Skew line ray model of nonparaxial Gaussian beam

Shuhe Zhang; Jinhua Zhou; Lei Gong

doi:10.1364/OE.26.003381

1. Introduction

Gaussian beams are widely used in optical imaging [1, 2], optical communications [3, 4] and optical manipulations [5, 6]. The physical expression of Gaussian beams is of great significance for theoretically investigating the beams’ propagation or inter-actions between beam and other objects. Since nonparaxial form of Gaussian beams provides a universal way to describe beams’ propagation, it has been studied for years. Lots of approaches have been proposed to describe the propagations of nonparaxial Gaussian beams, such as the perturbation method [7], power-series expansion [8, 9], angular spectrum representation [10, 11] and integral solution of the wave equation [12, 13]. All of them involve complicated mathematical algorithms based on wave optics. Notably, geometrical optics based description of beam propagation provides a convenient and intuitive explanation for many aspects of light beams, such as “self-healing” [14, 15], “self-accelerating” [16, 17] as well as the Gouy and Berry phases (also, geometrical phases). Thus, the geometrical optical field is of continuous interest, adding to the momentum of the rapidly growing field of structured light beams. Meanwhile, a ray-optics model of nonparaxial Gaussian beam offers a rather convenient way to reveal the beam’s propagation behaviors.

Ray-optics models of paraxial Gaussian beam have been studied for years, including the complex ray representation of Gaussian beam [18, 19], the skew line ray (SLR) model of Gaussian beam [20, 21], the Poynting vector model of Laguerre-Gauss beam and Bessel-Gauss beam [22], and the Poincaré sphere ray-optics model of structured Gaussian beams [23]. Notably, the SLR model regards the skew lines (straight generatrixes) of one-sheet hyperboloid as the trajectories of light rays. However,those ray-optics models are based on the paraxial beam and fail when paraxial condition is broken. Paraxial beam is the solution of scalar Helmholtz equation on paraxial approximation. In other words, nonparaxial beam is the exact solutions of scalar Helmholtz equation. Note that the exact solutions of scalar Helmholtz equation have been given in oblate spheroidal coordinates [24–26]. On the paraxial approximation, the zero order of the solutions tends to be a fundamental Gaussian beam. Undoubtedly, this beam can be regarded as a reference of nonparaxial Gaussian beam. Considering that one-sheet hyperboloid is a component of the oblate spheroidal coordinates. In the spirit of SLR model, here we present a SLR model of nonparaxial Gaussian beam. This model is based on skew lines of one-sheet hyperboloid in the oblate spheroidal coordinates.

Since the skew lines are regarded as trajectories of light rays, the geometrical optical field of nonparaxial Gaussian beam, including amplitude and phase, is traveling along these rays. Our mathematical derivations demonstrate that the expression of geometrical optical field, derived by SLR model, is consistent with the result obtained by wave optics for both nonparaxial and paraxial conditions. The free-space evolutions of amplitude and phase of beam are given in terms of rays. Further study shows that the transverse distributions of ray density change with beam propagation in the nonparaxial case, leading to the corresponding changes of intensity distribution in the transverse plane. By contrast, on the paraxial condition, the change of the ray density in the transverse plane is so small that it can be ignored. Therefore, our SLR model enables precise description of the propagation behaviors of nonparaxial light fields without explicit diffraction calculations, and provides new insight into the physics of structured light beams.

The structure of this paper is arranged as follows. In the next section, we give a brief introduction of the nonparaxial Gaussian beam under oblate spheroidal coordinates. In Section 3, the geometrical optical fields of the nonparaxial Gaussian beam are deduced via SLR model. In Section 4, the evolution of SLR model on the paraxial approximation is demonstrated. An explanation of the broken self-similar property on nonparaxial regime is given in terms of rays in Section 5. The relationship between ellipsoid wavefront and rays is revealed in Section 6. The last section includes our conclusions.

