1. Introduction

An optical beam that is altered by a partial obstruction propagates showing certain degree of self-reconstruction within and near the obstruction domain. This phenomenon, usually referred to as self-healing of the beam, has been demonstrated and studied in propagation invariant beams (PIBs), such as Bessel [1–4], Airy [5], Mathieu and Weber [6], and Caustic [7] beams. The effect has been also studied in scaled propagation invariant beams [8,9] and spatially variant polarization beams [10,11]. The beam self-healing, has been explained employing either geometrical [12,13] or wave-optics arguments [9,14,15].

Contrasting and complementing previous approaches, we analyze the self-healing by decomposing the obstructed beam into two mutually orthogonal components, one of which is an exact copy of the unobstructed beam, attenuated by a weighting factor. Our analysis provides a simple formula to compute this factor. The second component, which is orthogonal respect to the first one, cannot contribute to the self-healing process; i.e., it represents a pure distortion field. We establish a quantitative measure of the beam damage due to the obstruction. Then, the degree of self-healing is defined, as the relative reduction of the beam damage in a particular area, during propagation of the obstructed field.

There are several interesting results derived from our approach: The total damage, which corresponds to the damage computed in the whole propagation plane, is invariant respect to the propagation distance z. On the other hand, the self-healing degree at the whole propagation plane is null. However, the beam damage tends to reduce within and near the area where the beam is affected by the obstruction. In relation to these facts we obtain that if A and B are disjoint domains filling the propagation plane, the damage reduction in A is only possible if the damage increases in B. In consequence, a positive degree of self-healing in A is only possible if the degree of self-healing is negative in B.

The second orthogonal component of the obstructed beam, i. e. the distortion, is usually confined to a small area of the beam domain, at the obstruction plane. Thus, the spread of the distortion during propagation is faster than that of the beam, propitiating the partial self-healing of the distorted field, at least in the obstruction area. However it will be proved that in the far field propagation regime, the relative amplitudes of the unobstructed beam and the distortion stagnate, causing the suspension of the self-healing.

Although this self-healing process occurs for most of the beams and obstruction types, in our analysis we prove that some finite phase obstructions, whose extension is smaller than the beam, can annihilate it completely. It means that the weighting factor of the first orthogonal component of the obstructed beam becomes zero.

2. Mathematical analysis of self-healing

If the complex amplitude of the beam is denoted as f(x,y,z), the obstructed field can be represented, at the plane z = 0, by the expression

The first term, at the right side of Eq. (2), is an exact copy of the unobstructed beam, and the second term, o(x,y,0) f(x,y,0), is usually understood as the field that distorts the beam. It has been assumed that the beam self-healing, during propagation, is possible due to the diffraction, and consequent attenuation, of such a distortion term.

Although the expression for the obstructed field in Eq. (2) is mathematically valid, its use in the analysis of self-healing presents problems. Considering the restriction |t(x,y,0)|≤1, it is noted that the optical power of the obstructed beam must be smaller than that of the beam itself. Thus, the obstructed field cannot generate the beam f(x,y,0) without attenuation. Moreover, it can be shown that (in general) the two terms at the right side of Eq. (2) are not mutually orthogonal. It means that the second term, o(x,y,0) f(x,y,0), cannot be considered a pure distortion or noise term.

With the aim of developing a consistent mathematical analysis of self-healing, we have found that a convenient representation of the obstructed field is

By free propagation of the fields in Eq. (3), to a distance z, we establish the relation

The distortion term e(x,y,0) at the obstruction plane (z = 0) in general presents a reduced spatial extension in comparison to the beam size. Therefore, it is expected that the propagated distortion e(x,y,z) diffracts faster than the beam component βf(x,y,z). It means that the transverse extension of e(x,y,z) increases faster than the beam extension, meanwhile its relative amplitude (compared to the beam amplitude) tends to reduce. The beam self-healing corresponds to this relative reduction of the distortion amplitude.

There is a crucial question in this context: can the relative distortion amplitude (respect to the beam amplitude) be attenuated to the point of disappearing? Our answer, based on our theory, is no. We note that at the far field domain the functions e(x,y,z) and f(x,y,z) are respectively given by the Fourier transforms of e(x,y,0) and f(x,y,0), subject to a common re-scaling in terms of the distance z, and attenuated by a common factor |λz|⁻¹. Thus, the relative amplitude of e(x,y,z), respect to the amplitude of the beam f(x,y,z), stagnates at the far field propagation domain. Several examples illustrating this result, which correspond to a Gaussian beam affected by different obstructions, are presented in section 3.

