Propagation Dynamics of Spatio-Temporal Wave Packets

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PROPAGATION DYNAMICS OF SPATIO-TEMPORAL WAVE PACKETS

Thesis

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulﬁllment of the Requirements for

The Degree of

Master of Science in Electro-Optics

Qian Cao

UNIVERSITY OF DAYTON

Dayton, Ohio

August, 2014 PROPAGATION DYNAMICS OF SPATIO-TEMPORAL WAVE PACKETS

Name: Cao, Qian

APPROVED BY:

Andy Chong, Ph.D. Joseph Haus, Ph.D. Advisor Committee Chairman Committee Member Assistant Professor, Department of Professor, Electro-Optics Graduate Physics Engineering Program

Partha Banerjee, Ph.D. Committee Member Professor, Electro-Optics Graduate Engineering Program

John G. Weber, Ph.D. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean Dean, School of Engineering School of Engineering

ii c Copyright by

Qian Cao

2014 ABSTRACT

PROPAGATION DYNAMICS OF SPATIO-TEMPORAL WAVE PACKETS

Name: Cao, Qian University of Dayton

Advisor: Dr. Andy Chong

We measured the three-dimensional (3D) propagation dynamics of the Airy-Bessel wave packet, inculding its intensity and phase evolution. Its non-diffraction and non-dispersive features were veriﬁed. Meanwhile, we built a spatial light modulator (SLM) based wave packet shaping system to generate other types of wave packets such as Airy-Airy-Airy and dual-Airy-Airy-rings. These wave packets were also measured in 3D. The abrupt 3D autofocusing effect was observed on dual-Airy-

Airy-rings.

iii To my family, my advisor and committee members

and the time in University of Dayton.

iv ACKNOWLEDGMENTS

First of all, I want to express my thank to my advisor, Prof. Andy Chong. Although the time when I became his student it was the second year of his professor career, he taught and guided me in the world of science in such an experienced way. Without his guidance cannot I become so interested in physics. His enthusiasm in science, his broad knowledge and the encouragement is invaluable in my research. At the same time, I want to thank my committee members, Prof. Joseph

Haus and Prof. Partha Banerjee for their guidance and patience.

It was my fortune and pleasure to work in Dr. Chong’s ultrafast laser group. I sincerely thank

Chenchen Wan, Xin Huang and Peiyun Li, the group members, who have contributed a lot in my research. Without their help and company, I will not have the courage to accomplish this work.

Finally I sincerely thank my family. I am indebted forever for their priceless love and support.

They have been asking about the progress of my thesis for long. I believe as I ﬁnish the thesis writing, the story will come to a happy ending.

v TABLE OF CONTENTS

ABSTRACT ...... iii

DEDICATION ...... iv

ACKNOWLEDGMENTS ...... v

LIST OF FIGURES ...... vii 1. INTRODUCTION ...... 1 1.1 Organization of the thesis ...... 1 1.2 Introduction ...... 2 2. THEORETICAL BACKGROUND ...... 4 2.1 Optical Airy wave ...... 4 2.1.1 Airy pulse ...... 5 2.1.2 Airy beam and other types of Airy wave ...... 9 2.2 Bessel beam ...... 11 3. THREE-DIMENSIONAL MEASUREMENT SYSTEM ...... 14 3.1 Mechanism and mathematical background ...... 14 3.2 Experimental conﬁguration ...... 17 3.3 Temporal phase retrieval ...... 20 4. MEASUREMENT RESULTS ...... 22 4.1 Propagation dynamics of Airy-Bessel wave packet ...... 22 4.1.1 Experiment parameters ...... 23 4.1.2 3D intensity before propagation ...... 24 4.1.3 3D intensity after propagation ...... 26 4.1.4 Temporal phase retrieval ...... 27 4.2 Measurement of Airy-Airy-Airy wave packet ...... 28 4.3 Measurement of dual-Airy-Airy-rings wave packet ...... 30 5. CONCLUSIONS ...... 34

BIBLIOGRAPHY ...... 35

vi LIST OF FIGURES

2.1 Airy pulse simulation ...... 7

2.2 Airy pulse propagation simulation under the inﬂuence of normal dispersion . . . . 8

2.3 Airy beam simulation ...... 9

2.4 dual Airy pulse simulation ...... 10

2.5 Bessel beam simulation ...... 12

2.6 Axicon to generate Bessel beam ...... 13

3.1 Conceptual sketch of the measurement system ...... 15

3.2 Experimental setup for Airy-Bessel wave packet ...... 18

3.3 Experimental setup for general spatio-emporal wave packet ...... 19

3.4 Conceptual sketch of temporal phase retrieval ...... 21

4.1 Experiment parameters for Airy-Bessel measurements ...... 23

4.2 Results of the Airy-Bessel wave packet before propagation ...... 25

4.3 Results of the Airy-Bessel wave packet after propagation ...... 26

4.4 Temporal phase results for Airy-Bessel wave packet ...... 27

4.5 Experimental parameters for Airy-Airy-Airy measurements ...... 28

4.6 Results of the Airy-Airy-Airy wave packet ...... 29

vii 4.7 Temporal phase retrieval of Airy-Airy-Airy wave packet ...... 30

4.8 Experimental parameters for dual-Airy-Airy-rings measurements ...... 31

4.9 Results of the dual-Airy-Airy-rings wave packet at the Fourier plane ...... 32

4.10 Results of the dual-Airy-Airy-rings wave packet at the autofocus plane ...... 32

viii CHAPTER 1

INTRODUCTION

The main theme of this thesis is the study of the propagation dynamics of spatio-temporal wave packets. The Airy-Bessel wave packet is the main wave packet to be studied in the thesis. Other wave packets such as Airy-Airy-Airy and dual-Airy-Airy-rings are also investigated.

1.1 Organization of the thesis

There are be ﬁve chapters in this thesis.

Chapter 1 is the introduction to the research on optical spatio-temporal wave packets. It ex- plains why they are an important research subject. This chapter gives an initial description of the measurement techniques used in this thesis and how they differ from conventional pulse measurement techniques. Finally, it ends with a brief description of wave packets speciﬁc to those studied in this thesis.

Chapter 2 is the theoretical background for the spatio-temporal wave packets. Airy pulses,

Bessel beams and other types of optical Airy waves are discussed in this chapter. For an Airy pulse, its mathematical form and unique properties such as self-healing and self-acceleration is explained.

All the contents are in the words of optics to make it straightforward to understand how to realizes

Airy pulses optically. For Bessel beams, the origin of the non-diffracting property is explained.

Other optical realizations of Airy waves are introduced in the last section of this chapter.

