation electric field.^{12, 13} Also, we realize that collagen fiber has its own physical properties compared to the uniformly distributed dipole on surface, and in practice, the polarization of the excitation light relative to the collagen fiber axis is indeterminable because of the unknown sample-laid position. It is therefore desirable for the theoretical exploration of SHG emission under the polarized focused beam on collagen fibrils.

Moreover, it is well known that the structure of collagen plays an important role in various pathological processes and diseases, such as wound healing, malignancy, or development,¹⁴ and SHG microscopy can be accomplished in vivo. Thus, this theoretical analysis exhibits promise for biomedical assessment of tissue structure, especially in pathological processes where the collagen is being actively remodeled.

Therefore, in this paper, we aim to explore the effects of polarized focused excitation beam on SHG in collagen. A single fibril is applied as the collagen model in this paper.

2. Theory of SHG Field Propagation under a Linearly Polarized Focused Beam

We consider here an illumination beam focusing on to the collagen fiber through a microscopic objective with the numerical aperture of $NA = n_{ω} \sin ϕ$ . The light is linearly polarized in the $x - y$ plane $({\overset{⃑}{E}}_{ω})$ with the angle of $α$ from the $\overset{⃑}{x}$ direction and propagates in the $\overset{⃑}{z}$ direction with an angular frequency of $ω$ . Collagen fiber is assumed along the $\overset{⃑}{x}$ direction. The fibrils within the fiber are generally aligned to the fiber axis, and the nonlinear scatterers (dipoles) are distributed uniformly within the collagen fibrils. The diagram of coordinate system for calculating the SHG field at an observation point $(r, θ, φ)$ is shown in Fig. 1 .

Fig. 1

Schematic diagram defining SHG emission from collagen under linearly polarized focused beam. Collagen fiber is assumed along the $\overset{⃑}{x}$ direction. The fibrils within the fiber are generally aligned to the fiber axis. The light is linearly polarized in $x - y$ plane $({\overset{⃑}{E}}_{ω})$ with the angle of $α$ from $\overset{⃑}{x}$ direction and propagates in the $\overset{⃑}{z}$ direction. SHG propagates in the emission plane, defined by $(θ, φ)$ .

2.1.

Electric-field ${\overset{⃑}{E}}_{ω}$ Profile of the Linearly Polarized Focused Beam

The local field distribution of excitation beam within the focus is very important for the determination of the propagation direction of the SHG field. For nonlinear applications, a linearly polarized, $\overset{⃑}{z}$ -directed beam at the focus can be well approximated by a three-dimensional Gaussian profile as follows:¹⁵

1. Eq. 1a

{\overset{⃑}{E}}_{ω} (x, y, z) = - i {\vec{E}}_{ω}^{(0)} \exp (- \frac{x^{2} + y^{2}}{w_{x y}^{2}} - \frac{z^{2}}{w_{z}^{2}} + i ξ k_{ω} z) \hat{ε},

Eq. 1b

{\vec{E}}_{ω}^{(0)} = {\vec{E}}_{ω} (0, 0, 0),

where

{\overset{⃑}{E}}_{ω} (x, y, z)

is the complex amplitude of the driving field at the point

(x, y, z)

,

\hat{ε}

represents its polarization direction (

ε_{z} = 0

in our case).

k_{ω} = 2 π n_{ω} ∕ λ_{ω}

is the wave vector in a specimen with a refractive index of

n_{ω}

.

ξ

is the wave-vector reduction factor due to focusing that accounts for a reduction in axial momentum by conversion to lateral momentum components within the focus.

w_{x y}

and

w_{z}

are the

1 ∕ e

radii of the focal ellipse in the lateral and axial directions, respectively.¹

2. Eq. 2a

w_{x y} = {\begin{matrix} \frac{0.320 λ_{ω}}{NA} & NA ⩽ 0.7 \\ \frac{0.325 λ_{ω}}{{NA}^{0.91}} & NA > 0.7 \end{matrix},

Eq. 2b

w_{z} = 0.532 λ_{ω} [\frac{1}{n_{ω} - \sqrt{n_{ω}^{2} - {NA}^{2}}}] .

2.2.

Induced Dipole Moment of SHG

The total dipole moment $\vec{μ}$ induced by the field ${\vec{E}}_{ω}$ is often written as a Taylor series¹⁶

Eq. 3

\vec{μ} = {\vec{μ}}_{0} + α \cdot {\vec{E}}_{ω} + \frac{1}{2!} β \cdot {\vec{E}}_{ω}^{2} + \frac{1}{3!} γ \cdot {\vec{E}}_{ω}^{3} + \dots .

Here,

{\vec{μ}}_{0}

is the permanent dipole moment, the frequency dependent

α

,

β

, and

γ

are the linear polarizability, and the first and second hyperpolarizability, respectively. SHG is related to the third term, and the total local dipole moment per unit scatterer concentration can be expressed in more detail as⁸

Eq. 4

{\vec{μ}}_{2 ω, i} (x, y, z) = \frac{1}{2} {\vec{E}}_{ω}^{2} (x, y, z) \sum_{j, k} ⟨ β_{i j k} ⟩ {\hat{ε}}_{j} {\hat{ε}}_{k},

where

⟨ β ⟩

symbolizes a local ensemble average of the fibril hyperpolarizabilities. Under the assumption of Kleinman and cylindrical symmetry (please see the Appendix { label needed for app[@id='xA'] }), dipole moment introduced by electric field

{\vec{E}}_{ω}

along the

\vec{x}

and

\vec{y}

direction is respectively to be

5. Eq. 5a

μ_{2 ω, x} (x, y, z) = \frac{1}{2} E_{ω}^{2} (x, y, z) (β_{x x x} ε_{x} ε_{x} + β_{x y y} ε_{y} ε_{y}),

Eq. 5b

μ_{2 ω, y} (x, y, z) = \frac{1}{2} E_{ω}^{2} (x, y, z) (β_{x y y} ε_{x} ε_{y} + β_{x y y} ε_{x} ε_{y}) .

