Influence of the higher-orders of diffraction on the pattern evolution for tightly focused beams

Daquan Lu; Zhenjun Yang; Wei Hu

doi:10.1364/OE.19.011170

1. Introduction

The Maxwell’s equations, which govern the propagation of optical beams, are mathematically complicated to get analytical solutions. For optical beams whose beam waist is much larger than the wave length, the paraxial approximation is satisfied and the Maxwell’s equations can be simplified to the paraxial equation. Based on the paraxial equation which is mathematically simple, various types of beam solutions with different transverse profiles have been obtained and deeply investigated (e.g., [1–3] and references therein). However, for tightly focused beams, which are important for many areas such as microscopy [4], particle trapping [5], and electron acceleration [6, 7], the paraxial approximation is not satisfied. Therefore the paraxial solutions become invalid; and the effect of nonparaxiality should be taken into account for the beam evolution. In the past decades, to deal with the problem of nonparaxial propagation, many approaches, such as the perturbative method [8–11], the angular spectrum analysis method [12,13], and the complex-source-point method [14,15], have been developed. Here we note that, beyond these various mathematical methods for getting the approximate nonparaxial solution, the mechanism of the nonparaxial propagation for tightly focused beams remains unexplored, which motivates us to do this work.

In addition, as will be shown in the following, the higher-orders of diffraction (HOD) differ the nonparaxial propagation from the paraxial propagation, which is governed only by the 2^nd-order diffraction. For the management of the 2^nd-order diffraction, as well as its counterpart in temporal domain, i.e., the 2^nd-order dispersion, there are various approaches developed. It is the standard technique the use of diffraction gratings (lenses) to compress a broadened pulse (beam). In nonlinear media, the self-focusing balances the 2^nd-order diffraction (dispersion) so that during propagation the beam keeps invariant and becomes a soliton [16]. In linear and dispersive medium, specific transverse beam profiles (temporal pulse forms) have been developed to exhibit pseudo-diffraction-free (pseudo-dispersion-free) properties [17]. Additionally, if the transverse profile of a pulsed beam is suitably designed, the angular dispersion would induce a group velocity dispersion which plays against the material group velocity dispersion [18–20]; and a dispersion-free and diffraction-free localized propagation will be resulted in [21, 22]. With proper angular dispersion, even the transmission of a subcycle pulse through fused silica is possible [19]. During the process of degenerate optical parametric amplification, the nonlinear interaction modifies the spatial and temporal phase distribution of the signal and idler pulsed beams. Therefore the spatial diffraction and the temporal dispersion can be suppressed and a simultaneous temporal and spatial localization is reached [23]. All these approaches and results about the management of the 2^nd-order diffraction or dispersion indicate that the management of the HOD would induce novel phenomena and might be of interest for real applications.

In this paper, we note that the HOD play important roles in the nonparaxial propagation. The HOD induce characteristics which are crucially different from those predicted by the traditional paraxial theory. Based on the theory of Fourier optics, we propose an approach to manage the HOD. By introducing a pre-added chirp in spectrum domain, the nonparaxial propagation of the pre-chirped field approximates well to the result of the propagation of the unpre-chirped field predicted by the paraxial theory; and one can get a focused spot which is expected by applications and identical with that predicted by the paraxial theory.

2. Preliminaries

It is well known that, outside the realm of paraxial approximation, the field have to be handled as a vectorial one. In free space, the transverse part E _⊥ and the longitudinal component E _‖ connect with each other based on the relation [12]

\nabla \cdot E = 0 .

E _⊥,‖ can be regarded as the superposition of various angular spectrum components, each of which represents a planar wave propagating in a certain direction. Every spectrum Ẽ _⊥,‖ experiences a corresponding phase shift when it propagates from z = z ₁ to z = z ₂, i.e.,

{\tilde{E}}_{⊥, | |} (z_{2}) = {\tilde{E}}_{⊥, | |} (z_{1}) exp (i k_{r} \cdot r + i k_{z} Δ z),

where

k_{z} = \sqrt{k^{2} - k_{r}^{2}}

, k_r = k_x e _x + k_y e _y,

k_{r}^{2} = k_{x}^{2} + k_{y}^{2}

, Δz = z ₂ – z ₁. According to Eq.(1), at every plane, the longitudinal spectrum component can be evaluated from the transverse spectrum component with the relation

{\tilde{E}}_{| |} = - \frac{1}{k_{z}} k_{⊥} \cdot {\tilde{E}}_{⊥} e_{z} .