1 Chapter 3 is the discussion around the three-dimensional (3D) measurement system. Firstly, the mathematics and the experimental conﬁguration of this 3D measurement technique is brieﬂy introduced. There are two different setups presented. One of them is used to generate the Airy-

Bessel wave packet and measure its intensity proﬁle in 3D. The other one is used to generate wave packets such as Airy-Airy-Airy and dual-Airy-Airy-rings. These two wave packets are measured in 3D and presented in the measurement results chapter. Finally the process of temporal phase retrieval is explained. This temporal phase retrieval technique is accurately a byproduct of the 3D measurement. Although some assumption must be made to access the temporal phase retrieval, this method can reveal the temporal phase signature with a good agreement with theoretical predictions.

The temporal phase results are also presented in the measurement results chapter.

Chapter 4 presents the 3D measurement results, as well as the temporal phase retrieval of the

Airy-Bessel wave packet and the Airy-Airy-Airy wave packet. As the main theme of this thesis, the propagation dynamics of Airy-Bessel wave packet is studied in details covering from the experimental parameters to the 3D intensity profile and the temporal phase profile. For an Airy-Airy-Airy wave packet, its 3D intensity profile and its temporal phase is shown. For a dual-Airy-Airy-rings wave packet, to the best of our knowledge, it is the first time that such wave packet is experimentally generated. The 3D autofocusing effect is observed and measured in 3D.

Chapter 5 states the conclusions, as the last chapter of this thesis.

1.2 Introduction

Spatio-temporal wave packets have demonstrated their potential in the field of nonlinear science, plasma generation [1, 2], laser machining [3] and other scientific researches [4]. Combining some specific pulse shape and beam profile, these wave packets can have unique properties such as self-acceleration [5], self-healing [6], lateral acceleration [7], non-diffracting [8–10] and abrupt

2 autofocusing [11]. With these properties, spatio-temporal wave packets can serve as a powerful tool in various applications. Itself has also become an interesting research topic.

To utilize and study those wave packets, it is necessary to learn how to measure them. Conven- tional measurement techniques include auto-correlation [12], cross-correlation and charge-coupled devices (CCD) camera. They are easy to implement. However, spatio-temporal coupled wave packets cannot be resolved by these methods. To overcome this, methods based on frequency-resolved optical gating (FROG) [13] and some other techniques with nonlinear processes [14] are developed.

But these methods have restrictions on the type of the wave packets, wave packets’ energy and their spectrum bandwidth. These restrictions make the measurement scenarios highly limited.

Later on, an interferometric measurement system based on noncollinear ﬁrst order cross correlation is suggested [15–17]. This system can measure 3D intensity proﬁle of spatio-temporal wave packets at various wavelengths. Recently its capability to measure the temporal phase is demonstrated [18]. The 3D measurements in this thesis are all achieved by using this measurement technique.

Conventional methods will be used as a supplement and for comparison purpose.

The wave packets to be studied in this thesis are Airy-Bessel, Airy-Airy-Airy and dual-Airy-

Airy-rings. An Airy-Bessel wave packet has been studied as a “linear light bullet” since it can propagate without dispersion or diffraction in the linear regime [19]. An Airy-Airy-Airy wave packet also belongs to the family of “linear light bullet”s. It has been studied both in the linear and the nonlinear regime [20]. A dual-Airy-Airy-rings wave packet is a wave packet which has been just recently proposed [21]. It can autofocus in an abruptly way which can be useful in some applications such as bio-imaging.

The next chapter discusses the theories for Airy waves and Bessel beams.

3 CHAPTER 2

THEORETICAL BACKGROUND

In this chapter, the theoretical background for optical spatio-temporal wave packets is covered. Airy pulses, Bessel beams and other types of optical Airy waves are discussed. The three- dimensional (3D) measurement system is introduced in the next chapter and the 3D measurement results are presented in the fourth chapter.

2.1 Optical Airy wave

An Airy wave is initially studied in the context of quantum mechanics. It has been proved that a wave function (ψ) in the form of an Airy function can propagate in free space without distortion and with constant acceleration [22]. It has also been proved that the Airy packet can preserve its integrity under the inﬂuence of a force F (t). Since the Schrodinger¨ equation used in quantum mechanics is entirely analogous to the paraxial approximation of the Helmholtz wave equation,

Airy waves can also studied in the context of optics. It has been proposed that a two dimensional

(2D) Airy beam can propagation free of diffraction and it can self-accelerate in free space. After the

ﬁrst realization of 2D Airy beam [23], an Airy beam has been studied in its unique properties such as self-acceleration [23] and self-healing [6], in the paraxial regime [5] and the non-paraxial regime as well [24]. Later on, Airy pulses have been suggested [7] and it is pointed out that an Airy pulse can keep its shape under the inﬂuence of dispersion. These days, optical realizations of Airy waves

4 has become a big research topic. In the following sections, the mathematics behind Airy pulses and other type of Airy waves are explained.

2.1.1 Airy pulse

Before explaining the mathematics of Airy pulses, it is necessary to introduce some physical parameters which have been used to characterize the optical pulse. Due to the intrinsic Fourier transform property, a broad optical spectrum is needed to have an ultrafast optical pulse. The optical spectrum is determined by two physical parameters, its amplitude and its phase. These two parameters are deﬁned through

Ee(ω) = |A(ω)| · exp(i · φ(ω)), (2.1) where A(ω) is the amplitude term and φ(ω) is the phase term. It is of note that the carrier frequency of an optical ﬁeld is usually at the PHz level (1 PHz = 1015 Hz). For example, the carrier frequency of light at 1 µm wavelength has an angular carrier frequency of 1.88 PHz. For simplicity, we assume

ω = ω1 − ω0 in Eqn. 2.1, where ω1 is the real optical frequency and ω0 is the center frequency. The phase term φ(ω) can be Taylor expanded around the center frequency [25]. Mathematically it can be written as

1 1 φ(ω) = β(ω) · z = β + β ω · z + β ω2 · z + β ω3 · z + ··· , (2.2) 0 1 2 2 6 3 where dmβ β = (m = 0, 1, 2,...). (2.3) m dωm

β(ω) in Eqn. 2.2 is the propagation constant of the optical ﬁeld, which has been widely used in 2π waveguide optics and ultrafast optics. In a dispersive medium β(ω) = · n(ω), where n(ω) is the λ

refractive index of the medium and λ is the wavelength of the optical ﬁeld. βm in Eqn. 2.3 is the

mth order Taylor expansion coefﬁcient of the propagation constant around the center frequency.

5 β1 is related to the group velocity, which describes how fast ultrafast pulse propagates in the medium. When a localized coordinate is used (which means the observer moves with the pulse), β1

does not play a signiﬁcant role analyzing the evolution of the ultrafast pulse (unless two pulses with

different center frequency travel together, but this is not the case in this thesis).

