Propagation equation for tight-focusing by a parabolic mirror

A. Couairon; O. G. Kosareva; N. A. Panov; D. E. Shipilo; V. A. Andreeva; V. Jukna; F. Nesa

doi:10.1364/OE.23.031240

1. Introduction

Modeling and simulation of ultrashort laser pulse propagation in solid state materials and gases underwent significant progress over the last decade [1, 2]. This is partly due to the availability of high power computing facilities, allowing for numerical resolution of otherwise intractable propagation equations. Simultaneously, progress in laser technology has opened up to possibility to explore the domain of extreme nonlinear optics, characterized by pulses with broad temporal or/and angular spectra [3–8], abrupt variations in the field amplitude [9–11], or high intensities leading to a significant material modification during laser-matter interaction [12]. These extreme regimes exhibit challenging modeling and computational problems.

The evolution of petawatt (PW) femtosecond laser pulse systems leads to the intensities in the focal spot of 10²³–10²⁴ W/cm² [3, 13, 14]. This intensity range can be reached by tightly focusing the beam in vacuum using a parabolic mirror. In these fields, the f -number characterizing the focusing optics can easily be smaller than unity, leading to nonparaxial effects during propagation as well as vectorial effects affecting the polarization direction of the incident beam. Tight focusing geometries with f -number about 1 are also used in other fields from microscopy to laser-matter interaction in the regime of classical nonlinear optics, opening the way to the development of new radiation sources based on the generation of a micro-plasma [8]. Motivated by the development of laser systems of PW class, Jeong et al. note the existence of a general demand for precisely describing focused electromagnetic fields under tight focusing conditions [15]. To tackle this problem, they have proposed a model for describing on-axis focusing of an aberrated laser beam with low f -number by a parabolic mirror, based on the Stratton-Chu integral solution to the Maxwell equations [16] and its application to vectorial diffraction by Varga and Török [17,18].

Propagation effects are relatively well described in the case of classical nonlinear optics when the laser beam is collimated or loosely focused, by using unidirectional propagation equations, (e.g. the nonlinear Schrödinger equation), and appropriate solvers. Their main advantage lies in their validity over long propagation distances due to the use of a frame moving with the propagating pulse and a natural separation of the evolution or longitudinal coordinate, z, from transverse coordinates (x and y). Starting from an input field distribution E(x,y,z = z_i) given at a certain plane z = z_i, the field distribution E(x,y,z) in the transverse plane is obtained by propagating the input field up to the distance z. This procedure means that reflection from curved surfaces, including the case of focusing by parabolic mirrors, requires additional modeling of the input field distribution in a plane close to the reflecting surface. So far, the question about the applicability of propagation equations approach for tight focusing is still open.

Our investigation aims at developing a general beam (pulse) propagation model which keeps the advantages of unidirectional propagation equations while being capable of handling nonparaxial and vectorial effects. Previous investigations in this direction were based on envelope propagation equations and a perturbative approach for nonparaxial and vectorial effects [19,20]. We develop a method based on the propagation of the Hertz vector potential and on the angular spectrum representation of electromagnetic fields to solve beam propagation numerically. We call it Unidirectional Hertz vector Propagation Equation (UHPE).

We model low f -number focusing by a parabolic mirror by simulations of the UHPE. We show that results are accurate provided suitable input conditions are used. To numerically simulate beam focusing with unidirectional propagation equations, the input condition is usually defined as a parabolic phase and a beam amplitude, both defined in the transverse plane of input conditions. This becomes invalid to handle low f -number focusing by a parabolic mirror. We propose a practical criterion to determine the critical f -number, below which an inaccurate solution to unidirectional propagation equations is obtained with a parabolic input phase. Below critical f -number, we applying a diffraction integral method between the mirror and the plane of input conditions, to determine suitable input conditions.

2. Solutions to Maxwell’s equations with the Hertz vector

2.1. Hertz vector potential

Maxwell’s equations in a medium with free charge density ρ and free current density J read

\begin{array}{l} \nabla \cdot D = ρ \\ \nabla \cdot B = 0 \\ \nabla \times E = - \frac{\partial B}{\partial t} \\ \nabla \times H = J + \frac{\partial D}{\partial t} \end{array}

where the polarization and magnetization are included in D ≡ ε₀E + P and µ₀H ≡ B − µ₀M. We will deal with nonmagnetic dielectric media satisfying M = 0, µ = µ₀ and D = ε(ω)E +P_n, where P_n denotes the nonlinear polarization and ε(ω) the permittivity of the medium describing dispersion effects. For the sake of generality, we will work in the frequency domain. All quantities are thus frequency dependent even if it is not explicitly specified.