At the plane z = z ₂, the linear superposition of all angular spectrums yields the field in spatial domain, i.e.

{\tilde{E}}_{⊥, | |} (z_{2}) = \int \int {\tilde{E}}_{⊥, | |} (z_{1}) exp (i k_{r} \cdot r + i k_{z} Δ z) d k_{x} d k_{y} .

Equation (4) is tantamount to Ẽ _⊥,‖ (z ₂) = F̂ ⁻¹ {F̂ {Ẽ _⊥,‖ (z ₁) } exp (ik_zΔz) }, where operators F̂ and F̂ ⁻¹ represent the forward and the inverse Fourier transform, respectively. Therefore, in the view of Fourier optics, the nonparaxial propagation from z = z ₁ to z = z ₂ results from three steps: i) transform E _⊥,‖ (z ₁) to the angular spectrum domain; ii) add a phase k_zΔz on the angular spectrum E _{⊥,_‖} (z ₁); and iii) inversely transform the angular spectrum at z = z ₂ to the spatial domain.

During propagation, the second step plays an important role since different angular spectrum components experience different phase shifts k_zΔz, which results in the evolution of the beam pattern. In fact, if we make a Taylor expansion on k_z, we have

k_{z} = k + \frac{1}{2!} β_{2} k_{r}^{2} + \frac{1}{4!} β_{4} k_{r}^{4} + \dots,

where β_n = −n!|n − 3|!!/n!!kⁿ⁻ ¹ (n ≥ 2). As shown in Eq. (5), during propagation the beam experiences different orders of diffraction (in the following we call β_n the n^th order diffraction), which is similar to the different orders of dispersion for a laser pulse in the fiber. However, there are two differences between the spatial diffraction and the temporal dispersion: i) During propagation in free space the beam experiences only even-order diffraction, whereas the pulse in the fibre experiences both even- and odd-order dispersion; ii) For different media and different carrier wavelengths, the temporal dispersion can be positive or negative, and the dispersion with different orders can be with different signals. However for the diffraction of the beam in free space, the signals of all orders of diffraction are negative (β_n < 0), which means that the higher spatial frequency experiences a smaller phase-shift, and all orders of spatial diffraction together cause a negative spatial chirp during propagation.

Under the paraxial condition, the width of the angular spectrum is so narrow that only the lowest-order diffraction β ₂ is necessary to be taken into account. Then Eq. (4) reduces to

E_{⊥, | |}^{(p)} (z_{2}) = \int \int {\tilde{E}}_{⊥, | |} (z_{1}) exp (i k_{r} \cdot r + ik Δ z + \frac{i}{2} β_{2} Δ z k_{r}^{2}) d k_{x} d k_{y},

which is the well-known Fresnel integral that governs the evolution of an arbitrary paraxial beam. In fact, under the paraxial condition, because the angular spectrum is very narrow, the longitudinal component is so weak that can be neglected; and only the transverse component is necessary to be taken into account.

However, if the beam width is comparable with the wavelength or the divergency angle is very large so that the width of the angular spectrum Δk is no longer small enough to justify the truncation of the expansion (5) after the β ₂ term, the HOD should be included [12], i.e.,

E_{⊥, | |}^{(n p)} (z_{2}) = \int \int {\tilde{E}}_{⊥, | |} (z_{1}) exp (i k_{r} \cdot r + ik Δ z + \frac{i}{2!} β_{2} Δ z k_{r}^{2} + \frac{i}{4!} β_{4} Δ z k_{r}^{4} + \dots) d k_{x} d k_{y},

which describes the propagation of nonparaxial beams, such as the tightly focused beams.

3. Influence of the HOD on the pattern evolution for tightly focused beams

In this section, we will discuss the influence of different orders of diffraction on the propagation of the tightly focused beams based on Eqs. (6) and (7). The propagation properties are illustrated with the example of the Gaussian beam, which is frequently used in application. But it should be noted that the analysis approach and most of the propagation properties are applicable for other beams such as the Hermite-, Laguerre-, and Ince-Gaussian beams, etc.