In order to establish a metrics for the self healing process it is convenient to compute the relation for the powers of the three fields in Eq. (6). Considering that f(x,y,z) is orthogonal to e(x,y,z), it is proved that such powers obey the identity

It is noteworthy that the quotients at the left side in Eq. (7) correspond to the relative powers of the fields βf(x,y,z) and e(x,y,z), respectively. Since these relative powers are complementary, we can analyze the self-healing process considering only one of them, e.g. the second one, which is rewritten as

This quotient D_T, which evaluates the power of the deviation field e(x,y,z), relative to the power of the damaged field f_O(x,y,z), is referred to as the total beam damage, due to the obstruction transmittance. It is called “total” because the inner products in Eq. (8) are computed in the complete propagation plane (x-y). Considering the nature of the distortion field, we note that the damage measure in Eq. (8) implicitly takes into account both amplitude and phase errors. The extreme value of the field damage D_T = 0 occurs if the deviation e(x,y,z) is null. In this case there is no damage at all since f_O(x,y,z) = f(x,y,z). In the other extreme case, i. e. when β tends to 0, the total damage attains its maximum value, which depends on the type of beam and obstruction transmittance. An example is considered in section 3.

Since both inner products in Eq. (8) are independent of z, the defined total damage D_T is invariant during propagation of the obstructed field. Thus, we anticipate that the self-healing of this field can be possible only in a subdomain of the propagation plane. In order to assess such a finite domain self-healing, we establish the modified damage metrics

Now, we establish that the self-healing occurs in the domain Ω_Z when the beam damage D_Ωz(z) reduces during the propagation of the obstructed field. To formalize this idea we propose a metrics for the degree of self-healing, given by

An interesting fact is that the degree of self-healing in the whole propagation plane (denoted as SH_T) is necessarily zero, regardless the distance z. This result is obtained from Eqs. (9) and (10) assuming that both domains Ω₀ and Ω_Z are equal to the whole propagation plane. In consequence, if the whole propagation plane is formed by two disjoint regions A and B, we can establish the identity SH_T = SH_A(z) + SH_B(z) = 0. It means that a positive self-healing in a given region A, of the propagation plane, is only possible if the self-healing is negative in the complementary region B. In a given domain, the self-healing degree can be negative, positive, or zero, with obvious meanings in each case. Indeed, the possible values for SH(z) are in the range [−1, 1].

We have noted above that if f₀(x,y,0) is orthogonal respect to f(x,y,0) (i. e. β = 0), then the total damage reaches its maximum possible value. Let us briefly discuss how it is possible to obtain β = 0 for different complex obstructions transmittances. Considering Eq. (1) and the complex form of the transmittance t(x,y) = a(x,y) exp[iϕ(x,y)], for a positive defined function a(x,y), we obtain that a necessary and sufficient condition for the orthogonality between f₀(x,y,0) and f(x,y,0)〉 is the identity

On the other hand, it can be proved that for any functions f(x,y,0) and a(x,y) it is possible to find different non-constant phases ϕ(x,y) that fulfill Eq. (11). To consider a simple case we define I_Ω = ∫∫_Ω |f(x,y,0)|²a(x,y)dxdy. This integral is denoted by I_Σ if Ω is replaced by the whole plane Σ. Since the argument in this integral is non-negative, it is possible to find disjoint domains A and B that fill the plane Σ such that I_A = I_B = I_Σ/2. Therefore, as a particular case, it is easy to show that Eq. (11) is fulfilled if ϕ(x,y) is equal to 0 in A and equal to π in B. Consequently, for an arbitrary beam f(x,y,0) it is possible to find complex obstruction transmittances, for which β = 0, without necessity of obstructing the complete beam f(x,y,0). According to Eq. (6) this kind of complex obstruction generates an obstructed field [f₀(x,y,z) = e(x,y,z)], which is unable to reconstruct the beam f(x,y,z). At the end of section 3 we consider an example of a relatively small phase obstruction that produces the annihilation of a Gaussian beam.

3. Self-healing of a Gaussian beam

The discussed theory can be applied to any type of optical field f(x,y,z) and obstruction transmittance t(x,y). Next, we illustrate our approach analyzing the self-healing of a Gaussian field. The 3D Gaussian field is expressed, omitting the phase factor exp(ikz), as

Following Ref [15], we assume that the beam at z = 0 and the obstruction transmittance are given, respectively, by the functions

By propagation of the fields in Eqs. (13), (15) and (17) we obtain the 3D fields

We employ Eqs. (18)-(20) to compute the fields βf(x,y,z), f₀(x,y,z), and e(x,y,z), consi-dering different parameters. As first example we assume that the waist w_B in Eq. (14) is equal to w_A/4. In this case, the attenuation coefficient, obtained with Eq. (16), is β≅0.889. We computed the fields for propagation distances cZ_R(w_A), where Z_R(w_A) is the beam Rayleigh range and c takes the values 0, 1/4, 1/2, 1, 10, and 20. The amplitudes of the computed fields are depicted in Fig. 1. As second example we assume w_B = w_A/2. In this case the attenuation coefficient is β≅0.667, and the computed fields, for the propagation distances already considered, have the transverse amplitudes displayed in Fig. 2. The transverse position (x) in these Figs. is normalized by the z-dependent beam radius w(z,w_A).