A convenient representation of the electromagnetic field for the resolution of Maxwell’s equations when vectorial effects are involved is the Hertz vector potential Π [21], defined from the scalar and vector potentials, ϕ and A, so that

\begin{matrix} E = i ω A - \nabla ϕ, & B = \nabla \times A \\ ϕ = - \nabla \cdot \prod, & A = - i ω μ_{0} ε \prod . \end{matrix}

The scalar and vector potentials satisfy the Helmholtz wave equations (the Lorentz gauge was chosen)

\begin{array}{l} Δ ϕ + k^{2} (ω) ϕ = \frac{1}{ε} \nabla \cdot (Φ - ε (ω) \nabla l) \\ Δ A + k^{2} (ω) A = i μ_{0} ω (Φ - ε (ω) \nabla l), \end{array}

where k²(ω) ≡ µ₀ε(ω)ω², Φ = Q + P_n describes the nonlinear material response, l denotes a gauge potential to be specified below and the stream vector potential Q is consistently defined from the free sources to satisfy the continuity equation ρ = ∇ · Q, J = −iωQ. The Hertz vector satisfies the wave equation

Δ \prod + k^{2} \prod = - \frac{1}{ε} (Φ - ε \nabla l) .

The gauge potential l is chosen to set the z component of Π to zero and therefore satisfies

\frac{\partial l}{\partial z} = \frac{Φ_{z}}{ε} .

If the Hertz potential is known from a solution to Eq. (4), the electric and magnetic fields are obtained from the relations

E = \nabla (\nabla \cdot \prod) + k^{2} \prod - \nabla l, B = μ_{0} ε \nabla \times (- i ω \prod) .

The angular spectra of Hertz vector components ${\prod^{\sim}}_{x} (k_{x}, k_{y}, z)$ and ${\prod^{\sim}}_{y} (k_{x}, k_{y}, z)$ are obtained by a two-dimensional Fourier transform from the two components Π_x(x,y,z) and Π_y(x,y,z) of the Hertz vector potential. Therefore, Eq. (6) in the Fourier domain reads

(\begin{matrix} {\tilde{E}}_{x} \\ {\tilde{E}}_{y} \\ {\tilde{E}}_{z} \end{matrix}) = (\begin{array}{l} k^{2} - k_{x}^{2} & - k_{x} k_{y} \\ - k_{x} k_{y} & k^{2} - k_{y}^{2} \\ - k_{x} k_{z} & - k_{y} k_{z} \end{array}) (\begin{matrix} {\prod^{\sim}}_{x} \\ {\prod^{\sim}}_{y} \end{matrix}) - (\begin{matrix} i \tilde{l} k_{x} \\ i \tilde{l} k_{y} \\ {\tilde{Φ}}_{z} / ε \end{matrix}),

where

k_{z} (ω, k_{x}, k_{y}) \equiv \sqrt{k^{2} (ω) - k_{x}^{2} - k_{y}^{2}} .

The components for the Hertz vector potential are obtained as

(\begin{matrix} {\prod^{\sim}}_{x} \\ {\prod^{\sim}}_{y} \end{matrix}) = \frac{1}{k^{2} k_{z}^{2}} (\begin{matrix} k^{2} - k_{y}^{2} & k_{x} k_{y} \\ k_{x} k_{y} & k^{2} - k_{x}^{2} \end{matrix}) [(\begin{matrix} {\tilde{E}}_{x} \\ {\tilde{E}}_{y} \end{matrix}) + i \tilde{l} (\begin{matrix} k_{x} \\ k_{y} \end{matrix})] .

The constitutive relations for the nonlinear response of the medium Φ(E) are usually represented in temporal domain. For simple atoms, they can be obtained from the direct numerical solution of the time-dependent Schrödinger equation [22, 23], describing bound-bound, bound-free and free-free transitions between the energy levels of the atoms. More frequently, the nonlinear response is separated into the Q term describing nonlinear current and the P_n term describing nonlinear polarization. The nonlinear current can include the transient photocurrent in ionization conditions [24], the ponderomotive effect [25], the light pressure, the magnetic component of Lorenz force [26], etc. The nonlinear polarization P_n is usually described phenomenologically [27] as a series expansion of powers of the electric field. In isotropic media, only odd terms are non-zero. The general expression for the third order nonlinear polarization is given in [28] for isotropic media with spatial and frequency dispersion. In the case of a local and instantaneous response it can be reduced to P_n = χ⁽³⁾|E|²E, where χ⁽³⁾ is a nonlinear susceptibility [29]. In [30], the instantaneous response is generalized for high-order Kerr terms.

