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Abstract
Using the asymptotic formula of the Hermite polynomials for higher-orders $n \gg 1$, an elegant mathematical expression that makes Hermite–Gaussian beams and cosine beams equivalent is obtained. Two factors of merit, the similarity and the power content ratio, are used to quantify the degree of equivalence between the two beams. These results yield a new nondiffracting Hermite–Gaussian beam in one dimension (1D) and that is easily extended to 2D.
© 2020 Optical Society of America
Corrections
Abdelhalim Bencheikh and Andrew Forbes, "Non-diffracting nature of truncated Hermite–Gaussian beams: publisher’s note," J. Opt. Soc. Am. A 37, 1906-1906 (2020)https://opg.optica.org/josaa/abstract.cfm?uri=josaa-37-12-1906
23 June 2020: A typographical correction was made to the Fig. 6 caption.
1. INTRODUCTION
Since the introduction of nondiffracting Bessel beams by Durnin and Eberly [1,2], much attention has been devoted to the subject, both theoretically and experimentally [3–10]. In particular, their peculiar property to propagate without spreading has attraction renewed attention in fields such as micromachining [11,12], imaging [13,14], quantum [15,16], and telecommunications [17–20]. The first introduced nondiffracting beam was the Bessel beam [1,2], which has a cylindrical symmetry and could be obtained by a superposition of plane waves propagating on a cone. Since then, many other geometries have been exploited [21–23], including arbitrarily shaped nondiffracting fields [24]. Later, one-dimensional (1D) and two-dimensional (2D) cosine beams were introduced as a nondiffracting beam in Cartesian coordinates [25,26] and exploited in various optical systems [27–32] including free space, turbulence, plasmas, and crystals to name a few. Such beams could be obtained by interfering two oppositely tilted plane waves so that the resulting beam remains invariant in shape and in intensity over some region [33–35], and they have subsequently been extended to other domains, for example, plasmonic cosine beams [36].
Although Bessel and cosine beams are nondiffracting, they are not eigenmodes of free-space propagation: they just exist in a region of overlapping skewed waves, and beyond that region they change their structure, which is undesirable in many applications. Recently, an elegant framework has been outlined showing the resemblance between nondiffracting Bessel beams and Laguerre–Gaussian (LG) beams [37,38]. The authors showed that under some considerations, the LG beams behave as a nondiffracting Bessel beam. Consequently, by choosing a good candidate from LG beams, the latter could replace Bessel beams in their nondiffracting character while retaining the property of shape invariance of LG beams.
Here we present a new kind of nondiffracting beam that we refer to as a nondiffracting truncated Hermite–Gaussian beam (NDTHGB). The latter resulted from a peer-to-peer comparison between 1D nondiffracting cosine beams and Hermite–Gaussian (HG) beams. We show that by using the asymptotic formula for Hermite polynomials for higher orders $n \gg 1$, the HG beam behaves like a nondiffracting cosine beam. Since the latter results from the interference between two oblique plane waves, the range of the resulting nondiffracting cosine beams is limited. In contrast, HG beams are eigenmodes of free-space propagation and so they propagate infinitely without changing their structure. We will see that despite the asymptotic approach, HG and cosine beams are not perfectly similar to each other. Thus, we introduce truncation as a means to increase the similarity, which we explore in this work. Further, this comparison yields a simple approach to extend the results from 1D to 2D. We believe that these results will be useful in many applications requiring nondiffracting beams in rectangular symmetry and over extended distances.
2. TRUNCATED COSINE AND HG BEAMS IN 1D
HG beams are solutions of the paraxial wave equation in Cartesian coordinates $(x,y,z)$, whose amplitude can be expressed in one dimension as
Let us recall the expression of the field distribution of the nondiffracting cosine beam, which is given by [27]
where $\sqrt {k_y^2 + k_z^2} = k = 2\pi /\lambda$, and $\lambda$ is the wavelength. As implied in Eq. (2), the intensity distribution of the cosine beam is independent of the propagation distance $z$, and hence the cosine beam is considered to be nondiffracting. In order to find a relationship between cosine and HG beams, we follow the asymptotic treatment of the Hermite polynomials [39], and because cosine beams are 1D beams, the comparison will be achieved with the 1D HG beams. We have the asymptotic behavior of the Hermite polynomials when $n \gg 1$ [39]:After some algebra, and using the variable change $y = \sqrt 2 y/w(z)$ to move from mathematical functions to optical beams with widths $w(z)$ of the Gaussian basis, we find the final expression of the electric field amplitude for a given HG beam ${{\rm HG}_n}$ of order $n$ and its equivalent cosine beam
Equation (4) provides an elegant relationship between cosine (sine) beams and 1D HG beams: for even $n$, the right side of Eq. (4) remains a cosine function and corresponds to the even HG beams (symmetric about origin), while for odd $n$ we have a sine function, corresponding to odd HG beams (inverted about the origin).