2. Gaussian Beam in oblate spheroidal coordinates

The oblate spheroidal coordinates are orthogonally built on confocal oblate spheroids and one-sheet hyperboloids in three-dimensions. The z-axis is defined as the rotation axis, and the distance between foci is set to be 2D. Therefore, the scalar Helmholtz can be separated as the product of the spheroidal radial functions and spheroidal angular functions [24, 26]. The set of coordinates $(ξ, η, θ)$ describes a unique point in Cartesian coordinates $(x, y, z)$ with.

\begin{array}{l} x = D \sqrt{(1 + ξ^{2}) (1 - η^{2})} \cos θ, \\ x = D \sqrt{(1 + ξ^{2}) (1 - η^{2})} \sin θ, \\ z = D ξ η, \end{array}

where

- \infty < ξ < + \infty

,

0 \leq η \leq 1

, and

0 \leq θ \leq 2 π

. Note that all points with a nonzero constant η form a one-sheet hyperboloid, and all points with nonzero constant ξ form an oblate spheroid.

A family of exact solutions of the scalar Helmholtz equation in the oblate spheroidal coordinates have been derived by Landesman and Barrett [24]. The zero-order solution is given by

ψ (ξ, η, θ) = \frac{\exp [- k D (1 - η)]}{k D \sqrt{ξ^{2} + η^{2}}} \exp (i k D ξ) \exp (- i arc \tan \frac{ξ}{η}),

where

k = 2 π / λ

is the wave number, λ is the wavelength in vacuum. On paraxial approximation, Eq. (2) corresponds to a fundamental Gaussian beam. The relationship between parameter D, beam waist radius

ω_{0}

and half divergence angle ε is determined by [24, 26]

D = \frac{ω_{0}}{\sin ε}, ω_{0} = \frac{1}{k} \sqrt{2 (- 1 + \sqrt{1 + k^{2} D^{2}})} .

As mentioned in [24], this solution is a nonparaxial Gaussian beam. The wave front is an oblate ellipsoid (ξ = nonzero constant). When $ξ = 0$ , the beam has a plane wavefront at $z = 0$ .

3. SLR model of nonparaxial Gaussian beam

3.1 The skew lines of a hyperboloid

According to Eq. (1), one-sheet hyperboloid with $η = η_{Q}$ can be determined in Cartesian coordinates by

\frac{x^{2} + y^{2}}{D^{2} (1 - η_{Q}^{2})} - \frac{z^{2}}{D^{2} η_{Q}^{2}} = 1.

As shown in Fig. 1, considering that an arbitrary point $Q (x_{Q}, y_{Q}, 0)$ is located in waist plane, the parameter $η_{Q}$ of the hyperboloid, containing point Q, is derived by Eq. (4) as

Fig. 1 Two skew lines of point $Q (x_{Q}, y_{Q}, 0)$ . $| Q S_{1} | = | Q S_{2} |$ .

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η_{Q} = \sqrt{1 - \frac{x_{Q}^{2} + y_{Q}^{2}}{D^{2}}} .

While there are two skew lines cross the same point Q, we define the QS₁ (orange line) as the negative skew line $l_{Q}^{-}$ of point Q, and QS₂ (blue line) as the positive skew line $l_{Q}^{+}$ . These two skew lines are given as

l_{Q}^{+} : x = x_{Q} - y_{Q} \frac{τ}{D η_{Q}}, y = y_{Q} + x_{Q} \frac{τ}{D η_{Q}}, z = τ,

l_{Q}^{-} : x = x_{Q} + y_{Q} \frac{τ}{D η_{Q}}, y = y_{Q} - x_{Q} \frac{τ}{D η_{Q}}, z = τ,

τ is a parameter. We assume that the beam propagates in a linear, homogeneous and isotropic medium, so that rays are straight lines. The skew lines can be regarded as the trajectories of rays of Gaussian beam. Both the amplitudes and phases of optical fields travel along those rays.

