APPLICATION OF SPACE-TIME STRUCTURED LIGHT TO CONTROLLED HIGH-INTENSITY LASER MATTER INTERACTIONS IN POINT AND
LINE TARGET GEOMETRIES
by
c Copyright by Amanda K. Meier, 2015 All Rights Reserved
A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctorate of Philosophy (Applied Physics).
Golden, Colorado Date Signed: Amanda K. Meier Signed: Dr. Charles G. Durfee Thesis Advisor Golden, Colorado Date Signed: Dr. Je↵ Squier Professor and Head Department of Physics
ABSTRACT
With ultrashort laser pulses, nonlinear e↵ects can be observed with low energy in each pulse. The broad bandwidth that makes it possible to produce a short pulse also introduces new degrees of freedom for manipulating the beam. The di↵erent frequency components that make up the bandwidth can be thought of as individual Gaussian beamlets that travel at di↵erent directions through optical elements due to dispersion or spatial chirp. These distortions are usually minimized in laser alignments yet manipulation of the of the Gaussian beamlets can be useful in axial localization and pulse front tilt (PFT), which is called simultaneous spatial and temporal focusing (SSTF) and is useful in many applications.
In order to use SSTF, we need to characterize the pulse in both the spatial and spectral domains. We have developed a novel Sagnac shearing interferometer which combines spatial and spectral interference. The combination of divergence and spatial shear results in a local angle between the beams which can be extracted from the interference pattern using spatially resolved spectral interferometry by Fourier analysis. A spatial inversion allows our design to be extended to characterize a coupled spatio-temporal distortion, spatial chirp.
We have developed techniques to control relative PFT for focused beams. We use a single pass grating compressor as a passively stable pump-probe experiment stemming from di↵ractive optics, where the +/-1 di↵racted orders from a transmission grating pair are focused by an o↵-axis parabola which crosses the pump beams to form an index grating at the focus that is probed by the zero order. The experiment can be aligned for overlap of the pulse front tilt across the entire focal spot. This PF overlap can be applied in nonlinear mixing processes, such as harmonic generation or four wave mixing, to characterize semiconductor samples or ionization dynamics.
We have also extended SSTF to a cylindrical geometry in Bessel-Gauss and vortex beams. Our novel setup for producing radial SSTF double passes a Gaussian beam through an axicon to produce a collimated ring beam that is then focused to a Bessel zone. Including a vortex mask in the beam gives a phase singularity on axis which creates a higher order Bessel-Gauss, corresponding to the vortex mode order. Circular gratings were also designed to extend SSTF to a cylindrical geometry as well as utilize the pulse front matching technique mentioned above. The Bessel zone with vortex singularity allows for high intensity walls that ionize causing the index of refraction to be higher in the core than the cladding, therefore allowing beam guiding. We modeled the waveguide geometry to optimize mode coupling. Radial SSTF could be used to guide high intensity beams with application to guide high harmonics.
TABLE OF CONTENTS
ABSTRACT . . . iii
LIST OF FIGURES . . . vi
LIST OF TABLES . . . xii
LIST OF ABBREVIATIONS . . . xiii
ACKNOWLEDGMENTS . . . xiv
DEDICATION . . . xv
CHAPTER 1 GENERAL INTRODUCTION . . . 1
1.1 Ultrafast and nonlinear optics background . . . 1
1.1.1 Harmonic conversion (frequency mixing) . . . 2
1.1.2 Nonlinear beam propagation . . . 4
1.1.3 Ionization . . . 5
1.2 Chirped Pulse Amplification . . . 5
1.3 Characterization of ultrashort pulses in space and time . . . 10
1.3.1 Spectral Interferometry . . . 11
1.4 References Cited . . . 15
CHAPTER 2 SPATIAL CHIRP INTRODUCTION . . . 17
2.1 Spatial chirp . . . 17
2.2 Optical arrangements utilizing spatio-temporal focusing . . . 21
2.3 Shearing interferometer techniques . . . 22
2.4 References Cited . . . 24
CHAPTER 3 BROADBAND INTERFEROMETRIC CHARACTERIZATION OF DIVERGENCE AND SPATIAL CHIRP . . . 25
3.1 Optics letters accepted manuscript . . . 25
3.2 Supplemental information . . . 33
CHAPTER 4 PULSE FRONT ACHROMATIC BEAM CROSSING IN LINEAR GEOMETRY . . . . 42
4.1 Pulse front matching to avoid temporal smearing . . . 43
4.2 PFT of beam di↵racted from grating . . . 46
4.3 Our approach to PF matching . . . 47
4.4 Experimental design and implementation . . . 49
4.4.1 Confirmation of PFT overlap in semiconductor sample . . . 53
4.4.2 Plasma grating application . . . 55
4.4.3 Application to THG . . . 58
4.5 References Cited . . . 61
CHAPTER 5 RADIALLY CHIRPED SIMULTANEOUS SPATIAL AND TEMPORAL FOCUSING . . 63
5.1 Bessel and Bessel-Gauss beams . . . 63
5.1.1 Bessel beams . . . 63
5.1.2 Bessel-Gauss beams . . . 64
5.1.3 Higher order B-G beams . . . 69
5.2 Radial SSTF . . . 70
5.3 Plasma waveguide with B-G beams . . . 75
5.3.1 Plasma generation . . . 77
5.4 High-mode selective coupling to multimode waveguides . . . 78
5.5 References Cited . . . 83
APPENDIX - PUBLICATION IN PEER REVIEWED JOURNAL AND COPYRIGHT PERMISSION . . . 85
LIST OF FIGURES
Figure 1.1 Wave-vector diagrams for SHG showing two frequencies (red) adding to emit a second harmonic frequency (blue) and THG showing three frequencies (red) adding to emit a third harmonic frequency (purple), the solid line is the ground state and the dashed