β2 is called the group velocity dispersion (GVD) coefﬁcient. When there is no phase variation

on φ(ω) (φ(ω) = constant), an ultrafast pulse with a ﬁnite spectrum is called a Fourier transform- limited (TL) pulse. As a TL pulse propagates through a dispersive medium, the temporal pulse width broadens due to the inﬂuence of GVD. The broadening occurs whether the pulse is in a normal dispersive medium, β2 > 0, and in an anomalous dispersive medium, β2 < 0; the distinction between

the two GVD cases is in the advancement or lagging of the pulse’s frequency spectrum (higher

frequency components “blue” lags lower frequency components “red” in normal GVD media and

“red” lags “blue” in anomalous GVD media).

β3 is the third order dispersion (TOD) coefﬁcient. When the spectrum of an optical pulse is

broad enough, TOD effect cannot be ignored. Higher order dispersion effects such as the fourth

order dispersion play little roles so that they are not taken into consideration in this thesis.

For a rigorous Airy wave, its spectrum consists an uniform amplitude and a cubic phase, which

can be written as

i 3 Ee(ω) = |1| · exp( · β30ω · z0), (2.4) 6

where β30 and z0 are constants. However the wave form as in Eqn. 2.4 is not square integrable, i.e.

it does not have any physical meaning. An “aperture” is needed to bound the total energy. In the

form of an Airy pulse, the aperture is the optical spectrum. Thus a truncated Airy pulse has a ﬁnite

spectrum which can be written as

i 3 Ee(ω) = |A(ω)| · exp( · β30ω · z0). (2.5) 6

6 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 ) (a.u.)

ω 0.4 0.4 0.4 A(

0.2 Intensity (a.u.) 0.2 Intensity (a.u.) 0.2

0 0 0 1.7 1.8 1.9 −0.5 0 0.5 −1.5 −1 −0.5 0 0.5 ω (PHz) Time (ps) Time (ps) (a) Angular frequency spectrum (b) Fourier transform-limited pulse (c) Airy pulse

Figure 2.1: Airy pulse simulation.

More detailed derivation can be found in this reference [7].

Fig. 2.1 gives simulation results to help to understand Airy pulses. Fig. 2.1(a) is the spectrum of the optical ﬁeld. It is centered at 1.83 PHz (corresponding to 1.03 µm) and has a Gaussian distribution. Fig. 2.1(b) is the TL form pulse of this spectrum. By adding TOD phase, Airy pulse can be formed as shown in Fig. 2.1(c). It is of note that there is a π phase shift between each lobe of the Airy pulse. This is a signature of Airy pulse. More generally speaking, this is the signature of Airy waves.

When an Airy pulse propagates through a dispersive medium with GVD coefﬁcient β2 for a distance z, an extra GVD is added. The phase of the spectrum φ(ω) becomes

1 3 1 2 Ee(ω) = |A(ω)| · exp(i · ( β30ω z0 + β2ω z + β1ωz)) 6 2 2 (2.6) 1 β2z 3 1 β2 z β2z = |A(ω)| · exp(i · ( β30z0(ω + ) + (β1 − )(ω + ) · z + C)), 6 β30z0 2 β30z0 β30z0 where C is some constant phase. This constant is not of our interest in this thesis. 2 1 β2 z Some conclusions can be made from Eqn. 2.6. Firstly, the linear phase term (β1 − )(ω + 2 β30z0 β z 2 ) · z will result in a shift in the time domain. This property is called self-acceleration. The β30z0 direction of self-acceleration depends on the sign of β30z0, which is the initial TOD.

7 1 1

0.5 0.5

Intensity (a.u.) 0 Intensity (a.u.) 0 0 5 2 5 −2 0 Time (ps) −4 0 L Time (ps) −2 0 L D D (a) Pulse evolution with initial positive TOD (b) Pulse evolution with initial negative TOD

Figure 2.2: Airy pulse propagation simulation under the inﬂuence of normal dispersion.

β z Secondly, the center frequency will shift by an amount of ∆ω = − 2 . When the new center β30z0 frequency is within the bandwidth of the optical spectrum, the Airy pulse can partially keep its shape. In other word, the inﬂuence of dispersion is reduced. Since ∆ω is inversely proportional to

the initial TOD, adding more initial TOD can enhance such dispersion-free property. Equivalently,

having a broader spectrum will beneﬁt it as well (an inﬁnite broad spectrum can have this dispersion-

free property forever).

Some other simulation results about the pulse evolution under GVD are shown in Fig. 2.2. The

z-axis of this graph is the normalized intensity. The x-axis is a localized time scale. And the y-axis

is plotting with respect to the dispersion length. The dispersion length, similar with the diffraction

length (Rayleigh range), is a parameter to characterize the dispersion effect. It is deﬁned as

2 t0 LD = , (2.7) β2

8 (a) Initial Gaussian beam (b) Phase mask in the k-space (c) 2D Airy beam

Figure 2.3: Airy beam simulation.

t2 where t0 is the width of the ﬁeld via E(t) = exp(− 2 ) and β2 is the GVD coefﬁcient of the t0

dispersive medium where pulse propagates. Here t0 is determined by the main lobe of the Airy

pulse.

From the graphs, the self-acceleration effect is evident. With different sign of initial TOD, an

Airy pulse will self-accelerate towards different directions. The Airy pulse can maintain its shape

within ∼3 times the dispersion length. After propagation for 3 dispersion lengths, the Airy pulse

starts to lose its shape. Finally the pulse evolution gets dominated by GVD effect.

2.1.2 Airy beam and other types of Airy wave

Similarly a phase mask can be designed and applied to convert the transverse spatial proﬁle of

an optical ﬁeld to approximate a 2D Airy beam [5, 26]. Fig. 2.3 illustrates the progression how

a Gaussian beam will evolve when a cubic phase mask is added to its reciprocal space (k-space).

Fig. 2.3(a) shows the intensity proﬁle of a Gaussian beam with a beam diameter of 1 mm. Fig. 2.3(b)

is the cubic phase mask added in the k-space of such Gaussian beam. The mathematical expression

9 1 1 1 2 0.8 0.8 0.8

0.6 0.6 0.6 1 0.4 0.4 0.4 Intensity (a.u.)

Intensity (a.u.) Intensity (a.u.) Intensity (a.u.) 0 0.2 0.2 0.2 5 4 0 0 0 0 2 −1 0 1 −5 0 5 −5 0 5 Time (ps) −5 0 3 2 Time (ps) Time (ps) Time (ps) GVD (×5×10 fs ) (a) Initial Gaussian (b) Dual Gaussian (c) Dual Airy (d) evolution of dual Airy under normal GVD

Figure 2.4: dual Airy pulse simulation.

of this phase mask can be written as

3 3 ϕ(kx, ky) ∝ (kx + ky), (2.8)

where kx and ky are the coordinate in the reciprocal space. Reciprocal means in the Fourier domain.

The Fourier transform of a Gaussian beam still has a Gaussian distribution. Fig. 2.3(c) shows the 2D

Airy beam when such phase mask is applied. The optical Fourier transform can be accomplished

by a single lens system. This property can be found in Fourier optics textbooks [27, 28].