 

Fig. 1 Transverse amplitudes of the fields βf(x,y,z) (blue), f₀(x,y,z) (red), and e(x,y,z) (yelow) for w_B/w_A = 1/4. The propagation distances are cZ_R(w_A) where c is (a) 0, (b) 1/4, (c) 1/2, (d) 1, (e) 10, and (f) 20.

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Fig. 2 Similar transverse amplitudes to those in Fig. 1, for w_B/w_A = 1/2.

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There are several facts that we learn in the displayed field amplitudes. An obvious issue is the fastest damage reduction during propagation of the less damaged beam [Fig. 1]. It is interesting to note that the amplitude |f_O(x,y,z)|, at different propagation distances, is a distorted version of the beam amplitude |βf(x,y,z)|. This result is related to the orthogonal character of the error e(x,y,z) respect to the beam.

Another interesting observation, evident in Figs. 1 and 2, is that the relative amplitudes of the fields are indistinguishable at the distances z = 10 Z_R(w_A) and z = 20 Z_R(w_A) (ignoring changes in the transverse and vertical scales). This result indicates that the relative forms of the fields f(x,y,z), f₀(x,y,z) and e(x,y,z) stagnate at the far field propagation domain. In this propagation regime, the amplitudes of these fields become invariant under propagation, being subject only to a common scale change, proportional to λz. This result motivates the calculation of the limit values for the beam damage and the self-healing degree, at the far-field propagation domain. The approach is specially meaningful for scaled propagation invariant beams. Later, we present formulas for such limits, in the discussed Gaussian beam case, in terms of the parameters that specify the beam and the obstruction function at z = 0.

To assess the evolution of the beam damage, during propagation of the obstructed beam, it is necessary to specify the domain Ω_Z [in Eqs. (9) and (10)] for each distance z. The simple approach is to take the domain Ω_Z independent of the distance z. However, since the transverse scale of the Gaussian beam f(x,y,z) is proportional to the beam radius w(z,w_A), at the propagation distance z, it can be pertinent to adopt (as a second option) an assessment domain Ω_Z whose scale is also proportional to w(z,w_A) [9]. As a particular case, we assume that Ω_Z is a circle of radius α w(z,w_A), for a constant α. A consequence of this scaling of Ω_Z, which will be noted later, is that the stagnant amplitudes of the fields f(x,y,z) and e(x,y,z) in the far field [e. g. the amplitudes in Figs. 1(e) and 1(f) and Figs. 2(e) and 2(f)], produce also stagnant values for the beam damage and the degree of self-healing.

For the Gaussian example under consideration, with the assumptions in the previous paragraph, the different terms required to obtain the beam damage and self-healing degree can be computed by formulas that are only dependent on the attenuation factor β and the constant α that determines Ω_Z. An example is the total damage, computed by the simple expression

To obtain the limit value for the damage in the far field, the numerator in Eq. (9) is replaced by 〈E(u,v,0)|E(u,v,0)〉_Ωρ, where E(u,v,0) is the Fourier transform of the deviation e(x,y,0), (u,v) are spatial frequency coordinates, and Ω_ρ is a circle of radius α (πw_A)⁻¹, in the Fourier domain. After some algebra the damage limit value is expressed as

Assuming α<<1, the exponentials in Eq. (22) can be replaced by the first order approximation of their Taylor series, and the limit error reduces to the simple formula D_lim = α²(1−β)²(1 + β)⁻¹. The limit value for the degree of self-healing is obtained by taking D_Ωz(z) = D_lim in Eq. (10). The damage D_Ω0(0), and other terms required in Eqs. (9) and (10), can also be computed by formulas that are similar to Eq. (22); but for brevity they are not developed here. After all, the damage and self-healing can be directly computed considering the functions in Eqs. (19) and (20).