In particular, for a linearly polarized beam at an angle

α

with respect to the collagen fiber produces the SHG dipole moment as follows:

Eq. 6

{\vec{μ}}_{2 ω} (x, y, z) = (\begin{matrix} μ_{2 ω, x} (x, y, z) \\ μ_{2 ω, y} (x, y, z) \\ μ_{2 ω, z} (x, y, z) \end{matrix}) = \frac{1}{2} {\vec{E}}_{ω}^{2} (x, y, z) (\begin{matrix} β_{x x x} ε_{x} ε_{x} + β_{x y y} ε_{y} ε_{y} \\ β_{x y y} ε_{x} ε_{y} + β_{x y y} ε_{y} ε_{x} \\ 0 \end{matrix}) = \frac{1}{2} {\vec{E}}_{ω}^{2} (x, y, z) (\begin{matrix} β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α \\ β_{x y y} \sin 2 α \\ 0 \end{matrix}) = \frac{1}{2} {\vec{E}}_{ω}^{2} (x, y, z) β .

2.3.

Induced Electric Field ${\vec{E}}_{2 ω}$ of SHG

Following the achievement of the induced SHG dipole moment, we proceed to calculate the second-harmonic far field radiated in the direction $(θ, φ)$ based on the coordinate system illustrated in Fig. 1. Far from the dipole, where it is called the wave or radiation zone, the configuration of ${\vec{E}}_{2 ω}$ is as follows:¹⁷

Eq. 7

{\vec{E}}_{2 ω} (ψ) = \frac{{\vec{μ}}_{2 ω} ω^{2}}{π ε_{0} c^{2} r} \sin (ψ) \exp (- i {\vec{k}}_{2 ω} \cdot \vec{r}) \hat{ψ},

where

ψ

represents the angle between the

\overset{⃑}{x}

-axis and the emission direction

\overset{⃑}{r}

of SHG.

\sin ψ = {(\sin^{2} θ \sin^{2} φ + \cos^{2} θ)}^{1 ∕ 2}

illustrates the projection relationship between the direction of emission SHG dipole moment

{\vec{μ}}_{2 ω}

and the excitation electric field

{\vec{E}}_{2 ω}

of SHG. We define

η = ω^{2} ∕ π ε_{0} c^{2}

, where

ε_{0}

is the spree-space permittivity,

c

is the speed of light, and

ω

is the frequency of fundamental beam.

The total radiated second-harmonic signal in the far field is the integrated one from all scatterers that have the dipole volume density defined by $N_{V}$ . Presented in polar coordinates, ${\vec{E}}_{2 ω}$ is described as follows:

Eq. 8

{\vec{E}}_{2 ω} (θ, φ) = \frac{η N_{V}}{r} \int \int \int \sin ψ \cdot {\vec{μ}}_{2 ω} (x, y, z) \exp [- i k_{2 ω} (z \cos θ + y \sin θ \sin φ + x \sin θ \cos φ)] d x d y d z = \frac{η N_{V}}{r} {(\sin^{2} θ \sin^{2} φ + \cos^{2} θ)}^{1 ∕ 2} \int \int \int \frac{1}{2} β {\vec{E}}_{ω}^{(0) 2} \exp (- 2 \frac{x^{2} + y^{2}}{w_{x y}^{2}} - 2 \frac{z^{2}}{w_{z}} + 2 i ξ k_{ω} z) \times \exp [- i k_{2 ω} (z \cos θ + y \sin θ \sin φ + x \sin θ \cos φ)] d x d y d z .

(Derivation from Eq. 8, 9, see Appendix { label needed for app[@id='xB'] }).

Eq. 9

{\vec{E}}_{2 ω} (θ, φ) = {(\sqrt{\frac{π}{2}})}^{3} N_{V} w_{x y}^{2} w_{z} \frac{η}{r} {(\sin^{2} θ \sin^{2} φ + \cos^{2} θ)}^{1 ∕ 2} \cdot {\vec{μ}}_{2 ω}^{(0)} \times \exp {- \frac{k_{2 ω}^{2}}{8} [w_{x y}^{2} {(\sin θ \cos φ)}^{2} + w_{x y}^{2} {(\sin θ \sin φ)}^{2} + w_{z}^{2} {(\cos θ - ξ^{'})}^{2}]} = N {\vec{E}}_{2 ω}^{(0)} A (θ, φ),

where the introduced parameters are

{\vec{μ}}_{2 ω}^{(0)} = (\begin{matrix} μ_{2 ω, x}^{(0)} \\ μ_{2 ω, y}^{(0)} \\ μ_{2 ω, z}^{(0)} \end{matrix}) = \frac{1}{2} {\vec{E}}_{ω}^{2} (0, 0, 0) β = \frac{1}{2} {\vec{E}}_{ω}^{(0) 2} β,

which represents the contribution from the local induced polarization per unit volume density to the radiated electric field at the flocal center only [indicated by the superscript (0)].

Eq. 10

N = {(\sqrt{\frac{π}{2}})}^{3} w_{x y}^{2} w_{z} N_{V}

Eq. 11

{\vec{E}}_{2 ω}^{(0)} = \frac{η}{r} {(\sin^{2} θ \sin^{2} φ + \cos^{2} θ)}^{1 ∕ 2} \cdot {\vec{μ}}_{2 ω}^{(0)}

Eq. 12

A = \exp {- \frac{k_{2 ω}^{2}}{8} [w_{x y}^{2} {(\sin θ \cos φ)}^{2} + w_{x y}^{2} {(\sin θ \sin φ)}^{2} + w_{z}^{2} {(\cos θ - ξ^{'})}^{2}]} = \exp {- \frac{k_{2 ω}^{2}}{8} [w_{x y}^{2} \sin^{2} θ + w_{z}^{2} {(\cos θ - ξ^{'})}^{2}]} .

A

is the angular modulation term.

$N = V N_{V}$ , and $V = {(\sqrt{π ∕ 2})}^{3} w_{x y}^{2} w_{z}$ is the active SHG volume.¹⁸ $N$ defines as the effective total number of dipoles that contribute to the generation of second-harmonic light.

2.4.

Emission Direction of Electric Field of SHG

Emission direction is confined by the emission plane $(θ, φ)$ . Electric field of SHG is normal to the observed direction $\overset{⃑}{r}$ . The radiated SHG with polarization components parallel $(p)$ to and perpendicular $(s)$ to this plane can be derived as follows:

Eq. 13

{\vec{E}}_{2 ω}^{(0)} (θ, φ) = (\begin{matrix} E_{2 ω}^{(0) p} (θ, φ) \\ E_{2 ω}^{(0) s} (θ, φ) \end{matrix}) = \frac{η}{r} \vec{M} \cdot {\vec{μ}}_{2 ω}^{(0)} .

\vec{M}

is the projection matrix, which permutes coordinate

(x, y, z)

to

(θ, φ)

. It is defined by

Eq. 14

\vec{M} (\begin{matrix} \vec{θ} \\ \vec{φ} \end{matrix}) = (\begin{matrix} \cos θ \cos φ & \cos θ \sin φ & - \sin θ \\ - \sin φ & \cos φ & 0 \end{matrix}) .

