Propagation properties of chirped Airy hollow Gaussian wave packets

Accepted Manuscript Propagation properties of chirped Airy hollow Gaussian wave packets Shijie Chen, Xinyi Zheng, Youwei Zhan, Shudan Ma, Dongmei Deng

PII: DOI: Reference:

S0030-4018(18)30991-X https://doi.org/10.1016/j.optcom.2018.11.039 OPTICS 23624

To appear in:

Optics Communications

Received date : 6 August 2018 Revised date : 10 November 2018 Accepted date : 15 November 2018 Please cite this article as: S. Chen, X. Zheng, Y. Zhan et al., Propagation properties of chirped Airy hollow Gaussian wave packets, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.11.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Propagation properties of chirped Airy hollow Gaussian wave packets Shijie Chen1, Xinyi Zheng1, Youwei Zhan1, Shudan Ma1, and Dongmei Deng1,* 1

Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal

University, Guangzhou 510631, China *

Corresponding author: E-mail address: [email protected] (Dongmei Deng).

Abstract: Based on the analytic expression of the chirped Airy hollow Gaussian (CAiHG) wave packets derived from the (3+1)-dimensional ((3+1) D) Schrödinger equation, their propagation properties are investigated in detail for the first time. The results show that a focusing of the chirped Airy Gaussian pulses is found, which leads to the same focusing of the propagation trajectory and the maximum scattering force. The distribution factor rings at

can modulate the convergence of the side wave

. During the propagation process, with the increase of the , when the

side wave rings rise along the main wave axis but their moving is opposite when the When the beam order

is negative, the is positive.

, the intensity, the angular momentum and the gradient force gradually

tend to the center but they move radially when

. The direction of the peripheral gradient force

points to the center but the direction is opposite in the center. The beam orders only affect the magnitude of the normalized maximum scattering force but have little effects on their distribution. Keywords: Mathematical model in optics; Chirping; Wave propagation.

1. Introduction In 1979, by solving the linear Schrödinger equation exactly, the Airy wave packets were predicted theoretically for the first time [1]. And then the existence of Airy wave packets was confirmed experimentally in 2007[2,3]. The Airy wave packets have many unique propagation properties such as self-acceleration [2,4], self-healing [5], weak diffraction [6], which aroused the strong interest of the optical researchers. At the same time, because of these extraordinary propagation properties, Airy wave packets have a wide range of applications in optics such as micro-manipulation [7-9], Airy beam manipulation [10], Laser filamentation [11] and vacuum electron acceleration [12,13]. Related researches in linear [14,15] and nonlinear [16,17] fields have made the Airy wave packets more deeply understood. Recently, some studies about Airy beams with an initial frequency chirp were investigated. In general, the unique propagation properties of a chirped Airy beam under different conditions [18-21] have attracted researchers to explore it in more depth. On the other hand, with zero central intensity, dark-hollow beams (DHBs) have drawn massive attention owing to their growing application [22-26]. A new, widely accepted beam model, called

the hollow Gaussian beam (HGB), accurately describes DHBs [27]. The propagation properties of

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

HGB have aroused great interest of researchers. The researches of the HGB such as spatial filtering generation technology [28], the analytic vectorial structure in the far field [29], the propagation and polarization properties in a uniaxial crustal [30] and the infinite series non-paraxial correction expressions [31] have been investigated. However, based on the resources available to date, almost all of the studies on HGB do not include the Airy distribution, nor do they include the chirped Airy distribution. Based on the particular propagation properties of the chirped Airy beam, the propagation of the chirped Airy hollow Gaussian (CAiHG) wave packets will be investigated in this paper. In this article, first, in the second section, we combine the chirped Airy Gaussian pulses with HGB to obtain the CAiHG wave packets by solving (3+1) D Schrödinger equation. In the third part, the CAiHG wave packets and their propagation properties are described and analyzed in detail. Finally, we summarize the whole article in the fourth section.