Recalling Fig. 2.2, Airy pulses with different signs of initial TOD will self-accelerate towards

different directions under normal dispersion (β2 > 0). This phenomenon is evident in Eqn. 2.6 since 2 1 β2 z the sign of the change of the linear term ∆β1 = is dependent on the sign of β30. One may 2 β30z0 ask: what would happen when two Airy pulses co-propagate in a dispersive medium? To answer this question it is necessary for us to know how to generate such dual Airy pulse.

An illustration showing the progression from Gaussian to dual Gaussian to dual Airy pulses are shown in Fig. 2.4. A Gaussian pulse is presented in Fig. 2.4(a). A linear phase in the frequency domain of such pulse delays the pulse in the time domain by a certain shift. Similarly, a “triangle” shape phase in the frequency shifts two pulses in different directions, as shown in Fig.2.4(b). By adding cubic phase with different signs in each half of this spectrum, dual Airy pulses can be 10 generated, as presented in Fig. 2.4(c). Now it is possible to numerically study the evolution of such dual Airy pulses under the inﬂuence of normal GVD. The result is shown in Fig. 2.4(d). Two Airy pulses will counter self-accelerate towards the center. At some point, a higher peak intensity can be formed. The peak intensity can be as high as ∼2 times the initial peak intensity. This feature might be useful in some applications.

A radially symmetric phase third-order phase proﬁle in the optical beam’s Fourier domain can also be examined. The phase mask can be mathematically written as

ϕ(r) = a · r + b · (r − ∆)3, (2.9)

where r is the radius in a polar coordinate, ∆ is a constant displancement, a and b are phase co-

efﬁcients. When this phase mask is applied and the beam propagates, the linear phase term in this

equation transform the initial beam into a ring and the cubic phase term forms the Airy distribution

in the radial direction. With an Airy proﬁle in the radial direction forming Airy ring beams, the

beam will autofocus from all angles towards the center [11, 21]. It has been proposed that a wave

packet which combines the dual Airy pulses in time and the Airy ring beams in space can autofocus

abruptly in a 3D fashion. The intensity contrast at the focus can be as high as ∼400 times [21].

2.2 Bessel beam

A Bessel beam is a solution of the Helmholtz wave equation [29]. A Bessel beam is superposed

by plane waves whose propagation constants lie on a cone. Thus sometimes a Bessel beam is called

a conic beam. The Fourier transform of a rigorous Bessel beam is an inﬁnitely thin ring. This

property is apparent when the Hankel transform (Fourier Bessel transform) has been recalled [30].

A rigorous Bessel beam requires an inﬁnite energy, which is not physical. For a truncated Bessel

beam, its Fourier transform is still a ring, but with a ﬁnite width.

11 (a.u.) y Y (a.u.) K Intensity (a.u.)

X (a.u.) K (a.u.) R (a.u.) x (a) Radially Bessel distribution (b) Bessel beam in x-y space (c) Bessel beam in k-space

Figure 2.5: Bessel beam simulation.

Some simulation results of a Bessel beam are presented in Fig. 2.5 to help to understand the truncated Bessel beam. Fig. 2.5(a) is the radial intensity distribution of a Bessel beam. To have a physical meaning, the beam has to be truncated to bound the total energy. The truncated Bessel beam proﬁle is shown in Fig. 2.5(b). This multiple ring structure is a signature of a Bessel beam.

The Fourier transform of a truncated Bessel beam is shown in Fig. 2.5(c), which is a ring with a

ﬁnite width in the k-space.

To explain the non-diffracting property of the Bessel beam, it is necessary to recall the spatial impulse response. The spatial impulse response is used in the paraxial regime, which is a linear system in the context of Fourier optics. It is written as [31]

" 2 2 # Ψe ∆ i · (kx + ky) = H(kx, ky; z) = exp · z , (2.10) Ψe0 2k0 where Ψe and Ψe0 are the Fourier transform of the optical ﬁeld at the point of interest and the initial

point respectively. H(kx, ky; z) is called the spatial transfer function. z is the propagation distance,

k0 is the propagation constant and kx and ky are the reciprocal coordinates.

Due to the fact that the Fourier transform of a Bessel beam has a ring structure. There is

no reciprocal phase variation accumulated in the process of propagation. This means the Bessel 12 beam

Bessel beam axicon

Figure 2.6: Axicon to generate Bessel beam.

beam can propagate without any diffraction effect. If the Bessel beam is truncated, this diffraction- free property can be still kept for a ﬁnite distance since the reciprocal phase variation is little.

Experimentally, such Bessel beam can be easily generated with the help of an optical axicon [10,32].

An axicon bends the propagation constant of the incident light radially so that the output becomes the superposition of plane waves in a conic fashion. This process is illustrated in Fig. 2.6.

13 CHAPTER 3

THREE-DIMENSIONAL MEASUREMENT SYSTEM

In previous chapters, theories of Airy pulse, Bessel beam and other optical form of Airy waves including Airy-Airy beam, Airy rings and dual Airy pulses have been briefly discussed. In this chapter, the mechanism and the theory of the three-dimensional (3D) measurement system are firstly explained. What follows is the experimental setup for generating and measuring spatio-temporal wave packets. At the end of this chapter, the process of temporal phase retrieval is discussed. Using this system, the 3D intensity and phase profiles of complicated spatio-temporal wave packets can be diagnosed. The capability of this system is demonstrated in the next chapter by studying the propagation dynamics of certain wave packets.

3.1 Mechanism and mathematical background

Spatio-temporal wave packets have been utilized in various applications and scientiﬁc researches [1–4]. To use these wave packets, knowing how to measure them is necessary. Researchers use cross-correlation (XC), charge-coupled devices (CCD) camera to measure those wave packets.

Methods based on frequency-resolved optical gating (FROG) [13] and some other techniques with nonlinear processes [14] have also been proposed and developed. XC and CCD camera measure the wave packets separately in the time domain and in the spatial domain, which make an assumption that the wave packets are spatio-temporal uncoupled. That is to say, the mathematical expression

14 θθ 15 between their propagation wave vector. The relative time delay between θ proposed an interferometric measurement system based on noncollinear first et al. The conceptual sketch of the measurement system is illustrated in Fig. 3.1. Pulsed laser output Finally Li is split into two, the reference arm and the object arm. The reference pulse is dechirped into its order cross correlation to measure the 3D17]. intensity profile The of layout of spatio-temporal the wave system packets is [15– simplewavelengths. and There it is can no be nonlinear process adopted involved to in measure the wave measurementlow packets so energy that at can wave be various packets well with diagnosed. Later on Li managed topulse measure using the this temporal technique phase [18]. of a chirped of the wave packet can be written asa a result, product spatio-temporal of coupled a wave function packets of cannot time be andFROG correctly [13] a diagnosed. uses function some Methods of specialized based space. diffractive on optical As component, makingized. the Other setup techniques highly involving special- nonlinear frequency conversion require highdetectable intensity signals. to generate Meanwhile the bandwidth of the optical spectrum is limited. the object and reference can be controlled by a micrometer stage. Figure 3.1: Conceptual sketch of theand measurement the system. reference Both the wave object packetwith wave (upper packet a (lower arm) small arm) are tilted sent angle to the CCD camera (the grey box in the right) transform-limited (TL) form in the time domain and remains its initial spatial profile, fundamental

Gaussian. The wave packet in the object arm is modulated to the desired shape. An optical delay line is introduced in the reference arm so that the relative time delay between two arms can be controlled. Then both wave packets are sent into a CCD camera with a small tilted angle θ between their propagation wave vectors. They are overlapped both in the time domain and the spatial domain.