On the basis of Eq. 9,

E_{2 ω}^{p} (θ, φ)

and

E_{2 ω}^{s} (θ, φ)

is then derived from

{\vec{E}}_{2 ω}^{(0)} (θ, φ)

.

E_{2 ω}^{p}

is the amplitude of

{\vec{E}}_{2 ω}

in the

\overset{⃑}{θ}

direction and

E_{2 ω}^{s}

is the amplitude of

{\vec{E}}_{2 ω}

in the

\overset{⃑}{φ}

direction.⁸

However, in practice, the measurements of components of $p$ and $s$ directions of SHG are difficult because they are not fixed as the emission plane changes. On the contrary, SHG are more easily to be detected along the parallel (∥) and perpendicular (⊥) direction to the excitation polarization $(α)$ . Therefore, we further transform the $(p, s)$ components to be (∥,⊥) components through the following relationship:

Eq. 15

(\begin{matrix} E_{2 ω}^{∥} \\ E_{2 ω}^{⊥} \end{matrix}) = [\begin{matrix} \cos (φ - α) & - \sin (φ - α) \\ \sin (φ - α) & \cos (φ - α) \end{matrix}] (\begin{matrix} E_{2 ω}^{p} \\ E_{2 ω}^{s} \end{matrix}) .

2.5.

Emission Power of SHG

On the basis of the above derivation, the relationship of total power distribution of SHG with the electric-field strength presented in Eq. 9 can be further described to be (see Appendix { label needed for app[@id='xC'] })

16.

P_{2 ω} (θ, φ) = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} A^{2} N^{2} E_{ω}^{(0) 4},

(1 - \sin^{2} θ \cos^{2} φ) {(\begin{matrix} β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α \\ β_{x y y} \sin 2 α \\ 0 \end{matrix})}^{2} .

By integrating Eq. 16 in solid angle, total power of SHG is derived, which only depends on the polarized direction

α

Eq. 17

P_{2 ω} = \frac{1}{2} n_{2 ω} ε_{0} c η^{2} N^{2} Θ (μ_{2 ω, x}^{(0) 2} + μ_{2 ω, y}^{(0) 2} + μ_{2 ω, z}^{(0) 2}) = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} Θ E_{ω}^{(0) 4} [{(β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α)}^{2} + {(β_{x y y} \sin 2 α)}^{2}],

where

Θ = \int \int A^{2} (θ, φ) (1 - \sin^{2} θ \cos^{2} φ) \sin θ d θ d φ,

Accordingly, the power distribution in parallel (∥) and perpendicular (⊥) components are as follows (see Appendix { label needed for app[@id='xD'] }):

18. Eq. 18a

P_{2 ω}^{∥} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} {∣ E_{2 ω}^{∥} (θ, φ) ∣}^{2} = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} A^{2} (θ, φ) E_{ω}^{(0) 4} {\cos (φ - α) [\cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α] - \sin (φ - α) [- \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α]}^{2},

Eq. 18b

P_{2 ω}^{⊥} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} {∣ E_{2 ω}^{⊥} (θ, φ) ∣}^{2} = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} A^{2} (θ, φ) E_{ω}^{(0) 4} {\sin (φ - α) [\cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α] + \cos (φ - α) [- \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α]}^{2} .

SHG power in these two directions (∥,⊥), respectively, can be calculated by integrating in solid angle as follows:

19. Eq. 19a

P_{2 ω}^{∥} = \int \int P_{2 ω}^{∥} (θ, φ) \sin θ d θ d φ,

Eq. 19b

P_{2 ω}^{⊥} = \int \int P_{2 ω}^{⊥} (θ, φ) \sin θ d θ d φ .

3. Simulation Study of Parameters of Linearly Polarized Focused Beam on SHG Emission

3.1.

Effects of Illumination Polarization on SHG Emission Power under Focused Beam

In this section, we aim to explore the effects of illumination polarization on SHG emission power under focused beam. From the theoretical expression of total, parallel, and perpendicular SHG power in Eqs. 17, 19a, 19b, we note that there remain two valid hyperpolarizability elements $β_{x x x}$ and $β_{x y y}$ . $β_{x y y}$ is added in this situation due to the introduce of the illumination polarization $α$ , which is not consistent with the fiber axis $(α = 0 \deg)$ . $ρ = β_{x x x} ∕ β_{x y y}$ has been evaluated to vary between $- 3$ to 3,¹⁹ in this section, we take $ρ$ to be 2.6 (type I collagen in rat liver) and further assume $β_{x y y} = 1$ to simplify the model. The effect of $ρ = β_{x x x} ∕ β_{x y y}$ is to be investigated in more detail in Sec. 3.3.

Fig. 2 shows the total $P_{2 ω}$ , parallel $P_{2 ω}^{∥}$ , and perpendicular $P_{2 ω}^{⊥}$ SHG emission power from collagen fiber as the change of polarization angle $α$ with respect to fiber axis. All data are normalized by $(1 ∕ 8) n_{2 ω} ε_{0} c η^{2} N^{2} E_{ω}^{(0) 4} = 1$ . We can see from Fig. 2 that $P_{2 ω}$ , $P_{2 ω}^{∥}$ , and $P_{2 ω}^{⊥}$ SHG emission power change as the polarization angle $α$ periodically. When polarization direction locates at $α = 0$ , 180, $360 \deg$ , indicating that the polarization of the excited beam has the same direction as the fibrils axis, the total power $P_{2 ω}$ , and parallel $P_{2 ω}^{∥}$ power component achieve their maximum value, while the perpendicular $P_{2 ω}^{⊥}$ power component has its lowest value. On the contrary, we find that when the polarization of excited beam is perpendicular on the fibril axis ( $α = 90$ , $270 \deg$ ) the total power $P_{2 ω}$ and parallel $P_{2 ω}^{∥}$ power component have their lowest value and perpendicular $P_{2 ω}^{⊥}$ component is the reverse. $P_{2 ω}$ and $P_{2 ω}^{∥}$ power component keep the similar configuration in amplitude and $P_{2 ω}^{∥}$ stands predominance in contribution to the SHG total power.

Fig. 2

Effects of polarization angle $α$ on total $P_{2 ω}$ , parallel $P_{2 ω}^{∥}$ and perpendicular $P_{2 ω}^{⊥}$ components of SHG emission power.