3.2 Amplitude propagation along the SLR

Since two skew-line rays are associated with point $S (x, y, z)$ , these two rays launch from different points in the waist plane. The geometrical optical field at point S is the superposition of the fields carried by these rays. As shown in Fig. 2, assuming that rays $l_{Q_{1}}^{+}$ and $l_{Q_{2}}^{-}$ pass through the observation point S, the equation of $l_{Q_{1}}^{+}$ and $l_{Q_{2}}^{-}$ can be derived by Eqs. (6-1) and (6-2). In the following text, the subscript “GO” denotes the optical field derived by geometrical optics. Then the geometrical optical field Ψ_go(S) at point S is the sum of fields of the rays associated with point S [15, 27]. It can be expressed as

ψ_{GO} (S) = ψ_{GO}^{+} (S) + ψ_{GO}^{-} (S),

where

ψ_{GO}^{+} (S)

and

ψ_{GO}^{-} (S)

are the geometrical optical fields carried by rays

l_{Q_{1}}^{+}

and

l_{Q_{2}}^{-}

, respectively. Here, we choose the waist plane of

z = 0

as the initial plane, so the

ψ_{GO}^{+} (S)

and

ψ_{GO}^{-} (S)

are given by [27]

ψ_{GO}^{+} (S) = \frac{1}{2} A_{Q_{1}} \sqrt{\frac{J_{Q_{1}}}{J_{S}}} \exp [i (φ_{Q_{1}} + φ_{l})],

ψ_{GO}^{-} (S) = \frac{1}{2} A_{Q_{2}} \sqrt{\frac{J_{Q_{2}}}{J_{S}}} \exp [i (φ_{Q_{2}} + φ_{l})] .

Where

A_{Q_{1}}

(

A_{Q_{2}}

) and

φ_{Q_{1}}

(

φ_{Q_{2}}

) are the amplitude and phase at initial point Q₁ (Q₂) of ray, respectively. According to the symmetry of fundamental Gaussian beam,

A_{Q_{1}} = A_{Q_{2}}

, and

φ_{Q_{1}} = φ_{Q_{2}} = 0

. Phase

φ_{l}

accumulates with the ray’s traveling length l. Variable J denotes the determinant of Jacobian matrix and is determined by

J = det [\partial (x, y, z) / \partial (x_{Q}, y_{Q}, τ)]

[19, 27]. Note that variables x, y and

η_{Q}

are the functions of

x_{Q}

,

y_{Q}

, thus J can be derived from Eqs. (5), (6-1) and (6-2) as

J = 1 + {(\frac{τ}{D η_{Q_{n}}^{2}})}^{2}, n = 1, 2.

The subscript index in Eq. (9), of

η_{Q}

denotes the specific point.

Fig. 2 Sketch of the rays arriving at point S outside the waist plane.

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Since there are two skew lines from a same initial point (Fig. 1), both of them play the same role in wave propagation. The weight of field’ amplitude at initial point should be divided equally to two rays. A factor of 1/2 should be multiplied at initial amplitude in Eqs. (8-1) and (8-2).

3.3 Phase propagation along the SLR

Assuming geometrical optical field propagates a tiny length $d l$ along a SLR of QS in Fig. 3, the phase changes $d φ_{l}$ . From Eq. (2), we can obtain that

\frac{d φ_{l}}{d l} = \frac{d \arg (U)}{d l} = \frac{d [k D ξ_{S} - arc \tan (ξ_{S} / η_{S})]}{d l},

where

ξ_{S}

and

η_{S}

are the oblate spheroidal coordinates of point S. Note that the optical path length

l

(Eikonal) is equal to the distance between points Q and S. From Eqs. (6-1) and (6-2),

l = τ / η_{Q}

. According to Eq. (1),

z = D ξ_{S} η_{S} = τ

and

η_{Q} = η_{S}

(points S and Q are on the same hyperboloid), then

ξ_{S} = \frac{l}{D} .