2.2. Unidirectional propagation equations

We consider propagation problems for which a preferential direction (forward) exists and restrict our analysis to the case of negligible backward propagation. In these conditions, the Unidirectional Pulse Propagation Equation (UPPE) is derived from Maxwell’s equations [31] and reads in Fourier space $\tilde{E} (k_{x}, k_{y}, ω, z) \equiv \tilde{ℱ} [E (x, y, t, z)]$ :

\frac{\partial \tilde{E}}{\partial z} = i k_{z} \tilde{E} + i \frac{ω^{2}}{k_{z} c^{2}} \frac{\tilde{Φ}}{2 ε_{0}} .

To solve the UPPE (9), the incident light field in plane z = z₀ is decomposed into a set of harmonics, each of which is propagated to the image plane with its own propagation constant k_z(ω,k_x,k_y). In the image plane the harmonics are recombined together and the multifrequency field distribution is obtained.

Equation (9) describes nonparaxial effects. Often, the UPPE can be used in the scalar approximation, assuming that the incident light field is linearly polarized and this polarization is preserved through the propagation. Circularly or elliptically polarized fields can also be described by the vectorial equation (9). In this case, the matter response encoded in a relation Φ(E) usually introduces cross-polarization effects. However, not all vectorial effects originally present in Maxwell’s equations are retained since vectorial effects arising because of the ∇ (∇ · E) term were neglected in the derivation of Eq. (9). When these effects are not negligible, the UPPE is generalized as the g-UPPE [1,2]

\frac{\partial \tilde{E}}{\partial z} = i k_{z} \tilde{E} + i \frac{ω^{2}}{k_{z} c^{2}} \frac{\tilde{Φ}}{2 ε_{0}} - \frac{i}{ε k_{z}} k_{⊥} (k_{⊥} \cdot \frac{\tilde{Φ}}{2 ε_{0}}),

which takes into account nonparaxial diffraction and vectorial effects.

By following the same procedure as that leading from the wave equation to the UPPE (see [31]), Equation (4) is factorized in the same way as the wave equation to derive a Unidirectional Hertz vector Propagation Equation (UHPE)

\frac{\partial \prod^{\sim}}{\partial z} = i k_{z} \prod^{\sim} + \frac{i}{2 k_{z}} (\frac{\tilde{Φ}}{ε} - i \tilde{l} k_{⊥}),

where k_⊥ = (k_x,k_y). The vectorial g-UPPE (10) can be simply derived from the UHPE (11) if one introduces Eq. (8) into Eq. (11).

Equation (11) constitute one of the main results in our paper. It describes the unidirectional propagation of an electromagnetic field undergoing nonparaxial and vectorial diffraction as well as nonlinear effects. It is valid for pulses with broad spectra. It has the same structure as the UPPE and can therefore be solved numerically by means of the same tools. Vectorial effects are actually described through the use of the Hertz vector rather than the electric field and through the coupling between nonlinearity and vectorial effects encoded in the nonlinear polarization source and in the gauge potential. Equation (11) can be seen as an extension of the paraxial propagation equation obtained by Milsted and Cantrell [32].

We can now detail the steps to be performed in order to propagate a field harmonic E(x,y,z,ω) from z to z + Δz by means of the UHPE (11), with the angular spectrum method including nonlinear, nonparaxial and vectorial effects.