Parallel $P_{2 ω}^{∥}$ and perpendicular $P_{2 ω}^{⊥}$ power distribution along $(θ, φ)$ , using Eqs. 18a, 18b, at polarization angles $α = 0$ , $α = 30$ , $α = 60$ , and $α = 90 \deg$ , respectively have been demonstrated in three-dimension in Fig. 3 . Basically, the parallel and perpendicular distribution of SHG emission possess two types of patterns. Under $α = 0$ , $α = 30$ , and $α = 60 \deg$ conditions, these three parallel distribution has similar patterns, and the perpendicular one has another similar patterns; however, under the $α = 90 \deg$ situation, the distribution shows reverse patterns in parallel and perpendicular components. As the polarization angle increases from $α = 0$ to $α = 90 \deg$ , SHG parallel emission power in the $\vec{z}$ -axis $(θ = 0 \deg)$ decreases, while the perpendicular one increases. Although the emission distribution pattern of the parallel component at $α = 90 \deg$ and that of perpendicular components at $α = 0 \deg$ are very similar, their amplitude is still slight different, which indicates that although the relative direction of measurement direction and fibril axis is the same as the above two conditions (which is perpendicular to the fiber axis), the polarization direction relative to the fiber axis has an important influence on the SHG emission power.

Fig. 3

Effects of polarization angle $α$ on SHG emission power distribution $(θ, φ)$ of parallel $P_{2 ω}^{∥}$ and perpendicular $P_{2 ω}^{⊥}$ components.

3.2.

Effects of the Numerical Aperture (NA) on SHG Emission

The most important parameter to characterize the focused beam is the numerical aperture, $NA = n_{ω} \sin ϕ$ , where $ϕ$ represents the incident angle of the illumination light (please refer to Fig. 1).

The focus of the light causes the phase retardation near the focus center along the axial direction. This phase retardation is called Guoy phase shift and can be represented by a wave-vector reduction factor $ξ$ . The change of NA will induce the change of $ξ$ as approximately by $ξ \approx \cos (ϕ ∕ \sqrt{2})$ (when $NA < 1.2$ at most⁸).

In our single-fibril model, the phase match term, including the Guoy phase shift and the dispersion due to the refraction index change between the incident $(n_{ω})$ and second harmonic frequency $(n_{2 ω})$ , can be expressed as

Eq. 20

Δ \overset{⃑}{k} = {\overset{⃑}{k}}_{2 ω} - 2 ξ {\overset{⃑}{k}}_{ω} (k_{ω} = 2 π n_{ω} ∕ λ_{ω} and k_{2 ω} = 2 π n_{2 ω} ∕ λ_{2 ω}) .

$ξ$ appears in the axial direction of the focal ellipse $(w_{z})$ exclusively. Momentum conservation along the axial direction then forces the SHG $(k_{2 ω})$ to propagate at an off-axis angle $θ$ . This angle $θ$ is $θ = \cos^{- 1} [ξ (n_{ω} ∕ n_{2 ω})]$ to satisfy the phase-match condition, $k_{2 ω} \cos θ - 2 ξ k_{ω} = 0$ , thus achieves the max SHG emission intensity. Figure 4a illustrates the relationship between the phase match terms. The solid arrow stands for the condition of phase match, and the dash arrow represents for condition of the phase mismatch. $ξ^{'} = ξ n_{ω} ∕ n_{2 ω}$ has been introduced in Eq. 33c in Appendix { label needed for app[@id='xB'] } thus $θ = \cos^{- 1} (ξ)$ . If $n_{ω} = n_{2 ω}$ , where dispersion effect has been ignored, then $θ = \cos^{- 1} (ξ)$ .

Fig. 4

Phase-match condition makes SHG propagate in the direction of $k_{2 ω}$ , which deviates an angle of $θ$ with respect to incident beam $k_{ω}$ due to the Guoy phase shift $(ξ)$ and dispersion. (b) Effects of NA on SHG electrical radiation pattern (on the right is the enlargement part from the square center of the image on the left).

In this paper, the refractive index of SHG and fundamental excitation light in collagen are correspondingly taken to be $n_{2 ω} = 1.503$ , $n_{ω} = 1.453$ based on the research results from Bashkatov.²⁰ To demonstrate the influence of NA under the consideration of dispersion, SHG emissions under $NA = 0.8$ and $NA = 0.2$ has been compared as those shown in Fig. 4b. We note that due to the influence of dispersion, no matter at high NA or low NA, SHG emission presents two off-axis lobes. However, at higher NA, the SHG emission lobes are much more distributed and detached, the max value of SHG emission appears at the angle of $θ = 27 \deg$ . On the other hand, at lower NA, the SHG emission concentrates along small angle $θ = 16 \deg$ and has much weaker intensity.

To investigate how NA under polarized beam influences the total power, parallel, and perpendicular components, $P_{2 ω}$ , $P_{2 ω}^{∥}$ , and $P_{2 ω}^{⊥}$ [all data are normalized by $(1 ∕ 8) n_{2 ω} ε_{0} c η^{2} N^{2} E_{ω}^{(0) 4} = 1$ ] along NA, which vary from 0–1.0 under polarization angle $α = 0, 30, 60, 90 \deg$ is demonstrated in Fig. 5 . In general, we note that the total emission power of SHG increases as the increase of NA, but the change does not show linearity. As the increase of NA, SHG parallel emission at $α = 0, 30, 60 \deg$ has the similar trend of change but different degree and at $α = 90 \deg$ , there is no change. Whereas SHG perpendicular emission also increases at different degree when $α = 30, 60, 90 \deg$ and the slight increase at $α = 0 \deg$ can be ignored. As the polarization angle $α$ changes from $0 to 90 \deg$ , the total and parallel components of SHG emission power decrease, while the perpendicular one increases.

Fig. 5

Effects of NA on total $P_{2 ω}$ , parallel $P_{2 ω}^{∥}$ , and perpendicular $P_{2 ω}^{⊥}$ components of SHG emission power under different polarization angle $α$ .

3.3.

Effects of the Ratio $ρ = β_{x x x} ∕ β_{x y y}$ under Different Polarization Direction

$β$ is the hyperpolarizability of the collagen fibrils which is related to the electronic transition in the material, thus reflects its biological structure and chemical features.^{21, 22} In our specific collagen model with the polarized illumination beam, two elements $β_{x x x}$ , $β_{x y y}$ remain and have the contribution to SHG. The ratio $ρ = β_{x x x} ∕ β_{x y y}$ plays very important role in characterizing the collagen features. It has been proved that $ρ$ increases with the increasing of age.^{9, 11} As mentioned above, $ρ = β_{x x x} ∕ β_{x y y}$ has been predicted to vary between $- 3$ to 3.¹⁹ The minus value of $ρ$ suggests the polarity of the fibrils reverse from the excited electrical field. The study of the effect of $ρ$ on SHG emission power may provide us significant information for tissue characterization.