Combining with Eq. (10), we get

φ_{l} = k l - arc \tan \frac{l}{D η_{Q}} .

Equation (12) shows that the phase shift along a ray can be divided into two parts. One is the Berry phase shift of

φ_{b} = k l

. The other is the Gouy phase shift of

φ_{g} = arc \tan [l / (D η_{Q})]

which is related with rotation angle of l in terms of the initial point. A geometrical explanation of Gouy phase shift has been given by Landesman [20], as shown in Fig. 3. The edge ray (QS) from point Q has arrived at point S on a wave front

ξ_{S}

. The azimuth angle from point Q to S is α. However, the central ray (OC) arrives at point C at the same time. If the edge ray passes through plane C, it should travel an additional path

Δ l_{0}

. The

Δ l_{0}

will lead to the azimuth angle difference

Δα

between S’ and S. It can be find that

α + Δ α = arc \tan \frac{l}{D η_{Q}} .

From Eqs. (7) to (13), the geometrical optical field of point

S (x, y, z)

should be

ψ_{GO} (S) = \sum_{n = 1}^{2} \frac{1 / 2 A_{Q_{n}}}{\sqrt{1 + {[z / (D η_{Q_{n}}^{2})]}^{2}}} \exp [i (k l - arc \tan \frac{l}{D η_{Q_{n}}})] .

Here,

A_{Q_{n}}

is the amplitude at initial points, and

A_{Q_{1}} = A_{Q_{2}}

. According to Eq. (2), we get the initial amplitude

A_{Q_{n}} = exp [- k D (1 - η_{Q_{n}})] / (k D η_{Q_{n}})

. If we make the substitution of

z = D ξ_{S} η_{S}

,

η_{Q_{n}} = η_{S}

and Eq. (11) into Eq. (14), it can be rewritten as

ψ_{GO} (S) = \frac{\exp [- k D (1 - η_{S})]}{k D \sqrt{ξ_{S}^{2} + η_{S}^{2}}} \exp [i (k D ξ_{S} - arc \tan \frac{ξ_{S}}{η_{S}})] .

This equation has the same form as Eq. (2).

Fig. 3 Wave front of SLR. The increment $Δ α$ corresponds to the segment $Δ l_{0}$ .

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An intriguing phenomenon is that the parameter $η_{Q}$ in Eq. (5) must have a real root if the equation of skew line is to be determined. That means the initial points of skew line are restricted in a circle area with radius equal to D. In the subsequent of this paper, we call this circle area the “foci disk”. This feature indicates that fields of nonparaxial Gaussian beam in other transverse planes ( $z \neq 0$ ) are traced by rays from the confined foci disk.

4. Paraxial form of SLR model

As shown in Fig. 4, points $S (x, y, z)$ and $C_{1} (0, 0, z)$ are in the same transverse plane. Supposing that a ray launching from point Q to S, the optical path length l between points Q and S is $| Q S |$ , and can be separated into two parts as

l = | Q S | = | Q S_{1} | + | S_{1} S |,

where

| Q S_{1} |

is the distance between points Q and S₁,

| S_{1} S |

is the distance between points S₁ and S. Point S₁ is located on spheroidal wavefront

ξ_{S_{1}}

(vertex at point C₁, blue curve); and point S is located on wavefront

ξ_{S_{2}}

(vertex at point C₂, green curve). If the parameters k, D and η of the nonparaxial Gaussian beam satisfy the following condition [28]

k D > > 1, η \to 1,

then the beam propagates on the paraxial approximation. The condition of

k D ≫ 1

means that the size of beam waist is large enough compared with its wavelength. The condition of

η \to 1

makes those one-sheet hyperboloids tend to be cylinders which are very close to z-axis.

Fig. 4 Sketch of the ray tracing along the skew line.

