Analytic theory of photoacoustic wave generation from a spheroidal droplet

Yong Li; Hui Fang; Changjun Min; Xiaocong Yuan

doi:10.1364/OE.22.019953

1. Introduction

As fundamental problems in photoacoustics, the photoacoustic wave generation in one dimension from an infinite layer, in two dimensions from an infinite cylinder, and in three dimensions from a sphere have all been extensively studied [1] and their analytical solutions under the optically-thin-body condition are well-known [2,3]. The analytical solutions dutifully convey the characteristic spectral and temporal information embedded in the photoacoustic radiation, which are tightly correlated to the specific geometries of these bodies.

In this paper, we embark on the derivation of the complete analytic solution in three dimensions for a droplet with a more general shape other than sphere—a spheriodal droplet which can be either a prolate spheroid or an oblate spheroid. To the best of our knowledge, such theoretical investigation has not been taken account before. While such an analytic solution should be broadly applicable, we anticipate that it will first find important applications in biomedical photoacoustic imaging, considering the rapid progress in this research field to detect individual cells or organelles [4–8] and that these light absorbing biological capsules such as red blood cells and cell nuclei are close to spheroids in shape.

Our derivation is based on solving the photoacoustic Helmholtz equation by employing the separation-of-variables method in spheroidal coordinates. Working with the spheroidal coordinate systems (see Fig. 1, including the prolate spheroidal coordinate system and the oblate spheroidal coordinate system) is a natural choice because a spheroidal droplet can be fitted into such a typical coordinate system and the boundary conditions can be easily defined and applied. The resulted analytic formulas are expressed as the sum of a serials of $S_{m n} R_{m n}$ with the weighting coefficients to be determined by the boundary conditions, where $S_{m n}$ and $R_{m n}$ are defined respectively as the angular and radial spheroidal wave functions (SWFs) [9].

Fig. 1 Illustration of photoacoustic (PA) wave generation of a prolate spheroidal droplet or an oblate spheroidal droplet. In the left panel, on the top a prolate spheroid along with its prolate spheroidal coordinate system is plotted and at the bottom an oblate spheroid along with its oblate spheroidal coordinate system is plotted, where the two black dots represent the foci and the z-axis represents the revolution axis for each system. In the right panel. the three geometries, infinitely long cylinder, sphere, and infinitely large layer, represent respectively the asymptotic spheroids under the extreme conditions of infinite or zero interfocal distance. For these three geometries, the analytic solutions of the photoacoustic waves are well-known [1–3].

Download Full Size | PDF

The method of solving a Helmholtz equation in spheroidal coordinates has its roots in the problems of sound scattering as well as light scattering by a spheroidal particle [10,11], and these researches still undergo some updates [12–15]. Due to the specific rotational symmetry of the oblate spheroidal coordinates, this method also incurs new applications in describing paraxial Gaussian-Laguerre laser beam [16,17] and in calculating the first-order correctness of Fabry-Perot-resonator eigenmodes [18]. We find that introducing this approach into photoacoustics is very attractive because only the $m = 0$ modes $S_{0 n} R_{0 n}$ contribute to the analytic solution due to the fact that the photoacoustic wave generation does not depend on the laser incident direction relative to the spheroidal droplet under the conventional optically-thin-body assumption. This is contrary to the cases of sound or light scattering where this simplicity only exists in the axisymmetric solution when the sound or light incident direction is restricted to being along the rotational axis of the spheroid.

In order to validate the analytic solution we reached, we present two extended theoretical examinations. The first is the asymptotic analyses of the analytic solution under three extreme conditions: when the interfocal distance of a prolate spheroid and an oblate spheroid diminishes to zero, when the interfocal distance of a prolate spheroid stretches to infinity, and when the interfocal distance of an oblate spheroid expands to infinity. It is found that the analytic solution asymptotically degenerates respectively to the standard formulas for a sphere, an infinite cylinder and an infinite layer. The second is the decomposition analysis on the time domain solution from the perspective of photoacoustic wave partial transmission and reflections at the droplet boundary. We demonstrate that the photoacoustic pulse outside the spheroidal droplet can be decomposed to successively transmitted waves which experience multiple internal reflections before passing through the boundary, while the photoacoustic pulse formed inside can be decomposed to successively reflected waves plus the source terms. This type of decomposition is consistent with the physics picture previously revealed in the studies of the three special geometries [2].

Around these major interrogations, this paper is organized as follows. Section 2 details the derivation of the general solution, first in the frequency domain and then in the time domain. Section 3 then analyzes the asymptotic behaviors of the solution by using the asymptotic properties of angular and radial SWFs. Section 4 along with Appendix provides the proof for how to fit the time domain solution into the frame of the partial transmission and reflection model where the Wronskian relation of radial SWFs plays an essential role. Finally, Section 5 summarizes our major findings and also directs the future works.

2. General solution

The problem is to resolve the photoacoustic waves generated by a spheroidal droplet due to laser illumination, as illustrated in Fig. 1. The light absorbing droplet is either a prolate spheroid or an oblate spheroid which is immersed in a fluid absorbing little light. The laser beam can be in the form of intensity-modulated continuous wave (cw) or short pulse. Because of the laser heating, a temperature change and subsequently a pressure variation will be introduced both inside the droplet and in the surrounding fluid. Putting into formulas, this process is governed by the following two equations for nonviscous fluid [1]:

\frac{\partial}{\partial t} T = \frac{κ}{ρ C_{P}} \nabla^{2} T + \frac{H}{ρ C_{P}},

\nabla^{2} p - \frac{1}{v^{2}} \frac{\partial^{2}}{\partial t^{2}} p = - \frac{α}{v^{2}} \frac{\partial^{2}}{\partial t^{2}} T,

where

κ

is the thermal conductivity,

ρ

is the mass density,

C_{p}

is the specific heat capacity at constant pressure,

H

is the heat energy per unit volume and time deposited by the laser,

v

is the sound speed (Only the longitudinal sound speed is considered here since for nonvisous fluid the effect of the shear sound speed can be neglected.), and

α

is the pressure expansion coefficient at constant volume defined as

α = {(\partial p / \partial T)}_{V}

. The pressure variation

p

is the photoacoustic (PA) wave. In Fig. 1, the photoacoustic wave outside of the droplet is shown.

Given that the thermal conduction term $\frac{κ}{ρ C_{P}} \nabla^{2} T$ is usually small compared to $\frac{\partial}{\partial t} T$ , its effect can be neglected. This treatment is valid not only for the nanosecond laser pulse typically employed [19] but also for the cw laser with its modulation frequency in the ultrasound range [20]. Therefore, the photoacoustic wave functions of Eq. (1) and Eq. (2) can be reduced to [1]

\nabla^{2} p - \frac{1}{v^{2}} \frac{\partial^{2}}{\partial t^{2}} p = - \frac{β}{C_{p}} \frac{\partial}{\partial t} H,

where

β = (1 / V) {(\partial V / \partial T)}_{p}

is the thermal expansion coefficient related to

α

as

α = β ρ v^{2}

. To explicitly express that the laser heating happens only in the droplet and the sound speeds in the droplet and in the surrounding fluid are generally different, we divide Eq. (3) into

{\begin{cases} \nabla^{2} p_{s} - \frac{1}{v_{s}^{2}} \frac{\partial^{2}}{\partial t^{2}} p_{s} = - \frac{β}{C_{p}} \frac{\partial H}{\partial t} (inside the droplet) \\ \nabla^{2} p_{f} - \frac{1}{v_{f}^{2}} \frac{\partial^{2}}{\partial t^{2}} p_{f} = 0 (outside the droplet) \end{cases}

where the subscript “s” denotes quantities of the spheroidal droplet while “f” denotes those of the surrounding fluid.