From the input electric field E(x,y,z,ω), calculate angular spectrum Ẽ(k_x,k_y,z,ω), the nonlinear response Φ(x,y,z,ω) by a constitutive relation for the medium (an additional pair of Fourier transforms from the temporal to the frequency domains may be used) and its angular spectrum $\tilde{Φ} (k_{x}, k_{y}, z, ω)$ . The gauge potential is assumed to be known (zero for the initial step).
Calculate the angular spectrum for the Herzt vector $\prod^{\sim} (k_{x}, k_{y}, z, ω)$ by means of Eq. (8). The gauge potential l is known from the previous step.
Propagate $\prod^{\sim} (k_{x}, k_{y}, z, ω)$ from z to z + Δz with the UHPE (11) to obtain $\prod^{\sim} (k_{x}, k_{y}, z + Δ z, ω)$ and the gauge potential by means of Eq. (5) to obtain $\tilde{l} (k_{x}, k_{y}, z + Δ z, ω)$ .
From $\prod^{\sim} (k_{x}, k_{y}, z + Δ z, ω)$ , calculate the angular spectrum for the field $\tilde{E} (k_{x}, k_{y}, z + Δ z, ω)$ by means of Eq. (7).
Fourier transform back to x,y space to get E(x,y,z + Δz,ω).

3. Applicability conditions of the propagation equations for tight-focusing with a parabolic mirror

In this section, we consider the problem of tight-focusing with a parabolic mirror to test the applicability conditions of propagation equations. Based on the results of numerical simulations we formulate the limitations on numerical aperture (f -number) for correct calculations of beam focusing by a parabolic mirror using propagation equations initialized with a parabolic spatial phase. We propose a method to overcome this limitation.

Although it does not describe vectorial effects, we purposefully use the Unidirectional Pulse Propagation Equation [31] in scalar approximation for three reasons: (i) The UPPE describes nonparaxial diffraction, as required in tight-focusing geometries. (ii) The UPPE can be integrated numerically in reduced dimensions in the case of axial symmetry $(r = \sqrt{x^{2} + y^{2}})$ , significantly decreasing the numerical costs. (iii) Our purpose here is to reveal the limit of applicability of the parabolic phase as input condition for the scalar UPPE. This limit holds for the UHPE as well.

Vector diffraction integrals (VDI) based on the Stratton-Chu solution to Maxwell’s equations [16] will serve as a reference for benchmarking the results obtained by propagation equations. More precisely, we use vector diffraction integrals adapted for the focusing by parabolic mirror by Varga and Török [17]. Figure 1 summarizes the schemes used for the simulation of tight-focusing based on the different methods. Vector diffraction integrals require the field distribution on the focusing parabola [Fig. 1(a)] whereas a propagation equation requires the definition of input conditions within a transverse plane [Figs. 1(b), 1(c)].

Fig. 1 Focusing geometry with a parabolic mirror. (a) case of calculations using vector diffraction integrals, (b) propagation equations and (c) propagation equations with suitable input conditions. Black solid lines (a, c) and red solid lines (b, c) show the beam path simulated by using vector diffraction integrals and propagation equations, respectively. Dashed lines state the case f # = 2 and solid lines state the case f # = 0.5. Blue vertical lines locate the planes where initial conditions for the UPPE are set (b) and calculated using VDI (c).

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We consider a top hat beam of cylindrical shape with radius r₀ and intensity I₀. The beam is monochromatic with the wavelength of 800 nm. The input condition for the propagation equation is

E_{x} (r, z = - f) = Θ (r_{0} - r) \times \exp (- \frac{i k_{0} r^{2}}{2 f}),

where f denotes the mirror paraxial focal length, k₀ the wave-number and Θ the step function. For the calculations using vector diffraction integrals, the paraboloid mirror is introduced as z = r²/4f − f, or, in spherical coordinates (ρ_s,θ_s, φ_s), ρ_s = 2 f/(1 − cosθ_s). Input conditions read

E_{x} (θ_{s}, φ_{s}) = Θ (θ_{s} - θ_{m}),

where θ_m denotes a beam edge, r₀ = 2f sinθ_m/(1 − cosθ_m). An expression for the f -number is

f # = - \frac{1}{2} \cot θ_{m} = \frac{f}{2 r_{0}} - \frac{r_{0}}{8 f} .

First, we consider the case of a focal length f = 1 cm. Figure 2(a)–2(d) shows a comparison of the on-axis intensity of the radiation for the two calculation methods as a function of propagation distance z. The methods give reasonably close results for f -numbers down to f # = 5. In the case of f # = 5, the only minor difference is the longitudinal position of the best focus. The transverse distributions of the field at best focus loci match with a high accuracy (see Fig. 3). In the case of f # = 3, the difference becomes significant. This difference could originate from vectorial effects, which are accounted for in the vector diffraction integral but are not considered in the scalar UPPE. However, vectorial effects are still negligible at f # = 3; we show below that the difference mainly comes from the setting of the input field.