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From the conditions of (17), we have divergence angle $ε \to 0$ . Since $ε = λ / π ω_{0}$ [29], the parameter D tends to be

\lim_{ε \to 0} D = \lim_{ε \to 0} \frac{ω_{0}}{\sin ε} = \frac{π ω_{0}^{2}}{λ} = z_{r},

where

z_{r}

is the Rayleigh distance. On this approximation, we have

| Q S_{1} | \approx | O C_{1} |

. Length

| S_{1} S |

can be replaced by the distance

Δ d

between the transverse plane of S and spheroidal surface

ξ_{S_{1}}

. For the condition of

η \to 1

, the spheroidal surface

ξ_{S_{1}}

can be regard as a spherical surface at vicinity of the vertex

C_{1}

. The radius of this spherical surface is

R (z)

at point

C_{1} (0, 0, z)

. Combining Eqs. (1) with (18), we can derive

R (z) = z (1 + \frac{D^{2}}{z^{2}}) \approx z (1 + \frac{z_{r}^{2}}{z^{2}}),

then the distance

Δ d = (x^{2} + y^{2}) / (2 R)

[29]. Therefore, the optical path length l in Eq. (16) tends to be

l \approx z + \frac{x^{2} + y^{2}}{2 R (z)} .

Meanwhile, the skew lines in Eqs. (6-1) and (6-2) have the paraxial form of

l_{Q}^{+} : x = x_{Q} - y_{Q} \frac{τ}{z_{r}}, y = y_{Q} + x_{Q} \frac{τ}{z_{r}}, z = τ,

l_{Q}^{-} : x = x_{Q} + y_{Q} \frac{τ}{z_{r}}, y = y_{Q} - x_{Q} \frac{τ}{z_{r}}, z = τ .

Equations (21-1) and (21-2) are consistence with the skew line equation in [2,21]. Then the determinant J of Jacobian is

J = 1 + {(\frac{z}{z_{r}})}^{2} .

Since Δα→0 and

\tan α = | S_{0} Q | / | O Q |

, according to Eqs. (13), (21-1) and (21-2), the Gouy phase is given by

φ_{g} = arc \tan (z / z_{r})

. In this case, the total phase shift

φ_{l}

along a skew line is

φ_{l} = k [z + \frac{x^{2} + y^{2}}{2 R (z)}] - arc \tan \frac{z}{z_{r}} .

We choose the

A_{0} = exp [- (x_{Q}^{2} + y_{Q}^{2}) / ω_{0}^{2}]

for the initial amplitude, the paraxial geometrical optical field at point

S (x, y, z)

is

\begin{array}{l} ψ_{GO} (S) = \frac{1}{\sqrt{1 + {(z / z_{r})}^{2}}} \exp (- \frac{x_{Q}^{2} + y_{Q}^{2}}{ω_{0}^{2}}) \\ \times \exp {i k [z + \frac{x^{2} + y^{2}}{2 R (z)}]} \exp (- i arc \tan \frac{z}{z_{r}}) . \end{array}

For paraxial Gaussian beam,

ω (z) = ω_{0} \sqrt{1 + {(z / z_{r})}^{2}}

, there is a relation of

ω_{0}^{2} / ω^{2} (z) = (x_{Q}^{2} + y_{Q}^{2}) / (x^{2} + y^{2})

according to Eqs. (21-1) and (21-2). Then Eq. (24) can be rewritten as

\begin{array}{l} ψ_{GO} (S) = \frac{ω_{0}}{ω (z)} \exp [- \frac{x^{2} + y^{2}}{ω^{2} (z)}] \\ \times \exp {i k [z + \frac{x^{2} + y^{2}}{2 R (z)}]} \exp (- i arc \tan \frac{z}{z_{r}}) . \end{array}

Obviously, Eq. (25) is the field of the well-known paraxial Gaussian beam [29].