The spherical lenses, which provide positive linear spatial chirp, are frequently used to focus a beam. Before and in the focusing lens, the beam experiences paraxial propagation, and at the plane next to the focusing lens (because it is the beginning of the nonparaxial propagation, we call it the entrance plane, and correspondingly the field there is called the input field), the transverse component (assumed to be polarized in x direction) of a Gaussian beam can be written as

E_{⊥} (z_{1}) = A_{0} exp [- \frac{1 + iC}{w_{z 1}^{2}} (x^{2} + y^{2}) + i ϕ_{z 1}] e_{x},

where w_z ₁ is the beam width,

C = - k w_{z 1}^{2} / 2 R

, R is the radius of spherical cophasal surface, whose signal is negative because of the concave cophasal surface. ϕ_z ₁ is the initial phase. Correspondingly, the spectrum of the transverse and the longitudinal components are

{\tilde{E}}_{⊥} (z_{1}) = A (k_{x}, k_{y}) exp [i \frac{C w_{z 1}^{2}}{4 (1 + C^{2})} k_{r}^{2} + i ϕ_{z 1}] e_{x},

{\tilde{E}}_{| |} (z_{1}) = - \frac{k_{x}}{k_{z}} A (k_{x}, k_{y}) exp [i \frac{C w_{z 1}^{2}}{4 (1 + C^{2})} k_{r}^{2} + i ϕ_{z 1}] e_{z},

where

A (k_{x}, k_{y}) = \frac{A_{0} w_{z 1}^{2}}{2 (1 + i C)} exp [- \frac{w_{z 1}^{2}}{4 (1 + C^{2})} k_{r}^{2}] .

As will be shown in the following, under the tightly focusing condition, the HOD would induce novel propagation characteristics which are crucially different from those predicted by the traditional paraxial theory. Therefore at first a discussion on the paraxial propagation property of the beam would be constructive.

3.1. Paraxial results

In the framework of the paraxial theory, when a beam is transmitted to the plane z ₂, the spectrums of the transverse and the longitudinal components become

{\tilde{E}}_{⊥}^{(p)} (z_{2}) = A (k_{x}, k_{y}) exp [i (\frac{C w_{z 1}^{2}}{4 (1 + C^{2})} + \frac{1}{2} β_{2} Δ z) k_{r}^{2} + i ϕ_{z 1} + ik Δ z] e_{x},

{\tilde{E}}_{| |}^{(p)} (z_{2}) = - \frac{k_{x}}{k} A (k_{x}, k_{y}) exp [i (\frac{C w_{z 1}^{2}}{4 (1 + C^{2})} + \frac{1}{2} β_{2} Δ z]) k_{r}^{2} + i ϕ_{z 1} + ik Δ z]] e_{z},

where we have adopt the approximation k _⊥/k_z ≃ k _⊥/k.

There are two key characteristics of the paraxial propagation for a focused beam:

I) The evolution of the beam is induced only by the 2^nd order diffraction, which does not vary the angular spectrum distribution, but varies the parameter

Ω_{p} (Δ z) = \frac{C w_{z 1}^{2}}{4 (1 + C^{2})} + \frac{1}{2} β_{2} Δ z

and therefore the radius of the spherical phase distribution in the angular spectrum domain during propagation. In the view of Fourier optics, the Hermite-, Laguerre-, and Ince-Gaussian beams with spherical cophasal surface are the eigen functions of the Fourier transform in different coordinate systems, i.e.

\hat{F} {Λ (k_{r} / a_{1}) exp (- i b_{1} k_{r}^{2})} = Λ (r / a_{2}) exp (- i b_{2} r^{2}),

where Λ(·) represents an arbitrary Hermite-, Laguerre-, or Ince-Gaussian function. Therefore, an arbitrary beam, which is resulted from the linear superposition of these three types of beams with the same waist location and Rayleigh distance, would remain shape-invariant during propagation under the paraxial approximation. The paraxial propagation only varies the beam width and the radius of the cophasal surface (as shown in row 1 in Fig. 1).