Since cosine beams theoretically are infinite beams and cannot be realized, so to deal with meaningful and practical beams, we use an apertured version of cosine beams. The size of the limiting aperture of cosine beams depends on the width of the corresponding HG beams: the Gaussian envelope width ${w_G}$ results in second moment width of ${w_{\rm HG}} = {w_G}\sqrt {2n + 1}$. In addition, from Eq. (4) ones can calculate the transverse frequency of the cosine (HG) beams where ${k_y} = \sqrt 2 \sqrt {2n + 1} y/w(z)$.
Besides, Eq. (4) allows us to investigate the propagation of nontruncated HG and cosine beams; however, to assess intensities’ distributions at the transverse and longitudinal planes of truncated beams, the numerical calculation of Fresnel–Kirchhoff diffraction integral is used throughout the paper, its simplified mathematical expression is given by
3. NUMERICAL RESULTS
The following numerical simulations for 1D beams are based on incident beam waists of ${w_0} = 1\; {\rm mm}$ in the $x$ and $y$ directions at a wavelength of $\lambda = 1064\; {\rm nm}$. Figure 1(a) shows a simulated cosine beam along with its ${{\rm HG}_{10}}$ equivalent in (b), with a cross section of both shown in (c). We noticed that the two beams differ substantially at the wings; consequently, we apply an aperture to both beams to study the similarities of apertured versions of the beams. What should the size of the aperture $a$ be? We suggest two figure of merit (FoM) factors to answer this. The first is a similarity factor [40] that quantifies the degree of self-healing of any obstructed beam
Fig. 1. (a) Intensity pattern of an infinite cosine beam and (b) its equivalent ${{\rm HG}_{10}}$ beam. The cross-sections of these intensities profiles are shown in (c).
Fig. 2. Two FoMs as a function of the truncation parameter $\beta$ for a cosine and equivalent HG beam both of order $n = 10$.
Figure 3 generalizes these results to beams orders $n = 10$ and $n = 40$. We see that the truncated HG beams with lower orders are more similar to cosine beams than those with higher orders, seemingly in contradiction to Eq. (4). We point out that here we are considering truncated HG beams and so we do not need to use a very high order to ensure the equivalence. Further, higher orders does not mean that we have better similarities because when applying a truncation the situation changes such that the similarity may be better for lower orders that for higher ones. The results in Fig. 3 are consistent with those plotted in Fig. 2 but offer a more visual representation of the beam similarities: at low truncation parameters we have high similarity between the beams, decreasing as the truncation parameter is increased.
Fig. 3. FoMs as function of $\beta$ for $n = 10$ and $n = 40$ showing (a) the similarity factor and (b) the power content ratio factor.
Depending on the application of the beams, curves presented above allow us to find a compromise between the beam order and the similarity degree between HG and cos beams.
To continue studying the effect of the truncation on HG and cos beams we have chosen three different values of the truncation parameter: $\beta = 0.32$, $\beta = 0.55$, and $\beta = 1$. The intensity profiles for two different beam orders $n = 10$ and $n = 21$ (even and odd) are presented in Fig. 4. It is noticeable that the truncation parameter $\beta$ significantly affects the resemblance between the two beams, and throughout the paper we show that the truncation parameter $\beta$ is the main key to produce a quasi-nondiffracting beam from HG beam.
Fig. 4. Intensity profiles for even $(n = 10)$ and odd $(n = 21)$ beams: (a) truncated $H{G_{10}}$ and its equivalent cosine beam with $\beta = 0.32$, (b) same beams with $\beta = 0.55$, and similarly in (c) and (d) for $n = 21$. For a truncation parameter $\beta = 1$, the degree of similarity is low ($S \lt 0.25$), resulting poor visual similarity, shown in (e) and (f) for $n = 10$ and $n = 21$, respectively.
Figure 5 shows the propagation behavior in the $y^\prime - z$ plane of truncated HG beams at the left side and their equivalent cosine beams at the right side for three beam orders: $n = 10$ (first row), $n = 21$ (second row), and $n = 40$ (third row), all truncated at $\beta = 0.32$. At first glance, the HG beams seem to be identical to the cosine beams in all space, confirming the similarly of Fig. 2, which increases with beam order (see later).
Fig. 5. Propagation behavior on the $y^\prime - z$ plane of truncated HG beams (left) and their equivalent cosine beams (right) for $\beta = 0.32$ with (a) and (b) $n = 10$, (c) and (d) $n = 21$, and (e) and (f) $n = 40$. Skewed dashed lines show the formation of nondiffracting structure due to the interference of two oblique plane waves, the transverse dashed lines show the range of resulting nondiffracting beams.
