We perform a theoretical analysis by considering a particle crossing the fringes of a Bessel beam with a constant velocity normal to the longitudinal axis and passing through the center of the beam. For simplicity, we use a very basic model for the light scattered from this particle and collected by the detector. We consider that the particle size is smaller than the fringes, and we assume that the intensity of this light is proportional to the intensity of the fringes it crosses. It is well established that from an incoming Gaussian beam, a regular axicon of base angle $\alpha $ produces an intensity proportional to $J02(k\beta r)$, where $J0$ is the Bessel function of the first kind of order 0, $r$ is the radius from the center of the beam, $k$ is the wave number, and $\beta $ is the refraction angle. The wave number is given by $\u2009\u2009k=2\pi /\lambda $, and the refraction angle is approximated for small angles in terms of $\alpha $ by $\beta \u2248(n\u22121)\alpha $, where $\lambda $ is the wavelength and $n$ is the index of refraction of the axicon. Thus, as the particle crosses the fringes through the center, it scatters light with intensity proportional to $J02(k\beta r)$ and a photodetector collecting this light produces a proportional electrical signal (particle 1 in Fig. 1). If the particle is moving with a velocity $v$ parallel to the $y$ axis, we can substitute $r=vt$ and the intensity as a function of time, $t$, is proportional to $\u2009\u2009J02(k\beta \nu t)$. The spectrum, $F(f)$, as a function of the frequency $f(Hz)$, of this signal is obtained by calculating its Fourier transform, $F$, which is given using the distribution theory.^{12}Display Formula
$F(f)=F([J0(k\beta vt)]2)={2fb\pi 3K(1\u2212f2fb2),f<fb0,otherwise.$(1)
$K()$ is the complete elliptic integral^{13} of the first kind given for a parameter $m$ by $K(m)=\u222b01dt/(1\u2212t2)(1\u2212mt2)$ and $fb=k\beta v/\pi $, which can be written as Display Formula$fb=2\beta v\lambda .$(2)
$F(f)$ is plotted if Fig. 2 for $\alpha =5\u2009\u2009deg$, $\lambda =658\u2009\u2009nm$, and $n=1.52$ and different velocities. It has a half-hut shape, high at low frequencies, and decreases until $fb$, after which it becomes zero. The expression of the frequency, $fb$, which is called the Bessel frequency,^{4} is similar to the expression of the Doppler frequency shift, $fD$, for the LDV dual beam mode, where $\beta $ is the half angle between the two intersecting beams. The difference is that in the case of the LDV, the spectrum shows a peak at $fD$, while in this case $fb$ marks the edge of the hut-shaped frequency distribution.