The solution of $p_{s}$ in Eq. (4) can be further divided into two parts:

p_{s} = p_{s1} + p_{s2},

where

p_{s1}

is the special solution for

\frac{1}{v_{s}^{2}} \frac{\partial^{2}}{\partial t^{2}} p_{s1} = \frac{β}{C_{p}} \frac{\partial H}{\partial t}

and

p_{s2}

satisfies

\nabla^{2} p_{s2} - \frac{1}{v_{s}^{2}} \frac{\partial^{2}}{\partial t^{2}} p_{s2} = 0.

This solving method is appropriate because the quantity of

H

is uniform across the whole droplet under the “optically-thin-body” assumption. This assumption also implies that the photoacoustic wave generation does not depend on the laser beam illumination angle relative the spheroidal droplet.

Since it is trivial to solve $p_{s1}$ in Eq. (6), the problem is thus focused on to solve the two Helmholtz equations: one in Eq. (4) for $p_{f}$ , and the other Eq. (7) for $p_{s2}$ . We coin them together hereafter the “photoacoustic Helmholtz equation”.

2.1 Frequency domain solution

The frequency domain solution corresponds to the photoacoustic waves produced by a cw laser beam with the intensity function of $I = I_{0} e^{- i ω t}$ where $ω$ represents the intensity modulation frequency, under which the heating function $H$ is

H = ε_{th} μ_{a} I_{0} e^{- i ω t}

where

μ_{a}

represents the light absorption coefficient of the droplet, and

ε_{th}

the percentage of the absorbed light energy being converted to heat which takes account the possibility of other energy-consumption routes such as fluorescent emission.

The frequency domain solution of $p_{s1}$ corresponding to Eq. (6) can be obtained easily by substituting $H$ of Eq. (8) into Eq. (6) and taking out the time dependent part $e^{- i ω t}$ , as

p_{s1} (ω) \equiv p_{0} = i \frac{ε_{th} μ_{a} β I_{0} v_{s}^{2}}{ω C_{p}} .

To get the frequency domain solution of

p_{s2}

from Eq. (7) and

p_{f}

from Eq. (4), the standard separation-of-variables method of solving a Helmholtz equation in spheroidal coordinates

(ξ, η, ϕ)

can be applied [9], resulting in

p_{s2} (ω, ξ, η, ϕ) = p_{0} \sum_{m = 0}^{\infty} \sum_{n \geq m}^{\infty} p_{m n}^{s} S_{m n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{m n}^{(1)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases}) e^{\pm i m ϕ},

p_{f} (ω, ξ, η, ϕ) = p_{0} \sum_{m = 0}^{\infty} \sum_{n \geq m}^{\infty} p_{m n}^{f} S_{m n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}, η) R_{m n}^{(3)} ({\begin{cases} c_{f}, ξ \\ - i c_{f}, i ξ \end{cases}) e^{\pm i m ϕ} .

Here the dimensionless variables

c_{s}

and

c_{f}

are defined as

c_{s} = \frac{ω}{v_{s}} \frac{d}{2} = {\hat{k}}_{s} \frac{d}{2}, and c_{f} = \frac{ω}{v_{f}} \frac{d}{2} = {\hat{k}}_{f} \frac{d}{2}

where

d

is the interfocal distance of the spheroid droplet (therefore the angular SWFs and radial SWFs all depend on

ω

),

p_{m n}^{s}

and

p_{m n}^{f}

are the weighting coefficients to be determined by applying boundary conditions. We have used the branch brackets to work as the denotation symbol to differentiate the variables in angular SWFs

S_{m n}

and the radial SWFs

R_{m n}^{(1)}

and

R_{m n}^{(3)}

(where

R_{m n}^{(1)}

are called the first kind and

R_{m n}^{(3)}

the third kind) depending on whether these SWFs are considered in the prolate spheroidal coordinate system or in the oblate spheroidal coordinate system: when switching between the former system to the latter system, the replacements of

c \leftrightarrow - i c

(

c

represents either of

c_{s}

and

c_{f}

) and

ξ \leftrightarrow i ξ

are needed. We will use this type of expression throughout this paper.

As plotted in Fig. 1, the boundary of the spheroidal droplet fits to a spheroidal surface with a constant $ξ$ , i.e. $ξ_{0}$ . With the conventional notation of $a$ and $b$ respectively as the semi-major and semi-minor axes of the revolving ellipse that forms the spheroid, the expression for determining $ξ_{0}$ will be different depending on the droplet is a prolate spheroid or an oblate spheroid, respectively as

ξ_{0} = \frac{a}{d / 2} \geq 1 (prolate spheroid),

ξ_{0} = \frac{b}{d / 2} \geq 0 (oblate spheroid) .

These expressions can be directly extracted from the following coordinate transformation relations between spheroidal coordinates and rectangular coordinates (by looking at the z axis along which

η = 1

):

x = \frac{d}{2} \sqrt{(1 - η^{2}) (ξ^{2} - 1)} \cos ϕ, y = \frac{d}{2} \sqrt{(1 - η^{2}) (ξ^{2} - 1)} \sin ϕ, z = \frac{d}{2} η ξ (prolate spheroid),

x = \frac{d}{2} \sqrt{(1 - η^{2}) (ξ^{2} + 1)} \cos ϕ, y = \frac{d}{2} \sqrt{(1 - η^{2}) (ξ^{2} + 1)} \sin ϕ, z = \frac{d}{2} η ξ (oblate spheroid) .

The boundary conditions that shall be applied here is identical to that employed in the classic solutions [1,2], and is just the same as that applied to solve the problem of ultrasound scattering from a penetrable spheroid [21], namely, that the pressure and the normal acceleration must both be continuous across the boundary, i.e.

p_{s} (ω, ξ_{0}, η, ϕ) = p_{f} (ω, ξ_{0}, η, ϕ), \frac{\partial p_{s} (ω, ξ, η, ϕ)}{ρ_{s} \partial ξ} |_{ξ = ξ_{0}} = \frac{\partial p_{f} (ω, ξ, η, ϕ)}{ρ_{f} \partial ξ} |_{ξ = ξ_{0}} .

As shown in Fig. 1, since the

η

confocal hyperbolas are orthogonal to the

ξ

confocal ellipses, the normal direction of the boundary is actually along the direction of the

η

hyperbolas. It can be deduced from Eq. (15) and Eq. (16) that

when ξ \to \infty, η \to \cos θ,

where

θ

is the polar angle defined in the spherical coordinates. The spheroidal coordinate

ϕ

is defined in the conventional sense, as the azimuthal angle.

Based on Eq. (17), before going for the solution of $p_{m n}^{s}$ and $p_{m n}^{f}$ needed in Eq. (10) and Eq. (11), we first arrive at the following constraint conditions

m = 0, n = 2 k .