Fig. 6. Propagation behavior on the $y^\prime - z$ plane of (a) the truncated CG beam and (b) its equivalent HG beam for $\beta = 1$.
As mentioned before, the frequency of HG and cosine beams depends on the beam order, and using a simple geometry of truncated cosine beam, we can obtain the maximum propagation distance ${Z_{\max}}$ that defines the range of the nondiffracting beam. After some algebra we get ${Z_{\max}} = \sqrt 2 \beta {z_R}$. An interesting result is that the range of the beams is independent of the beam order, it depends only on the truncation parameter and on the Rayleigh length of the Gaussian beam. This expression of ${Z_{\max}}$ is consistent with the simulations shown in Fig. 5, where at a fixed $\beta = 0.32$ and different beam orders $(n = 10)$, $(n = 21)$, and $(n = 40)$, we have the same range, obtained from the plots equal to $0.45{z_R}$ and calculated using the expression of ${Z_{\max}}$, it is also ${Z_{\max}} = 0.45{z_R}$.
Figure 6 shows plots in the $y^\prime - z$ plane of cosine beam at (a) and its equivalent ${{\rm HG}_{10}}$ beam for a truncation parameter $\beta = 1$; the similarity degree is low ($S \lt 0.25$), and the two beams evolve differently during propagation. An interesting result for the case of $\beta = 1$ is that the beam range ${Z_{\max}} = \sqrt 2 {z_R}$, which is greater than ${Z_{\max}} = {z_R}$ of the usual Gaussian beam. It seems contradictory, but it is not; for that reason, cosine beams are nondiffracting, and also it is obvious from Fig. 4 that the central intense line (fringe) keeps its intensity constant for a long distance compared to HG beams.
Figure 7 gathers the intensity patterns in the $y^\prime - z$ and $x^\prime - y^\prime $ planes with the intensity profiles shown. With $\beta = 0.55$ and $S \approx 0.8$, the two beams are almost similar in the region before the plane $z = 0.5{z_R}$, but they differ in the far field, where the HG beams keeps its shape, while the cosine beam evolves to become two separate spots. This is corroborated with the cross-sectional plots at $z = 0.2{z_R}$ and at $z = {z_R}$, the former showing high similarity, particularly near the center of the beams, while the latter does not.
Fig. 7. Density plot and profiles for ${{\rm HG}_{10}}$ beam and its equivalent cos beam for a given truncation parameter $\beta = 0.55$. Propagation in the $y^\prime - z$ plane at (a) and (b), respectively. (c) and (d) Their corresponding transverse patterns at the $z = {z_R}$ plane. (e) and (f) Their intensity profiles at two planes, $z = 0.2{z_R}$ and $z = {z_R}$, respectively.
Fig. 8. On-axis intensity of the truncated cosine beam and ${{\rm HG}_{10}}$ beams for (a) $\beta = 0.32$ and (b) $\beta = 0.55$.
To further investigate the nondiffracting nature of HG and cosine beams we use the well-known on-axis intensity decay, shown in Fig. 8 for two truncation parameter values. Both beams display the well-known fast oscillations in on-axis intensity that are indicative of nondiffracting beams.
Fig. 9. 2D non-diffracting truncated HG beams with (a) ${{\rm HG}_{10,10}}$ and (b) its equivalent cosine beam.
4. EXTENSION TO 2D
It is worth noting that one of the applications that we suggest for such nondiffracting truncated HG beams is the generation of nondiffracting array beams by the generalization of the result from 1D to 2D. Figure 9 shows an example of the similarity between 2D cosine beams and the 2D version of our truncated HG beams for a truncation parameter $\beta = 0.55$. We achieve this by multiplication of the two separable 1D solutions to form a 2D solution with two indices, $n$ and $m$, i.e., ${{\rm HG}_{n,m}}(x,y) = {{\rm HG}_n}(y) \times {{\rm HG}_m}(x)$, each given by Eq. (4). From the notion of separability and the superposition principle in wave optics, we immediately have that this beam will also be nondiffracting.
5. CONCLUSION
We have demonstrated that cosine beams and HG beams are practically the same under some circumstances. Numerical results showed that HG beam could be better than cosine beams; the former propagates for longer distances, keeping their shape, while the latter loses its shape to become an arbitrary beam. Since the cosine beam is a 1D nondiffracting beam, our results are easily extended to 2D nondiffracting beams with Cartesian symmetry: a non-diffracting truncated HG beam.
Disclosures
The authors declare no conflicts of interest.
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