In paraxial condition, the restriction of Eq. (5) vanishes for the radius of foci disk is much larger than beam waist. Although the paraxial SLR model can be enforced to trace the propagation of rays in nonparaxial regime, the initial points of rays might be out of the foci disk. Besides, we note that the optical path length calculated by Eq. (20) is an approximate treatment in paraxial condition. In practical ray tracing methodology, the optical path length is always rigorously equal to the distance between the initial and terminal points of a ray. In order to distinguish these two models, we define that the ray trajectories traced by Eqs. (6-1) and (6-2) is nonparaxial SLR model and by Eqs. (21-1) and (21-2) are paraxial SLR model, respectively.

5. Ray-based explanation on broken self-similar property

In previous works on nonparaxial Gaussian beam, the beam profiles are quite different from those of the conventional transverse pattern of paraxial beam [12, 13]. This phenomenon indicates that self-similar property of beam is broken in nonparaxial region. However, none explanation is given. Combining with ray tracing methodology, here we give an explanation in terms of rays. In the following simulations, we choose $λ = 1.064$ μm and take $k D = 100$ ( $ω_{0, 1} \approx 2.38$ μm) for a paraxial beam. In this case, the half divergence angle $ε \approx 8^{°}$ which ensures the parameter $η \to 1$ . By contrast, we choose $k D = 1$ ( $ω_{0, 2} \approx 0.15$ μm) for a nonparaxial beam where $ε \approx {65.5}^{°}$ . The optical fields are observed in the plane of $z = 5 z_{r}$ .

Trajectories of rays of the nonparaxial SLR model are presented in Fig. 5. Those rays are traced by Eqs. (6-1) and (6-2) and the initial points of rays are arranged in equidistant concentric circles. The coordinates are nondimensionalized by $1 / ω_{0}$ for x-/y-coordinates and by $1 / z_{r}$ for z-coordinate. In the case of $k D = 100$ , the outer envelope is one-sheet hyperboloids as shown in Fig. 5(a). Trajectories of rays of paraxial SLR model for both $k D = 100$ and $k D = 1$ (not shown) are consistent with those in Fig. 5(a). For $k D = 1$ , as shown in Fig. 5(b), the outer envelope is similar to that in Fig. 5(a) because of the nondimensionalization of coordinates. However, the distribution of rays’ tips is more concentrated in central zone than that in Fig. 5(a).

Fig. 5 Rays’ trajectories of nonparaxial SLR model with different parameter. (a) $k D = 100$ ; (b) $k D = 1$ .

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For detailed discussions, we set the rays with uniform distribution in the initial plane ( $z = 0$ ), and trace the rays’ distribution in an observation plane as shown in Fig. 6. Gray images denote the ray’s distribution pattern. The gray value at each pixel represents the ray’s density of that area. Both x and y axes are nondimensionalized by $1 / ω_{0}$ . Figure 6(a) shows the distribution of rays in the initial plane ( $z = 0$ ). The rays are uniformly distributed within the circle radius of $ω_{0}$ . Figures 6(b)-6(e) are distributions of rays in the observation plane ( $z = 5 z_{r}$ ). For the beam with $k D = 100$ in Figs. 6(b) and 6(c), ray distributions are traced by paraxial and nonparaxial SLR model, respectively. In this case, the gray values are still uniformly distributed in a circle with radius approaches to $5 ω_{0}$ . When $k D = 100$ , the rays’ distribution after propagation is similar to the initial distribution of rays. This property ensures the self-similar feature with beam propagation. Hence, the intensity distributions are modulated by rays’ distributions. In this condition, the intensity patterns traced by the two models should be similar.

Fig. 6 Ray’s distribution patterns in an observation plane ( $z = 5 z_{r}$ ). (a) Initial plane $z = 0$ ; (b) and (d) are traced by paraxial SLR model for $k D = 100$ and $k D = 1$ , respectively. (c) and (e) are traced by nonparaxial SLR model for $k D = 100$ and $k D = 1$ , respectively.