Fig. 2 Comparison of beam propagation through the focus (peak intensity vs propagation distance) with vector diffraction integrals (dotted curves) and with numerical simulation of the UPPE (solid curves). The input condition of Fig. 1(b) corresponds to a parabolic phase (gray curves). The results obtained with suitable input condition of Fig. 1(c) are shown by the solid orange curve.

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Fig. 3 Distribution of |E_x|² at the best focus loci (z = 0 for the calculation with vector diffraction integrals and z = −25 µm for the simulation with UPPE) for the case of f # = 5. Values on the colorbar indicates I/I₀.

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In order to demonstrate directly the crucial role of input conditions, we modify the input field for the UPPE so as to reduce the difference in the results calculated with the two methods. The following example is provided for the beam with f # = 3 and f = 1 cm. We first use vector diffraction integrals to calculate the field distribution in the plane z = −8 mm, i.e. 8 mm before the focus. (The following remains true for any other choice in the range −8 mm ≤ z ≤ −100 µm). We then solve numerically the UPPE using this new field distribution as an input condition [see Fig. 1(c)]. The resulting dependence of the on-axis intensity upon propagation distance z is shown in Fig. 2(d) by the orange curve, in very good agreement with the calculation entirely performed with vector diffraction integrals.

Therefore, the difference between the results obtained using vector diffraction integrals and UPPE is due to the approximation made in modeling the input beam at the plane z = −f as a cylindrical beam with a parabolic phase. The UPPE can be effectively used for calculations in tight focusing conditions provided a suitable input field is preliminarily obtained. Our proposed method (UPPE with suitable input conditions) consists in performing a first propagation step for the field from the curved surface of the parabolic mirror to the suitable input plane, with a vector diffraction integral. Since the UPPE can be used in the nonlinear regime, the method remains valid to describe nonlinear processes in the case of tight focusing provided nonlinear effects during the first propagation step can be neglected.

Let us trace the propagation of the radiation starting from a plane (UPPE) and from a parabola (VDI) in terms of geometric optics. The periphery of the beam has the same propagation direction in both approaches. However the positions of the emitting points are different (z = r²/4 f − f for the diffraction integral and z = −f for the propagation equation). At the edge of the cylindrical beam, the gap between the mirror and the plane z = − f is $δ z = r_{0}^{2} / 4 f$ . To give a practical criterion for the validity limit of the input condition that consists in a parabolic phase in the plane z = − f, we will show that it becomes inaccurate when the gap δz becomes larger than the confocal parameter. The red curve in Fig. 4 represents the gap as a function of the f -number, obtained by eliminating the beam radius by means of Eq. (14). The black curve marked by squares in Fig. 4 represents the dependence of the confocal parameter upon f -number, i.e., the length of the focal region b^{(top hat)} calculated from the vector diffraction integral. The curves intersect at f #_crit ≈ 3.3. Above this critical f -number, simulations with the UPPE and a parabolic input phase correctly describe parabolic focusing (see Fig. 2). On the other hand, the input condition must be modified for f # < f #_crit (or b^{(top hat)} < δz). Therefore, for f # > f #_crit (or b^{(top hat)} > δz), the parabolic phase in the plane z = −f can be satisfactorily used with the UPPE, otherwise for f # < f #_crit a suitable input condition must be obtained to simulate the focusing by the parabolic mirror with the UPPE.

Fig. 4 Distance δz from the mirror to the plane z = −f (input condition) at the beam edge (red solid curve, analytical result) as a function of f -number. Confocal parameter b^{(top hat)} for a cylindric beam (curve marked by black squares, result of a simulation with the vectorial diffraction integral). Confocal parameter b^(Gauss) for the paraxial propagation of a Gaussian beam (black solid curve, analytical expression). The focal length is f = 1 cm for all curves.