As a result, interference fringe pattern can be detected. The angle θ is chosen so that the spatial period of the interference pattern is small but measurable by the camera. The repetition rate of the pulsed laser source is ∼40 MHz, and the exposure time of the camera is ∼100 ms. Within one shot of the camera, ∼4 million wave packets are measured. Thanks to the stability of the laser source and the robustness of the measurement system, the interference pattern can be well kept.

A TL pulse has the shortest duration when the spectrum width is finite. Compared to the object pulse, the reference pulse is short enough to be regarded as a δ-function (the Dirac delta function, which is an infinitely high, infinitely thin spike at the origin). The 3D measurement system is like a

“tomography” system. By calculating the modulating depth of the interference pattern, the intensity of the object wave packet at certain time delay can be reconstructed. Similarly the 3D intensity proﬁle can be measured when the delay line is changing. Equivalently it is like using the reference wave packet to “scan” the object wave packet.

When the reference wave packet and the object wave packet are sent to the camera with certain time delay τ, the two-dimensional ﬂuence proﬁle I(~r, τ) as detected by the camera can be written in a time integral form [17], Z I(~r, τ) = dt|Ao(~r, t) exp[iφo(t) + i~ko · ~r − iω0t]

2 + Ap(~r, t − τ) exp[iφp(t − τ) + i~kp · ~r − iω0(t − τ)]| (3.1) Z ∗ = Io(~r) + Ip(~r) + 2 cos{ω0[τ + δ(~r)]} × dtAo(~r, t)Ap(~r, t − δ(~r) − τ)

× cos{φo(t) − φp[t − δ(~r) − τ]},

16 where A(~r, t) and φ(~r, t) are the amplitude and the phase of the wave packet; ~k is the propagation wave vector; the subscript o and p is referred as the object and probe (alias for reference); ω0 is the center angular frequency of the optical spectrum; Io(~r) and Ip(~r) are the ﬂuences of the object and probe respectively; The last term corresponds to the interference pattern between the them; function

δ(~r) denotes the transverse phase difference between the object and probe, which takes the tilted angle θ into consideration.

Equation 3.1 can be further simplified by the following approximations. Firstly the temporal profile of the probe is approximated by a δ-function (the Dirac delta function, not the δ(~r) in E- qn. 3.1). Secondly, it is assumed that there is no phase variation on the spatial profile of the probe, c which is valid when the probe beam size is much larger than its center wavelength λ0 = . With ω0 these approximations, same formula as in [17] can be reached, which is written as

q q I(~r, τ) ≈ Io(~r) + Ip(~r) + 2 cos{ω[τ + δ(~r)]} ∆tpIo(~r, τ) Ip(~r), (3.2)

where ∆tp is the duration of the probe pulse and Io(~r, τ) is the object intensity at certain time delay

τ. The object pulse intensity can be thus retrieved via

2 Io(~r, τ) ∝ C (~r, τ)/Ip(~r), (3.3)

where C(~r, τ) is the correlation between the object and probe, or equivalent the modulation depth

of the interference pattern. One main advantage of using a noncollinear conﬁguration is that by

translating the object on a spatial carrier frequency, the information of the object can be recorded.

3.2 Experimental conﬁguration

The experimental setup to generate Airy-Bessel wave packets and implement 3D measurement

in the previous section is shown in Fig. 3.2. Pulses are generated from the nonlinear-polarization

17 Spatial light modulator (SLM) Confocal Glass telescope Camera

Grating Pairs Axicon Laser Delay line Beam Splitter

Autocorrelation / Cross correlation (AC) / (XC)

Figure 3.2: Experimental setup for Airy-Bessel wave packet.

evolution (NPE) ejection port of a fiber laser oscillator based on a dispersion-managed (DM) soliton [33]. The beam profile is fundamental Gaussian with 1 mm in diameter. Firstly, the light is split into two arms, the object arm (upper one) and the reference arm (lower one). Due to the scheme of the fiber laser oscillator, the pulse is initially positively chirped. A grating pair is introduced in the reference arm to dechirp the pulse into TL. This process is checked by auto-correlation (AC).

On the other hand, the object pulse is modulated by a SLM-based pulse shaper. Its temporal profile is checked by XC. Since a larger beam will be beneficial to generate higher quality Bessel beam, a confocal telescope configuration is used to expand the beam by a factor of 3. What follows is an axicon, thus Airy-Bessel wave packets can be generated from this configuration. Finally, to implement the 3D measurement, an optical delay line is employed in the reference arm. Both the object and the reference are sent to a CCD camera with a small tilted angle to accomplish the measurement technique discussed in the previous section. The space of dashed line box in Fig. 3.2 is reserved for

18 Spatial light modulator (SLM) Confocal telescope Fourier lens Camera

Grating Pairs SLM Laser Delay line Beam Splitter

Autocorrelation / Cross correlation (AC) / (XC)

Figure 3.3: Experimental setup for general spatio-temporal wave packet.

the dispersive glass. By adding extra dispersion and diffraction, the propagation dynamics of the

Airy-Bessel wave packet can be thus studied, as to be discussed in the next chapter.

Another experimental scheme for generating more general spatio-temporal wave packet is illustrated in Fig. 3.3. The axicon used in the previous setup has limited the spatial proﬁle of the object packet to be a Bessel distribution. Having another SLM after the beam expander would enable more general beam shaping. A lens of ∼40 cm focal length will achieve the optical Fourier transform. The distance between the SLM (the red box after the beam expander), the Fourier lens and the camera is chosen to be the focal length of the lens. The reason for this conﬁguration and the Fourier- transforming property of lenses can be found in Fourier optics textbooks [27, 28]. This system can be used to generate and measure the Airy-Airy-Airy wave packet and the dual-Airy-Airy-rings wave packets, which will be covered in the next chapter.

19 3.3 Temporal phase retrieval

Phase and amplitude are intrinsic properties of an optical field. The intensity information (amplitude square) of an ultrafast pulse has become accessible after the invention of auto-correlator [12], although the shape of the pulse needs to be assumed. Efforts have been put in studying the phase information of an ultrafast pulse. Finally techniques such as FROG [34,35], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [36] are developed. These two techniques can obtain one-dimensional phase and amplitude information of an ultrafast optical field. To extend the measurement technique to a higher dimension, Li presents a temporal phase retrieval method based on the 3D measurement system discussed in Sec. 3.1 and demonstrates the capacity by measuring the temporal phase of a chirped pulse [37].

