Conversion of any finite bandwidth optical field into a shape invariant beam

V. Arrizón; F. Soto-Eguibar; D. Sánchez-De-La-Llave; H. M. Moya-Cessa

doi:10.1364/OSAC.1.000604

1. Introduction

Propagation of a coherent or partially coherent light beam is an important process in applied and theoretical optics. Researchers have devoted special attention to light beams that preserve their intensity profile during propagation. The so called non-diffracting (ND) beams, as Bessel [1–4], Mathieu [5], and Weber [6] beams, maintain their transverse intensity profiles without a scale change. Unfortunately, in the practical implementation of these beams, which include a finite envelope [4,7–10], the invariant structure is only present in a finite region. Other optical fields, which we refer to as scaled propagation invariant (SPI) beams, preserve their shapes in an unbounded propagation range, at the expense of a re-scaling in their transverse extensions. The Hermite-Gaussian and Laguerre-Gaussian beams are examples of such optical fields [11].

Interesting methods, intended to extend the existence domain of Bessel beams, have been previously reported [12–15]. These beams are modified in such a way that they also present a re-scaling in their transverse widths.

Here we discuss a method to transform any optical field, with finite frequency bandwidth, into a SPI beam. The principle of the method is quite simple. It consists in modulating the field with a quadratic phase of appropriate curvature radius. This phase modulation can be obtained (for example) by passing the original beam through a divergent lens. As a particular result, we will show that the method allows the extension of the existence region of a ND beam to an unlimited propagation range.

It is interesting to note that the modification of a beam by a quadratic phase may occur (as a particular case) in the generalized Airy-Gauss beams, proposed in Ref [9]. However, the possible extension of the existence domain of the beam, due to the presence of the quadratic phase, was not analyzed in that work.

The theoretical support of our proposal is discussed in section 2. The relevance and implications of the proposal is illustrated by numerical examples in section 3. Finally, conclusions and remarks are presented in section 4.

2. Theory

To start our discussion, let us consider an optical field f(x,y), defined at the plane z = 0, and its Fourier transform F(u,v). The Fraunhofer propagation of the field f(x,y), to a distance z₀, is given by h(x,y) = E(x,y,z₀) F(x/λz₀, y/λz₀) [16], where λ is the field wave-length and

E (x, y, z_{0}) = \exp [i k (x^{2} + y^{2}) / (2 z_{0})]

is a quadratic phase of curvature radius z₀ and wave number k = 2π/λ. An interesting fact, little considered in optics literature, is that h(x,y) is a field whose transverse shape is invariant for propagation to any distance z>z₀, with a z-dependent scale change. A crucial fact that we have noticed is that this invariance is possible by the presence of the quadratic phase factor in the complex amplitude of h(x,y). This fact motivated us to analyze the propagation of an arbitrary optical field when it is modulated by a quadratic phase factor.

Let us consider a monochromatic field of wavelength λ whose complex amplitude, at the plane z = 0, is g₀(x,y). We will study the phase modulated version of this field, given by

g (x, y) = E (x, y, R) g_{0} (x, y) .

For the sake of compactness, in the following mathematical analysis, we omit some constant complex factors. To compute the propagation of g(x,y) we first obtain its Fourier transform. Using the convolution theorem in Eq. (2), and defining the vectors V = (u,v) and V₀ = (u₀,v₀), the Fourier transform of g(x,y) is given by the integral

G (u, v) = \iint_{Ω} G_{0} (u_{0}, v_{0}) \exp (- i π λ R | V - V_{0} |^{2}) d u_{0} d v_{0},

where G₀(u,v) is the Fourier transform of g₀(x,y). From now on we consider that the non-zero values of G₀(u,v) appear in a finite circle of radius ρ_C, denoted as Ω, which corresponds to the integration domain in Eq. (3). On the other hand, we assume the restriction

λ R ρ_{C}^{2} = A < < 1.

Under such restrictions, imposed to G₀(u,v), the exponential exp[−iπλR(u₀² + v₀²)] that appears by developing the argument of the integral in Eq. (3), is approximated to 1; and this integral is solved as

G (u, v) = \exp [- i π λ R (u^{2} + v^{2})] g_{0} (λ R u, λ R v) .