2. Model and analytical expressions In the anomalously dispersive system, the beams specifically obey the equation i0

 1  2 2 u  x, y , z ,     2  2 z 2  x y

where

,

 1 2  u  x, y, z,    0  2 2 u  x, y, z,  =0, 2  

,

the group speed, and

,

is the inverse of

is the dispersion index in the case of anomalous

dispersion. The above equation can be expressed in dimensionless form by introducing the new coordinates length.

, and

and

, where

is the diffraction

denote the beam waist width and the wave number, respectively.

Furthermore, the following change must be made :

. Here,

is the

dispersion length. In order to obtain a class of the CAiHG wave packets, we equalize the effects of the diffraction and dispersion for simplifying the analysis. In the normalized coordinate system and for an anomalously dispersive system [3,32], without any loss of generality, the spatiotemporal propagation of the CAiHG wave packets with the amplitude U along the direction Z can be expressed in the form [33-36]:

i where

U 1   2U  2U  2U        0, Z 2  X 2 Y 2 T 2 

(1)

describes the amplitude distribution of the light field.

To obtain the solutions to Eq. (1) in the normalized coordinate system, we use the method of separation of variables and assume that the solution to Eq. (1) satisfies the following form:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

U  X , Y , Z , T   A T , Z   X , Y , Z  .

(2)

Substituting Eq. (2) into Eq. (1), we can obtain the following equations:

i

i

A T , Z 

  X , Y , Z  Z

Z



2 1  A T , Z   0, 2 T 2

(3a)

2 2 1     X ,Y , Z     X ,Y , Z        0. 2 2  X Y 2 

(3b)

As shown above, it is obvious that Eq. (3a) is a 1D Schrödinger equation without any potential. And one of the solutions to Eq. (3a)—Airy Gaussian pulses without the temporal chirp parameter, have been widely studied [2,3]. Here, it is not difficult to obtain the finite-energy Airy Gaussian pulses with the temporal chirp parameter function,

denotes the decay factor in

, where direction,

is the Airy

is the distribution factor and

represents the temporal chirp parameter. By solving Eq. (3a) with the Fourier transform and inverse Fourier transform method and using the identity relation [37]:

 Ai  

  1   2   exp     d   Ai  4 2  2  16    

 2  1   exp   3 2  ,  96 6 3    4 8 

the following analytical equation can be derived:

A T , Z  

  1 c  c2 c 1  Ai   exp    2 4 4 2 4 4 2 2  2 2 3 6 6  16 b  d 2 b  d 4 b  d 8 b  d 96 b  d bd    

i

   1 a a2  1  1   exp i   3   Z   a  3 T  , 6 2 6 3 2 128  d 8  d 8  d 768  d      

here,

(4)

determines the new acceleration direction of the chirp Airy function envelope. and

.

Next, we concentrate on the solution to Eq. (3b). As an initial spatial input to Eq. (3b), the normalized paraxial hollow Gaussian beam at the

plane can be expressed as [31]:

n  X   X , Y , 0   C0  X 2  Y 2  e 

2

Y 2



,

(5)

where

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denotes the order of the paraxial hollow Gaussian beam (HGB),

constant. We apply the identity [38]

, where

factorial of an integer ,

is the

represents a binomial coefficient and

is a is the th-order

Laguerre polynomial. So Eq. (5) can correspondingly be expressed in the following form:

n

  X , Y , 0   C0 n !   1   Lm  X 2  Y 2  e m0 m n

m



 X 2 Y 2



. (6)

Equation (6) shows that the distribution of the light field of the

th-order HGB across the

plane is a superposition of elegant Laguerre-Gaussian beams [39-42] of

cyclical

symmetry. Applying the Fourier transform to Eq. (3b), the following expression can be obtained:

  f X , fY , Z     f X , fY ,0  e using

,

,

,



,

,

Fourier transform of the wave function on

 



2i 2 f X 2  fY 2 Z

(7) ,

, the

is given by:

 n 

 



  f  ,0   C0 n !   1   2   Lm   2  e  J 0  2 f    d    ,  m0   m  0   n

m

2

(8)

Using the identity relation [37]: 

 2  e Lm   J 0   d  2

0

where

22 m 1 2 m 4  e , m! 2

(9)

represents a zero order Bessel function, we can get the following formula:



 n  1

  f  , 0   C0 n !   1    2 m 1 f  2 m e m0   m   m! n

m

 2 f  2

  . 