To implement the temporal phase retrieval, some experimental condition must be met. Firstly, the reference pulse is close to TL, so that there is no phase left in the time domain. Secondly, there should no phase variation in the transverse plane of the reference wave packet, at least within a small region. This would be valid given the reference wave packet has a Gaussian-Gaussian (Gaussian in both time domain and space domain) distribution and it has only propagated ∼1.2 Rayleigh ranges from its beam waist plane to the camera plane. Near the center of the Gaussian beam, the phase variation can be neglected. Moreover, within a small region of the transverse plane of the object, there is no phase variation either. The size of this region can be as small as ∼0.1 mm.

The conceptual sketch for the mechanism of the temporal phase retrieval is shown in Fig. 3.4.

Given a small tilted angle between the probe (reference) and the object, which is ∼0.04 rad in experiment, interference pattern can be detected by the CCD camera (the grey box). Since there is no phase variation in a small region of the probe and the object, the phase embedded on the spatial carrier frequency would be purely the temporal phase. The “camera” size is chosen to let

20 Probe

Object θ ⋅ L t = c

Figure 3.4: Conceptual sketch of temporal phase retrieval.

the corresponding step size ∆t in the time domain match with the scanning step size. For example, when the scanning step size is 13.3 fs and the tilted angle is 0.04 rad, the corresponding “camera” ∆t · c size L = = 0.1 mm, which is small enough to meet all the requirements in the previous θ context.

After all the effort explaining the theory of these spatio-temporal wave packets and the measurement system, the measurement results are presented in the next chapter.

21 CHAPTER 4

MEASUREMENT RESULTS

In previous chapters, the theoretical background of the optical realization of Airy waves including their unique properties, the diffraction-free feature of Bessel beam and the three-dimensional

(3D) measurement system has been covered. In this chapter, the measurement results are presented including the propagation dynamics of an Airy-Bessel wave packet, the 3D intensity proﬁles of

Airy-Airy-Airy and dual-Airy-Airy-rings.

4.1 Propagation dynamics of Airy-Bessel wave packet

An Airy pulse has its unique dispersion-free and self-acceleration property. It has been applied in many nonlinear science researches such as supercontinuum generation [38] and soliton generation [39, 40]. Meanwhile, as an old research topic [29], a Bessel beam is probably the most famous non-diffraction beam due to its conical feature. A Bessel beam can be easily generated with the help of an axicon [32], a conical optical component. It can be put into various applications, for example, medical imaging [41] and generation of plasma channel [1].

The Airy-Bessel wave packet combines the Airy pulse in the time domain and the Bessel beam in the space domain. It has been studied as a “linear light bullet” due to its propagation invariant property under linear propagation (i.e. dispersion and paraxial diffraction) [11, 19]. Later on the 3D intensity proﬁle and the temporal phase proﬁle of the Airy-Bessel wave packet has been

22 1 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 0.5 0.4 0.4 0.4 0.2 0.2 Intensity (a.u.) 0.2 XC signal(a.u.) 0 Measured AC(a.u.) Simulated AC (a.u.) 0 0 0 −0.2 −0.2 950 1000 1050 1100 −400−200 0 200 400 −500 0 500 −500 0 500 1000 1500 Wavelength (nm) Delay (fs) Delay(fs) Delay(fs) (a) Laser spectrum (b) Simulated AC (c) Measured AC (d) Measured XC

Figure 4.1: Experiment parameters for Airy-Bessel measurements. The laser spectrum is shown in Fig. 4.1(a). The simulated and measured auto-correlation signal of the reference pulse is shown in Fig. 4.1(b) and Fig. 4.1(c). Fig. 4.1(d) shows the XC signal between the object pulse and the reference pulse.

measured [18]. In this section, the propagation dynamics of the Airy-Bessel wave packet is studied in a 3D fashion. The Airy-Bessel wave packet is generated from a chirped Gaussian-Gaussian wave packet. The Airy pulse is generated by adding third-order-dispersion (TOD) in the frequency

(spectrum) domain using a spatial light modulator (SLM) based pulse shaper. The Bessel beam is generated by using an axicon. To see the wave packet evolution under the inﬂuence of dispersion and diffraction, a 4-inch-long (10.16 cm) Schott SF11 glass rod is inserted. Both sides of the glass rod has been polished. The propagation invariant property is veriﬁed in 3D.

4.1.1 Experiment parameters

The experiment’s setup is the same as the conﬁguration in Fig. 3.2. The spectrum of the pulsed laser source is shown in Fig. 4.1(a). The laser is pumped by a 978 nm semiconductor diode laser, and the center of the output spectrum is located at ∼1030 nm, which is the center of Ytterbium’s

(Yb) gain bandwidth. The laser spectrum has ∼80 nm width at its base, as shown in Fig. 4.1(a).The pumping current is 600 mA, the average power is 69 mW and the repetition rate is 40 MHz. The energy of each pulse would be ∼1.7 nJ given there is no multi-pulsing inside the ﬁber laser oscillator.

23 To have a short pulse, a broad spectrum is needed. The spectrum has ∼80 nm width at its

base. The theoretical auto-correlation (AC) of the transform-limited (TL) pulse with such spec-

trum is shown in Fig. 4.1(b). As shown in Fig. 3.2, the reference pulse is ﬁrstly sent to a dechirp

stage (grating pair) and then measured by a homemade auto-correlator. The AC signal is shown in

Fig. 4.1(c). The full-width-half-maximum (FWHM) of the simulated AC is 69 fs and the FWHM

of the measured AC is 83 fs, which indicates the duration of the reference pulse is 54 fs. The dis-

crepancy between the theoretical value and the experimental value might be caused by the residual /

introduction of higher order dispersion. But it is clear that most energy is conﬁned in the main lobe

of the reference pulse. Compared with the duration of object pulse (∼2 ps), the reference pulse is

short enough to be regarded as a δ-function.

The object pulse is modulated by a SLM based pulse shaper. A negative group velocity dispersion (GVD) is ﬁrstly added to dechirp the pulse. TOD is added as well to generate the Airy pulse.

The object pulse and the reference pulse are sent to the cross-correlator. The cross-correslation (XC) signal is shown in Fig. 4.1(d). The FWHM of the main lobe of the Airy pulse’s XC signal is ∼170

fs, which corresponding to an intensity duration of ∼140 fs. A factor of 1.2244 is used because there is a conversion ratio between the XC signal and the real intensity envelope. The separation between the peak of the main lobe and the peak of the second lobe is 174 fs. From Fig. 4.1(d), ∼9

side lobes which can be clearly recognized from XC. It is evident that the object pulse has an Airy

wave form in the time domain.

