## Abstract

Quantitative description of the self-healing ability of a beam is very important for studying or comparing the self-healing ability of different beams. As describing the similarity by using the angle of two infinite-dimensional complex vectors in Hilbert space, the angle of two intensity profiles is proposed to quantitatively describe the self-healing ability of different beams. As a special case, quantitative description of the self-healing ability of a Bessel-Gaussian beam is studied. Results show that the angle of two intensity profiles can be used to describe the self-healing ability of arbitrary beams as the reconstruction distance for quantitatively describing the self-healing ability of Bessel beam. It offers a new method for studying or comparing the self-healing ability of different beams.

© 2014 Optical Society of America

## 1. Introduction

Nondiffracting waves which have many interesting properties and potential applications have attracted more attentions [1–6]. Self-healing ability of beams is one of the most interesting properties which make them very useful in many areas. In recent years, much works concerning the self-healing properties of nondiffracting waves has been carried out [7–12]. For example, S. Vyas et al. have studied the self-healing of tightly focused scalar and vector Bessel–Gauss beams at the focal plane [7]; J. D. Ring et al. shows that Pearcey beams have the self-healing property [8]; R. Cao et al. have found that optical pillar array also has the self-healing ability [9]; M. Anguiano-Morales et al. have focused his study on the self-healing property of a caustic optical beam [10]; Pravin Vaity et al. have analyzed the self-healing property of optical ring lattice [11]; J. Broky et al. have investigated the self-healing properties of optical Airy beams [12].

Besides the properties of self-healing ability, the mechanism of many nondiffracting waves have been investigated [13–16]. For example, the phenomenon of reconstruction for a Bessel beam was explained by considering the dynamics of the conical waves [13, 14]; the explanations of the self-healing of an Airy beam are given by using the method of uniform geometrical optics and catastrophe optics [15, 16]. From the explanations we can see that the mechanism of the self-healing ability for different beams is different. In addition, which self-healing ability of a beam is stronger in practice is also very important. However, to the best of our knowledge, the comparison of the self-healing ability between different beams has not been made until now.

Self-healing ability of a beam describes that a beam shape partially blocked by an opaque obstacle can be reconstructed during propagation. It means that the beam shape partially blocked by an opaque obstacle will become more and more similar to that of the beam without obstruction. Similarity can be used to estimate the difference between beams, for example, Martinez-Herrero et al. in Ref. [17] have studied the difference between the closest field and the target function by using the similarity. Therefore, we can use similarity to describe the self-healing process. In present paper quantitative description of the self-healing ability is proposed. As an example, quantitative description of the self-healing ability for Bessel-Gaussian beam is investigated.

## 2. Quantitative description of the self-healing ability

To describe the self-healing procession, the optical fields of a beam with and without obstacle are denoted by ${E}^{\prime}\left(x,y,z\right)$ and $E\left(x,y,z\right)$in the following. In Hilbert space ${E}^{\prime}\left(x,y,z\right)$ and $E\left(x,y,z\right)$ can be regarded as two infinite-dimensional complex vectors. The angle of two infinite-dimensional vectors can be defined as [18]

*S*is named as similarity of two functions in general.

## 3. Special case: self-healing ability of a Bessel-Gaussian beam

Self-healing of a Bessel-Gaussian beam has been studied based on geometrical optics [13, 14]. By considering
the dynamics of the conical waves, there is a minimum distance (reconstruction distance) behind
an obstacle of radius *R* before reconstruction occurs [13, 14]

*α*is the axicon angle (see Fig. 1). For comparison with the existing results in a special case, as an example, quantitative description of the self-healing ability of a Bessel-Gaussian beam is studied in the following. As shown in Fig. 1, Gaussian beam passing through an axicon is used to generate the Bessel-Gaussian beam [13].

The angular spectrum of an axicon is

where*δ*() is the DiracDelta function and $\kappa =\sqrt{{u}^{2}+{v}^{2}}$ is the radial coordinate (

*u*and

*v*being the component of the angular spectrum along

*x*and

*y*-axis),

*k*= 2

*π/λ*is the wavenumber.

