The pressure field at a large distance from the center of the uniformly illuminated region composed of an ensemble of absorbers with identical physical properties and equal radii becomes,^{22}^{,}^{23}Display Formula
$pfensemble(r,kf)\u2248i\mu \beta I0\upsilon sa2CPr\u2062[sin\u2009q^\u2212q^\u2009cos\u2009q^]eikf(r\u2212a)q^2[(1\u2212\rho ^)(sin\u2009q^/q^)\u2212cos\u2009q^+i\rho ^\upsilon ^\u2009sin\u2009q^]\u2211n=1Ne\u2212ikf\xb7rn.$(2)
In the above derivation, the linear superposition principle has been used generating the resultant pressure field from the acoustic waves emitted by the individual absorbers. Here, $rn$ indicates the position vector of the nth absorber and $kf$ denotes the direction of observation as shown in Fig. 1. In this figure, small filled circles represent hemoglobin molecules, which are confined inside the cells. The optical absorption coefficient for each cell depends upon the concentration and oxygen saturation states of these molecules. Each big circle (solid line) outlines the cellular boundary. The region containing uniformly illuminated absorbers is denoted by the biggest circle (dashed line). Further, $N$ number of absorbers are present in the illuminated region and their contributions are summed up. The presence of a collection of OA sources is denoted by the superscript ensemble. This derivation essentially utilizes the single particle approach, which provides a simple way to model the generated pressure field. The single particle model works well for a sparse medium and has been successfully utilized to explain experimental results in light and ultrasound scattering problems.^{26}^{,}^{27} Equation (2) clearly shows that the OA signal amplitude is linearly proportional to the cellular absorption coefficient. The time dependent pressure field for the uniform illumination of the absorbers by a delta function heating pulse can be obtained by employing the Fourier transformation of Eq. (2) as, Display Formula$pfensemble(r,t)\u2248i\mu \beta F\upsilon sa22\pi CPr\u2062\u222b\u2212\u221e\u221ed\omega [sin\u2009q^\u2212q^\u2009cos\u2009q^]eikf(r\u2212a\u2212vft)q^2[(1\u2212\rho ^)(sin\u2009q^/q^)\u2212cos\u2009q^+iq^\upsilon ^\u2009sin\u2009q^]\u2062\u2211n=1Ne\u2212ikf\xb7rn,$(3)
where $F$ is the fluence of the incident optical radiation. In Eq. (3), contributions from all possible frequencies are summed up generating the time dependent pressure field from an ensemble of OA sources. However, in practice, only positive frequencies are considered and for such a case Eq. (3) provides an analytic signal.^{28} In case of an analytic signal, the imaginary part of the signal is the Hilbert transform of the real part. Therefore, at any time point the real part of the complex valued function provides the instantaneous RF signal and the corresponding signal amplitude can be obtained from its magnitude.