Fig. 1 Rows 1 and 2: the evolution of the intensity distribution of the transverse component of a tightly focused Gaussian beam in spatial domain resulted from the paraxial theory and the nonparaxial theory, respectively. Rows 3 and 4: the evolution of the intensity (dashed line), the phase (dotted line), and the chirp (solid line) in the angular spectrum domain resulted from the paraxial theory and the nonparaxial theory, respectively. The distance between the entrance plane and the pseudo-waist plane is z_s = 30z_R. The beam width of the pseudo-waist resulted from the paraxial theory is w ₀ = λ = 1μm.

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For example, for the Gaussian beam, according to Eq. (6), the field in the spatial domain becomes [12]

E_{⊥}^{(p)} (z_{2}) = \frac{w_{0}}{w} exp (- \frac{r^{2}}{w^{2}}) exp (ik \frac{r^{2}}{2 R} - i ϕ) e_{x},

E_{| |}^{(p)} (z_{2}) = \frac{2 i w_{0} x}{w^{3} k (1 + i Δ z / z_{R})} exp (- \frac{r^{2}}{w^{2}}) exp (i k \frac{r^{2}}{2 R} - i ϕ) e_{z},

where

w = w_{0} \sqrt{1 + {(Δ z - z_{s})}^{2} / z_{R}^{2}}

,

R = (Δ z - z_{s}) [1 + {(Δ z - z_{s})}_{R}^{2} / Δ z^{2}]

,

z_{R} = k w_{0}^{2} / 2

, ϕ = ϕ_z ₁ + arctan(z_s/z_R) + arctan [(Δz – z_s)/z_R],

w_{0} = w_{z 1} / \sqrt{1 + C^{2}}

,

z_{s} = C w_{z 1}^{2} / [2 (1 + C^{2}) β]

. As shown in Eqs. (11) and (12), the transverse and longitudinal components are the (0,0) and (1,0) mode Hermite-Gaussian beams, respectively.

II) Because only the 2^nd order diffraction is taken into account and the HOD are neglected, the diffraction-induced negative linear chirp can completely compensate the initial positive linear chirp when it propagates a distance Δz = z_s, i.e.,

Ω_{p} (z_{s}) = 0.

According to Eq. (4), at that plane the beam width is Fourier-transform-limited and arrives its minimum w ₀, thus that plane becomes the waist of the focused beam; and the intensity distribution is symmetric about the waist (we will call it the pseudo-waist in the following, because it is not really the waist for a tightly focused beam), as shown in Eqs. (14) and (15).

3.2. Influence of the HOD

However, for the tightly focused beams, not only the 2^nd-order diffraction but also the HOD play critical roles; and the paraxial theory becomes invalid. According to Eq. (7), the vectorial field in the framework of nonparaxial propagation becomes:

E_{⊥}^{(np)} (z_{2}) = \int \int A (k_{x}, k_{y}) exp (i Ω_{np} + i ϕ_{z 1} + ik Δ z) e_{x} d k_{x} d k_{y},

E_{| |}^{(np)} (z_{2}) = \int \int \frac{k_{x}}{k_{z}} A (k_{x}, k_{y}) exp (i Ω_{np} + i ϕ_{z 1} + ik Δ z) e_{z} d k_{x} d k_{y} .

where

Ω_{np} (k_{r}, Δ z) = [\frac{C w_{z 1}^{2}}{4 (1 + C^{2})} + \frac{1}{2!} β_{2} Δ z] k_{r}^{2} + \frac{1}{4!} β_{4} Δ z k_{r}^{4} + \dots

represents the interaction between the initial chirp and all orders of diffraction. It is the HOD that differs the paraxial from nonparaxial propagation. Because of the mathematical complexity, the analytical expressions of the integral equations (16) and (17) are too difficult to get. The result of the example is based on the numerical simulation of the integral equations.

