1. Introduction

Helical beam is a kind of vortex beams with helical wavefront, and hence it carries the orbital angular momentum (OAM). Allen et al presented that the helical beam with a helical phase term e^ilφ, e.g. the Laguerre-Gaussian beam (LGB), possesses OAM quantified with lh̄ per photon [1], where l is an integer called angular quantum number (AQN). In past a decade, the helical beam is paid great attention in many applications, such as optical spanner [2], high density data propagating [3], quantum information processing [4], and etc. Now, several work groups study intensively the optical communication using the helical beams [3,5–11] by virtue of the infinite eigen-states of the OAM operator −ih̄∂_φ [12].

Recently, the ultrafast optics was paid great attention and the pulse width was shorten to the order of a wavelength. This leads to the femtosecond optics, which play significant roles in ultrafast optical switch and the future all optical network. The ultrafast optics may instil new vitality to the OAM division multiplexing (OAM-DM) technology, which is an important channel multiplexing technology first presented by Wei [13]. The OAM-DM technology constructs channels by using the eigen-modes of the OAM operator and transmitting information in each channel. The ultrashort pulse beam operating on the fundamental Gaussian mode was studied intensively in the past decades, while that operating on the helical modes appeared only in numbered works [14, 15]. The time-varying and propagating characteristics of the pulse helical beam are still undiscovered.

This work propose a pulse helical beam model by chopping the incident continuous wave LGB into ultrashort pulse like that used by Porras et al [16], and expand the pulse LGB into a series of LGBs with the same AQN but different angular quantum numbers (RQNs). The expansion coefficients of these decomposed modes show the time-varying and propagating characteristics of the pulse beam. This work details these characteristics and is important for the optical communication by using the OAM-DM technology.

2. Theoretical analysis

2.1. Solution of pulse wave equation in paraxial condition

The paraxial envelope equation of the pulse beam is familiar as [16]

where ∆_⊥ = ∂² _x + ∂² _y is the two dimensional Laplace operator and k ₀ = ω ₀/c the wave number with ω ₀ and c the central angular frequency and light velocity, respectively. In case of monochromatic wave, the third term in the right hand of Eq. (1) can be omitted and Eq. (1) can be simplified as

The solution of Eq. (2) may be the familiar normalized LGBs ψ_m,n with RQN m nonnegative integer and AQN n integer.

An simple method to solve the pulse equation Eq. (1) is the perturbation method [16], which considers the third term in Eq. (1) as the perturbation to the second term. But in this work we expand the pulse beam as the superposition of LGBs and obtain the expansion coefficients, which contain the time-varying and propagating characteristics of the pulse beam.

The incident beam can be expand as the superposition of LGBs

Substitute Eq. (3) into Eq. (1) and consider Eq. (2), we can obtain

Multiply the complex conjugation ψ*_m′,n′ on Eq. (4) and integrate on the transverse plane, then

where H_m′,m = 〈ψ*_m′,n∂_zψ_m,n〉, with the symbol 〈*〉 denotes the integral on the transverse plane. Equation (5) can also be rewritten as matrix equation

where X = (a _0,n,a _1,n, ⋯,a_m,n, ⋯)^T and H = (H_m′,m).

The matrix equation Eq. (6) can be converted to a group of second order partial differential equations with two variables and several functions. Its solution relies on the properties of the matrix H. According to Zauderers’ work [17] and the property of Laguerre polynomials [18], we can obtain the matrix elements H_m′,m and then the properties of the matrix H:

There must be an invertible matrix U to convert H to a diagonal matrix, i.e. U ⁻¹ HU = −iΛ, where the diagonal matrix Λ has positive real diagonal elements λ _0,n, λ _1,n, ⋯, λ_m,n, ⋯ according to the properties (b), (c) and (d). Then the matrix equation [Eq. (6)] can be simplified as

where Y = (b _0,n,b _1,n, ⋯,b_m,n, ⋯)^T = U ⁻¹ X.

We note that Λ is a diagonal matrix, then the coupled equations in Eq. (5) can be decoupled to some separate equations. Each separate equation reads

The solution of the Eq. (8) relies on the boundary condition and initial condition which relate to the pulse model.

2.2. Pulse model, boundary condition and initial condition

In optical communication, fast modulation is needed. The future applications of the OAM in high date rate communication and quantum information processing will deep into the domain of ultrafast optics. A simple model ultrashort LGB can be constructed by chopping a single mode continuous wave LGB into slice.

As shown in Fig. 1, a traveling LGB is chopped by a planner shutter perpendicular to the beam axis and large arbitrarily in transverse plane. The transmittance of the planner shutter is equal in transverse plane and changes quickly.

 

Fig. 1. Model of ultrashort pulse LGBs generation.

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It’s convenient to let the transmittance or wave envelope f(t) be Gaussian, i.e.

where T is related to the pulse duration by $\Delta t=\sqrt{2\ln 2}T$ .

Consider the case of single incident LGB mode ${\psi }_{m_{0,}n}$ , the boundary condition of Eq. (1) reads

and therefore the boundary condition of Eq. (7) reads

where ${\overline{U}}_{m,m_0}={\left({\mathbf{U}}^{-1}\right)}_{m,m_0}$ denote the matrix element of U ⁻¹.

Similarly, the initial condition of Eq. (7) also reads

According to Eqs. (8), (11) and (12), the problem of pulse LGB can be formulated as

In this work, the numeric results are given for further discussions.