From Eqs. (2) and (5), it is noticed that g(x,y) and its Fourier transform G(u,v) have similar structures, except by the phase conjugation in Eq. (5). Thus, the field g(x,y) can be considered a quasi-self-Fourier function [17–19].

In the Fresnel approximation, the propagation of the optical field g(x,y), to the plane z = z₀, is given by

g_{p} (x, y) = ℑ^{- 1} {G (u, v) e x p [- i π λ z_{0} (u^{2} + v^{2})]},

where ℑ⁻¹ is the inverse Fourier transform operator. Considering Eqs. (5) and (6), the propagated field g_p(x,y) can be expressed by means of E(x,y,R + z₀)⊗G₀ (−x/λR,−y/λR) where ⊗ is the convolution operator and E is the quadratic phase defined in Eq. (1). By developing the explicit integral form of this convolution, applying again the restrictions imposed to the Fourier spectrum G₀, we obtain the propagated field

g_{p} (x, y) = E (x, y, M R) g_{0} (\frac{x}{M}, \frac{y}{M}),

with the scaling factor

M = 1 + z_{0} / R .

According to Eq. (7), the propagated field g_p(x,y) is identical to the field g(x,y), except by a re-scaling in the coordinates of the factor g₀(x,y) and the curvature radius of the divergent phase. The field g_p(x,y), in Eq. (7), provides a good approximation to the exact propagated field if the curvature radius R fulfills the condition in Eq. (4). According to this relationship the approximation improves by decreasing A (or R).

Since the expression for the field in Eq. (7) corresponds to the Fresnel propagation of the field g(x,y), the distance z₀ should be (in principle) large enough to satisfy this propagation regime. However, we can prove that (under the adopted assumptions) Eq. (7) is a good approximation for any positive propagation distance z₀. First, we note that the field g(x,y) at z = 0 [Eq. (2)] represents the Fraunhofer propagation of the field G₀(−x/λR, −y/λR), when it is located at the plane z = −R [Fig. 1

Fig. 1 Generation of the propagated fields in the far field domain of the field G₀(−x/λR,−y/λR).

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]. For the validity of this result it is only required to fulfill the band-width restriction imposed to the function G₀. But in this context, it is obvious that the far field propagated from the plane z = −R to the plane z = z₀ (for any distance z₀≥0) is also given by Eq. (7). Therefore, Eq. (7) provides a good approximation for the propagation of the field g(x,y) to the plane z = z₀, regardless the distance z₀.

3. Numerical examples

To develop the first example we define the function

g_{1} (x) = \sin c (x / p) e x p [- {(x / w_{0})}^{2}],

where w₀ is the waist radius of the Gaussian factor, p is the distance between adjacent roots of g₁(x), and the ‘sinc’ function is defined as sinc(α) = (πα)⁻¹sin(πα). Next, we define g₀(x,y) = g₁(x)g₁(y), i. e.

g_{0} (x, y) = \sin c (\frac{x}{p}) \sin c (\frac{y}{p}) e x p (- \frac{x^{2} + y^{2}}{w_{0}^{2}})

that we name as sinc-Gaussian (SG) field. The Fourier transform of g₀(x,y) is G₀(u,v) = G₁(u)G₁(v), where G₁(u) = rect(pu)⊗exp(−π²w₀²u²) and rect(pu) is a rectangular pulse of width 1/p (in the u-domain). Now we assume the waist radius w₀ = 6p. In this case, the width of the Gaussian term, in the above convolution that defines G₁(u), is smaller than the width of the rectangular pulse. Thus, the radius of the circular domain Ω, which contains the non-zero values of G₀ (u,v), is approximately ρ_C = (2)^−1/2p⁻¹. In addition, assuming A = 1/20 in Eq. (4), we obtain R = p²/(10λ). Thus, by employing Eq. (2), we obtain the complex amplitude of the phase modulated sinc-Gaussian (PMSG) field

g (x, y) = \exp [i 10 π (x^{2} + y^{2}) / p^{2}] g_{0} (x, y) .