These constraints can be understood more intuitively from the symmetry consideration. The

m = 0

constraint condition corresponds to the requirement that

p_{s2}

and

p_{f}

should be both rotationally invariant with respect to

ϕ

which means

\frac{\partial p_{s2}}{\partial ϕ} = 0, \frac{\partial p_{f}}{\partial ϕ} = 0,

while the

n = 2 k

constraint condition corresponds to the parity requirement of these quantities respect to

η

which is

p_{s2} (- η) = p_{s2} (η), p_{f} (- η) = p_{f} (η)

which further requests that

S_{0 n} ({\begin{cases} c \\ - i c \end{cases}, - η) = S_{0 n} ({\begin{cases} c \\ - i c \end{cases}, η)

in Eq. (10) and Eq. (11). However, the angular SWFs

S_{0 n}

are related to the Legendre functions

P_{l} (η)

as [9]

S_{0 n} ({\begin{cases} c \\ - i c \end{cases}, η) = \sum_{l = 0, 1}^{\infty}' d_{l}^{0 n} ({\begin{cases} c \\ - i c \end{cases}) P_{l} (η)

where

d_{l}^{0 n}

are called the spheroidal coefficients and the prime over the summation sign here indicates that the sum is over only even or odd

l

depending on

n

is even or odd, thereby finally resulting in the condition that

n

must be an even number, i.e.

n = 2 k

. Therefore, the expressions for

p_{s2}

and

p_{f}

in Eq. (10) and Eq. (11) are much simplified, being reduced to

p_{s2} (ω, ξ, η, ϕ) = p_{0} \sum_{n = 0}^{\infty} p_{0 n}^{s} S_{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{0 n}^{(1)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases}), n = 2 k,

p_{f} (ω, ξ, η, ϕ) = p_{0} \sum_{n = 0}^{\infty} p_{0 n}^{f} S_{0 n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}, η) R_{0 n}^{(3)} ({\begin{cases} c_{f}, ξ \\ - i c_{f}, i ξ \end{cases}), n = 2 k .

Now, it is the time to applying the boundary conditions by substituting the expressions in Eq. (9), Eq. (23) and Eq. (24) into Eq. (17). We further substitute the expansion for $S_{0 n}$ as shown in Eq. (22) and remove the dependence on $η$ by employing the orthogonal property of the Legendre function, subsequently obtaining

\begin{array}{l} {\begin{cases} δ_{0, l} + \sum_{n = 0}^{\infty} p_{0 n}^{s} d_{l}^{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}) R_{0 n}^{(1)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}) = \sum_{n = 0}^{\infty} p_{0 n}^{f} d_{l}^{0 n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}) R_{0 n}^{(3)} ({\begin{cases} c_{f}, ξ_{0} \\ - i c_{f}, i ξ_{0} \end{cases}) \\ \sum_{n = 0}^{\infty} p_{0 n}^{s} d_{l}^{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}) {R^{'}}_{0 n}^{(1)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}) = \frac{ρ_{s}}{ρ_{f}} \sum_{n = 0}^{\infty} p_{0 n}^{f} d_{l}^{0 n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}) {R^{'}}_{0 n}^{(3)} ({\begin{cases} c_{f}, ξ_{0} \\ - i c_{f}, i ξ_{0} \end{cases}) \end{cases}, \\ n = 2 k, l = 2 k^{'} \geq 0, \end{array}

where

{R^{'}}_{0 n}^{(1)}

and

{R^{'}}_{0 n}^{(3)}

both denote the first derivative with respect to

ξ

. Note that Eq. (25) actually contains a series of such linear equations with the total number determined by the dimension of

l

. It will be much more convenient to cast these linear equations into a square-matrix form by setting the dimension of

l

equal to that of

n

, thereby arriving at

{\begin{cases} A + D_{s} R_{1s} P^{s} = D_{f} R_{3f} P^{f} \\ D_{s} {R^{'}}_{1s}^{} P^{s} = \frac{ρ_{s}}{ρ_{f}} D_{f} {R^{'}}_{3 f}^{} P^{f} \end{cases} .

Here, all of

P^{s}

,

P^{f}

, and

A

are column-vectors where the elements of

P^{s}

and

P^{f}

are the weighting coefficients

p_{0 n}^{s}

and

p_{0 n}^{f}

to be determined, and the element of

A

is simply defined as (here

δ_{i, j}

is the Kronecker delta symbol)

A_{i} = δ_{i, 1} .

On the other hand, all of

D_{s}

,

D_{f}

,

R_{1 s}

,

{R^{'}}_{1 s}

,

R_{3f}

,

{R^{'}}_{1 s}

represent square-matrixes with their elements predetermined respectively as:

{{(D}_{s})}_{i j} = d_{2 i - 2}^{0 (2 j - 2)} ({\begin{matrix} c_{s} \\ - i c_{s} \end{matrix}), {{(D}_{f})}_{i j} = d_{2 i - 2}^{0 (2 j - 2)} ({\begin{matrix} c_{f} \\ - i c_{f} \end{matrix}),

{{(R}_{1s})}_{i j} = δ_{i, j} R_{0 (2 i - 2)}^{(1)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}), {({R^{'}}_{1s}^{})}_{i j} = δ_{i, j} {R^{'}}_{0 (2 i - 2)}^{(1)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}),

{{(R}_{3f})}_{i j} = δ_{i, j} R_{0 (2 i - 2)}^{(3)} ({\begin{cases} c_{f}, ξ_{0} \\ - i c_{f}, i ξ_{0} \end{cases}), {({R^{'}}_{3f})}_{i j} = δ_{i, j} {R^{'}}_{0 (2 i - 2)}^{(3)} ({\begin{cases} c_{f}, ξ_{0} \\ - i c_{f}, i ξ_{0} \end{cases}) .

All of

R_{1s}

,

{R^{'}}_{1s}

,

R_{3f}

,

{R^{'}}_{3f}

are diagonal-matrixes.

In principle, $P^{s}$ , $P^{f}$ can be solved from Eq. (26) with matrix algebra, resulting in

P^{s} = {[\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} {({R^{'}}_{3f})}^{- 1} {{(D}_{f})}^{- 1} D_{s} {R^{'}}_{1s} - D_{s} R_{1s}]}^{- 1} A,

P^{f} = {[D_{f} R_{3f} - \frac{ρ_{s}}{ρ_{f}} D_{s} R_{1s} {({R^{'}}_{1s})}^{- 1} {{(D}_{s})}^{- 1} D_{f} {R^{'}}_{3f}]}^{- 1} A .

To this end, by incorporating the results of Eq. (31) and Eq. (32) respectively back into Eq. (23) and Eq. (24), and further combining this result with Eqs. (9) and (5), the frequency domain solution for

p_{s}

and

p_{f}

in Eq. (4) is then finally determined.

2.2 Time domain solution

Once the frequency domain solution has been settled, the time domain solution for short laser pulse excitation can be readily reached by taking Fourier transform based on the spectral analysis method [20].

Let us first consider decomposing the laser pulse intensity into its spectral components as

I (t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \tilde{I} (ω) e^{- i ω t} d ω .

Then the time domain solution is just

p_{s} (t, ξ, η, ϕ) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \frac{\tilde{I} (ω)}{I_{0}} p_{0} e^{- i ω t} [1 + \sum_{n = 0}^{\infty} p_{0 n}^{s} S_{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{0 n}^{(1)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases})] d ω, n = 2 k,

p_{f} (t, ξ, η, ϕ) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \frac{\tilde{I} (ω)}{I_{0}} p_{0} e^{- i ω t} \sum_{n = 0}^{\infty} p_{0 n}^{f} S_{0 n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}, η) R_{0 n}^{(3)} ({\begin{cases} c_{f}, ξ \\ - i c_{f}, i ξ \end{cases}) d ω, n = 2 k,

where

p_{0}

has the form as shown in Eq. (9) and

p_{0 n}^{s}

and

p_{0 n}^{f}

have already been determined as shown in Eq. (31) and Eq. (32).