line is the excited state . . . 3 Figure 1.2 An input Gaussian pulse shape (red) with a nonlinear frequency shift adds new
frequencies to give the self-phase modulation broadened spectrum on the right in blue . . 4 Figure 1.3 An input Gaussian going through a material with an intensity dependent refractive
index (red) causes self-focusing or collapse that curves the wavefront like a beam
through a lens . . . 5 Figure 1.4 A chirped pulse amplification (CPA) system process schematic showing temporal pulse
shape after each system component, where the colors represent spectral bandwidth. A short pulse is generated in the oscillator, spectrally dispersed in the stretcher, amplified for more intensity and dispersion compensated for in the compressor giving a short
pulse with all spectral components overlapped. Figure adapted from Fig 1.3 of . . . 6 Figure 1.5 Results of dispersion orders (blue) on a Gaussian temporal pulse profile (red). First
order dispersion showing linear temporal shift in (a), second order dispersion showing increase in pulse duration in (b), third order dispersion showing non-symmetric shift and increase in pulse duration in (c) and fourth order dispersion showing increased
pulse duration symmetrically and decreased intensity in (d). . . 8 Figure 1.6 Double pass grating compressor design with parallel, equal groove density gratings and
a roof mirror to retro-reflect the beam back at a vertical o↵set allowing recombination of all spectral components back at the first grating . . . 9 Figure 1.7 Scanning FROG schematic showing beamsplitter (BS) to have a translation stage delay
(mirror pair) in one arm and then recombining in a focusing lens in to the SH crystal where the SH mixing signal is collected in a spectrometer. Figure adapted from . . . 10 Figure 1.8 Two wave vectors (k1, k2) crossed with half crossing angle ✓, beam waist radius x and
resulting delay between the pulses ⌧ in (a) and the resulting interference pattern of
these two crossed pulses showing the fringe smearing outside overlap area in (b) . . . 11 Figure 1.9 Dispersion scan done by scanning SH crystal through focus in z that maps the axial
position to second order phase that allows dispersion information to be gathered for each frequency beamlet to align the single-pass compressor with fourth-order limited
dispersion. Figure from conference presentation . . . 11 Figure 1.10 Fourier analysis block diagram for phase retrieval, starting with a spatial-spectral
interferogram we Fourier transform in the frequency domain to give the spatio-temporal domain that shows the DC (amplitude) and two AC (phase) peaks. Selection of an AC peak or sidelobe and re-centering to x = 0, T = 0. We then Fourier transform back to the spatial-spectral domain that yields the phase information of the interferogram. . . . 13 Figure 1.11 With the phase information in the spatial-spectral domain, we can extract Fourier analysis
Figure 2.1 Spatio-spectral intensity profile from Figure 3 in showing transverse chirp in the line of circles and angular chirp in the line of triangles . . . 18 Figure 2.2 A schematic describing pure lateral chirp from spectral chirp is shown in (a). The
spectral spread after the grating is focused yet the frequency components are not overlapped and therefore result in a larger pulse duration. A schematic describing pure angular chirp is shown in (b). All the frequency components with a strong angular
dependence overlap at focus to shorten the pulse duration. . . 19 Figure 2.3 Combining spatial and spectral chirp results in the shown PFT in z. With a
compressed pulse, the pulse duration is the shortest at z = 0 and PF direction does not change through the focus. With an under-compressed or over-compressed pulse, the pulse is shortest where there is no overall spectral phase and the angular chirp controls the chirp. . . 20 Figure 2.4 A schematic showing simultaneous spatial and temporal focusing (SSTF). The
wavelengths only overlap at the focus, which means there is only a short pulse at the focus and a longer pulse outside of it. The result is a large pulse front tilt from the
angular chirp of the frequency components. Figure adapted from Block . . . 21 Figure 2.5 Crossing wavefronts in solid black with each wavefront radius of curvature R, half the
distance between the wavefronts xsand crossing angle between the beams ✓ . . . 22 Figure 2.6 Di↵erent shearing interferometer designs with the shearing plate interferometer
utilizing the o↵set of the front and back surfaces of the glass plate in (a), the air-wedge shearing interferometer that similarly uses the reflections of the front and back surface but of two glass wedges creating a variable angle in (b) and the Sagnac cyclic shearing interferometer with counter-rotating beams in (c). Figure adapted from . . . 23 Figure 3.1 Two corner cube interferometer configuration (C1) for divergence measurement and
roof prism and corner cube configuration (C2) for spatial chirp measurement in (a), resulting divergence interferogram in (b) and resulting spatial chirp interferogram in
(c). . . 28 Figure 3.2 Inverse radius of curvature versus change in lens position z (points) and fit line gives
slope to extract the focal length of the lens . . . 30 Figure 3.3 (a) Spatial Fourier transform of a Configuration 2 interferogram with zero time delay.