The normalized transverse amplitude of the SG (or PMSG) field is displayed in Fig. 2(a)

Fig. 2 (a) Transverse amplitudes of the SG field in Eq. (10) and (b) its Fourier spectrum. In (c) we show the transverse amplitude of the Fourier transform of the PMSG field defined in Eq. (11).

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. In addition, the normalized transverse amplitudes of the Fourier spectra of the SG and PMSG fields are shown in Figs. 2(b) and 2(c). As expected, the Fourier spectrum of the SG field is a rectangular pulse (of width 1/p), softened by the convolution with a narrow Gaussian function. On the other hand, the amplitude shape in Fig. 2(c) confirms the striking result in Eq. (5), i. e. the Fourier spectrum G(u,v) inherits the shape of g₀(x,y).

According to Eq. (7), it is also expected that the propagated PMSG field inherits the transverse shape of the SG field, for any propagation distance z₀>0. To verify it we computed the propagated PMSG beam, using the exact propagation operator. The amplitude of this beam, for z₀ in the range [0, 2R], is depicted in Fig. 3

Fig. 3 Transverse amplitude of the propagated PMSG field, at the range [0, 2R] for z₀.

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. In this result, the linear dependence of the transverse magnification, respect to z₀, is in agreement with Eq. (8). On the other hand, the field attenuation for z₀>0 is due to the normalization of the computed amplitudes, respect to the peak amplitude at z₀ = 0.

In the example under discussion, the curvature radius R was computed with Eq. (4) using A = 1/20, which is much smaller than 1. Under this condition, the propagated PMSG beam preserve the shape of the SG field g₀(x,y), with good approximation. Even when we depicted the field amplitudes (instead of intensities), the errors of the field shapes in Figs. 2(c) and 3, respect to the amplitude of the field g₀(x,y), are not distinguished. Such errors are more significant when the restriction A<<1 [Eq. (4)] is transgressed. This situation is considered in the next example.

As second case we consider the Bessel-Gaussian (BG) beam

g_{0} (x, y) = J_{0} (2 π ρ_{0} r) \exp (- r^{2} / w_{0}^{2}),

where r is the radial coordinate, ρ₀ is the asymptotic radial frequency, and w₀ is the waist radius. By making the reasonable assumption w₀>>ρ₀⁻¹ (where ρ₀⁻¹ is the asymptotic beam period), we establish that the radius (ρ_C) of the circle that contains the non-zero values of the beam Fourier spectrum G₀(u,v) is approximately ρ₀. Using this result in Eqs. (2) and (4), we obtain the phase modulated BG (PMBG) beam

g (x, y) = \exp (i π ρ_{0}^{2} r^{2} / A) g_{0} (x, y) .

Let us consider specific BG and PMBG beams with the parameters w₀ = 3ρ₀⁻¹ and A = 1/4. The amplitude of the BG (or PMBG) beam, at z = 0, is shown in Fig. 4(a)

Fig. 4 (a) Transverse amplitudes of the BG field in Eq. (12) and (b) its Fourier spectrum. In (c) we show the transverse amplitude of the Fourier transform of the PMBG field g(x,y) obtained with A = 1/4.

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; and the amplitudes of the normalized Fourier spectra G₀(u,v) and G(u,v) are depicted in Figs. 4(b) and 4(c). The transverse amplitude of G₀(u,v) corresponds, as expected, to an annular field of radius ρ₀, whose transverse section is a Gaussian. On the other hand, the plot in Fig. 4(c) confirms that the Fourier spectrum G(u,v) inherits the shape of g₀(x,y).

Now we establish the longitudinal size of the existence zone of the BG beam, roughly given by z_C = w₀/λρ₀. This length is obtained geometrically, assuming that the beam radius is w₀. We computed the propagation of the BG and PMBG beams, for z₀ at the range [0, 1.5z_C], using exact propagation. The amplitudes of the propagated fields g_0p(x,y) and g_p(x,y) in this range are depicted in Figs. 5(a) and 5(b)

Fig. 5 Transverse amplitudes of the propagated (a) BG beam and (b) PMBG beam (for A = 1/4) at the propagation range [0,1.5z_C]. The transverse amplitudes at three specific propagation distances for the BG beam and the PMBG beam are displayed in (c, d), respectively.