For the ideal delta-pulse excitation, there is

I (t) = I_{0} δ (t) \Rightarrow \frac{\tilde{I} (ω)}{I_{0}} = 1,

under which Eq. (34) and Eq. (35) can be further simplified.

3. Asymptotic analyses

As illustrated in the right panel of Fig. 1, when the interfocal distance $d \to 0$ , the prolate spheroid and the oblate spheroid will be both rounded respectively to a sphere; on the other hand, when $d \to \infty$ , the prolate spheroid will be elongated to an infinitely long cylinder, while the oblate spheriod will be stretched to an infinitely wide circular-plate. Therefore, it is a natural conjecture that the general frequency domain solution derived in Section 2.1 (thus also the time domain solution) will asymptotically degenerate into the standard solutions for a sphere, an infinite cylinder, and an infinite layer respectively at these extreme limits. At follows, we shall provide the proofs. For the sake of brevity, here we only describe the analyses for the photoacoustic waves generated outside the droplet.

3.1 Sphere

For a prolate spheriod or an oblate spheroid, the condition of $d \to 0$ is equivalent to the requests that the dimensionless variables $c_{s}, c_{f}$ reach the limit of [refer to Eq. (12)]

c_{s} \to 0, c_{f} \to 0.

Since there is [9]

S_{0 n} ({\begin{cases} c \\ - i c \end{cases}, η) ~ P_{n} (η), n = 2 k, for c \to 0,

we can first deduce that [refer to Eq. (22)]

d_{2 i}^{0 (2 j)} ({\begin{cases} c \\ - i c \end{cases}) ~ δ_{i, j}, for c \to 0.

Therefore,

D_{s}

,

D_{f}

in Eq. (28) are both reduced to the unit matrix

D_{s} \sim E, D_{f} \sim E .

By substituting the results of Eq. (40) and Eq. (27) into Eq. (32), it is straightforward to reach

{(P^{f})}_{i} ~ δ_{i, 1} \times \frac{1}{{{(R}_{3f})}_{11} - \frac{ρ_{s}}{ρ_{f}} {{(R}_{1s})}_{11} \frac{{({R^{'}}_{3f})}_{11}}{{({R^{'}}_{1s})}_{11}}},

which means that only the single mode

S_{00} R_{00}

contributes.

Now let us focus on determining the value of ${{(R}_{1s})}_{11}$ , ${({R^{'}}_{1s})}_{11}$ , ${{(R}_{3f})}_{11}$ , ${({R^{'}}_{3f})}_{11}$ which are related to the radial SWFs as defined in Eq. (29) and Eq. (30). We need to apply the following asymptotic relations [9]

R_{0 n}^{(1)} ({\begin{cases} c, ξ \\ - i c, i ξ \end{cases}) ~ j_{n} (\hat{k} r), n = 2 k, for c \to 0,

R_{0 n}^{(3)} ({\begin{cases} c, ξ \\ - i c, i ξ \end{cases}) ~ h_{n}^{(1)} (\hat{k} r), n = 2 k, for c \to 0,

where

c ξ \sim \hat{k} r, for c \to 0

has been used, which comes from the coordinate transformation in Eq. (15) and Eq. (16) by setting

η \sim \frac{z}{r}

. Here,

j_{n} (\hat{k} r)

is the spherical Bessel function and

h_{n}^{(1)} (\hat{k} r)

is the spherical Hankel function of the first kind. The asymptotic expression of

{R^{'}}_{0 n}^{(1)}

and

{R^{'}}_{0 n}^{(3)}

can be derived in the similar way, after which we can move a further step from Eq. (41) to get

{(P^{f})}_{i} ~ δ_{i, 1} \times \frac{1}{h_{0} ({\hat{k}}_{f} a) - \frac{ρ_{s}}{ρ_{f}} \frac{{\hat{k}}_{f} h_{1}^{(1)} ({\hat{k}}_{f} a)}{{\hat{k}}_{s} j_{1} ({\hat{k}}_{s} a)} j_{0} ({\hat{k}}_{s} a)} .

As the final step, by substituting Eq. (45) into Eq. (24) and by noting the expression of $S_{00}$ in Eq. (38) and the expression of $R_{00}^{(3)}$ in Eq. (43), we arrive at

p_{f} (ω, ξ, η, ϕ) ~ p_{0} \frac{j_{1} ({\hat{k}}_{s} a) h_{0}^{(1)} ({\hat{k}}_{f} r)}{j_{1} ({\hat{k}}_{s} a) h_{0} ({\hat{k}}_{f} a) - \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} h_{1}^{(1)} ({\hat{k}}_{f} a) j_{0} ({\hat{k}}_{s} a)} .

If we write out the expression for

j_{0} (\hat{k} r)

,

j_{1} (\hat{k} r)

,

h_{0}^{(1)} (k r)

, and

h_{1}^{(1)} (k r)

, the standard solution for a sphere [1] will be recovered:

p_{f} (ω, ξ, η, ϕ) ~ \frac{p_{0}}{r / a} e^{- i \hat{q} τ} \times \frac{(\sin \hat{q} - \hat{q} \cos \hat{q}) / \hat{q}}{[(1 - \frac{ρ_{s}}{ρ_{f}}) (\sin \hat{q} / \hat{q}) - \cos \hat{q} + i \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} \sin \hat{q}]},

where

\hat{q} = {\hat{k}}_{s} a

and

τ = - (v_{s} / a) (r - a) / v_{f}

.

3.2 Infinite long cylinder

For a prolate spheroid, the condition of $d \to \infty$ is equivalent to the requests that the dimensionless variables $c_{s}, c_{f}$ reach the limit of [refer to Eq. (12)]

c_{s} \to \infty, c_{f} \to \infty .

This will lead to

D_{s} {~D}_{f},

which will in turn reduce the expression of

P^{f}

in Eq. (32) to

P^{f} ~ {[R_{3f} - \frac{ρ_{s}}{ρ_{f}} R_{1s} {({R^{'}}_{1s})}^{- 1} {R^{'}}_{3f}]}^{- 1} {{(D}_{f})}^{- 1} A .

For this time, we need to use the following asymptotic forms for the radial SWFs [22]:

R_{0 n}^{(1)} (c, ξ) ~ \sqrt{\frac{π}{2 c}} J_{0} (c \sqrt{ξ^{2} - 1}), n = 2 k, for c \to \infty,

R_{0 n}^{(3)} (c, ξ) ~ \sqrt{\frac{π}{2 c}} H_{0}^{(1)} (c \sqrt{ξ^{2} - 1}), n = 2 k, for c \to \infty,

where

J_{0} (c \sqrt{ξ^{2} - 1})

is zero-order Bessel function and

H_{0}^{(1)} (c \sqrt{ξ^{2} - 1})

is zero-order Hankel function of the first kind.