As shown in Eqs. (13) and (18), what differs the paraxial and nonparaxial propagation is the HOD. Because of the HOD, a tightly focused beam presents interesting evolutions which are critically different from those predicted by the paraxial theory:

The HOD induce fourth- and higher-order phase factors in the angular spectrum domain. Mathematically, each Hermite-, Laguerre-, or Ince-Gaussian function with such phase factors is no longer the eigen function of Fourier transform, i.e., $\hat{F} {Λ (k_{r} / a_{1}) exp (- i b_{1} k_{r}^{2} - i c_{1} k_{r}^{4} + ...)} \neq Λ (r / a_{2}) exp (- i b_{2} r^{2} - i c_{2} r^{4} + ...) .$ Therefore any paraxially shape-invariant beam becomes shape-variant under the nonparaxial condition (e.g. the fundamental Gaussian beam shown in row 2 in Fig. 1).
At the pseudo-waist, the linear initial positive chirp induced by the focusing lens only balances the 2^nd-order-diffraction-induced negative chirp, and the HOD are not eliminated, i.e., $Ω_{np} (k_{r}, z_{s}) = \frac{1}{4!} β_{4} z_{s} k_{r}^{4} + \frac{1}{6!} β_{6} z_{s} k_{r}^{6} + \dots .$ Therefore the HOD induce a nonlinear negative chirp in angular spectrum domain, i.e., $Δ r (z_{s}) = \frac{\partial}{\partial k_{r}} Ω_{np} (k_{r}, z_{s}) = \frac{1}{3!} β_{4} z_{s} k_{r}^{3} + \frac{1}{5!} β_{6} z_{s} k_{r}^{5} + \dots .$ which broadens the beam size. Therefore the beam width is larger than the Fourier-transform-limited one predicted by the paraxial theory. The farther the distance between the entrance plane and the pseudo-waist is, the larger the chirp will be, and in turn the larger the beam width will be resulted in (Fig. 2).
There exists a plane where the beam width is the smallest (however it is still larger than the Fourier-Transform-limited one). In the following we call it the real-waist. The real-waist becomes farther and father from the pseudo-waist with the increase of the distance between the entrance plane and the pseudo-waist (Fig. 2).
The phase distribution in angular spectrum domain, i.e., Ω_np(k_r, Δz), is asymmetric about the pseudo-waist, therefore neither the pattern nor the size of the beam is symmetric about the pseudo-waist. Because Ω_np(k_r, Δz) is asymmetric about the pseudo-waist, the chirp in angular spectrum domain, i.e., $Δ r (Δ z) = \frac{\partial}{\partial k_{r}} Ω_{np} (k_{r}, Δ z) = 2 [\frac{C w_{z 1}^{2}}{4 (1 + C^{2})} + \frac{1}{2!} β_{2} Δ z] k_{r} + \frac{1}{3!} β_{4} Δ z k_{r}^{3} + \dots$ is also asymmetric about the pseudo-waist. At the planes after the pseudo-waist, Ω_np(k_r, Δz) decreases monotonically with |k_r| and results in a negative [i.e., Δr(Δz) = −|Δr(Δz)|] and nonlinear chirp. Therefore the size would be larger than that predicted by the paraxial theory; and the shape is approximate to the paraxial result. However, very differently, at some planes before the pseudo-waist, the phase in angular spectrum domain (Ω_np) varies nonmonotonically with |k_r| and results in a s-like distribution of the chirp Δr. Therefore, at such planes there is the same chirp occurring at three values of k_r. These three angular spectrum components interfere constructively or destructively, depending on their relative phase difference, just like the mechanism for the changes in the pulse spectrum induced by the self-phase modulation in nonlinear fiber [16]. This property is validated by the example of the transverse component of the Gaussian beam (rows 2 and 4 in Fig. 1): at the planes after the pseudo-waist the pattern remains bell-like and the size is larger than that predicted by the paraxial theory; but at some planes before the pseudo-waist, the interference among the three angular spectrum components results in a multi-ringed distribution of the beam in the spatial domain.

4. An approach on controlling the intensity pattern: managing the HOD

Although phenomena of the tightly focused beams induced by the HOD are interesting, they frequently occur negative impact on real applications of the tightly focused beams. In many applications, the tightly focused beam is expected that: i) the spot size is the smallest one to have the highest intensity or grad; ii) the real-waist is located at the pseudo-waist; and iii) the pattern at the real-waist is the same as that predicted by the paraxial theory. Then a question arises: can we design the evolution of the tightly focused beam to satisfy the above requirements? The answer is yes. In fact, we can do this by managing the HOD.