(10)

Next, Eq. (7) is converted into a polar form and Eq. (10) is used. Fourier inversion in polar coordinates yields the expression: n

n 1



   , Z   C0 n !   1    2 m 1   f  2 m 1e m0  m  m! 0 m

applying the following identity relation to Eq. (11) [42]:

 2 f  2  2 iZ 1

2 J 0  2 f    df  ,

(11)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65



2 m 1

exp   2 2  J 0   d 

0

 2   2  m ! 2 m 1  exp   2  Lm  2  , 2  4   4 

(12)

results in a solution to Eq. (3b), which satisfies the following form: n



n

   , Z   C0 n !   1    2iZ  1 m0   m m

  m 1

  2    2  exp    Lm   ,  2iZ  1   2iZ  1  

(13)

converion of this expression to Cartesian coordinates results in the solution: n



n

  X , Y , Z   C0 n !   1    2iZ  1 m0   m m

  m 1

 X 2  Y 2   X 2  Y 2  exp    Lm   .  2iZ  1   2iZ  1  

(14)

Substituting Eq. (4) and Eq. (14) into Eq. (1), we can obtain the exact solution to Eq. (2) as the following expression:

U  X ,Y , Z ,T  

C0i bd

 i  2iZ  L n 1 n   2iZ  1  2Z n

n!

X 2 Y2   2iZ  1 

 X 2 Y2    1 c  c2 c 1  Ai  2 4 4  exp  exp   2 4 4    2 2  2 2 3 6 6  2b d  8b  d 96b  d  16b  d  2iZ  1   4b d

   1 a a2  1  1  exp i    .  Z   a  3 T  6 2 3 6 3 8 d 2  8 d  768 d     128 d

(15)

3. Analysis Based on the specific analytical solution in the previous section, we can analyze the propagation properties of the CAiHG wave packets. In Eq. (4), we can easily find the propagation focusing of temporal chirp parameter

. When the

is negative, the focusing position of the chirped Airy function

envelope will shift toward the positive direction of the temporal chirp parameter

when

axis. Besides, due to the dispersion effect,

can change the frequency component of the wave packets in the

temporal domain, which leads to the deflection of the direction of propagation and a change of the speed in an anomalously dispersive system. When the temporal chirp parameter

changes

from negative to positive, the acceleration direction of the chirped Airy function envelope will change from positive to negative in the

axis. According to Eq. (4), the chirped Airy pulse in the

temporal domain with different temporal chirp parameters can be showed in Fig. 1. Interestingly, in Figs. 1(a1) and 1(b1), first, due to the differences in the optical path and propagation time between wavelets, the wavelets overlap. Hence, the Airy pulse is compressed and accelerates in

the positive direction of

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axis and then the pattern of Airy pulse breaks up with further

increasing of the propagation distance. Finally, a new Airy pulse pattern along the original direction is regenerated. As Figs. 1(a2)-1(a4) and 1(b2)-1(b4) show, the Airy pulse accelerates in the positive direction of the

axis with the increase of propagation distance, and its

acceleration gradually decreases with

increasing. In Figs. 1(a5) and 1(b5), we can clearly find

that the Airy pulse accelerates in negative direction of propagation distance.

axis with further increasing of the

Fig. 1. Propagation properties of Airy pulse with different temporal chirp parameters. (a1)-(a5) present the propagation intensity

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distributions of finite-energy Airy wave packets with different temporal chirp parameters. (b1)-(b5) present the intensity distributions of Airy wave packets in different propagation distances. Decay factor , (a3)-(b3)

, (a4)-(b4)

, (a5)-(b5)

. Distribution factor

Fig. 2. Snapshots describing the initial shape of the CAiHG wave packets at (a3)-(b3)

(a4)-(b4)

(a1)-(a4)

05, (b1)-(b4)

2.