Before moving forwards, we need first to analyze the asymptotic behavior of the angular SWFs which is [9]

S_{0 n} (c, η) ~ h_{0}^{n} 2^{- n / 2} e^{- c η^{2} / 2} H_{n} (\sqrt{c} η), n = 2 k, for c \to \infty,

where

H_{n} (\sqrt{c} η)

is the Hermite polynomials and

h_{0}^{n}

presents constants only depending on

n

. From Eq. (53), we can conclude that the solution of Eq. (24) with physical meaning will only exist for

η \sim 0,

i.e. the photoacoustic waves are generated only along the direction perpendicular to the revolution axis (the z axis, refer to Fig. 1). This is amount to reducing the problem down to 2 dimensions. Let r denote the distance perpendicular to the z axis, we obtain from Eq. (15) that

c \sqrt{ξ^{2} - 1} \sim \hat{k} r .

Now by incorporating Eq. (55) into Eq. (51) and Eq. (52), and then by using them (also the expressions for ${R^{'}}_{0 n}^{(1)}$ and ${R^{'}}_{0 n}^{(3)}$ ) in Eq. (50), we reach that

P^{f} ~ \frac{J_{1} ({\hat{k}}_{s} b)}{\sqrt{\frac{π}{2 c_{f}}} H_{0}^{(1)} ({\hat{k}}_{f} b) J_{1} ({\hat{k}}_{s} b) - \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} \sqrt{\frac{π}{2 c_{f}}} J_{0} ({\hat{k}}_{s} b) H_{1}^{(1)} ({\hat{k}}_{f} b)} {{(D}_{f})}^{- 1} A .

Finally, by going back to Eq. (24) we can get

p_{f} (ξ, η) ~ p_{0} \frac{J_{1} ({\hat{k}}_{s} b) H_{0}^{(1)} ({\hat{k}}_{f} r)}{H_{0}^{(1)} ({\hat{k}}_{f} b) J_{1} ({\hat{k}}_{s} b) - \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} J_{0} ({\hat{k}}_{s} b) H_{1}^{(1)} ({\hat{k}}_{f} b)} \sum_{k = 0}^{\infty} {[{(D_{f})}^{- 1}]}_{(k + 1) 1} S_{0 (2 k)} (c_{f}, η) .

However, due to

\begin{array}{l} \sum_{k = 0}^{\infty} {[{(D_{f})}^{- 1}]}_{(k + 1) 1} S_{0 (2 k)} (c_{f}, η) = \sum_{k = 0}^{\infty} {[{(D_{f})}^{- 1}]}_{(k + 1) 1} \sum_{k^{'} = 0}^{\infty} d_{2 k^{'}}^{0 (2 k)} P_{2 k^{'}} (η) \\ = \sum_{k^{'} = 0}^{\infty} \sum_{k = 0}^{\infty} {[{(D_{f})}^{- 1}]}_{(k + 1) 1} {(D_{f})}_{(k^{'} + 1) (k + 1)} P_{2 k^{'}} (η) = δ_{k^{'}, 0} P_{2 k^{'}} (η) = 1, \end{array}

the standard solution for an infinite cylinder [1] is therefore recovered with b representing the radius of the infinite cylinder:

p_{f} (ω, ξ, η, ϕ) ~ p_{0} \frac{J_{1} ({\hat{k}}_{s} b) H_{0}^{(1)} ({\hat{k}}_{f} r)}{H_{0}^{(1)} ({\hat{k}}_{f} b) J_{1} ({\hat{k}}_{s} b) - \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} J_{0} ({\hat{k}}_{s} b) H_{1}^{(1)} ({\hat{k}}_{f} b)} .

3.3 Infinite large layer

As the last asymptotic analysis, we consider the oblate spheroid with $d \to \infty$ . In this case the dimensionless variables $c_{s}, c_{f}$ also satisfy

c_{s} \to \infty, c_{f} \to \infty .

The results expressed same as Eq. (49) and Eq. (50) will be reached. However, it is now working under the oblate spheridal coordinate system rather than in the prolate spheroidal coordinate system. The asymptotic forms of the radial SWFs will not be the same as those in Eq. (51) and Eq. (52), but rather are [9]

R_{0 n}^{(1)} (- i c, i ξ) ~ \frac{\cos (c ξ)}{c} \times e^{- i \frac{n}{2} π}, n = 2 k, for c \to \infty,

R_{0 n}^{(3)} (- i c, i ξ) ~ \frac{e^{i c ξ}}{c} \times e^{- i \frac{n}{2} π}, n = 2 k, for c \to \infty .

Here, the asymptotic expression for the angular SWFs to be used is [9]

S_{0 n} (c, η) ~ A_{0}^{0 n} {e^{- c (1 - η)} L_{n / 2} [2 c (1 - η)] + e^{- c (1 + η)} L_{n / 2} [2 c (1 + η)]}, n = 2 k, for c \sim \infty,

where

L_{n / 2} [2 c (1 - η)]

and

L_{n / 2} [2 c (1 + η)]

are the Laguerre polynomials, and

A_{0}^{0 n}

are constants only depending on

n

. In order to have a solution with physical meaning, Eq. (63) requires that

η \sim \pm 1.

This is clearly different from that of Eq. (54). The constraint of Eq. (64) reduces the problem in the oblate spheroidal coordinate system down to 1 dimension (refer to Fig. 1), i.e. the photoacoustic waves are produced only along the direction of z axis. From the coordinate transformation of Eq. (16), it can be deduced that

c ξ \sim \hat{k} | z | .

Now, by incorporating the results of Eq. (61), Eq. (62) and Eq. (65) into the expression of $P^{f}$ [which is same as Eq. (50)], we can further arrive at

P^{f} ~ \frac{c_{f} \sin ({\hat{k}}_{s} b) e^{- i {\hat{k}}_{f} b}}{\sin ({\hat{k}}_{s} b) + i \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} \cos ({\hat{k}}_{s} b)} \bar{E} {{(D}_{f})}^{- 1} A,

where

\bar{E}

denotes the diagonal matrix with the element

{\bar{E}}_{k k} = e^{i (k - 1) π}

.

Finally, by substituting Eq. (66) and Eq. (62) into Eq. (24), and by processing the similar derivation of Eq. (58), as expected, we recover the standard solution for an infinite layer [1] with b representing here the half thickness of the infinite layer:

p_{f} (ω, ξ, η, ϕ) ~ p_{0} \frac{\sin ({\hat{k}}_{s} b) e^{i {\hat{k}}_{f} (| z | - b)}}{\sin ({\hat{k}}_{s} b) + i \frac{ρ_{s} v_{s}}{ρ_{f} v_{f}} \cos ({\hat{k}}_{s} b)} .

4. Decomposition analysis on time domain solution

As for the time domain solutions of a sphere, an infinite cylinder and an infinite layer, the model of photoacoustic wave partial transmission and reflection at the boundary has been applied successfully to explain the characteristic photoacoustic temporal responses, which appear as being composed by successive pulses [1,2]. After having demonstrated in Section 3 that the solution of a spheroidal droplet actually contains all of the standard solutions for these three special geometries as the asymptotic representations, we expect that this model should also be applicable to the time domain solution of a spheroidal droplet, which is illustrated in Fig. 2.