Fig. 2 (a) The comparison between the evolution of the beam width for the transverse component of the Gaussian beam (defined based on the second-order moment) resulted from the nonparaxial theory and from the paraxial theory for various z_s. (b) Solid lines: the beam width at the real-waist (Δz = z_f) and the pseudo-waist (Δz = z_s) vs z_s. Dashed line: the distance between the real-waist and pseudo-waist vs z_s. Parameters are the same as those in Fig. 1 except z_s.

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In previous investigation, there are many approaches developed to manage the 2^nd order diffraction. For instance, during the process of degenerate optical parametric amplification, the nonlinear interaction modifies the signal and idler pulsed beams on the phase distribution in angular spectrum and frequency spectrum domain, and therefore on the spatial and temporal distribution. In suitably designed case, the spatial diffraction and the temporal dispersion can be suppressed and a simultaneous temporal and spatial localization is reached [23]. The method in Ref. [23] for the management of the 2^nd order diffraction is based on the 2^nd order nonlinear phase modulation; it shows that the controlling of the phase distribution in angular spectrum domain would be a feasible approach to influence the beam profile in spatial domain. Here we propose a method to manage the HOD with linear optical process based on the theory of Fourier optics.

Specifically, if the distance between the entrance plane and the pseudo-waist is z_s, we can pre-add a chirp on the angular spectrum of the input field at the entrance plane, i.e.

{\tilde{E}}_{⊥, | |}^{'} (z_{1}) = {\tilde{E}}_{⊥, | |} (z_{1}) exp (- Δ β z_{s}),

where

Δ β = k_{z} - k - \frac{1}{2!} β_{2} k_{r}^{2} = \frac{1}{4!} β_{4} k_{r}^{4} + \frac{1}{6!} β_{6} k_{r}^{6} + \dots

represents all orders of the HOD. Therefore Ẽ′ _⊥,‖ (z ₁) is connected with

{\tilde{E}}_{⊥, | |}^{(p)} (z_{1} + z_{s})

by the relation

{\tilde{E}}_{⊥, | |}^{'} (z_{1}) = {\tilde{E}}_{⊥, | |}^{(p)} (z_{1} + z_{s}) exp (- ik z_{s} - \frac{i}{2!} β_{2} z_{s} k_{r}^{2} - \frac{i}{4!} β_{4} z_{s} k_{r}^{4} - ...) .

And thus in spatial domain the nonparaxial propagation of Ẽ′ _⊥,‖ (z ₁) becomes

\begin{array}{l} E_{⊥, | |}^{' (np)} (z_{2}) & = & \int \int {\tilde{E}}_{⊥, | |}^{'} (z_{1}) exp (i k_{r} \cdot r + i k Δ z + \frac{i}{2!} β_{2} Δ z k_{r}^{2} + \frac{i}{4!} β_{4} Δ z k_{r}^{4} + ...) d k_{x} d k_{y} \\ = & \int \int {\tilde{E}}_{⊥, | |}^{(p)} (z_{1} + z_{s}) exp (i Ω_{np}^{'}) exp (i k_{r} \cdot r) d k_{x} d k_{y}, \end{array}

where

Ω_{np}^{'} (k_{r}, Δ z) = (ik + \frac{i}{2!} β_{2} k_{r}^{2} + \frac{i}{4!} β_{4} k_{r}^{4} + ...) (Δ z - z_{S}) .

Now we analyze the propagation of pre-chirped field, i.e., E′ ⁽ ^np ⁾, based on Eqs. (23) and (24). First, at the pseudo-waist plane where Δz = z_S, we have

Ω_{np}^{'} = 0

; and the field becomes

E_{⊥, | |}^{' (np)} (z_{1} + z_{s}) = E_{⊥, | |}^{(p)} (z_{1} + z_{s}),

which means that the beam pattern of the pre-chirped field at the pseudo-waist plane in the framework of nonparaxial theory becomes the same as that of the unpre-chirped field at the pseudo-waist plane in the framework of paraxial theory, and also does the beam width. At the plane (z ₁ + z_s), not only the second-order but also the higher-orders of diffraction are balanced by the pre-added initial chirp in Ẽ′ _⊥,‖ (z ₁), the beam width arrives its minimum which is the Fourier transform limited one, and the pseudo-waist of

E_{⊥, | |}^{(n p)} (z_{1} + z_{s})

becomes the real waist of

E_{⊥, | |}^{(np)} (z_{1} + z_{s})

.