From the convolution of the angular spectrum of an axicon and a Gaussian beam, the angular spectrum at the initial plane can be obtained as

*I*

_{0}is the zeroth-order modified Bessel function of the first kind,

*w*

_{0}is the waist width of the Gaussian beam. By using the transfer function of the angular spectrum in free spacewe can obtain the angular at

*z*-plane as

*J*

_{0}is the zeroth-order Bessel function of the first kind, and $r=\sqrt{{x}^{2}+{y}^{2}}$ [(

*x*,

*y*) being the transverse coordinates]. When we set

*z*= 0 or ${w}_{0}\to \infty $from Eq. (17) one can find

For simplicity to quantitatively study the self-healing ability, the expression of the transmission for an opaque obstacle which block the initial optical field is given by a Gaussian function as

where*R*is the radius of the obstacle. With the same method as in Eq. (17), the optical field of a Bessel-Gaussian beam at z-plane with an obstacle can be given as

To see the self-healing process of Bessel-Gaussian beam in free space we set$\alpha =\pi /12$, $\lambda =512nm$in the following calculation. Intensity distributions of a Bessel-Gaussian beam with and without an opaque obstacle at initial plane are plotted in Fig. 2.

It can be seen that the central spot of a Bessel-Gaussian beam in Fig. 2(a) is obstructed. To see the self-healing process, the evolution of the intensity distribution of a Bessel-Gaussian beam with and without opaque obstacle is shown in Fig. 3 where the beam travel along z-axis and y = 0.

From Fig. 3 we can see that the central spot obstructed by
an opaque obstacle gradually reconstructs during propagation. Because of the circular symmetry
of the Bessel-Gaussian beam with and without an opaque obstacle in present paper, only
one-dimensional similarity *S* is investigated. From Eqs. (11), (17) and (21) the similarity can be
calculated. Figure 4 shows the variation of the
similarity *S* during propagation with different parameters.

Figure 4(a) shows the variation of the similarity *S* with different *w*_{0} where *R* = 0.8*μ*m. It can be seen that the similarity is large with large *w*_{0} when the propagation distance is short. With the increase of propagation distance the difference of the similarity corresponding different *w*_{0} become small. For comparison with the reconstruction distance in Eq. (12), ${d}_{\mathrm{min}}$ is also denoted in the Figs. We can see that the similarity is about 0.95 when the propagation distance equal to ${d}_{\mathrm{min}}$. Namely, the reconstruction distance can be used as a metric to estimate the self-healing ability. Small ${d}_{\mathrm{min}}$ means fast speed of the self-healing process. Even though ${d}_{\mathrm{min}}$ denotes the distance where the reconstruction occurs in geometrical optics, the self-healing process almost complete when we consider diffraction of the beam. Figure 4(b) shows the variation of the similarity *S* with different *R* where *w*_{0} = 10*μ*m. We can see that the speed of the self-healing is different with different *R*. When R is small, the distance is small to reconstruct its shape. We also can see from Fig. 4(b) that can also be used to describe the distance where the reconstruction has completed.

It should be pointed out that the reconstruction distance in Ref. [13] which is obtained from the geometrical optics is only used to describe the self-healing of Bessel beam generated by an axicon. From the study we can see that Eq. (11) can be used to describe the evolution of the similarity of arbitrary beams during propagation. As comparing with the reconstruction distance we can see that Eq. (11) can be regarded as a metric to estimate the self-healing ability of Bessel-Gaussian beam. Because the self-healing ability means that a beam which is partially blocked by an opaque obstacle will become more and more similar to that without obstacle during propagation, Eq. (11) can be used to describe the self-healing ability of arbitrary beam.

## 4. Conclusion

In Hilbert space, optical field can be regarded as an infinite-dimensional complex vector. Using the angle of two infinite-dimensional complex vectors, the similarity of two intensity distribution is used to describe the self-healing ability. As an example, the self-healing ability of a Bessel-Gaussian beam is studied. Study shows that similarity can be used to quantitatively describe the strength of the self-healing ability of any beams liking reconstruction distance for Bessel beam. However, reconstruction distance is only used to describe the self-healing of Bessel beam generated by an axicon, similarity can be used to quantitatively describe the self-healing ability of arbitrary beams. With the help of the similarity for two intensity profiles, the self-healing ability of different beams can be compared.

## Acknowledgment

The project was supported by National Natural Science Foundation of China (No. 11374264).

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