Fig. 2 Illustration for the model of outgoing photoacoustic wave partial transmission and reflection at the boundary of the spheroidal droplet. The left side figure illustrates the case for a prolate spheroidal droplet, and the right side figure illustrates the case for an oblate spheroidal droplet. In each figure, the orange contour plots the droplet boundary, and the black curves plot the spheroidal coordinate system where the thick black line corresponds to the area with the minimum $ξ$ . The red arrows inside the boundary represent the outgoing photoacoustic waves, the red arrows outside the boundary represent the partially transmitted photoacoustic waves, and the green arrows represent the partially reflected photoacoustic waves. All of these photoacoustic waves propagate along the direction normal to the $ξ$ surfaces.

Download Full Size | PDF

It is very interesting to find that for the ideal $I_{0} δ (t)$ laser pulse excitation usually considered for this model [refer to Eq. (36)], the solution of $p_{f} (t, ξ, η)$ and $p_{s} (t, ξ, η)$ in Eq. (35) and Eq. (34) indeed can be decomposed respectively into (refer to Appendix for the derivation)

\begin{array}{l} p_{f} (t, ξ, η, ϕ) = \\ \frac{1}{2 π} \int_{- \infty}^{+ \infty} p_{0} e^{- i ω t} \sum_{n = 0}^{\infty} {[ℑ (E + ℜ + ℜ^{2} + \dots) P^{f_i}]}_{(n/2+1)} S_{0 n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}, η) R_{0 n}^{(3)} ({\begin{cases} c_{f}, ξ \\ - i c_{f}, i ξ \end{cases}) d ω, n = 2 k, \end{array}

\begin{array}{l} p_{s} (t, ξ, η, ϕ) = \\ \frac{1}{2 π} \int_{- \infty}^{+ \infty} p_{0} e^{- i ω t} d ω + \frac{1}{2 π} \int_{- \infty}^{+ \infty} p_{0} e^{- i ω t} \sum_{n = 0}^{\infty} {[P^{s_i}]}_{(n/2+1)} S_{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{0 n}^{(1)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases}) d ω + \\ \frac{1}{2 π} \int_{- \infty}^{+ \infty} p_{0} e^{- i ω t} \sum_{n = 0}^{\infty} {[2 (E + ℜ + ℜ^{2} + \dots) ℜ P^{f_i}]}_{(n/2+1)} S_{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{0 n}^{(1)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases}) d ω, n = 2 k . \end{array}

Here,

ℑ

,

ℜ

represent respectively the transmission matrix and the reflection matrix as defined in Appendix [Eq. (81)], and

P^{f_i}

,

P^{s_i}

are respectively defined as the initial

P^{f}

vector and the initial

P^{s}

vector which are calculated respectively from Eq. (32) and Eq. (31) under the conditions of

ρ_{f} = ρ_{s}, v_{f} = v_{s}

(i.e. when the surrounding medium has the same acoustic properties as the spheroidal droplet under which

ℜ

will be a null matrix and

ℑ = E

). The expressions of

P^{f_i}

,

P^{s_i}

are

P^{f_i} = {[R_{3s} - {({R^{'}}_{1s})}^{- 1} R_{1s} {R^{'}}_{3s}]}^{- 1} {{(D}_{s})}^{- 1} A,

P^{s_i} = {[R_{3s} {({R^{'}}_{3s})}^{- 1} {R^{'}}_{1s} - R_{1s}]}^{- 1} {{(D}_{s})}^{- 1} A .

Now based on Eq. (68), the photoacoustic wave generation outside of the spheroidal droplet can be interpreted as follows. For each mode, the initial outgoing wave excited inside the droplet will undergo a series of successive partial transmissions and reflections. Integrating the processes of all of the modes completes the time domain solution. Note that since $ℑ$ , $ℜ$ are usually not diagonal matrixes, the mode hopping phenomenon will appear during partial transmission and reflection: one outgoing mode will smear into other modes. This is not unexpected, however, because in spheroidal coordinates the modes $S_{0 n} R_{0 n}^{(3)}$ are not orthogonal with each other.

The photoacoustic wave generation inside the spheroidal droplet can be interpreted similarly based on Eq. (69). The first term and the second term on the right side of this equation assemble the initial photoacoustic pulse generated across the spheroidal droplet which is generated even when there is no reflection at the boundary. The third term shows that each mode will undergo multiple reflections. It is worth to point out that the factor of 2 inside the integral of the third term is important. This is because this factor needs to be combined with $R_{0 n}^{(1)}$ to fulfill

2 R_{0 n}^{(1)} = R_{0 n}^{(3)} + R_{0 n}^{(4)},

where the radial SWF

R_{0 n}^{(3)}

and

R_{0 n}^{(4)}

(the fourth kind) are necessary respectively for describing outgoing waves and ingoing waves ([9], also refer to Appendix). Only through this step, the interpretation from the perspective of partial transmission and reflection can be justified: for the photoacoustic wave generation inside the spheroidal droplet, we need to consider not only the successive reflections for an initially outgoing wave but also the successive reflections for an initially ingoing wave; the initially ingoing wave will change to an outgoing wave when reaching the inner most area corresponding to the minimum

ξ

(

ξ = 1

for a prolate spheroid and

ξ = 0

for an oblate spheroid). The mode hopping also prevails during the reflection.

5. Conclusions and future works

We derived the analytic solution for describing the photoacoustic waves generated by a spheroidal droplet through two crucial steps. The first is to identify the photoacoustic Helmholtz equation, and the second is to solve it in spheroidal coordinates with separation-of-variable method which yields the expressions of Eq. (23) and Eq. (24). These expressions take the form of spheroidal modes summation with the weighting coefficients determined by Eq. (31) and Eq. (32), which are obtained by applying the boundary conditions as expressed in Eq. (17). We started our derivation in the frequency domain where the special solution of Eq. (9) also needs to be included into the complete solution. The time domain solution then can be established through Fourier transform as shown in Eq. (34) and Eq. (35).

Although the number of modes involved is theoretically unbounded and the compacted expressions of Eq. (31) and Eq. (32) in matrix format still seem complex, we were able to prove the analytic solution based on two approaches: one is to recover the standard solutions from the asymptotic analyses, and the other is to reconcile the partial transmission and reflection model by performing the decomposition analysis on the time domain solution. The former task was accomplished by exploiting the asymptotic behaviors of SWFs where the radial SWFs are asymptotically degenerated into spherical Bessel functions, Bessel functions, and sinusoidal functions which are required respectively in the solution for a sphere, an infinite cylinder and an infinite layer [23]. The key for accomplishing the latter task is to extensively apply the Wronskian relation for radial SWFs.

The information contained in the analytic solution is by no means limited by what we have demonstrated, and there is certainly more yet waiting to be explored. For example, the approximation solutions in three dimensions for a finite cylinder and for a finite plate should be reachable also through asymptotic analyses. Even more information can be extracted from the numerical realization of the analytic solution for practical uses. This is greatly promising because the hurdle of obtaining numerical results of SWFs has already been overcome with the ongoing development of the specialized computer programs [24–27]. We are currently working on this numerical realization.