Second, as shown in Eq. (24), the phase distribution $Ω_{np}^{'} (k_{r}, Δ z)$ of the angular spectrum becomes symmetrical about the real waist plane z = z ₁ + z_s. Therefore the field in spatial domain, which is the Fourier transform of the angular spectrum, is also symmetrical about z = z ₁ + z_s. In addition, for the pre-chirped field, the chirp in spectrum domain becomes

Δ r^{'} (Δ z) = \frac{\partial}{\partial k_{r}} Ω_{np}^{'} (k_{r}, Δ z) = (β_{2} Δ z k_{r} + \frac{1}{3!} β_{4} Δ z k_{r}^{3} + ...) (Δ z - z_{s}) .

Critically different from that for the unpre-chirped field [ shown in Eq. (19)], during the process of focusing (de-focusing) of the beam, i.e., at planes where Δz < z_s (Δz > z_s), Δr′ (Δz) always decreases (increases) monotonically with k_r, which means that the same chirp occurs only at one value of k_r. Therefore the influence of Δr′ (Δz) on the evolution of the beam is a weak broadening of the size, both before and after the waist.

And third, for the pre-chirped beam, not only the evolution of the pattern but also the beam width is in good approximation to that of the result of a un-chirped beam predicted by the paraxial theory. This property can by explained with the analysis of the parameter $Ω_{np}^{'} (k_{r}, Δ z)$ . On one hand, the coefficient β_n/n! before every order of diffraction $k_{r}^{n}$ , and in turn the ratio of the ⁿ ⁺² th order diffraction to the ⁿth order diffraction, i.e.,

Γ = \frac{\frac{β_{n + 2}}{(n + 2)!} k_{r}^{n + 2}}{\frac{β_{n}}{n!} k_{r}^{n}} = \frac{n - 1}{n + 2} \frac{k_{r}^{2}}{k^{2}}

keeps invariant during propagation, which is crucially different from that for the unpre-chirped beam [Eq. (19)]. On the other hand, according to the theory of Fourier transform, the ratio

Γ = (n - 1) k_{r}^{2} / [(n + 2) k^{2}]

in angular spectrum domain is corresponding to the operator

Γ = - (n - 1) \nabla_{⊥}^{2} / [(n + 2) k^{2}]

in spatial domain. For a smooth beam, we have the magnitude order relation Γ ∼ (n – 1)/[(n + 2)k ² w ²]. Even for a beam which is focused to the size of the wavelength λ, Γ ∼ (n – 1)/[(n + 2)4π ²] is still of the order of few percent. Therefore, all through the propagation the 2^nd order phase

β_{n + 2} / [(n + 2)! k_{r}^{n + 2}]

plays a key role for the evolution of the pattern and the beam width, and the higher orders of k_r only provide a little influence on the evolution; and thus the nonparaxial propagation of E′ _⊥,‖ approximates well to the result of the propagation of E _⊥,‖ predicted by the paraxial theory.

In Fig. 3, the nonparaxial propagation of the transverse component of the Gaussian beam with appropriate pre-added chirp is shown to illustrate the properties discussed in this section. As expected, the beam shape keeps bell-like during propagation, the width is in good approximation to the result for E _⊥ predicted by the paraxial theory (and at the waist plane Δz = z_s the two width are identical with each other), and the pseudo-waist becomes the real-waist.

Fig. 3 Dashed line and dash-dotted line: the evolution of the beam width (defined based on the second-order moment) for the transverse component of the focused Gaussian beam without the pre-added nonlinear chirp, resulted from the paraxial and nonparaxial theory, respectively. Solid line: the evolution of the width for the beam with an appropriate pre-added nonlinear chirp, resulted from the nonparaxial theory. Maps around are the corresponding intensity distributions. Parameters are the same as those in Fig. 1 except z_s = −60z_R.