. (a1)-(b1)

, (a2)-(b2)

2..

(a1)-(b1) Decay factor

(a2)-(b2) .

Fig. 3. Snapshots describing the shape of the CAiHG wave packets. Decay factor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a1)-(a5)

, (b1)-(b5)

(a4)-(d4)

, (a5)-(d5)

, (c1)-(c5)

, (d1)-(d5)

.

. (a1)-(d1)

. Distribution factor , (a2)-(d2)

, (a3)-(d3)

2. ,

.

Figure 2 represents the initial shape of the CAiHG wave packets with different distribution factors

. When

the CAiHG wave packets are hollow along the

2(a1)-2(b1). When rings.

When

the

axis as shown in Figs.

, the CAiHG wave packets have a main wave axis and side wave CAiHG

wave

packets

order

is

increased

and,

consequently,

mode-number-increasing Laguerre polynomials are superimposed, the ring number of side wave rings increases and the side wave rings become more complex as shown in Figs. 2(a2)-2(b2), 2(a3)-2(b3), 2(a4)-2(b4). In general, as the distribution factor

increases, the side wave rings

gradually converge upward. Equation (15) expresses the complex amplitude distributions of the CAiHG wave packets. In Fig. 3 we show snapshots describing the shape of specific spatiotemporal CAiHG wave packets with different temporal chirp parameters

and different beam orders

. In Figs. 3(a1)-3(a5),

we can clearly see that the thickness of the wave rings of the CAiHG wave packets is gradually getting smaller as when

increases. In Figs. 3(b1)-3(b3), 3(c1)-3(c3) and 3(d1)-3(d3), we see that

is negative, the side wave rings of the CAiHG wave packets gradually rise along the main

wave axis with

increasing. When

is positive, the side wave rings of the CAiHG wave packets

gradually decline along the main wave axis with an increase in

as seen in Figs. 3 (b4)-3(b5),

3(c4)-3(c5) and 3(d4)-3(d5). It is interesting to note that the wrinkles of main wave axis gradually split into lumps as shown in Figs. 3(b4)-3(b5). Meanwhile, the side wave rings begin splitting into more pieces, which is clearly shown in Figs. 3(c4)-3(c5) and 3(d4)-3(d5). Figure 4 shows a side view, on the X-Z plane, in the

-direction of the propagation trajectory

of the CAiHG wave packets with different temporal chirp parameters orders

and different beam

. In Figs. 4(a1)-4(d1), we can clearly observe that the CAiHG wave packets are

characterized by a hollow feature. During the propagation process of the CAiHG wave packets, focusing appears due to the chirped Airy Gaussian distribution, and its position exactly matches the real part

of

. The intensity of the CAiHG wave packets increases firstly and then

it weakens during the propagation process, and it is close to zero at

. However, after that, the

intensity keeps enhancing, hence a re-focusing effect of intensity appears.. Finally, the intensity of the CAiHG wave packets weakens with an increase of propagation distance due to the diffraction effect. In Figs. 4(a2)-4(a5), when

, it is interesting that the intensity on both sides of the

CAiHG wave packets first increases and then gradually weakens with an increase in propagation distance, and the intensity of original hollow parts of the CAiHG wave packets also focuses and enhances then weakens. With the temporal chirp parameters

increasing, the maximum

intensity focused into the hollow parts first increases and then weakens. Similarly, in Figs. 4(b1)-4(b4), 4(c1)-4(c4) and 4(d1)-4(d4), when

, the intensity of the CAiHG wave

packets first increases and then weakens. Besides, we find that mainly due to the coherent

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

superposition of the wavelets of the wave packets, the number of beams on both sides of the CAiHG wave packets increases obviously when

. When

increases to a certain extent, the

intensity of main lobe of the CAiHG wave packets weakens directly rather than first increasing as seen in Figs. 4(a5)-4(d5).