3. Results and discussions

In this work, the computation is carried out for the single cycle pulse LGBs, i.e. the pulse duration ∆t equals a wave cycle T ₀. The incident LGBs are ψ _0,0, ψ _0,3, ψ _0,6 and ψ _3,3, respectively. For a certain incident LGB, the pulse beam may be decomposed into many LGBs. These decomposed beams have the same AQN but different RQNs because of the orthogonal relationship of the LGBs. The following discuss bases on the analysis of the expansion coefficients |a_m,n(z, t)|², which shows the power distribution with the time t and along the beam propagation direction z. So |a_m,n(z, t)|² may give the time-varying and propagation characteristics of the pulse LGB.

 

Fig. 2. (color online)Time-varying and propagating of decomposed LGBs, |a_m,n(z, t)|². Incident LGBs (rows from top to bottom) are ψ _0,0, ψ _0,3, ψ _0,6, and ψ _3,3; RQN of decomposed LGBs (columns from left to right) are m = 0, 1, 2, but m = 2, 3 and 4 in the last row.

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Figure 2 gives the power distribution with t and along z of the decomposed modes of the pulsed LGBs, i.e. |a_m,n(z, t)|². The time axis is scaled by the wave cycle T ₀ and the z axis by Rayleigh range z_R as denoted in Fig. 2(a). Other subgraphs use the same setting. The inset in each subgraph shows the corresponding top view. Figs. 2(a)–2(c) in the first row are computed in case of pulsed LGB with (m,n) = (0,0), and show |a_m,n(z, t)|² of the decomposed LGBs with m = 0, 1 and 2, respectively. Although m is theoretically infinite, |a_m,n|² is rather small for m > 2 in range of about −5T ₀ ≤ t ≤ 5T ₀ and 0 ≤ z ≤ 10z_R. Therefore, the decomposed LGBs with m > 2 are not shown here. This also mean that only numbered terms in expansion Eq. (3) can give acceptable result. There are 8 expansion terms retained in this work and they are sufficient. Because of the orthogonal relation between the helical phase terms exp(inφ), the AQN does not change in the first row. The other rows from the top down are computed for incident LGBs with (m,n) = (0,3), (0,6) and (3,3),respectively. The RQNs of the decomposed LGBs from left to right in the upper three rows are 0, 1 and 2, respectively. But in the bottom row they are 2, 3 and 4, respectively.

In each row in Fig. 2, the power of the original mode is much bigger that others. For example, in the second row the original mode is shown in Fig. 2(d) with (m,n) = (0,3) and the power in the range of −5T ₀ ≤ t ≤ 5T ₀ and 0 ≤ z ≤ 10z_R is rather bigger than others in the same row. The original mode decays along the z axis while other modes arise. In near area with little z, the original mode is dominant. With increasing z, the power distribution spreads and delays in time axis. The mode dispersion also indicates that the pulse beam will spread in transverse plane, because higher order modes have bigger transverse dimension. From top to bottom, the incident mode with the same RQN m = 0 but larger AQN n shows bigger delay in time and decay in z direction, and larger pulse spread in time and transverse plane. In spite of the difference, they show similar time-varying and propagating characteristics. But for the incident LGBs with nonzero RQN m, even with the same AQN n, the original modes decay much faster. For example, the incident LGBs in the second and fourth rows are ψ _0,3 and ψ _3,3, respectively. The original mode ψ _3,3 in Fig. 2(k) decays much faster than ψ _3,3 in Fig. 2(d).

 

Fig. 3. Time-varying of decomposed LGBs, |a_m,n(z, t)|², at different distances. Incident LGBs (rows from top to bottom) are ψ _0,0, ψ _0,6, and ψ _3,3; Distances (columns from left to right) are z = z_R, 2z_R, and 4z_R.

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We also note that for the incident LGBs with zero RQN, the original mode is dominant in rather long range, e.g. z < 2z_R, along z axis. There the pulse beam may be processed as the original mode provided that the time delay and spread been dealt with properly. This is important for ultra fast modulation of the channels in the OAM-DM technology.

Figure 3 shows the time-varying of decomposed LGBs at different distances. The incident LGB modes in the rows from top to bottom are ψ _0,0, ψ _0,6 and ψ _3,3, respectively. The columns from left to right are obtained at z = z_R, 2z_R, and 4z_R, respectively. It’s clear that bigger AQN causes faster decay along z direction, and larger pulse spread and delay on the original modes (denoted by solid lines). For example, for incident LGB with m = 0 at z = 4z_R, the peak values of the original modes decay to about 0.8 [Fig. 3(c)] and 0.3 [Fig. 3(f)] for incident LGBs ψ _0,0 and ψ _0,6,respectively. The corresponding pulse widths are about 2.4T ₀ and 3.5T ₀, respectively. And the corresponding pulse delays are about 0.4T ₀ and 2.0T ₀, respectively. But for incident LGB with m = 3 in the lowest row, the original mode is flooded by other modes quickly with increasing z, e.g. in Fig. 3(i). In the optical communication using the OAM-DM technology large RQN should be avoid because of serious decay and pulse spread. The original mode with zero RQN will be dominant in rather long range, e.g. z < 2z_R.

4. Conclusion

This work expands the pulse beam as superposition of a series of LGBs. The expansion coefficients vary with time and longitudinal coordinate, and show the time-varying and propagating characteristics of the pulse beam. For the incident LGBs with nonzero RQN the original modes decrease seriously along longitudinal coordinate and are flooded by other modes. For the incident LGBs with zero RQN, the original modes are dominant in rather long range, e.g. z < 2z_R, and the pulse delay and spread increase gradually with the longitudinal coordinate increasing. This mean that in near area the original modes also can approximate the pulse LGB with zero RQN.