Naga

naga is special function that appears at the estimate of the loss of the pinhole waveguide in the paraxial approximation. naga is expressed as integral of the simple combination of the Bessel function BesselJ0 with elementary functions:

\(\displaystyle\mathrm{naga}(p)=2\int_0^\infty \frac{J_0(L_1 x)^2}{(1-x^2)^2} \, \exp(\mathrm i p x^2)\, x\, \mathrm dx\)

where \(L_1\!=\)BesselJZero[0,1]\(\approx\! 2.4\) is the first zero of the Bessel function; \(J_0(L_1)=0\).

Relation with other functions

For the efficient evaluation of naga defined above, especially for integration in vicinity of \(x\!=\!0\), various special functions are defined:

mori\((x)=\displaystyle \frac{ J_0 (L_1 x)}{1-x^2}~;\) \(~\displaystyle \mathrm{naga}(p)=2 \int_0^\infty \mathrm{mori}(x)^2 \, \exp(\mathrm i p x^2)\, x\, \mathrm d x\)

kori\((x)=\displaystyle \frac{ J_0 \big(L_1 \sqrt{x}\big)}{1-x}~;\) \(~\displaystyle \mathrm{naga}(p)= \int_0^\infty \mathrm{kori}(x)^2 \, \exp(\mathrm i p x)\, \mathrm d x\)

The corresponding limit at \(x\!=\!1\) is assumed for definitions of functions mori and kori.

Application

In the single mode approximation, the only Besselian mode is taken into account. Then, the loss of the pinhole waveguide is expressed with function maga,

maga\((p )=1-|\mathrm{naga}(p )|^2\)

at least for real values of \(p\). Then, maga\((p )\) can be interpreted as loss of waveguide made of equidistant pinholes of radius \(R L_1/k\) separated with distance \(2 p R^2/k^2\), for paraxial wave with wavenumber \(k\). For the physical interpretation and application to the paraxial optics, parameter \(R\) is supposed to be large real number, id est, \(R\!\gg\! 1~\).

For complex \(p\), expression

maga\((p )=1-\mathrm{naga}(p )\, \mathrm{naga}(p^*)^*\)

can be considered; however, the special care about the contour of integration in the expressions above is necessary to provide convergence of the integrals at infinity. Then, maga is expected to be holomorphic function.

Following the main ideology of TORI, the functions mentioned above are supposed to be analysed and implemented for the whole complex plane, even if the direct application mentioned above refers only to the real values of the argument. The analysis of the holomorphic properties of the functions involved is supposed to reveal the errors, mistakes in the deduction, if any, to allow refutation of the basic concept of the pinhole waveguide, if this concept is wrong.

Evaluation

For the representation of naga\((p )\) through function mori, at positive \(p\),

\(~\displaystyle \mathrm{naga}(p)=2 \int_0^\infty \mathrm{mori}(r)^2 \, \exp(\mathrm i p r^2)\, r\, \mathrm d r\)

the following contour of integration is suggested:

1. Integration along the real axis from zero to \(L_1/p\) .

2. Integration from \(r=L_1/p\) to \(r=L_1/p+\sqrt{\mathrm i} \infty\) along line \(\Im(r)=\Re(r)-L_1/p\) in the complex \(r\)–plane.

For the first part of the contour, the intelligent fit of function kori seems to be useful.

For the second part, at small \(p\), the asymptotic approximation of function mori seems to be useful,

mori\((z)\approx\,\)morias\(_m(z)~\), \(~z\rightarrow \infty\), with integer \(m\) and

\(\displaystyle \mathrm{morias}_m(z) = - \sqrt{\frac{2}{\pi L_1}} z^{-5/2} \, G_m(z^2)\, \cos\left( - \frac{\pi}{4} + L_1 z\, F_m(z^2) \right) \)

where
\(G_m(x) = \sum_{n=0}^{m} g_n x^{-n}\)
\(F_m(x) = \sum_{n=0}^{m} f_n x^{-n}\)

Approximations for the coefficients \(f\) and \(g\) follow from the asymptotic expansion of the Hankel function;

\(\begin{array}{l} f_0=1\\ f_1=- 0.021614383628830615865\\ f_2=0.0019465899152260872595\\ f_3=-0.0010834984344719114778\\ f_4=0.001464410164512283719\\ f_5=-0.003628876399615993660\\ f_6=0.01431760830195380729\\ f_7=-0.0824438982874790057\\ f_8=0.652747801052423657\\ f_9=-6.80376838070624330\\ f_{10}=90.322658904953727\\ f_{11}=-1487.942880868968812\\ f_{12}=29785.50901270392374\\ \end{array}\) \(~\) \(\begin{array}{l} g_0=1 \\ g_1=0.989192808185584692068 \\ g_2=0.99228788615079417081 \\ g_3= 0.989481317221334367489\\ g_4= 0.994709980602617872387\\ g_5= 0.97818700495778240956\\ g_6=1.0575251177784290263\\ g_7= 0.5188843197279991625\\ g_8=5.432808917007474985\\ g_9=-52.5640507009104629\\ g_{10}=807.429675670594971\\ g_{11}= -14844.4023379433794\\ g_{12}=328204.367306340176 \end{array}\)

Aproximation of the integrand through function morias appears as follows:

\(\mathrm{integrand}_m(p,r)=\mathrm{morias}_m(r)^2 \, \exp(\mathrm i p r^2)\, r\)

for \(p\!=\!0.1\) and \(m\!=\!11\), compex map of this approximation is expected to be shown in figure at right (sorry, is not loaded yet)
with lines \(u=\Re(\mathrm{integrand}_m(p,x\!+\!\mathrm i y))=\mathrm{const}\)
and lines \(v=\Im(\mathrm{integrand}_m(p,x\!+\!\mathrm i y))=\mathrm{const}\)
in the \(x\), \(y\) plane.

At the same figure, the suggested contour of integration described above is shown with thick green line.

For evaluation of function naga, descriptions of functions kori and mori are loaded.

Keywords

Bessel function, BesselJ0, mori, kori, Morinaga function, Pinhole waveguide