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We proceed with a simple analytical estimation for f #_crit. If we used the paraxial approximation for the focusing of a Gaussian beam, the confocal parameter would read $b^{(Gauss)} = 2 f^{2} / (k_{0} r_{0}^{2})$ . This parameter is shown in Fig. 4 as a function of the f -number (black curve). Our criterion δz(f#_crit) = b^(Gauss)(f#_crit) provides a solution

f #_{c r i t} = \frac{1}{4} (\sqrt[4]{2 k_{0} f} - \frac{1}{\sqrt[4]{2 k_{0} f}}) \approx \frac{1}{4} \sqrt[4]{2 k_{0} f},

or, in terms of the beam radius

r_{c r i t} = \sqrt[4]{8 f^{3} / k_{0}}

. We now introduce a multiplicative factor of 6 accounting for the difference between confocal parameters for the paraxial Gaussian beam and the nonparaxial cylindrical beam (compare marked and unmarked black curves in Fig. 4). The dependence of the critical f -number on focal length for the nonparaxial cylindrical beam is therefore similar to that for the paraxial Gaussian beam:

f #_{c r i t} \propto \sqrt[4]{f / λ}

, where λ denotes the laser wavelength. By using the critical value f #_crit ≈ 3.3 found for the nonparaxial propagation of a top-hat beam at the wavelength of 800 nm and the focal length of 1 cm, we obtain a practical formula to evaluate the dependence f#_crit(f,λ) as

f #_{c r i t} (f, λ) = 3.1 \times \sqrt[4]{\frac{f [cm]}{λ [μ m]}} .

Here the focal length f is measured in centimeters and wavelength λ in micrometers.

Our numerical simulation results summarized in Fig. 5 confirm the slow decrease of the critical f -number with the decrease of the focal length as stated in Equation (16). For f = 0.25 cm the value of f#_crit is 2.3, explaining that on-axis intensity profiles obtained by the UPPE with parabolic input phase and diffraction integrals for f # = 3 are similar and only differ by a longitudinal shift of about 17 µm [see Fig. 5(a)]. This shift decreases with the decrease of the focal length down to 600 µm [see Fig. 5(a)]. If we consider the numerical aperture f # = 2 [see Fig. 5(b)] a correct description of focusing by a parabolic mirror using the UPPE and input condition Eq. (12) can be achieved for f = 600 µm (f#_crit = 1.6) and shorter focal lengths. In the case of f# = 1, the UPPE and Eq. (12) provide satisfying results only for the focal length f = 40 µm [see orange line in Fig. 5(c)].

Fig. 5 Simulation results with the UPPE and input condition Eq. (12) for different focal lengths and f -numbers. For the same f -number, the suitability of the input condition improves with the decrease of the focal length. The vertical axis shows the dimensionless intensity. The intensity is normalized to the product of input intensity and squared focal length of the mirror, normalized to 1 cm. This is needed to ensure similar vertical scale for different focal lengths.

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The blue curves on Fig. 5 demonstrate the effect of peak intensity saturation with the decrease of f -number. This effect is similar to spherical aberration of optical elements: the beam periphery and the beam center focus into different points. The effect originates from the inappropriate use of a parabolic phase as a thin-lens model for tight nonparaxial focusing. The aberration-free parabolic mirror cannot be considered as thin. The parabolic phase factor defined on a plane models more closely the action of a Fresnel lens, which is actually thin and provides a quasi-parabolic phase, but harshly distorts the radiation at the discontinuities of the lens surface. Simulation results of the UPPE in the geometry displayed in Fig. 1(b) must be interpreted as representing the behavior of an experimental beam focused with a low f -number Fresnel lens.

The critical value for the f -number determined by Eq. (16) indicates when the parabolic phase in Eq. (12) is no longer suitable for simulations of tight focusing by a parabolic mirror based on the UPPE. For smaller f -numbers, we have shown that the UPPE can still be used provided a suitable input condition is derived, e.g., by using the vector diffraction integrals to propagate the field from the parabolic surface of the mirror to the chosen transverse plane of input condition. For a certain f -number, vectorial effects will become significant and the UPPE will have to be replaced by the UHPE.

4. Nonparaxial and vectorial diffraction

The unidirectional Hertz vector propagation equation allows us to describe vectorial effects which are significant for f -numbers smaller than 1. In contrast to the scalar UPPE, we demonstrate that a focused beam that is initially cylindrically symmetric undergoes symmetry breaking of the x-polarized component of the electric field when the UHPE is used. Besides we investigate the applicability of the UHPE for the case of focusing by a parabolic mirror by direct comparison with vector diffraction integrals. We consider linear propagation in a medium without free charges (Q = 0, P_n = 0). The input conditions for the UPPE and vector diffraction integrals are given by Eqs. (12), (13), respectively, while the input condition for the UHPE is Π_x(x,y,z = − f) = E_x(x,y,z = − f)/k² [32]. For the comparison we chose the case of f = 150 µm with f -numbers f # = 2 > f#_crit and f# = 1 < f#_crit (for f = 150 µm the critical f -number is about 1.15).