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. To establish the transverse scale in Fig. 5 we introduced the asymptotic BG beam period p = ρ₀⁻¹. The transverse amplitudes of the propagated BG beam g_0p(x,y), at the distances z₀ = 0, 0.75 z_C and 1.5 z_C, are displayed in Fig. 5(c). In this figure, the transverse amplitude at the distance z₀ = 1.5 z_C (black line) tends to adopt the annular form of the BG beam far field. The transverse amplitudes of the propagated PMBG beam, at the distances z₀ = 0, 0.12 zc, and 0.75 zc, are displayed in Fig. 5(d). The beams in Figs. 5(b) and 5(d), which results from the propagation of the PMBG beam, preserves the amplitude shape of this field. The expected transverse scaling is not visible in these figures because the transverse coordinate r has been normalized respect to the length M(z₀)p, where M(z₀) is the scale factor defined in Eq. (8). In Fig. 6

Fig. 6 Transverse amplitude of the propagated PMBG beam (for A = 1/4) at the propagation range [0, z_C/4]. In this case the transverse coordinate normalization is the same for every z₀.

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the transverse scaling of the propagated PMBG beam is made visible by the adopting the normalized transverse coordinate r/p (independent of the propagation distance). In this case, the propagation distance z₀ is limited to the range [0, z_C/4].

The arguments employed to obtain the PMBG beam in Eq. (13), can also be applied if the Bessel function J₀(2πρ₀r) is replaced by any ND beam with Fourier spectrum B(ϕ) δ(ρ−ρ₀), being (ρ,ϕ) the polar coordinates in the Fourier domain (u,v) and B(ϕ) an arbitrary azimuthal modulation. Therefore, the existence domain of any ND beam can also be extended by applying an appropriate quadratic phase factor to its complex amplitude.

Now we investigate the effects of transgressing the restriction A<<1. As first task, we compute the Fourier spectrum amplitudes of the PMGB beam defined in Eq. (13), by assuming A = 1, 2, 3 and 4. The computed Fourier spectrum amplitudes |G(u,v)|, displayed in Fig. 7

Fig. 7 Transverse amplitude of the Fourier transform of the PMBG field [Eq. (13)] obtained with A = 1, 2, 3, and 4.

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, show evident shape deviations respect to the field amplitude |g₀(x,y)| [in Fig. 4(a)]. These deviations are due to the fact that the required restriction A<<1 is no longer fulfilled for the adopted A values. In Fig. 8

Fig. 8 Normalized transverse amplitudes of the propagated PMBG beams for A = 3 and 4, at z₀ = 6z_C.

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we depict the normalized transverse amplitudes of the propagated PMBG beams for A = 3 and A = 4, at the propagation distance z₀ = 6z_C. As expected, the propagated PMBG beams tend to show the shapes of the Fourier spectrum amplitudes |G(u,v)|, respectively shown in Figs. 7(c) and 7(d).

The comparison of field amplitudes, made until now, is appropriate to enhance errors in small field levels. In the following computations we display the intensities of the fields, which enhance the similitude of low field values. In Fig. 9(a)

Fig. 9 Transverse intensities for (a) the BG beam and for the Fourier spectra of the PMBG beams, obtained with A equal to (b) 1, (c) 2, and (d)3.

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we display the intensity of the field g₀(x,y) under discussion [which corresponds to the BG beam amplitude in Fig. 4(a)]. On the other hand, in Figs. 9(b)–9(d) we show the intensities of the Fourier spectra of the PMBG beams, obtained with the parameters A = 1, 2, and 3, respectively. The Fourier spectra intensities (|G|²) in Figs. 9 provide (using the appropriate scaling) the intensity profiles of the far field propagated PMBG beams, for the different A values. The far field intensities of the PMBG beams for A = 1 and 2 [Fig. 9(b) and 9(c)] present the features of the unmodified BG beam intensity, with high fidelity. The far field PMBG beam intensity for A = 3 show clear errors beyond the first local minimum.