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For $k D = 1$ , rays’ distributions in Figs. 6(d) and 6(e) are traced by paraxial and nonparaxial SLR models, respectively. Although $k D = 1$ breaks the paraxial condition, Fig. 6(d) has little difference with Fig. 6(b) which comes from the paraxial SLR model. Figure 6(e) has a bright flare at the center and gradually dim toward outside. There is no obvious edge of the distribution. Compared with the initial rays’ distribution in Fig. 6(a), the distribution is obviously distorted after beam’s propagation. The self-similar feature of beam is therefore broken. The intensity pattern traced by nonparaxial SLR model should be more concentrated than that of paraxial SLR model. Since self-similar property requires that the transverse distribution of rays should also propagate in a self-similar way [23]. The change of distribution of rays’ tips in Fig. 6(e) indicates that self-similar property is broken in nonparaxial case.

To compare intensity patterns traced by two models, we ought to choose the same distribution of amplitude in the initial plane. Note that Eq. (2) has amplitude singularity in $z = 0$ plane which can cause intensity distortion in other planes after ray tracing. Although the modified amplitude has been proposed in [25], it is a superposition of an outgoing wave and an incoming wave, and it has infinite energy. For practical purposes, we use the Gaussian distribution $A_{0} = exp (- r^{2} / ω_{0}^{2})$ as initial amplitude in the waist plane, where r is the distance from the initial point of ray to the origin.

In the observation plane, the geometrical optical fields have been traced with nonparaxial [Eqs. (6-1) and (6-2)] and paraxial [Eqs. (21-1) and (21-2)] SLR models in Fig. 7. The intensity distributions $I_{1}$ and $I_{2}$ are obtained via ray tracing, while $I_{0, 1}$ and $I_{0, 2}$ that are calculated by Eq. (2) and Eq. (25), respectively. When $k D = 100$ in Fig. 7(a), $I_{1}$ is very closed to $I_{2}$ . Both of them are fitted with $I_{0, 1}$ and $I_{0, 2}$ . If we take $k D = 1$ [Fig. 5(b)], $I_{1}$ is narrower than $I_{0, 1}$ due to the difference of the initial amplitudes between Eq. (2) and Gaussian beam. The curve $I_{2}$ is fitted with $I_{0, 2}$ . The curve $I_{1}$ is obviously more concentrated in central zone than curve $I_{2}$ , and the peak of $I_{1}$ is also higher than the peak of $I_{2}$ . These appearances are consistent with discussions in paragraph above. While the paraxial condition is broken, the self-similar feature is vanished. The paraxial SLR model has a great deviation than nonparaxial SLR model, and makes the intensity distribution broaden and weaken.

Fig. 7 Comparison of the intensity distribution in the plane of $z = 5 z_{r}$ between nonparaxial and paraxial SLR models. (a) $k D = 100$ ; (b) $k D = 1$ .

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6. Ellipsoid wavefront as the envelope of equidistance rays

Considering that the wavefront can be regarded as an envelope of rays’ tips with equal length. We plot the rays’ trajectories for $k D = 100$ and $k D = 1$ , where the lengths of rays equal to $λ$ . The trajectories of rays in each case are separately traced by paraxial and nonparaxial SLR models, as shown in Fig. 8. The transverse coordinate is nondimensionalized by $ω_{λ} = ω_{0} \sqrt{1 + {(λ / z_{r})}^{2}}$ . When $k D = 100$ , the envelope of rays’ tips overlays on the ellipsoid with parameter $ξ = λ / D$ (the black dotted line). As shown in the Figs. 8(a) and 8(b), the ellipsoid can be regarded as a flat for its curvature radius is large enough compared with $λ$ . The wavefront tends to be a plane wave. For the case of $k D = 1$ , the envelope of tips, far from xoy plane, is still an ellipsoid as shown in Figs. 8(c) and 8(d). The curvature radius of the ellipsoid becomes large and the dotted line is obviously curved. However, in Fig. 8(c), the envelope near xoy plane appears to be distorted away from the origin. This would lead to a serious distortion on the wavefront. Since the paraxial SLR model is enforced in nonparaxial condition, the restriction that ray’s initial point must be located in foci disk of Eq. (5) disappears. The initial points of rays far away from z-axis are out of the foci disk, which leads to the distorted envelope.