The sloping represents the variation of beam angle with frequency. (b) Variation of the angular chirp rate with angular adjustment of the second grating in a double-pass compressor (points). The fit (line) can be used to determine the position where the gratings are parallel. The x-axis listed as change in grating angle is for an arbitrary
angle and therefore the zero angle does not correspond to zero angle between gratings. . 31 Figure 3.4 Lateral spatial chirp from SSTF compressor (a). Lens conversion from lateral to
angular spatial chirp with magnification to see fringe curvature (b). . . 32 Figure 3.5 Shearing interferometer with polarizing beamsplitter and birefringent plate for time delay
without spatial shear in (a) and introduction of spatial shear moving mirror 2 (M2) without
changing time delay in (b) . . . 34 Figure 3.6 Spatial shear between two interfering beams increasing from 1 mm to 5 mm. Increased
Figure 3.7 Spatio-spectral interferograms in configuration 1 for three settings of the input beam divergence: converging at a change in z-position of 2.5 mm, collimated or z = 0
position, and diverging at a change in z-position of 3 mm. . . 35 Figure 3.8 110 grooves/mm transmission grating in spatial chirp interferometer shows
interferogram without imaging the transmission grating to show lateral spatial chirp
(a) and imaging the transmission grating with a lens to show angular spatial chirp (b). . 35 Figure 3.9 Fourier transform the spatial-spectral interferogram in frequency gives the pulse front
tilt seen in (a) and Fourier transforming the interferogram in position along the slit
gives the spatial chirp in (b) . . . 36 Figure 3.10 Interferogram with rotated fringes showing diverging beam in (a) and the corresponding
extracted phase versus position data and fit line in (b) and retrieved angle versus frequency
(between the dashed lines there is decent spectral amplitude)in (c) . . . 37 Figure 3.11 Simulation of the quartic term spherical aberration with Fourier analysis measuring the
derivative of the wavefront or the phase front giving the cubic component . . . 38 Figure 4.1 Overlapping femtosecond pulses, shown in diagram as wave vectors k1 and k2, half the
crossing angle ✓, the beam waist radius x and the delay between the pulses ⌧ in (a) and the resulting intensity interferogram showing good fringe contrast where pulses overlap and temporal smearing outside overlap area . . . 43 Figure 4.2 4-f zero-dispersion grating stretcher (G1 and G2) with lenses (L1 and L2) separated
2 f apart in (a) and curved mirrors and retro mirrors for reflection instead of refraction in (b) . . . 44 Figure 4.3 Schematic showing the grating (G), grating di↵raction angle (✓d), grating separation to
next optical component (L), position of the pulse front (x(!)) and imaging lenses
(L1and L2) with imaging distances of the grating to the sample (L and f2). . . 45 Figure 4.4 PFT rotation (✓pf) with grating rotation or incident beam angle if incident beam is not
at normal incidence . . . 46 Figure 4.5 PF angle versus incident angle for m = 1 in blue and m = 1 in red. The solid line
shows the Littrow angle for the grating. . . 48 Figure 4.6 Wavelength and beamlet representation in 4-f imaging system with 2 lenses (left side
schematics) compared to single-pass compressor with grating replacing first lens (right side schematics). The PF is the the same in all depictions, as well as the wavelengths overlapping at the target, yet the beamlet focusing changes in the last depiction to focus on target instead of image to target . . . 48 Figure 4.7 PFT across a beam of radius 10 µm as a function of n = Lg/f or grating separation
di↵erent from focal length. When the grating separation equals the focal length, the
pulse front tilt change is zero. . . 50 Figure 4.8 Interference fringes across entire focal spot of 37 microns, with 1.04 micron fringe spacing.