Even when the condition in Eq. (4) has been transgressed by assuming relatively high values of A, the transverse scale of the Fourier spectra in Fig. 9 is still proportional to A⁻¹ (or R⁻¹) in agreement with Eq. (5). It can also be verified that the high quality far field shape preservation, in the cases A = 1 and A = 2, also occurs for the corresponding near field propagated PMBG beams.

To evaluate the shape deviation between two intensity profiles I₀(r) and I(r) we employ the formula

E = \frac{\int_{0}^{r_{p}} r {[I_{0} (r) - α I (r)]}^{2} d r}{\int_{0}^{r_{p}} r {[I_{0} (r)]}^{2} d r} .

We assume that the intensities I₀(r) and I(r) are defined in a circular domain of radius r_p and that α is a constant that allows the best fitting between I₀(r) and I(r). Thus, this constant is obtained from the local minimum condition δE/δα = 0. Due to the factor r in the integrands of Eq. (14), the integrals correspond to the circular 2-D domain. To measure the shape deviation of the propagated BG beam g_0p(x,y), I₀(r) and I(r) are assumed as the transverse intensities of g₀(x,y) and g_0p(x,y) respectively. On the other hand, to measure the shape preservation of the propagated PMBG beam g_p(x,y), I(r) corresponds to the transverse intensity of g_p(Mx,My). The factor M in this relation provides a compression in the field g_p that reverts its scaling attained during propagation. This procedure allows a fair comparison of the PMBG beam intensity shapes at different propagation planes.

The computed intensity profile deviation E, versus the propagation distance z₀, for the propagated BG beam, is depicted in Fig. 10(a)

Fig. 10 Intensity deviations (a) E and (b) ln(E) for the propagated BG beam (black) and for the propagated PMBG beams with parameter A equal to 1/4 (red), 1 (blue), and 2 (green).

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(black line). On the other hand, the deviation E, for different propagated PMBG beams (which are the phase modulated ones), is shown in the colored lines of Fig. 10(a). In the latter case we have considered three values for the parameter A (1/4, 1 and 2). In this figure, the shape error for the unmodified beam tends quickly to the maximum value (1) when the propagation distance tends to 1.5 z_C.

On the other hand, the shape errors for the modified beams are below 0.02 (2%) for the different A values. Alternate error plots, which allow the display of the full error ranges in all the cases (in the employed propagation domain), are shown in Fig. 10(b). In this case we displayed the natural logarithm of E.

4. Final remarks

We have proved that any optical field, whose Fourier spectrum is contained in a circle of finite radius (ρ_C), can be transformed into a beam with invariant shape during propagation, at the expense of a scaling in its transverse extension. The transformation consists in modulating the field by a quadratic phase factor, with a curvature radius R that fulfills the restriction λRρ_C²<<1. As a particular result, we proved that the Fourier transform of the phase modulated field inherits the mathematical form of this beam, except that the quadratic phase in the Fourier spectrum is conjugated. A straightforward method to modulate the original field g₀(x,y) with a quadratic phase consists in transmitting it through a divergent lens.

The analyzed propagation invariance is an approximation whose errors decrease by reducing the curvature radius in the employed quadratic phase modulation. Specifically, the restriction λRρ_C²<<1 ensures almost negligible errors in the analyzed field invariance.

However, the results are not affected too much if the restriction λRρ_C² = A<<1 is moderately transgressed. We illustrated this result in Fig. 9, considering PMBG beams obtained with A = 1, 2, and 3. The advantage of adopting relatively large A values is that the quadratic phase modulation, that modifies the BG beam, presents relatively large curvature radii. As a consequence, the re-scaling in the transverse field extension, during propagation, reduces its growth speed.

A plausible conclusion is that the existence of a beam in an unbounded propagation range is only possible if one accepts a z-dependent re-scaling of its transverse extension. However, it is pending to develop a general proof of this assertion.

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Conversion of any finite bandwidth optical field into a shape invariant beam

Abstract

1. Introduction

2. Theory

3. Numerical examples

4. Final remarks

References

Cited By

Figures (10)

Equations (14)

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