In both cases, peak intensities of the solutions to the UHPE are smaller by 10% (or less) than their counterparts obtained by using the UPPE (green curves at Fig. 5) since energy is transferred to the y- and z-components of the electric field. The longitudinal positions of the best focus (corresponding to the maximum along z for the peak intensity) are the same for the UHPE and the UPPE, z = −4 µm for f# = 2 and z = −9 µm for f# = 1. Therefore we compare the transverse intensity distribution at these propagation distances with the reference distribution obtained by vector diffraction integrals at z = 0.

Figures 6 and 7 represent the intensity patterns for the three components of the electric field, at the position of the best focus. Figures 6(a), 7(a) show the result obtained by the UPPE without vectorial effects. Rings surrounding the central lobe are typical of the far-field of a top-hat beam. The symmetry is broken when vectorial effects are taken into account, as shown in Figs. 6(b), 7(b), corresponding to the solution of the UHPE. The solution to the UHPE coincides with the solution to vector diffraction integrals for f# = 2 > f#_crit [cf. Figs. 6(b) and 6(c)], but beyond the applicability condition of the parabolic input phase with propagation equations [see Eq. (16)] these solutions differ significantly [cf. Figs. 7(b) and 7(c)]. However, this difference vanishes after initializing the UHPE with the suitable input condition, i.e. by translating the plane of input condition to z = −30 µm [cf. Figs. 7(c) and 7(d)]. Hence, we draw two important conclusions: (i) Eq. (16) equally set the applicability limit of a parabolic input phase for UPPE- and UHPE-simulations. (ii) When suitable input conditions are used, UHPE-simulations provide a generic approach to beam (pulse) propagation in extreme focusing conditions. It can be efficiently used for tight parabolic focusing while preserving the flexibility and advantages of unidirectional propagation equations.

Fig. 6 Intensity distributions in the waist for a x-polarized top-hat beam in case of f = 150 µm and f# = 2 > f#_crit. (a, b) Nonparaxial diffraction without and with vectorial effects (simulation by UPPE and UHPE respectively, z = −4 µm). (c) Simulation with vector diffraction integral (z = 0).

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Fig. 7 Intensity distributions in the waist for a x-polarized top-hat beam in case of f = 150 µm and f# = 1 < f#_crit. (a, b) Nonparaxial diffraction without and with vectorial effects (simulation by UPPE and UHPE respectively, z = −9 µm). (c) Simulation with vector diffraction integral (z = 0). (d) Nonparaxial diffraction with vectorial effects after translation of input conditions to the plane z = −30 µm.

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5. Conclusion

In conclusion, we have derived the Unidirectional Hertz vector Propagation Equation as an extension of the UPPE. The UHPE describes the propagation of laser beams and pulses, including few-cycle pulses, undergoing nonlinear, nonparaxial and vectorial effects. We compared the simulation results obtained with unidirectional propagation equations to calculations performed by vector diffraction integrals for the problem of beam focusing with a parabolic mirror. The comparison demonstrated that input conditions for unidirectional propagation equations can be defined by a parabolic phase [see Eq. (12)] within a transverse plane only for f -numbers above a critical value given by Eq. (16). For smaller f -numbers, Eq. (12) cannot be used to initialize unidirectional propagation equations. In this tight focusing regime, suitable input conditions are found by translation of the field from the parabolic mirror to a plane surface by vector diffraction integrals, thereby allowing us to simulate with a satisfying accuracy nonparaxial and vectorial effects during propagation with the UHPE.

Acknowledgments

O. G. Kosareva, N. A. Panov, D. E. Shipilo and V. A. Andreeva thank Russian Fund for Basic Research (grants 15-32-20966, 14-22-02021, 15-02-99630, 14-02-31379, 14-02-00979), the Council of RF President for Support of Young Scientists (grant MK-4895.2014.2), RF President Grant for Leading Scientific Schools (grants NSh-3796.2014.2), Dynasty Foundation.

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Propagation equation for tight-focusing by a parabolic mirror

Abstract

1. Introduction

2. Solutions to Maxwell’s equations with the Hertz vector

2.1. Hertz vector potential

2.2. Unidirectional propagation equations

3. Applicability conditions of the propagation equations for tight-focusing with a parabolic mirror

4. Nonparaxial and vectorial diffraction

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (16)

Optics Express