4.1.2 3D intensity before propagation

Using the 3D measurement system based on the conﬁguration in the previous chapter, the Airy-

Bessel wave packet is measured in a 3D fashion. The results are shown in Fig. 4.2. The step size

of the delay line is 2 µm, corresponding to 4 µm considering the reference go back and forth at the

24 1

0.8

0.6

0.4

Intensity (a.u.) 0.2

0 0.5 1 1.5 2 T (ps) (a) Intensity isosurface (b) Sagittal intensity (c) Temporal intensity

Figure 4.2: Results of the Airy-Bessel wave packet before propagation.

delay line point. So the temporal step size is 13.3 fs. The tilted angle between the object and the reference is ∼0.04 rad, which makes the period of interference pattern small but resolvable by the

CCD camera.

It is worth mentioning that the 3D result is plotted in three different forms. Fig. 4.2(a) plots the data in a isosurface form. The isovalue is chosen to be 0.01 the maximum intensity. Fig. 4.2(b) plots the sagittal intensity. The structure of the Airy-Bessel wave packet including the multiple rings of the Bessel beam and the oscillating lobes of the Airy pulse is evident. ∼8 side lobes can

be observed in this form of displaying data. Fig. 4.2(c) sums the transverse intensity to form a 1D

temporal intensity curve. The temporal intensity is plotted in an arbitrary unit scale (the maximum

equals to one). The curve has a clear Airy shape. From Fig. 4.2(c), the main lobe has an intensity

FWHM of 144 fs, and the separation between the peak of the main lobe and the peak of the second

lobe is 187 fs. Recall it that these two values are 140 fs and 174 fs when the same Airy pulse

is measured by XC. There is a good agreement between the 3D measurement technique and the

conventional method (XC). The main lobe conﬁnes the energy of the Airy-Bessel wave packet in

a small localized volume (88 µm in diameter, 144 fs in duration), and it is about 7% of the total

energy.

25 1

0.8

0.6

0.4

Intensity (a.u.) 0.2

0 0.5 1 1.5 2 T (ps) (a) Intensity isosurface (b) Sagittal intensity (c) Temporal intensity

Figure 4.3: Results of the Airy-Bessel wave packet after propagation.

4.1.3 3D intensity after propagation

To observe the evolution of the Airy-Bessel wave packet under the inﬂuence of dispersion and diffraction, a 4-inch-long SF11 glass rod is inserted into the optical path of the object wave packet.

2 The GVD coefﬁcient β2 of SF11 is 1,251 fs /cm. Considering the main lobe pulse duration is 140 fs, a single pass through the dispersive glass rod would correspond to ∼2 dispersion lengths. The

dispersion length, similar with the diffraction length (Rayleigh range), is a parameter to characterize

the dispersion effect. It uses the same deﬁnition as in Sec. 2.1.1.

To access the 3D measurement system, the delay line at the reference arm has been re-adjusted

to ensure the overlap between the object and the reference in the time domain. Using the same

measurement parameters in previous case, another 3D measurement is done and the results are

shown in Fig. 4.3. The results are plotted in the same fashion as in Fig. 4.2 (isovalue remains 0.01

maximum intensity). The FWHM duration of the main lobe was 134 fs. In the time domain, the

Airy pulse can keep its shape against dispersion. It is also apparent that the wave packet’s intensity

proﬁle has not signiﬁcantly changed. In previous measurement, the distance from the axicon to

the camera is 40 cm, which is ∼17 diffraction lengths (Rayleigh range). The extra optical path

26 10

8 15

(rad) (rad) 10 π π

(t)/ 4 (t)/ φ φ 5 2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 Time (ps) Time (ps) (a) Temporal phase before propagation (b) Temporal phase after propagation

Figure 4.4: Temporal phase results for Airy-Bessel wave packet.

provided by the glass rod was 3.23 diffraction lengths, so ∼20 diffraction lengths in total. After such dispersion and diffraction effect added, the main lobe (86 µm in diameter, 134 fs in duration) still consists 7.57% of the total energy. It is obvious that he peak power of the Airy-Bessel wave packet is conserved by comparing two measurement results of the Airy-Bessel wave packet before and after propagation.

4.1.4 Temporal phase retrieval

The temporal phase of the Airy-Bessel wave packet is also retrievable from the data of the 3D measurement. Assumption that the reference beam and the object beam has no phase variation within a small regime is valid for the Airy-Bessel case. The temporal phase before and after propagation through the glass rod are plotted in Fig. 4.4 respectively. φ(t) is displayed in the scale of π. From

Fig. 4.4(a), the Airy pulse signature is evident from the π phase shift between each lobes. After the

Airy-Bessel wave packet propagates through the dispersive glass (4-inch-long SF11), the π phase

shift gets blurred as shown in Fig. 4.4(b), which is a result from extra dispersion effect added. If

27 1 1 1 1

0.8 0.8 0.8 0.5 0.6 0.6 0.6

0.4 0.4 0.4 0 0.2 Measured AC(a.u.) 0.2 Intensity (a.u.) 0.2 XC signal(a.u.) Simulated AC (a.u.) 0 0 0 −0.2 −0.2 −0.5 950 1000 1050 1100 −500 0 500 −500 0 500 0 1000 2000 Wavelength (nm) Delay (fs) Delay(fs) Delay(fs) (a) Laser spectrum (b) Simulated AC (c) Measured AC (d) Measured XC

Figure 4.5: Experimental parameters for Airy-Airy-Airy measurements.

more dispersion effect is added, ﬁnally the pulse evolution will get dominated by GVD and the temporal phase will have a parabolic shape.

4.2 Measurement of Airy-Airy-Airy wave packet

Due to their self-acceleration, self-healing and lateral acceleration properties, Airy beams and

Airy pulses have been studied in many researches [2, 5, 7, 23]. An Airy-Airy-Airy wave packet

(AAA) has an Airy wave form in all three dimensions (Cartesian coordinate system), x,y and t. The spatial proﬁle of AAA is a 2D Airy beam and the temporal proﬁle is an Airy pulse. Since the Airy wave is the only 1D wave that can propagation without distortion, AAA can propagation without dispersion or diffraction as well. It means an AAA belongs to the family of “linear light bullet”s.

Daryoush et al. has studied the AAA both in linear and nonlinear regime [20]. In the following portion of this section, the AAA will be measured in 3D.

The experimental setup is same as the sketch in Fig. 3.3. The object beam is expanded by a factor of 3 using a telescope conﬁguration. Then it goes through a SLM based beam shaper. The phase added on the object beam is same as what is suggested in [5]. The laser spectrum, TL AC signal, experimental AC signal and XC signal are shown in Fig. 4.5 respectively. The laser spectrum, centered at 1030 nm, has ∼65 nm width at its base. The simulated TL AC has a FWHM of 89 fs.