How the value of $ρ = β_{x x x} ∕ β_{x y y}$ varies from $- 3$ to 3 influence the total power, parallel and perpendicular components at different polarization angle $α$ has been three-dimensionally presented in Fig. 6 . The overall configuration of the SHG emission pattern indicates that the influence of minus value and positive value of $ρ$ is not symmetric when $α \neq 0 \deg$ . For more details, two-dimensional figures under four particular polarization angles $α = 0, 30, 60, 90 \deg$ are selectively demonstrated in Fig. 7 . We note that no matter the total, parallel or the perpendicular components, at different polarization conditions, the change patterns as $ρ$ are quite different. And from minus value to positive value of $ρ$ , the change patterns are not symmetric except under the excitation condition of $α = 0 \deg$ . At polarization angle $α = 90 \deg$ , different $ρ$ makes $P_{2 ω}$ , $P_{2 ω}^{∥}$ , and $P_{2 ω}^{⊥}$ have no change. At polarization $α = 0 \deg$ , as the absolute value of $ρ$ increases, all SHG emission power (total, parallel, and perpendicular components) increases; however, the increase; of $ρ$ has minor contribution to the perpendicular one. At polarization $α = 30 \deg$ , as $ρ$ changes from $- 3$ to 3, the total, parallel, and perpendicular components of the SHG emission power change from high to low value and then increase again, but the lowest turning point does not locate at $ρ = 0$ . At polarization $α = 60 \deg$ , the curves increase slowing and have no obvious turning point along $ρ$ except in the perpendicular one, which slightly shows a turning.

Fig. 6

Three-dimensional demonstration of effects of ratio $ρ$ of hyperpolarizability elements of $β_{x x x}$ to $β_{x y y}$ on total, parallel, and perpendicular SHG power emission at different polarization angle $α$ .

Fig. 7

Two-dimensional demonstration of effects of ratio $ρ$ of hyperpolarizability elements of $β_{x x x}$ to $β_{x y y}$ on total, parallel, and perpendicular SHG power emission at different polarization angle $α$ .

4. Discussion and Conclusions

In this paper, a theoretical model of SHG on type I collagen under polarized focused beam has been established. On the basis of this model, we have investigated the total, parallel, perpendicular SHG emission power and their distribution along $(θ, φ)$ , induced by different polarization angles $α$ of the illumination light. The total, parallel, and perpendicular SHG emission power periodically change as the polarization angle $α$ , similar to the pattern produced from the uniformly distributed dye dipoles on the cell membrane. The distribution of parallel, perpendicular components along $(θ, φ)$ is complicated and possesses multiple peaks, but they symmetrically distribute around the plane, where fibrils locate ( $θ = 0 \deg$ , $φ = 180 \deg$ ). NA is another important factor that influences the illumination source. Therefore, the effects of NA on SHG emission are also investigated. At lower NA (for example, $NA = 0.2$ ), the SHG emission concentrates in two closer lobes (like two lines), while at relative higher NA, such as $NA = 0.8$ , SHG emission pattern has changed to be two obvious distributed lobes. At lower NA, where the light is loosely focused, it acts like a collimated light on the samples; thus, the influence from the Guoy phase shift can be ignored and dispersion alone plays role on SHG emission. According to $θ = \cos^{- 1} [ξ (n_{ω} ∕ n_{2 ω})]$ , when $ξ = 1$ (unfocused light by very low NA), the emission angle $θ$ is smaller than that at $ξ < 1$ (focused light by high NA). However, at relative higher NA, where the light is tightly focused, because of the Guoy shift (phase shift) induced by focusing as well as the dispersion effect, to keep the momentum equilibrium, SHG emission strays far away from the direction of illumination light. As NA increases, we realizes that the waists of $w_{x y}$ and $w_{z}$ decreases. On the basis of expression A in Eq. 12, we understand that NA not only affects the emission angle $θ$ , but also the distribution of the emission. At higher NA, SHG emission is much more distributed. The combination effect of NA under different polarization angles $α$ is also studied. We find the influence of NA on SHG emission under different excitation conditions of polarization angle $α$ is different, and the contribution of different NA to SHG emission is not linearity. However, our theoretical analysis results of NA are not applicable for linearly polarized light being focused with an objective with very high NA, because under such conditions, the polarization is no longer constant within the focus volume. At last, the ratio $ρ$ of two existing components of $β$ , which are $β_{x x x}$ and $β_{x y y}$ in our case is explored. Twenty-seven items of $β$ are reduced to seven nonzero elements (two independent) through cylindrical symmetry and rule of Kleinman symmetry. For the light linearly polarized in $x - y$ plane $(ε_{z} = 0)$ , those leave only two elements $β_{x x x}$ and $β_{x y y}$ . Here, the inconsistency of the polarization angle $α$ with respect to the fiber axis keeps one more valid hyperpolarizability element $β_{x y y}$ of collagen in the model rather than $β_{x x x}$ alone at $α = 0 \deg$ . As we have mentioned in Sec. 3.3, the significance of $ρ$ to collagen is obvious; therefore, SHG emission study of the changes of $ρ$ may offers us important clues of the relationship between SHG emission with collagen features, such as its maturity, polarity, as well as the pathologically induced characterization changes.

In conclusion, our theoretical simulation results provide useful clues for experimental study of microscopic SHG in collagen type I under linearly polarized beam.

Appendices

{ label needed for app[@id='xA'] }

Appendix A: Simplification of Nonlinear Optical Hyperpolarizability

${\vec{μ}}_{2 ω, i} (x, y, z)$ has three components, which are in the $x$ , $y$ , and $z$ directions respectively,

Eq. 21

{\vec{μ}}_{2 ω, i} (x, y, z) = \frac{1}{2} {\vec{E}}_{ω}^{2} (x, y, z) \sum_{j, k} ⟨ β_{i j k} ⟩ {\hat{ε}}_{j} {\hat{ε}}_{k} .