Fig. 4. Side-view,on the X-Z plane, describing the propagation trajectory of the CAiHG wave packets. (a1)-(a5) , (c1)-(c5) . Decay factor

, (d1)-(d5) .

. (a1)-(d1)

, (a2)-(d2)

. Distribution factor

, (a3)-(d3)

, (a5)-(d5)

2.

In order to discuss the CAiHG wave packets more thoroughly, we take cross sections at the ranges

, (a4)-(d4)

, (b1)-(b5)

, and choose three

in order to describe the intensity and phase of the

CAiHG wave packets with different propagation distances shown in Fig. 5 and Fig. 6, respectively. When

and different beam orders

as

, mainly affected by the chirped Airy pulse,

the intensity distribution of the CAiHG wave packets gradually tends to center with an increase of the propagation distance, as shown in Figs. 5(a1)-5(c1). When

, mainly due to the influence

of the HGB, the main intensity distribution of the CAiHG wave packets gradually presents a circular distribution with the propagation distance increasing, and the outline is clearer as shown in Figs. 5(a2)-5(c2). In Figs. 5(a3)-5(c3) and 5(a3)-5(c3), it is easy to see that when

, the

main intensity distribution is concentrated in the center of the CAiHG wave packets, and the intensity of the side lobe has a circular and more uniform distribution as the propagation distance increases. In general, with the propagation distance

increasing, the phase

distribution of the CAiHG wave packets becomes more complex due to the diffraction effect,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

meanwhile, with the increase of beam orders

, the trend of phase complexity becomes more

obvious as shown in Fig. 6.

Fig. 5. Sectional view of the intensity of the CAiHG wave packets. (a1)-(c1), (a2)-(c2), (a3)-(c3) and (a4)-(c4) denote the intensity of Airy Gaussian beams with different beam orders at different propagation distances. (a1)-(a4) , (c1)-(c4)

. (a1)-(c1)

. Distribution factor

, (a2)-(c2)

, (a3)-(c3)

, (a4)-(c4)

, (b1)-(b4)

. Decay factor

.

2

Fig. 6. Sectional view of the phase of the CAiHG wave packets. (a1)-(c1), (a2)-(c2), (a3)-(c3) and (a4)-(c4) denote the phase of Airy Gaussian beams with different beam orders at different propagation distances. (a1)-(a4) (c1)-(c4) Distribution factor

. (a1)-(c1)

, (a2)-(c2)

, (a3)-(c3)

, (a4)-(c4)

, (b1)-(b4)

. Decay factor

.

, .

2.

When further analyzing the propagation properties of the CAiHG wave packets, it is very valuable and necessary to discuss the longitudinal normalized angular momentum flow of the

CAiHG wave packets in HGB-like condition. The time-averaged angular momentum vector can be

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

written as [43]:

 J   r  E  B where

is the electric field and

(16)

is the magnetic field.

Using Eq. (16), the intensity and direction of the longitudinal normalized angular momentum vector of the CAiHG wave packets in HGB-like condition with different beam orders different propagation distances

and

can be numerically analyzed as shown in Fig. 7. The

background shows the intensity and shape of the angular momentum vector, the blue arrows indicate the direction of the angular momentum vector, and the coordinate arrows indicate the size of the angular momentum vector. Generally speaking, the angular momentum vector directions of the CAiHG wave packets for different propagation distances

and beam orders

are always in helical clockwise symmetrical distribution. From the perspective of angular momentum vector intensity, when

, due to the influence of the chirped Airy pulse, the main

intensity rings of the angular momentum vector of the CAiHG wave packets move closer to the center with the propagation distances contrary, when

increasing as shown in Figs. 7(a1)-7(c1). On the

, due to the influence of the HGB, the main intensity rings of the angular

momentum vector of the CAiHG wave packets stay radially further away as the propagation distance

increases as shown in Figs. 7(a2)-7(c2). When

increase of the propagation distances

, we can see that with the

, the intensity of the angular momentum vector of the

CAiHG wave packets in the central part weakens, and the surrounding intensity rings of the angular momentum vector broaden and distribute more evenly as shown in Figs. 7(a3)-7(c3) and 7(a4)-7(c4). It is special that despite the different beam orders distances

and the different propagation

of the CAiHG wave packets, the intensity of the angular momentum vector in the

center of the CAiHG wave packets is always zero. Next, we explore the gradient force and maximum scattering force of the CAiHG wave packets in HGB-like condition, which are closely related to the momentum changes of electromagnetic wave. First, we assume that the CAiHG wave packets which propagate along the strike a micro particle with refractive index

direction

. When the micro particle undergoes a steady

state, the time-averaged gradient force and scattering force can be expressed as [44]:

F grad  X , Y , Z  

1 2 n2 r03  m2  1    I  X , Y , Z dT , t  0.1, t 0 c  m2  2  t

(17)

2

t 1 8 n2 r06 k 4  m2  1  F scat  X , Y , Z     2  I  X , Y , Z  e Z dT , t  0.1, t0 3c m 2

(18)

with

I  X , Y , Z   cn2 0

U  X ,Y , Z  2

2

.

(19)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Here

represents the relative refractive index between the particles,

refractive index of the surrounding medium, the light velocity and take

presents the radius of the micro particle,

denotes the permittivity of vacuum.

,

, and

denotes the is

is gradient operator. Now we

.

Fig. 7. Sectional view of longitudinal normalized angular momentum density flow of the CAiHG wave packets in HGB-like condition with delay factor

.

(a4)-(c4)

, (b1)-(b4)

. (a1)-(a4)

. Distribution factor , (c1)-(c4)

2. (a1)-(c1) .

, (a2)-(c2)

, (a3)-(c3)

,

Fig. 8. Sectional view of the gradient force of the CAiHG wave packets in HGB-like condition with delay factor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

. Distribution factor (b1)-(b4)

, (c1)-(c4)

2. (a1)-(c1)

, (a2)-(c2)

, (a3)-(c3)

, (a4)-(c4)

.

. (a1)-(a4)

,

.

For different propagation distances

, the gradient force of the CAiHG wave

packets of different beam orders in HGB-like condition is displayed in Fig. 8. Likewise, the background shows the intensity and shape of the gradient force of the CAiHG wave packets, the red arrows represent the direction of the gradient force, and the coordinate arrows indicate the size of the gradient force. We can easily see that when

, the main intensity rings of the

gradient force of the CAiHG wave packets converge toward the center with an increase of the propagation distance as shown in Figs. 8(a1)-8(c1), which is similar to the distribution changes of the angular momentum intensity of the same order CAiHG wave packets. However, when , the main intensity rings of the gradient force of the CAiHG wave packets are radially away with the propagation distance increasing as shown in Figs. 8(a2)-8(c2). Compared with Figs. 8(a1)-8(c1) and 8(a2)-8(c2), when

, for the same propagation distance, we can see that

the main intensity ring of the gradient force of the CAiHG wave packets is always in the central region like in Figs. 8(a3)-8(c3) and 8(a4)-8(c4). Concerning the direction of the gradient force of the CAiHG wave packets, in general, the direction of the peripheral part of the gradient force of the CAiHG wave packets always points to the center of the circle, and the direction of the gradient force of the CAiHG wave packets which are in the central region is mainly along the radial direction, while the direction of the gradient force of the CAiHG wave packets located between the peripheral part and the central part of the beam is radial into different angles distribution. In order to discover the value of the acceleration and trapping of the particles, we plot in Fig. 9 the maximum scattering force of the CAiHG wave packets with the different propagation distances, the different temporal chirp parameters, and for different beam orders. Here Fig. 9(a) shows the normalized maximum scattering force. For chirp parameter , the magnitude of the maximum scattering force corresponds to magnitude of the maximum scattering force corresponds to of the maximum scattering force corresponds to maximum scattering force corresponds to force of a Gaussian (