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One may ask how to pre-add the chirp on the input field. In fact we can do this in a simple way based on the theory of Fourier optics. As shown in Fig. 4, the propagation before and after the input plane (z = z ₁) of the tightly focusing region are paraxial and nonparaxial, respectively. Therefore the propagation from z ₀ to z ₁ can be described by the theory of the Fourier optics. The propagation of the beam from z ₀ to $z_{1}^{'}$ is governed by the Fresnel integral, which can be written in an alternative form [24]:

E_{⊥, | |} (x_{1}, y_{1}, z_{1}^{'}) = \frac{exp (ikf)}{i λ f} \int \int E_{⊥, | |} (x_{0}, y_{0}, z_{0}) exp {i \frac{k}{2 f} [{(x_{1} - x_{0})}^{2} + {(y_{1} - y_{0})}^{2}]} dxdy .

The phase transmission effect of the lens is

E_{⊥, | |} (x_{1}, y_{1}, z_{1}) = t E_{⊥, | |} (x_{1}, y_{1}, z_{1}^{'}) = exp (i δ) exp [- i \frac{k}{2 f} (x_{1}^{2} + y_{1}^{2})] E_{⊥, | |} (x_{1}, y_{1}, z_{1}^{'}),

where δ is a constant phase shift. According to Eqs. (28) and (29), one has

\begin{array}{l} E_{⊥, | |} (x_{1}, y_{1}, z_{1}) & = & \frac{exp (ikf + i δ)}{i λ f} \int \int E_{⊥, | |} (x_{0}, y_{0}, z_{0}) exp [i \frac{k}{2 f} (x_{0}^{2} + y_{0}^{2})] \\ exp [- i \frac{k}{f} (x_{0} x_{1} + y_{0} y_{1})] d x_{0} d y_{0}, \end{array}

which means that the spectrum of E(z ₁) is connected with the field at the plane z ₀ by the relation

{\tilde{E}}_{⊥, | |} (k_{x}, k_{y}, z_{1}) = \frac{exp (ikf + i δ)}{i λ f} exp [i \frac{f}{2 k} (k_{x}^{2} + k_{y}^{2})] E_{⊥, | |} (k_{x}, k_{y}, z_{0}),

where

k_{x} = \frac{k}{f} x_{0}, k_{y} = \frac{k}{f} y_{0} .

Therefore, according to Eq. (30), when an appropriate phase –Δβz_s is induced by the phase plate at z ₀, i.e., E′ _⊥,‖ (k_x, k_y, z ₀) = E _⊥,‖ (k_x, k_y, z ₀) exp(−iΔβz_s), the spectrum of the field at z ₁ becomes Ẽ′ _⊥,‖ (k_x, k_y, z ₁) = Ẽ _⊥,‖ (k_x, k_y, z ₁) exp(−iΔβz_s) correspondingly, i.e., the chirp is pre-added.

Fig. 4 The sketch of an apparatus for managing the HOD.

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5. Conclusion

In conclusion, it is the HOD that play important roles in the evolution of the tightly focused nonparaxial beams. Because of the HOD, beams would present novel propagation characteristics which are crucially different from those predicted by the paraxial theory. For the tightly focusing case, a paraxially-shape-invariant beam (e.g., a fundamental Gaussian beam) becomes shape-variant; the real waist is larger than the Fourier-transform-limited one and deviates from the paraxially predicted waist (the pseudo-waist). By introducing a pre-added chirp in spectrum domain to manage the HOD, the nonparaxial propagation of the pre-chirped field Ẽ′ _⊥,‖ becomes in good approximation to the result of the propagation of the unpre-chirped field E _⊥,‖ predicted by the paraxial theory; and one can get a focused spot which is expected by applications and identical with that predicted by the paraxial theory.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant No. 10804033) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200805740002).

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Influence of the higher-orders of diffraction on the pattern evolution for tightly focused beams

Abstract

1. Introduction

2. Preliminaries

3. Influence of the HOD on the pattern evolution for tightly focused beams

3.1. Paraxial results

3.2. Influence of the HOD

4. An approach on controlling the intensity pattern: managing the HOD

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (38)

Optics Express