Its subscript

i

presents these three directions. On the basis of the definition of the second-order tensor, hyperpolarizabilities

β

owns 27 items

Eq. 22

(\begin{matrix} β_{x x x} & β_{x y y} & β_{x z z} & β_{x y z} & β_{x z y} & β_{x z x} & β_{x x z} & β_{x x y} & β_{x y x} \\ β_{y x x} & β_{y y y} & β_{y z z} & β_{y y z} & β_{y z y} & β_{y z x} & β_{y x z} & β_{y x y} & β_{y y x} \\ β_{z x x} & β_{z y y} & β_{z z z} & β_{z y z} & β_{z z y} & β_{z z x} & β_{z x z} & β_{z x y} & β_{z y x} \end{matrix}) .

Kleinman Symmetry

The assumption of Kleinman symmetry condition is applied to lossless media.²³ It is valid if the frequency $ω$ of excitation light is far from any natural frequencies of material. Because near-infrared light is usually used for SHG in biological tissue, in our case, if light with $800 - nm$ wavelength is assumed to be used, then it is far from the wavelength of the first electronic transition in collagen $(\sim 310 nm)$ ,²⁴ and therefore, Kleinman symmetry applies. Within in the validity of Kleinman symmetry,^{25, 26, 27, 28}

Eq. 23

β_{i j k} = β_{j k i} = β_{k i j} = β_{i k j} = β_{j i k} = β_{k j i} .

As a result, only ten independent elements of

β

remain, which are as follows:

24.

x x x, y y y, z z z, x y y = y x y = y y x, x z z = z x z = z z x, x x y = x y x = y x x,

x x z = x z x = z x x, y z z = z y z = z z y, z y y = y z y = y y z, x y z = x z y = y x z = y z x = z x y = z y x .

Cylindrical Symmetry

Collagen fiber possesses cylindrically symmetrical feature in structure^{29, 30, 31} (as shown in Fig. 8 ). Under such condition, the hyperpolarizability tensor can be further simplified. The cylinder symmetry indicates that by taking the changes of $x \to x$ , $y \to - y$ , $z \to - z$ on the subscripts of matrix $β_{i j k}$ in Eq. 22, the derived new matrix of $β_{i j k}$ in Eq. 5 should be the same as those in Eq. 22

Eq. 25

(\begin{matrix} x x x & x y y & x z z & x y z & x z y & - x z x & - x x z & - x x y & - x y x \\ - y x x & - y y y & - y z z & - y y z & - y z y & y z x & y x z & y x y & y y x \\ - z x x & - z y y & - z z z & - z y z & - z z y & z z x & z x z & z x y & z y x \end{matrix}) .

Therefore, the elements in Eq. 25 with minus sign should be zero and the elements of

β

now become

Eq. 26

(\begin{matrix} x x x & x y y & x z z & x y z & x z y & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & y z x & y x z & y x y & y y x \\ 0 & 0 & 0 & 0 & 0 & z z x & z x z & z x y & z y x \end{matrix}) .

Fig. 8

Schematic diagram of cylindrical symmetry of collagen fiber along $\vec{x}$ -axis.

Also, from Fig. 8, we get the relation between $y$ and $z$ because the cylinder symmetry by applying $x \to x$ , $y \to y^{'}$ , $z \to z^{'}$

Eq. 27

x = x, y = y^{'} \cos ϕ - z^{'} \sin ϕ, z = y^{'} \sin ϕ + z^{'} \cos ϕ .

A project matrix is used to present the relationship.

Eq. 28

(\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ϕ & - \sin ϕ \\ 0 & \sin ϕ & \cos ϕ \end{matrix}) (\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}) = (\begin{matrix} l_{x x} & l_{x y} & l_{x z} \\ l_{y x} & l_{y y} & l_{y z} \\ l_{z x} & l_{z y} & l_{z z} \end{matrix}) (\begin{matrix} x^{'} \\ y^{'} \\ z^{'} \end{matrix}) .

Or, further to be

Eq. 29

β_{i j k} = l_{i i^{'}} l_{j j^{'}} l_{k k^{'}} β_{i^{'} j^{'} k^{'}} .

According to the above expression, we have the following relationship:

β_{x x x} = l_{x x^{'}} l_{x x^{'}} l_{x x^{'}} β_{x^{'} x^{'} x^{'}}, β_{x y y} = l_{x x} l_{y z} l_{y z} β_{x z z} = \sin^{2} ϕ β_{x z z},

β_{x y z} = l_{x x} l_{y z} l_{z y} β_{x z y} = - \sin^{2} ϕ β_{x z y}, β_{y y x} = l_{y z} l_{y z} l_{x x} χ_{z z x} = \sin^{2} ϕ β_{z z x}

β_{y x y} = l_{y z} l_{x x} l_{y z} β_{z x z} = \sin^{2} ϕ β_{z x z}, β_{y z x} = l_{y z} l_{z y} l_{x x} χ_{z y x} = - \sin^{2} ϕ β_{z y x}

β_{y x z} = l_{y z} l_{x x} l_{z y} χ_{z x y} = - \sin^{2} ϕ β_{z x y}

The rotation between

y

and

z

,

ϕ = 90

or

270 \deg

, makes

\sin^{2} ϕ = 1

. Accoding to Eq. 4, we get

30. Eq. 30a

β_{x x x},

Eq. 30b

β_{x y y} = β_{x z z} = β_{y y x} = β_{z z x} = β_{y x y} = β_{z x z},

Eq. 30c

β_{x y z} = β_{x z y} = β_{y x z} = β_{y z x} = β_{z x y} = β_{z y x} = 0 .

As a result, there are only seven nonzero elements, in particular, two independent elements.

In our case, for the light linearly polarized in $x - y$ plane $({\overset{⃑}{E}}_{ω})$ , $ε_{z} = 0$ . It remains only two $(β_{x x x}, β_{x y y})$ of the above seven nonzero elements has the contribution to the dipole moment $μ$ , thus to the SHG emission.

{ label needed for app[@id='xB'] }

Appendix B: Integration of SHG emission electric field

In this part, we demonstrate the integration of the SHG emission electric field in $x$ , $y$ , $z$ direction from negative infinite to positive infinite.