) [45], Airy (

. When

and when

, and beam order . When

, the

, the magnitude , the magnitude of the

. Compared with the maximum scattering ) [46] and Bessel (

) [47] beams, the

maximum scattering force of the CAiHG wave packets is close to the value of the Airy beams but it is much smaller than the value of the Gaussian or Bessel beams due to the influence of the chirped Airy pulse and multi-beam combination. It is not difficult to find that although the order of the CAiHG wave packets increases, the normalized maximum scattering force of the CAiHG wave packets first increases and then decreases with increasing propagation distance. Before , the normalized maximum scattering force distribution of the CAiHG wave packets with

different orders is basically the same with only a slight shift. Likewise, after

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

, except for

, the normalized maximum scattering force of the CAiHG wave packets with other orders is also basically the same. When

, the downtrend of the normalized maximum scattering

force of the CAiHG wave packets is relatively flat. According to Fig. 9(a), we can easily see that changing orders of the CAiHG wave packets only changes the magnitude of the maximum scattering force of the CAiHG wave packets to a great extent, but has almost no effect on the distribution of the maximum scattering force. Therefore, we take the CAiHG wave packets with as an example, and show the maximum scattering force of the CAiHG wave packets with temporal chirp parameters at different times as shown in Fig. 9(b). It is interesting that when , due to the effect of the focusing of the chirped Airy Gaussian pulses, the maximum scattering force decreases sharply at the same focusing position, and then the maximum scattering force increases. A re-focusing effect occurs and finally the maximum scattering force decreases. When

, with the propagation distance increasing, the maximum scattering

force of the CAiHG wave packets first increases and then decreases. However, with the increase of temporal chirp parameters, the peak of maximum scattering force decreases and moves along the positive direction of the

axis. When

, the maximum scattering force of the CAiHG

wave packets always decreases with the propagation distance increasing. These phenomena may be helpful for particle trapping and particle acceleration.

Fig. 9. Maximum scattering force of the CAiHG wave packets in HGB-like condition with delay factor

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Distribution factor beam orders

.

.

2. (a) denotes the normalized maximum scattering force of the CAiHG wave packets with the different

. (b) presents the maximum scattering force of the CAiHG wave packets with the different temporal chirp

parameters .

4. Conclusion In conclusion, by solving the (3+1) D Schrödinger equation, the analytic expression of the CAiHG wave packets is derived. The propagation properties of the corresponding chirped Airy Gaussian pulses envelope, the shape and propagation trajectory of the CAiHG wave packets, the intensity and phase, the angular momentum in HGB-like condition, the gradient force and the maximum scattering force in HGB-like condition are analyzed and described in detail. It is shown that when

, the chirped Airy Gaussian pulses have a focusing, which leads to the same

focusing of the propagation trajectory and the maximum scattering force of the CAiHG wave packets. Moreover, after the focusing position, a re-focusing effect makes their intensity and maximum scattering force increase again. With the distribution factor wave rings of the CAiHG wave packets at as

increases, when

increasing, the side

gradually converge upward. It is interesting that

is negative, the side wave rings rise along the main wave axis but the

side wave rings decline when

is positive. The intensity along the propagation trajectory first

increases and then gradually weakens but when

increases to a certain extent, the intensity

weakens directly along the propagation trajectory. With the increase of the propagation distance, when

, the intensity, the angular momentum and the gradient force of the CAiHG wave

packets gradually tend to the center while they move radially to become a circular distribution when

. Interestingly, the directions of the angular momentum are always in helical

clockwise symmetrical distribution and the angular momentum in the center is always zero. The direction of the gradient force of the peripheral part points to the center but the direction of the central gradient force is opposite. The beam orders of the CAiHG wave packets only affect the magnitude of the normalized maximum scattering force but have little effects on the distribution. As

increases, the peak of maximum scattering force decreases and moves along the positive

direction of the

axis.

Our study not only enriches the understanding of the (3+1) D linear Schrödinger equation in optics, but also provides many special and interesting propagation properties of the CAiHG wave packets such as the re-focusing effect of intensity and maximum scattering force. Hence, we believe that our research can promote the potential applications of the CAiHG wave packets.

Funding National Natural Science Foundation of China (11775083, 11374108).

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