28 1

0.8

0.6

0.4

Intensity (a.u.) 0.2

0 0.5 1 1.5 2 T (ps) (a) Intensity isosurface (b) Sagittal intensity (c) Temporal intensity

Figure 4.6: Results of the Airy-Airy-Airy wave packet.

Measured AC has a FWHM of 103 fs. The higher order dispersion left on the reference pulse is acceptable since the reference pulse is considerably shorter than the object pulse. Fig. 4.5(d) shows the XC signal between the object pulse and the reference pulse, revealing a clear Airy pulse shape.

The object pulse duration is ∼2 ps so that the δ-function assumption is valid.

The 3D measurement uses almost the same experimental parameters in previous Airy-Bessel section, except the tilted angle between the object wave packet and the reference wave packet is

0.056 rad. The measurement results are shown in Fig. 4.6. From Fig. 4.6(a), the AAA structure has been nicely retrieved by this 3D measurement technique with Airy function distribution in all x, y and t directions. The sagittal plotting (Fig. 4.6(b)) shows clear 2D Airy proﬁle in y-t plane. And the

1D temporal intensity (Fig. 4.6(c)) agrees well with XC signals.

Meanwhile, based on the data collected, the temporal phase of an AAA can be retrieved. The result is shown in Fig. 4.7. The temporal phase φ(t) is plotted in the scale of π. The Airy pulse has a π phase shift signature, which is evident from the measurement result.

29 14 12 10 8 (rad) π 6 (t)/ φ 4 2 0 0 0.5 1 1.5 Time (ps)

Figure 4.7: Temporal phase retrieval of Airy-Airy-Airy wave packet.

4.3 Measurement of dual-Airy-Airy-rings wave packet

An Airy-Airy-Airy wave packet has an Airy distribution in all three dimensions of a Cartesian coordinate system. If the coordinate system is converted into a cylindrical coordinate system, Airy rings can be generated if there is an Airy distribution in the radial direction. Nikolaos et al. pointed it out that such radial wave can have abruptly autofocusing property [11]. Later on their group managed to generate such radially symmetric Airy beam [21]. It has been proposed that superposed with two facing Airy pulses in the time domain, such wave packet can have 3D autofocusing. In the following portion of this section, a dual-Airy-Airy-rings wave packet (dAAr) has been experimentally generated and measured in 3D. Its 3D autofocusing effect has also been observed.

The experimental parameters can be found in Fig. 4.8. The laser spectrum has ∼80 nm at its

base. The simulated AC FWHM is 62 fs. The experimental AC FWHM is 83 fs, corresponding to a

pulse duration of 54 fs.

Using the same phase pattern as suggested in Sec. 2.1.2, dual Airy pulses can be generated.

Fig. 4.8(d) shows the experimental XC signal of two Airy pulses. Under the inﬂuence of normal

30 1 1 1 1

0.8 0.8 0.8 0.5 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0 Intensity (a.u.) 0.2 XC signal(a.u.) Measured AC(a.u.) Simulated AC (a.u.) 0 0 0 −0.2 −0.2 −0.5 980 1000 1020 1040 1060 1080 −500 0 500 −500 0 500 0 2000 4000 Wavelength (nm) Delay (fs) Delay(fs) Delay(fs) (a) Laser spectrum (b) Simulated AC (c) Measured AC (d) Measured XC

Figure 4.8: Experimental parameters for dual-Airy-Airy-rings measurements.

dispersion, these two Airy pulses will accelerate to the center. In other words, they will collapse

(counter propagate and form a higher peak intensity) in the time domain under normal dispersion.

This effect can be also checked by applying normal GVD at the pulse shaper. The separation between two Airy waves’ peak can be adjusted by varying the amount of “triangle” phase. The delay in the time domain depends on the amount of linear phase in the spectrum (frequency) domain.

To observe the real autofocusing effect, the glass rod (4-inch-long SF11) has been put into the optical path of the dual Airy pulses. The linear phase is adjusted so that in the XC signal a highest peak is formed. When the glass rod is removed, the temporal proﬁle becomes dual Airy pulses with a modiﬁed separation between two peaks.

Using the conﬁguration shown in Fig. 3.3, Airy rings can be generated at the Fourier plane

(back focal plane of the Fourier lens). The 3D measurement results of dAAr at the Fourier plane are shown in Fig. 4.9. Fig. 4.9(c) gives the 1D temporal intensity. Compared with the previous XC signals, the separation between two peaks is smaller.

Then the glass rod is re-inserted into the object arm. The CCD camera is moved 20 cm away from the Fourier plane. At that position, tightly focused beam can be observed. Another 3D measurement is done and the results are shown in Fig. 4.10. It is evident that the dAAr can autofocus in

31 1

0.8

0.6

0.4

Intensity (a.u.) 0.2

0 1 2 3 T (ps) (a) Intensity isosurface (b) Sagittal intensity (c) Temporal intensity

Figure 4.9: Results of the dual-Airy-Airy-rings wave packet at the Fourier plane.

0.8

0.6

0.4

Intensity (a.u.) 0.2

0 0.5 1 1.5 2 T (ps) (a) Intensity isosurface (b) Sagittal intensity (c) Temporal intensity

Figure 4.10: Results of the dual-Airy-Airy-rings wave packet at the autofocus plane.

32 a 3D fashion. The focus contrast (the ratio between the peak intensity at the autofocus and the peak intensity at the Fourier plane) can be as high as ∼100 times. The dAAr autofocuses in an abrupt way.

Initially, there is no light in the on-axis regime and later on the on-axis intensity increases abruptly.

This feature can be useful in applications such as bio-imaging and nonlinear spectroscopy.

33 CHAPTER 5

CONCLUSIONS

Although also in an abrupt fashion, the theories about the Airy waves including Airy pulses, Airy beams, dual-Airy pulses and Airy rings, Bessel beams, the three-dimensional (3D) measurement system and the temporal phase retrieval are ﬁrstly revised in this thesis. Numerical simulations are provided to help understand these concepts in the context of optics. Two different experimental setups are presented. They can generate and measure the Airy-Bessel wave packet and general spatio-temporal wave packets. Both conﬁgurations are used in the experimental part of this thesis.

In the measurement results chapter, various spatio-temporal wave packets have been generated and measured in a 3D fashion. All the results have been compared with the results from conventional methods. There is a good agreement between them.

Firstly, the propagation dynamics of an Airy-Bessel wave packet are studied, as well as its temporal phase evolution. From the measurement results, the peak intensity of the wave packet is conserved after the propagation. Secondly, an Airy-Airy-Airy wave packet is generated. Its 3D intensity proﬁle and temporal phase are retrieved. Finally, a dual-Airy-Airy-rings wave packet is

ﬁrstly generated in the laboratory. Its 3D autofocusing effect has been observed.

We believe that having the 3D measurement capability is the ﬁrst step to study those spatio- temporal wave packets. It can pave the way for studying more complicated wave packets and utilizing these wave packets in the future.

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