As we know, the coherent state\(\left|\alpha \right.⟩\)has the following form[16],

$$\left|\alpha \right.⟩={e}^{-\frac{{\alpha }^{2}}{2}}{\sum }_{n=0}^{\infty }\frac{{\alpha }^{n}}{\sqrt{n!}}\left|n\right.⟩,$$

1

The expansion coefficients satisfy the Poisson distribution, the multi-photon states\(\left|n\right.⟩\)are involved. According to Eq. (1), we can easily derive the probability distribution \(p\left(n\right)\) and the mean photon number\(⟨n⟩\)

\(p\left(n\right)=\frac{{\alpha }^{2n}}{n!}{e}^{-{\alpha }^{2}},\) \(⟨n⟩=⟨\alpha \left|{a}^{†}a\right|\alpha ⟩={e}^{-{\alpha }^{2}}{\sum }_{n=0}^{\infty }\frac{{\alpha }^{2n}}{\left(n-1\right)!}={\alpha }^{2}{\sum }_{n=0}^{\infty }p\left(n\right)={\alpha }^{2},\) (2)

According to Ref.[19], the wave inside the fiber can be expressed as

$$u=y\left(\rho \right){e}^{-i\nu \phi }{e}^{-i\left(\omega t-{k}_{\zeta }\zeta \right)}=y\left(\rho \right){e}^{-i\nu \phi }{e}^{-i{\theta }_{l}},$$

Where\({\rho }_{0}\)is the optical fiber radius,\(\nu \text{i}\text{s} \text{t}\text{h}\text{e}\)Bessel function index,\(u\)satisfy the boundary condition\(\left|u\right|=\left|y\right|=0\), when\(\rho \ge {\rho }_{0}\). The solution of\(y\left(\rho \right)\)can be written by using the definition of a Bessel function

$$y\left(\rho \right)={J}_{\nu }\left({s}_{0n}k\rho w\right),y\left({\rho }_{0}\right)={J}_{\nu }\left({s}_{0n}k{\rho }_{0}w\right)=0,{s}_{0n}=\left(n+0.75\right)\pi ,$$

3

The index “0” represents the optical angular momentum (OAM) of the optical modes propagating inside the optical fiber, \({s}_{0n}\)are \(nth\) root of Bessel function with index “\(0\)”, which guarantees that

the boundary condition of \(u\) satisfied. Due to the total internal reflection is important in optical fiber, the root of Bessel function must satisfy the inequality\({s}_{0n}\le nk{\rho }_{0}\sqrt{{n}_{1}^{2}-{n}_{2}^{2}}\), here\(\lambda\),\({n}_{1}\),\({n}_{2}\)are the laser wave length, the optical fiber’s internal refraction index for \(\rho <{\rho }_{0}\) and the optical fiber’s outside refraction index for \(\rho >{\rho }_{0}\), respectively, \(nk\) is the \(n\)photons momentum. In the calculation of the diffraction, the wave field inside the fiber \(y\left(\rho \right)\), departure from the end of the fiber, the diffraction pattern onto the screen.

In order to calculate the spatial resolving power, we choice Bessel function index \(\nu =0\), the field on the screen \(P\left(x,y\right)\) being [1]. Eq. (4) gives out the Rayleigh diffraction limit integral.

\(U\left(P\right)=2\pi C{\int }_{0}^{{\rho }_{0}}{J}_{0}\left(k\rho w\right)\rho d\rho ,\) \(\frac{I\left(P\right)}{{I}_{0}}=\frac{{\left|U\left(P\right)\right|}^{2}}{{I}_{0}}={\left|\frac{{J}_{1}\left(k\rho w\right)}{k\rho w}\right|}^{2}={\left|\frac{{J}_{1}\left(x\right)}{x}\right|}^{2},x=k\rho w,\) (4)

\(w=\sqrt{{p}^{2}+{q}^{2}}\) represent the sin of angle\(\theta =arctg\frac{\sqrt{{\stackrel{̄}{x}}^{2}+{\stackrel{̄}{y}}^{2}}}{d}\) between the central principal ray direction\(\left(\text{0,0}\right)\)and diffraction direction\(\left(p,q\right), p=w{cos}[\phi ], q=w{sin}[\phi ]\)is the distance between the screen and the output surface,\(\left(\stackrel{̄}{x},\stackrel{̄}{y}\right)\)are the coordinates.

The diffraction integral formula of the Laser propagating through an optical fiber, comparing the Bessel functions \({J}_{0}\left({s}_{0n}k\rho w\right),{J}_{0}\left(k\rho w\right)\) in Eqs. (3)-(4), we can write out the diffraction integral mode through the fiber on the screen

$$\frac{I\left(P\right)}{{I}_{0}}={\left|\frac{{J}_{1}\left({s}_{0n}k\rho w\right)}{{s}_{0n}k\rho w}\right|}^{2}={\left|\frac{{J}_{1}\left({s}_{0n}{x}_{n}\right)}{{s}_{0n}{x}_{n}}\right|}^{2}, {s}_{0n}=\left(n+0.75\right)\pi ,$$

5

Referring to Eqs. (2)-(5), the diffraction pattern Intensity distribution \(I\left(x\right)\) can be obtained by taking the average value of the coherent state as

$$⟨\frac{I\left(P\right)}{{I}_{0}}⟩={\sum }_{n=0}^{\infty }p\left(n\right)(\frac{{J}_{1}\left({s}_{0n}{x}_{n}\right)}{{s}_{0n}{x}_{n}}{)}^{2},$$

6

We can write out the Rayleigh integral and Ultra integral

$$\frac{{J}_{1}\left({s}_{on}{x}_{n0}\right)}{{s}_{on}{x}_{n0}}=0, {s}_{on}{x}_{n0}={x}_{0}=k{\rho }_{0}w=3.833,\frac{{J}_{1}\left({x}_{0}\right)}{{x}_{0}}=0,$$

7

The \({x}_{n0}={x}_{0}/{s}_{on}\) can also read out from the zero point of the intensity distribution curves in the following Figs. 1–2.