Fig. 8 Rays’ trajectories with length equal to $λ$ . (a) and (c) are traced by paraxial SLR model for $k D = 100$ and $k D = 1$ . (b) and (d) are traced by nonparaxial SLR model for $k D = 100$ and $k D = 1$ , respectively.

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The phase patterns traced by two SLR models with $k D = 1$ are shown in Figs. 9(a) and 9(b). In Fig. 9(a), the shape of contour of isophase equal to $π$ is consistent with the shape of rays’ envelope in Fig. 8(a). In Fig. 9(b), the contour is an ellipsoid. The phase patterns calculated directly by Eqs. (25) and (2) are also shown in Figs. 9(c) and 9(d), respectively. Since Eq. (25) is not suit for $k D = 1$ condition, the pattern in Fig. 9(c) is obviously different from that in Figs. 9(b) or 9(d). The pattern in Fig. 9(b) is consistent with that of in Fig. 9(d), proving that nonparaxial SLR model can trace the evolution of phase precisely. We noticed that the phase pattern in Fig. 9(c) varies rapidly along x-axis with periodicity near $z = 0$ plane. This is mainly induced by the approximation of Eq. (20). This treatment is the paraxial approximation of optical path length. That leads to an increased optical path length.

Fig. 9 Axial patterns of phase. $k D = 1$ . (a) and (b) traced by paraxial and nonparaxial SLR model. (c) and (d) are calculated by Eq. (25) and Eq. (2) respectively.

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7. Concluding remarks

In this paper, we have presented a SLR model for geometrical description of nonparaxial Gaussian beams. This model enables us to precisely describe the propagation of nonparaxial Gaussian beam including the amplitude and phase. The optical field deduced by nonparaxial SLR model is consistent with that obtained from the wave equation. The explanations of both Berry phase and Gouy phase are given in term of ray optics. We demonstrate the rays’ trajectories of nonparaxial model in both paraxial and nonparaxial regimes. The simulation results show that our SLR model is suitable for both paraxial and nonparaxial cases. Moreover, on the paraxial approximation, the equations of skew line are consistent with the previous works.

Notably, this model depicts a ray by the initial point and the direction vector of ray. That makes it easy to combine with the ray-tracing methodology. Such a geometrical description provides a useful way to study beam’s behaviors after reflection/refraction for nonparaxial beam, or through the ABCD matrix on paraxial condition. Furthermore, apart from the Gaussian beam we focus on here, the SLR model would be further exploited for studying other kind of nonparaxial light beams, such as Laguerre-Gauss beam and Bessel-Gauss beam. Generally, positive and negative skew lines play the same role in presenting beam’ propagation. Considering that the SLR model can work as a ray-optics model for force calculation in optical trapping, it is necessary to choose both of positive and negative skew lines at the same times to counteract the artificial force torque.

Funding

The Key Project of Natural Science Foundation of the Anhui Higher Education Institutions (KJ2016A361); the Grants for Scientific Research of BSKY from Anhui Medical University (XJ201518); Scientific Research Foundation of the Institute for Translational Medicine of Anhui Province(2017zhyx25).

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Skew line ray model of nonparaxial Gaussian beam

Abstract

1. Introduction

2. Gaussian Beam in oblate spheroidal coordinates

3. SLR model of nonparaxial Gaussian beam

3.1 The skew lines of a hyperboloid

3.2 Amplitude propagation along the SLR

3.3 Phase propagation along the SLR

4. Paraxial form of SLR model

5. Ray-based explanation on broken self-similar property

6. Ellipsoid wavefront as the envelope of equidistance rays

7. Concluding remarks

Funding

References and links

Cited By

Figures (9)

Equations (28)

Optics Express