In the case of the Gaussian beam, under consideration, using the described approach we computed the beam damage D_Ωz(z), and the self-healing degree SH(z), considering α = 1 and different values for the ratio w_B/w_A. Thus, the evaluation domain Ω_Z, at the propagation distance z, is an axial circle of radius w(z,w_A). The plots for D_Ωz(z) and SH(z), depicted in Figs. 3(a) and 3(b), correspond to three values for w_B/w_A that are 1/4 (blue), 1/2 (red), and 3/4 (yellow). The z distances, at the horizontal axes in Fig. 3, are normalized by the Rayleigh range of the beam f(x,y,z). The limit values for D_Ωz(z) and SH(z) at the far field domain, computed with Eq. (22), are marked with dashed horizontal lines at the right side in each plot. The small vertical lines cutting each plot in Figs. 3(a) and 3(b) correspond to the so called minimum reconstruction distances (z₀), computed in base of the approach reported in Ref [15]. Such distances are obtained assuming that the radius of the obstruction at z = 0 corresponds to the waist w_B of the Gaussian function that determines the soft obstruction [Eq. (14)]. It is noted in each plot of Fig. 3 (b), that the SH value at the propagation distance z₀, is more than 80% of the far field limit value, which appears at z >> z₀.

 

Fig. 3 Beam damage (a, c) and self-healing degree (b, d) for a Gaussian beam obstructed by a soft transmittance [Eq. (14)]. The waist radii w_B/w_A is 1/4 (blue), 1/2 (red), and 3/4 (yellow). The dashed lines mark the far field limit values for damage and self-healing. We considered the cases of scaled (a, b) and fixed (c, d) assessing domains.

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For comparison, we now assume that the evaluation domain Ω_Z is a circle of constant radius w_A, for any propagation distance z. Under this assumption, we compute D_Ωz(z) and SH(z) for the already considered quotients w_B/w_A. The results are displayed in Figs. 3(c) and 3(d).

The damage plots in Fig. 3 (a) that correspond to the case of a scaled evaluation domain present different non-zero far field stagnation values, which appropriately reflect the stagnation process observed in the amplitude profiles of Figs. 1(e) and 1(f) and Figs. 2(e) and 2(f). In contrast, the damage plots in Fig. 3 (c), which corresponds to the case of a constant evaluation domain of radius w_A, converge to 0 at the far field. This result is explained noting that the numerator 〈e(x,y,z)|e(x,y,z)〉_Ωz [in Eq. (9)] tends to zero for large z, since Ω_Z remains constant meanwhile e(x,y,z) extends its transverse width and reduces its height. The zero value for D_lim, in the second case, explains the larger far field values for the self-heling degree in Fig. 3 (d). Such self-healing limits are computed by taking D_Ωz(z) = 0 in Eq. (10).

Now we compute the damage and self-healing degree for the evaluation domain which is complementary to Ω_Z. For brevity we only consider the case when Ω_Z is the scaled axial circle of radius R = w(z,w_A). The plots for the damage and self-healing degree, depicted in Fig. 4, clearly show the complementary nature of these parameters, respect to the ones obtained for the domain Ω_Z [Figs. 3(a) and 3(b)].

 

Fig. 4 Similar results to those in Fig. 3 (a,b), obtained for the evaluation domain which is complementary to Ω_Z. Again the dashed lines mark the far field limit values.

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Considering again the Gaussian beam given in Eq. (13) we now assume the binary obstruction transmittance t(x,y) = 1−circ(r/R), where circ(r/R) denotes the circle function of radius R (equal to 1 for r<R and zero otherwise). Assuming the obstruction radius R = w_A/4 we obtain the attenuation factor β = 0.882. The amplitudes of fields βf(x,y,z), f₀(x,y,z) and e(x,y,z), at the propagation distances already considered in previous examples, are shown in Fig. 5. It is interesting to note that the far field stagnant fields obtained in this case, are quite similar to the ones obtained with the soft obstruction transmittance in the first example.

 

Fig. 5 Similar results to those in Fig. 1, for the binary obstruction transmittance t(x,y) = 1−circ(r/R), with R = w_A/4.