31. Eq. 31a

{\vec{E}}_{2 ω} (θ, φ) = \frac{η N_{V}}{r} \int \int \int \sin ψ \cdot {\vec{μ}}_{2 ω} (x, y, z) \exp [- i k_{2 ω} (z \cos θ + y \sin θ \sin φ + x \sin θ \cos φ)] d x d y d z = η N_{V} ∕ r {(\sin^{2} θ \sin^{2} φ + \cos^{2} θ)}^{1 ∕ 2} \int \int \int \cdot \frac{1}{2} β {\vec{E}}_{ω}^{(0) 2} \exp (- 2 \frac{x^{2} + y^{2}}{w_{x y}^{2}} - 2 \frac{z^{2}}{w_{z}} + 2 i ξ k_{ω} z) \times \exp [- i k_{2 ω} (z \cos θ + y \sin θ \sin φ + x \sin θ \cos φ)] d x d y d z = N_{V} \frac{η}{r} {(\sin^{2} θ \sin^{2} φ + \cos^{2} θ)}^{1 ∕ 2} \cdot {\vec{μ}}_{2 ω}^{(0)}

Eq. 31b

\times \int \exp (- \frac{2 x^{2}}{w_{x y}^{2}} - i k_{2 ω} x \sin θ \cos φ) d x

Eq. 31c

\times \int \exp (- \frac{2 y^{2}}{w_{x y}^{2}} - i k_{2 ω} y \sin θ \sin φ) d y

Eq. 31d

\times \int \exp (- \frac{2 z^{2}}{w_{z}^{2}} + 2 i ξ k_{ω} z - i k_{2 ω} z \cos θ) d z .

According to the following integral equation,

Eq. 32

\int_{- \infty}^{+ \infty} \exp (- p^{2} x^{2} \pm q x) d x = \exp (\frac{q^{2}}{4 p^{2}}) \frac{\sqrt{π}}{p} (Re p^{2} > 0)

Equations 31b, 31c, 31d have the corresponding parameters of

p

and

q

, such as

33a.

p^{2} = \frac{2}{w_{x y}^{2}}, p = \frac{\sqrt{2}}{w_{x y}}, q = i k_{2 ω} \sin θ \cos φ, q^{2} = - k_{2 ω}^{2} {(\sin θ \cos φ)}^{2}

\int \exp (- \frac{2 x^{2}}{w_{x y}^{2}} - i k_{2 ω} x \sin θ \cos φ) d x = \exp [- \frac{k_{2 ω}^{2}}{8} w_{x y}^{2} {(\sin θ \cos φ)}^{2}] \sqrt{\frac{π}{2}} w_{x y}

33b.

p^{2} = \frac{2}{w_{x y}^{2}}, p = \frac{\sqrt{2}}{w_{x y}}, q = i k_{2 ω} \sin θ \sin φ, q^{2} = - k_{2 ω}^{2} {(\sin θ \sin φ)}^{2}

\int \exp (- \frac{2 y^{2}}{w_{x y}^{2}} - i k_{2 ω} y \sin θ \sin φ) d y = \exp [- \frac{k_{2 ω}^{2}}{8} w_{x y}^{2} {(\sin θ \sin φ)}^{2}] \sqrt{\frac{π}{2}} w_{x y}

33c.

p^{2} = \frac{2}{w_{z}^{2}}, p = \frac{\sqrt{2}}{w_{z}}, q = i k_{2 ω} \cos θ - 2 i ξ k_{ω}

k_{ω} = \frac{2 π n_{ω}}{λ_{ω}}, k_{2 ω} = \frac{2 π n_{2 ω}}{λ_{2 ω}}, ξ^{'} = ξ \frac{n_{ω}}{n_{2 ω}},

q^{2} = - k_{2 ω}^{2} {(\cos θ - ξ^{'})}^{2}

\int \exp (- \frac{2 z^{2}}{w_{z}^{2}} + 2 i ξ k_{ω} z - i k_{2 ω} z \cos θ) d z = \exp [- \frac{k_{2 ω}^{2}}{8} w_{z}^{2} {(\cos θ - ξ^{'})}^{2}] \sqrt{\frac{π}{2}} w_{z} .

{ label needed for app[@id='xC'] }

Appendix C: Total Power of SHG

The relation between total power distribution of SHG and electric field is¹⁸

Eq. 34

P_{2 ω} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} E_{2 ω}^{2} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} N^{2} A^{2} (θ, φ) E_{2 ω}^{(0) 2} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} N^{2} A^{2} (θ, φ) \frac{η^{2}}{r^{2}} (\sin^{2} θ \sin^{2} φ + \cos^{2} θ) μ_{2 ω}^{(0) 2} = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} A^{2} (θ, φ) E_{ω}^{(0) 4} (\sin^{2} θ \sin^{2} φ + \cos^{2} θ) β^{2} = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} E_{ω}^{(0) 4} A^{2} (θ, φ) (1 - \sin^{2} θ \cos^{2} φ) {(\begin{matrix} β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α \\ β_{x y y} \sin 2 α \\ 0 \end{matrix})}^{2} .

The total power of SHG then is derived from double integral on

θ

and

φ

in a solid angle,

35.

P_{2 ω} = \int \int P_{2 ω} (θ, φ) \sin θ d θ d φ = \int \int \frac{1}{2} n_{2 ω} ε_{0} c r^{2} N^{2} A^{2} (θ, φ) \frac{η^{2}}{r^{2}} (\sin^{2} θ \sin^{2} φ + \cos^{2} θ) μ_{2 ω}^{(0) 2} \sin θ d θ d φ = \frac{1}{2} n_{2 ω} ε_{0} c η^{2} N^{2} \int \int A^{2} (θ, φ) (\sin^{2} θ \sin^{2} φ + \cos^{2} θ) μ_{2 ω}^{(0) 2} \sin θ d θ d φ = \frac{1}{2} n_{2 ω} ε_{0} c η^{2} N^{2} Θ (μ_{2 ω, x}^{(0) 2} + μ_{2 ω, y}^{(0) 2} + μ_{2 ω, z}^{(0) 2}) = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} Θ E_{ω}^{(0) 4} [{(β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α)}^{2} + {(β_{x y y} \sin 2 α)}^{2}] .

Θ = \int \int A^{2} (θ, φ) (1 - \sin^{2} θ \cos^{2} φ) \sin θ d θ d φ

{\overset{⃑}{μ}}_{2 ω}^{(0)} = (\begin{matrix} μ_{2 ω, x}^{(0)} \\ μ_{2 ω, y}^{(0)} \\ μ_{2 ω, z}^{(0)} \end{matrix}) = \frac{1}{2} {\vec{E}}_{ω}^{(0) 2} (\begin{matrix} β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α \\ β_{x y y} \sin 2 α \\ 0 \end{matrix}) .