Appendix: derivation for the formulas of photoacoustic wave partial transmission and reflection at the boundary of a spheroidal droplet

We perform the derivation in the frequency domain by considering an initial outgoing photoacoustic wave with the weighting coefficients of every mode all equal to 1, i.e.

p_{I} (ω, ξ, η, ϕ) = p_{0} \sum_{n = 0}^{\infty} B_{(n / 2 + 1)} S_{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{0 n}^{(3)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases}), B_{(n / 2 + 1)} \equiv 1, n = 2 k

where we use the vector

B

to represent the weighting coefficients. Similar to establishing the partial transmission and reflection model for a sphere, an infinite cylinder, and an infinite layer [1,2], we seek the transmitted wave and reflected wave at the droplet boundary (refer to Fig. 2)

{\begin{cases} p_{T} (ω, ξ, η, ϕ) = p_{0} \sum_{n = 0}^{\infty} T_{(n / 2 + 1)} S_{0 n} ({\begin{cases} c_{f} \\ - i c_{f} \end{cases}, η) R_{0 n}^{(3)} ({\begin{cases} c_{f}, ξ \\ - i c_{f}, i ξ \end{cases}) \\ p_{R} (ω, ξ, η, ϕ) = p_{0} \sum_{n = 0}^{\infty} R_{(n / 2 + 1)} S_{0 n} ({\begin{cases} c_{s} \\ - i c_{s} \end{cases}, η) R_{0 n}^{(4)} ({\begin{cases} c_{s}, ξ \\ - i c_{s}, i ξ \end{cases}) \end{cases}, n = 2 k

which satisfying the boundary conditions [similar to Eq. (17)]:

{\begin{cases} p_{I} (ω, ξ_{0}, η, ϕ) + p_{R} (ω, ξ_{0}, η, ϕ) = p_{T} (ω, ξ_{0}, η, ϕ) \\ \frac{\partial [p_{I} (ω, ξ, η, ϕ) + p_{R} (ω, ξ, η, ϕ)]}{ρ_{s} \partial ξ} |_{ξ = ξ_{0}} = \frac{\partial p_{T} (ω, ξ, η, ϕ)}{ρ_{f} \partial ξ} |_{ξ = ξ_{0}} \end{cases} .

Here

R_{0 n}^{(4)}

represent the radial SWFs of the fourth kinds and are necessary for describing an ingoing wave [9]. The relation between the different kinds of radial SWFs are

R_{0 n}^{(3)} = R_{0 n}^{(1)} +i R_{0 n}^{(2)}, R_{0 n}^{(4)} = R_{0 n}^{(1)} - i R_{0 n}^{(2)},

where

R_{0 n}^{(2)}

are the second kind.

By using the method similar for obtaining Eq. (26), we find

{\begin{cases} D_{s} R_{3 s} B {+D}_{s} R_{4 s} R {=D}_{f} R_{3 f} T \\ D_{s} {R^{'}}_{3 s} B {+D}_{s} {R^{'}}_{4 s} R = \frac{ρ_{s}}{ρ_{f}} D_{f} {R^{'}}_{3 f} T \end{cases},

where the matrixes

D_{s}

and

D_{f}

have been defined in Eq. (28) and the matrixes

R_{3 f}

and

{R^{'}}_{3 f}

have been defined in Eq. (30), and the matrixes

R_{3 s}

,

{R^{'}}_{3 s}

,

R_{4 s}

,

{R^{'}}_{4 s}

are defined in the similar way, as

{{(R}_{3s})}_{i j} = δ_{i, j} R_{0 (2 i - 2)}^{(3)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}), {({R^{'}}_{3s})}_{i j} = δ_{i, j} {R^{'}}_{0 (2 i - 2)}^{(3)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}),

{{(R}_{4s})}_{i j} = δ_{i, j} R_{0 (2 i - 2)}^{(4)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}), {({R^{'}}_{4s})}_{i j} = δ_{i, j} {R^{'}}_{0 (2 i - 2)}^{(4)} ({\begin{cases} c_{s}, ξ_{0} \\ - i c_{s}, i ξ_{0} \end{cases}) .

Solving Eq. (77) with matrix algebra, we get

{\begin{cases} T = ℑ B \\ R = ℜ B \end{cases},

where we define

ℑ

as the transmission matrix and

ℜ

the reflection matrix which have the forms as

{\begin{cases} ℑ = [^{D_{f} R_{3f} - \frac{ρ_{s}}{ρ_{f}} D_{s} R_{4s} (^{{R^{'}}_{4s}) - 1} {(D}_{s}^{) - 1} D_{f} {R^{'}}_{3f}] - 1} [D_{s} R_{3s} - D_{s} R_{4s} (^{{R^{'}}_{4s}) - 1} {R^{'}}_{3s}] \\ ℜ = [^{\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{4s} - D_{s} R_{4s}] - 1} [D_{s} R_{3s} - \frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{3s}] \end{cases} .

1. Expressing $P^{f}$ in terms of $ℑ$ and $ℜ$

To construct the expected relation between $P^{f}$ in Eq. (32) and $ℑ {(E - ℜ)}^{- 1}$ , we first derive the expression for ${(E - ℜ)}^{- 1}$ . Assisted with the relation of Eq. (72), we can obtain from Eq. (81) that

\begin{array}{l} (^{E - ℜ) - 1} = \\ \frac{1}{2} [^{\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{1s} - D_{s} R_{1s}] - 1} [\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{4s} - D_{s} R_{4s}] . \end{array}

Furthermore, assisted with the Wronskian relation [9]

R_{0 n}^{(1)} {R^{'}}_{0 n}^{(2)} - {R^{'}}_{0 n}^{(1)} R_{0 n}^{(2)} = {\begin{cases} \frac{1}{c (ξ^{2} - 1)} (for prolate spheroid) \\ \frac{1}{c (ξ^{2} + 1)} (for oblate spheroid) \end{cases},

we can write

ℑ

as

ℑ = {[{R^{'}}_{4s} {{(D}_{s})}^{- 1} D_{f} R_{3f} - \frac{ρ_{s}}{ρ_{f}} R_{4s} {{(D}_{s})}^{- 1} D_{f} {R^{'}}_{3f}]}^{- 1} \times {\begin{cases} \frac{- 2 i}{c_{s} (ξ^{2} - 1)} \\ \frac{- 2 i}{c_{s} (ξ^{2} + 1)} \end{cases} .

Next, by noting the interesting relation

\begin{array}{l} [^{{R^{'}}_{4s} {(D}_{s}^{) - 1} D_{f} R_{3f} - \frac{ρ_{s}}{ρ_{f}} R_{4s} {(D}_{s}^{) - 1} D_{f} {R^{'}}_{3f}] - 1} [^{\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{1s} - D_{s} R_{1s}] - 1} = \\ [^{{R^{'}}_{1s} {(D}_{s}^{) - 1} D_{f} R_{3f} - \frac{ρ_{s}}{ρ_{f}} R_{1s} {(D}_{s}^{) - 1} D_{f} {R^{'}}_{3f}] - 1} [^{\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{4s} - D_{s} R_{4s}] - 1}, \end{array}

where the left side has undergone an interchange between

{R^{'}}_{4s}

and

{R^{'}}_{1s}

and also an interchange between

R_{4s}

and

R_{1s}

(which can be easily verified also by applying the Wronskian relation), we can move a further step to reach

ℑ {(E - ℜ)}^{- 1} = {[{R^{'}}_{1s} {{(D}_{s})}^{- 1} D_{f} R_{3f} - \frac{ρ_{s}}{ρ_{f}} R_{1s} {{(D}_{s})}^{- 1} D_{f} {R^{'}}_{3f}]}^{- 1} \times {\begin{cases} \frac{- i}{c_{s} (ξ^{2} - 1)} \\ \frac{- i}{c_{s} (ξ^{2} + 1)} \end{cases} .