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Now we consider that the binary phase obstruction transmittance t(x,y) = 1−2circ(r/R) is employed to alter the Gaussian beam g_wA(x,y,0). In this case the beam obstruction produces a phase dislocation of π radians within the axial circle of radius R. Under the idea discussed in the last paragraph of section 2, we search for the radius R of the phase dislocation for which β = 0. It is obtained that such a radius is R_C = w_A[ln(2)/2]^1/2, where the function 'ln' denotes natural logarithm. The amplitudes |f(x,y,z)| and |f₀(x,y,z)|, at the propagation distances 0, z_R(w_A) and 10 z_R(w_A), are displayed in Figs. 6(a)-6(c). We did not plot |βf(x,y,z)| which is zero. The sector of the beam f(x,y,z) which is affected by the phase dislocation is marked by the dashed lines in Fig. 6 (a). The orthogonality of f₀(x,y,z) respect to f(x,y,z) is also dependent on the phases of the fields. Such phases, at the considered propagation distances, are displayed in Figs. 6(d)-6(f). In Fig. 6 (a), the amplitude |f₀(x,y,z)| is equal to |f(x,y,z)| and in Fig. 6(d) the phase for f(x,y,z) is zero. The plots in Fig. 6 show that neither the amplitude nor the phase of the beam f(x,y,z) is recovered in the propagated obstructed field f₀(x,y,z), even in the far field propagation, which occurs (with good approximation) at the distance z = 10 z_R(w_A) [Figs. 6(c) and 6(f)].

 

Fig. 6 Amplitudes (a-c) and phases (d-f) of the Gaussian beam f(x,y,z) (blue) and the damaged field f₀(x,y,z) (red) generated by a binary phase obstruction t(x,y) = 1−2circ(r/R). The radius R is chosen to produce the attenuation factor β = 0. The propagation distance is 0 (a, d), z_R(w_A) (b, e), and 10 z_R(w_A) (c, f).

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4. Final remarks and conclusions

We have proposed a new method to analyze the self-healing of an optical beam that is altered by a complex transmittance obstruction. The method is based on the representation of the obstructed beam by two orthogonal components. The first component is an exact copy of the unobstructed beam, only attenuated by a constant factor, which is computed by a simple formula. The second component, which is orthogonal to the unobstructed beam, represents the beam distortion.

As far as we know, there are no previous reports that derive the right distortion component of the obstructed beam. The papers in Refs [9, 15], represent the obstructed beam by means of two components, the first of which is the beam f(x,y,z) (without attenuation), while the second one was conceived to represent the beam distortion term. However, as we pointed out in section 2, this second component is not orthogonal to the beam and cannot be rightfully considered as the distortion field.

The precise determination of the distortion component of the obstructed beam allowed us to establish, in a natural way, a quantitative measure of the beam damage due to the obstruction. We evaluated the degree of self-healing, as the relative damage reduction, in a given domain. According to our discussion, the total damage, which corresponds to the damage computed at the whole propagation plane, is invariant respect to the propagation distance z. On the other hand, the self-healing degree at the whole propagation plane is null. In connection with these facts we proved, for two disjoint domains A and B that fill the propagation plane, that the damage reduction in A is only possible if the damage increases in B. In consequence, a positive degree of self-healing in A is only possible if the degree of self-healing is negative in B.

We noted that in the far field propagation regime, the relative amplitudes of the beam f(x,y,z) and the distortion e(x,y,z) stagnate. In consequence, the higher relative spread of e(x,y,z) respect to f(x,y,z) no longer occurs in the far-field, and the self-healing process is suspended. This stagnation is noted, e. g. in Figs. 1(e) and 1(f) and Figs. 2(e) and 2(f). Under these conditions, it may be pertinent to obtain also a stagnant value for the far-field beam damage. In the case of a obstructed Gaussian beam, we noted that the stagnant beam damage is obtained only if the region Ω_Z, where the beam damage is evaluated, is scaled at the same rate that the z-dependent width of the unobstructed beam. In the particular case of a soft obstruction transmittance, defined in terms of another Gaussian function, we derived simple formulas to obtain the limit values for the damage and self-healing, computed in the far-field.

Our functions to measure self-healing, which depends on the determination of the pure distortion component of the obstructed beam, contrasts with other metrics introduced in previous papers, e. g. the similarity [14], the RMS [9], and the deviation [15]. As we obtained in the numerical simulation in Section 3 (see e. g. Figures 2 and 3), our damage and self-healing plots stagnate at the far field propagation range. In contrast, the self-healing plots in previous reports (see e.g. Fig. 4 in ref [15]) do not show the far field stagnation.

An important result obtained in our discussion is that the original unobstructed beam f(x,y,z) is not recovered at all, from the obstructed beam f₀(x,y,z), for certain phase obstructions. According to Eq. (6) in our manuscript, the obstructed field can be expressed as f₀(x,y,z) = β f(x,y,z) + e(x,y,z) where the distortion e(x,y,z) is orthogonal to the beam f(x,y,z) when β is computed by Eq. (5). The term β f(x,y,z), proportional to the beam, disappears when β = 0. At the end of section 2 we discussed different cases for which we can obtain β = 0.

We consider that the different results obtained from the representation of the obstructed field by two orthogonal components, one of which corresponds to a pure distortion field, improves in a significant way the description and comprehension of self-healing.