{ label needed for app[@id='xD'] }

Appendix D: Parallel and Perpendicular Components of SHG Emission Power

(\begin{matrix} E_{2 ω}^{∥} \\ E_{2 ω}^{⊥} \end{matrix}) = [\begin{matrix} \cos (φ - α) & - \sin (φ - α) \\ \sin (φ - α) & \cos (φ - α) \end{matrix}] (\begin{matrix} E_{2 ω}^{p} \\ E_{2 ω}^{s} \end{matrix}),

E_{2 ω}^{∥} = E_{2 ω}^{p} \cos (φ - α) - E_{2 ω}^{s} \sin (φ - α),

E_{2 ω}^{⊥} = E_{2 ω}^{p} \sin (φ - α) + E_{2 ω}^{s} \cos (φ - α) .

The parallel and perpendicular SHG emission power distributions are

36. Eq. 36a

P_{2 ω}^{∥} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} {∣ E_{2 ω}^{∥} (θ, φ) ∣}^{2} = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} {[\cos (φ - α) E_{2 ω}^{p} - \sin (φ - α) E_{2 ω}^{s}]}^{2},

Eq. 36b

P_{2 ω}^{⊥} (θ, φ) = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} {∣ E_{2 ω}^{⊥} (θ, φ) ∣}^{2} = \frac{1}{2} n_{2 ω} ε_{0} c r^{2} {[\sin (φ - α) E_{2 ω}^{p} + \cos (φ - α) E_{2 ω}^{s}]}^{2} .

While

{\vec{E}}_{2 ω}^{(0)} (θ, φ) = (\begin{matrix} E_{2 ω}^{(0), p} (θ, φ) \\ E_{2 ω}^{(0), s} (θ, φ) \end{matrix}) = \frac{η}{r} \vec{M} \cdot {\vec{μ}}_{2 ω}^{(0)} = \frac{η}{r} \vec{M} \cdot \frac{1}{2} {\vec{E}}_{ω}^{(0) 2} β = \frac{η}{2 r} E_{ω}^{{(0)}^{2}} (\begin{matrix} \cos θ \cos φ & \cos θ \sin φ & - \sin θ \\ - \sin φ & \cos φ & 0 \end{matrix}) (\begin{matrix} β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α \\ β_{x y y} \sin 2 α \\ 0 \end{matrix}) = \frac{η}{2 r} E_{ω}^{{(0)}^{2}} (\begin{matrix} \cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α \\ - \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α \end{matrix}) .

Therefore,

E_{2 ω}^{(0), p} (θ, φ) = \frac{η}{2 r} E_{ω}^{(0) 2} [\cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α],

E_{2 ω}^{(0), s} (θ, φ) = \frac{η}{2 r} E_{ω}^{(0) 2} [- \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α],

E_{2 ω} (θ, φ) = N E_{2 ω}^{(0)} A (θ, φ) = \frac{η N A (θ, φ)}{2 r} E_{ω}^{(0) 2} (\begin{matrix} \cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α \\ - \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α \end{matrix}),

E_{2 ω}^{p} (θ, φ) = \frac{η N A (θ, φ)}{2 r} E_{ω}^{(0) 2} [\cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α],

E_{2 ω}^{s} (θ, φ) = \frac{η N A (θ, φ)}{2 r} E_{ω}^{(0) 2} [- \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α] .

At last,

37. Eq. 37a

P_{2 ω}^{∥} (θ, φ) = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} A^{2} (θ, φ) E_{ω}^{(0) 4} {\cos (φ - α) [\cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α] - \sin (φ - α) [- \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α]}^{2} .

Eq. 37b

P_{2 ω}^{⊥} (θ, φ) = \frac{1}{8} n_{2 ω} ε_{0} c η^{2} N^{2} A^{2} (θ, φ) E_{ω}^{(0) 4} {\sin (φ - α) [\cos θ \cos φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos θ \sin φ \sin 2 α] + \cos (φ - α) [- \sin φ (β_{x x x} \cos^{2} α + β_{x y y} \sin^{2} α) + β_{x y y} \cos φ \sin 2 α]}^{2} .

Acknowledgment

The authors gratefully thank the National Natural Science Foundation of China (Grant No: 30470495) for its support.

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Citation Download Citation

Ying Chang, Changshui Chen, Jianxin Chen, Ying Jin, and Xiao-Yuan Deng "Theoretical simulation study of linearly polarized light on microscopic second-harmonic generation in collagen type I," Journal of Biomedical Optics 14(4), 044016 (1 July 2009). https://doi.org/10.1117/1.3174427

Published: 1 July 2009

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KEYWORDS

Second-harmonic generation

Polarization

Collagen

Chromium

Optical simulations

Phase shifts

Laser applications

1.

Introduction

2.

Theory of SHG Field Propagation under a Linearly Polarized Focused Beam

Fig. 1

2.1.

Electric-field E⃑ω Profile of the Linearly Polarized Focused Beam

1.

Eq. 1a

Eq. 1b

2.

Eq. 2a

Eq. 2b

2.2.

Induced Dipole Moment of SHG

Eq. 3

Eq. 4

5.

Eq. 5a

Eq. 5b

Eq. 6

2.3.

Induced Electric Field E⃗2ω of SHG

Eq. 7

Eq. 8

Eq. 9

Eq. 10

Eq. 11

Eq. 12

2.4.

Emission Direction of Electric Field of SHG

Eq. 13

Eq. 14

Eq. 15

2.5.

Emission Power of SHG

16.

Eq. 17

18.

Eq. 18a

Eq. 18b

19.

Eq. 19a

Eq. 19b

3.

Simulation Study of Parameters of Linearly Polarized Focused Beam on SHG Emission

3.1.

Effects of Illumination Polarization on SHG Emission Power under Focused Beam

Fig. 2

Fig. 3

3.2.

Effects of the Numerical Aperture (NA) on SHG Emission

Eq. 20

Fig. 4

Fig. 5

3.3.

Effects of the Ratio ρ=βxxx∕βxyy under Different Polarization Direction

Fig. 6

Fig. 7

4.

Discussion and Conclusions

Appendices

{ label needed for app[@id='xA'] }

Appendix A: Simplification of Nonlinear Optical Hyperpolarizability

Eq. 21

Eq. 22

Kleinman Symmetry

Eq. 23

24.

Cylindrical Symmetry

Eq. 25

Eq. 26

Fig. 8

Eq. 27

Eq. 28

Eq. 29

30.

Eq. 30a

Eq. 30b

Eq. 30c

Electric-field ${\overset{⃑}{E}}_{ω}$ Profile of the Linearly Polarized Focused Beam

Induced Electric Field ${\vec{E}}_{2 ω}$ of SHG

Effects of the Ratio $ρ = β_{x x x} ∕ β_{x y y}$ under Different Polarization Direction