Now, comparing Eq. (86) with Eq. (32), we can arrive at

P^{f} = ℑ {(E - ℜ)}^{- 1} {R^{'}}_{1s} {{(D}_{s})}^{- 1} A \times {\begin{cases} i c_{s} (ξ^{2} - 1) \\ i c_{s} (ξ^{2} + 1) \end{cases} .

However, noting that

P^{f_i}

in Eq. (70) is just equal to (also using the Wronskian relation)

P^{f_i} = {R^{'}}_{1s} {{(D}_{s})}^{- 1} A \times {\begin{cases} i c_{s} (ξ^{2} - 1) \\ i c_{s} (ξ^{2} + 1) \end{cases},

we then finally achieve

P^{f} = ℑ {(E - ℜ)}^{- 1} P^{f_i} .

2.Expressing $P^{s}$ in terms of $ℜ$

For this task, we work on the expression of $2 {(E - ℜ)}^{- 1} ℜ$ which can be obtained from Eq. (81) and Eq. (82) as that

\begin{array}{l} 2 (^{E - ℜ) - 1} ℜ = \\ [^{\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{1s} - D_{s} R_{1s}] - 1} [D_{s} R_{3s} - \frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{3s}] . \end{array}

We continue on by substituting the following expression (obtained by using the Wronskian relation) $R_{3s} = {({R^{'}}_{1s})}^{- 1} R_{1s} {R^{'}}_{3s} - {({R^{'}}_{1s})}^{- 1} \times {\begin{cases} \frac{i}{c_{s} (ξ^{2} - 1)} \\ \frac{i}{c_{s} (ξ^{2} + 1)} \end{cases}$ into Eq. (90), and can further reach
$\begin{array}{l} 2 (^{E - ℜ) - 1} ℜ = \\ - {R^{'}}_{3 s} (^{{R^{'}}_{1 s}) - 1} - [^{\frac{ρ_{f}}{ρ_{s}} D_{f} R_{3f} (^{{R^{'}}_{3 f}) - 1} {(D}_{f}^{) - 1} D_{s} {R^{'}}_{1 s} - D_{s} R_{1s}] - 1} D_{s} (^{{R^{'}}_{1 s}) - 1} \times {\begin{cases} \frac{i}{c_{s} (ξ^{2} - 1)} \\ \frac{i}{c_{s} (ξ^{2} + 1)} \end{cases} . \end{array}$
Now, by comparing Eq. (92) with Eq. (31), and by noting that $P^{s_i}$ in Eq. (71) is equal to (using the Wronskian relation again)
$P^{s_i} = {R^{'}}_{3 s} {{(D}_{s})}^{- 1} A \times {\begin{cases} i c_{s} (ξ^{2} - 1) \\ i c_{s} (ξ^{2} + 1) \end{cases},$ we can finally arrive at
$P^{s} = 2 {(E - ℜ)}^{- 1} ℜ P^{f_i} + P^{s_i} .$

Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grant No. 11074134, 61138003), the Program for New Century Excellent Talents in University (No. NCET-10-0502), the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No. 1139), and the Tianjin Municipal Science and Technology Commission under Grand number 14JCYBJC16600.

References and links

1. G. J. Diebold, “Photoacoustic monopole radiation: waves from objects with symmetry in one, two, and three dimensions,” in Photoacoustic Imaging and Spectroscopy, L. V. Wang, ed. (Taylor and Francis, 2009).

2. G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990). [CrossRef] [PubMed]

3. G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991). [CrossRef] [PubMed]

4. L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012). [CrossRef] [PubMed]

5. S. Hu and L. V. Wang, “Optical-resolution photoacoustic microscopy: auscultation of biological systems at the cellular level,” Biophys. J. 105(4), 841–847 (2013). [CrossRef] [PubMed]

6. E. I. Galanzha and V. P. Zharov, “Circulating tumor cell detection and capture by photoacoustic flow cytometry in vivo and ex vivo,” Cancers 5(4), 1691–1738 (2013). [CrossRef] [PubMed]

7. E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013). [CrossRef] [PubMed]

8. E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013). [CrossRef]

9. C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).

10. R. D. Spence and S. Granger, “The scattering of sound from a prolate spheroid,” J. Acoust. Soc. Am. 23(6), 701–706 (1951). [CrossRef]

11. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [CrossRef] [PubMed]

12. J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003). [CrossRef] [PubMed]

13. B. R. Rapids and G. C. Lauchle, “Vector intensity field scattered by a rigid prolate spheroid,” J. Acoust. Soc. Am. 120(1), 38–48 (2006). [CrossRef]

14. M. J. Mendes, I. Tobias, A. Marti, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

15. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes,” Opt. Express 19(17), 16207–16222 (2011). [CrossRef] [PubMed]

16. K. T. McDonald, “Gaussian laser beams via oblate spheroidal waves,” arXiv:physics/0312024v1 (2003).

17. M. Zeppenfeld, “Solutions to Maxwell's equations using spheroidal coordinates,” New J. Phys. 11(7), 073007 (2009). [CrossRef]

18. M. Zeppenfeld and P. W. H. Pinkse, “Calculating the fine structure of a Fabry-Perot resonator using spheroidal wave functions,” Opt. Express 18(9), 9580–9591 (2010). [CrossRef] [PubMed]

19. L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007), Chap. 12.

20. V. E. Gusev and A. A. Karabutov, Laser Optoacoustics (American Institute of Physics Press, 1993), Chap. 2.

21. A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys. 54(2), 153–167 (2008). [CrossRef]

22. W. M. John, “Asymptotic approximations for prolate spheroidal wave functions,” Stud. Appl. Math. 54, 315–349 (1975).

23. M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993). [CrossRef]

24. S. Zhang and J. Jin, Computation of Special Functions (Wiley, 1997).

25. W. J. Thompson, “Spheroidal wave functions,” Comput. Sci. Eng. 1, 84–87 (1999).

26. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

27. P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003). [CrossRef]

Analytic theory of photoacoustic wave generation from a spheroidal droplet

Abstract

1. Introduction

2. General solution

2.1 Frequency domain solution

2.2 Time domain solution

3. Asymptotic analyses

3.1 Sphere

3.2 Infinite long cylinder

3.3 Infinite large layer

4. Decomposition analysis on time domain solution

5. Conclusions and future works

Appendix: derivation for the formulas of photoacoustic wave partial transmission and reflection at the boundary of a spheroidal droplet

1. Expressing $P^{f}$ in terms of $ℑ$ and $ℜ$

2.Expressing $P^{s}$ in terms of $ℜ$

Acknowledgments

References and links

Cited By

Figures (2)

Equations (94)

Optics Express

Abstract

1. Introduction

2. General solution

2.1 Frequency domain solution

2.2 Time domain solution

3. Asymptotic analyses

3.1 Sphere

3.2 Infinite long cylinder

3.3 Infinite large layer

4. Decomposition analysis on time domain solution

5. Conclusions and future works

Appendix: derivation for the formulas of photoacoustic wave partial transmission and reflection at the boundary of a spheroidal droplet

1. Expressing Pfin terms of ℑand ℜ

2.Expressing Ps in terms of ℜ

Acknowledgments

References and links

Cited By

Figures (2)

Equations (94)

Optics Express

1. Expressing $P^{f}$ in terms of $ℑ$ and $ℜ$

2.Expressing $P^{s}$ in terms of $ℜ$