Line out shows very good contrast with little background pedestal. . . 51 Figure 4.9 New transmission grating design blazed for higher efficiency in the +/-1 orders than
the zero order. Second grating does not have a grating in the center to allow higher
Figure 4.10 Four wave mixing (4WM) schematic showing 3 wave vectors overlapping and the generated signal beam with phase matching condition ks= k3+ k1 k2. Figure
adapted from Figure 1 of . . . 53 Figure 4.11 The out-of-plane probe beam (in between k5 and k6) scattering o↵ two interfering pump
beams (below probe beam) in thin CdTe showing di↵raction of probe (k5 and k6),
self-di↵raction (circled spots, not labeled) and BOXCARS signal (k4) . . . 54
Figure 4.12 The out-of-plane probe beam (0 order) scattering o↵ two interfering pump beams (+/-1 orders) in thin CdTe showing di↵raction of probe (k5 and k6), self-di↵raction (circled spots, not labeled) and BOXCARS signal (k4)for two delay positions of rotating glass 1 degree apart with self-di↵raction and signal beams apparent when temporally overlapped and not with delay . . . 54 Figure 4.13 Extension of PF matching experimental apparatus with a BS to split the beam and
periscope to take it out of place. Then there are two sets of gratings vertically stacked to give the beam configuration listed in the Mask box and the Past Target box . . . 56 Figure 4.14 Lab picture of 4 gratings in final design, 2 at a low height and 2 at a vertical o↵set high
height . . . 56 Figure 4.15 Di↵raction orders of plasma grating at normal incidence in (a) and at the Bragg
condition in (b) . . . 57 Figure 4.16 Plasma fluorescence in blue compared to averaged fringe pattern across focal spot in
purple, showing that the visible spark is comparable in size to the focal spot . . . 57 Figure 4.17 Transient grating diagnostics schematic showing a Gaussian input intensity in red,
transport of electrons in green and recombination of electrons in blue . . . 58 Figure 4.18 Mismatched PF versus matched PF over full focal spot . . . 59 Figure 4.19 PF sweep in SH crystal imaged to CCD. The sweeping time delay allows quantification
of the PF angle mismatch. . . 59 Figure 4.20 Using the PF matching scheme to overlap pulses at a FS sample, the resulting THG
wave vectors result which can be measured using a PMT. Figure adapted from . . . 60 Figure 5.1 Bessel beam production from axicon with input beam size db, axicon aperture h and
angle ↵, crossing length l, ring thickness t, diameter of the ring drand projection to
the z-axis angle . . . 64 Figure 5.2 Bessel-Gauss geometry from tilted Gaussian beam with half cone angle , transverse
wave vector kr with projection angle to the x-axis ↵, and propagation constant k which component kz projects on the z-axis. Figure adapted from Bagini . . . 65 Figure 5.3 Bessel-Gauss vortex beam that pushes Bessel zone away from tip of axicon . . . 65 Figure 5.4 Gaussian beam waist showing the focused beam radius w0, the confocal parameter b
corresponding to twicep2w0and the width along the propagation direction w(z) . . . 66 Figure 5.5 Bessel-Gauss Vortex beam setup with mirror hole to allow beam input and reflection of
Figure 5.6 Lab pictures of B-G setup, top view with beams drawn in, side view to better see
labeled optics . . . 68 Figure 5.7 Imaged Bessel-Gauss focusing region for one z plane in x y plane showing bright
central core and rings of decreasing brightness around the core. . . 68 Figure 5.8 Hermite-Gaussian (HG) modes for l = 0, p = 1 in (a) HG modes for l = 1, p = 0 in (b)
and Laguerre-Gaussian mode for l = 1, p = 0 in (c) showing the vortex mode . . . 69 Figure 5.9 SLM masks: m = 4 phase ramp in (a) and phase screw or ramp with a phase grating in
(b) . . . 70 Figure 5.10 Interferogram of two vortex beams showing a fork in interference. 14 fringes can be
seen on the right and 6 fringes can be seen on the left of the red dashed line, where the di↵erence divided in half gives the m = 4 mode verification . . . 71 Figure 5.11 Circular gratings design of first grating giving higher efficiency in the zero order ring
and a collimating second grating. . . 71 Figure 5.12 Circular grating atomic force microscope (AFM) scan showing groove shape in both x
and y dimension . . . 72 Figure 5.13 Bessel zone from circular gratings in pulse front tilt matching geometry where grating
separation equals focal length of focusing optic (20 cm) with corresponding line out in y showing the zero-order Bessel FWHM of 5 µm . . . 72 Figure 5.14 The group delay dispersion of the achromatic Bessel beam versus the central beam
angle showing temporal focusing . . . 74 Figure 5.15 Radial PFT change as a function of grating separation di↵erent from focal length . . . . 74 Figure 5.16 Change of outer beam radius using two axicons with dotted line representative of the
input ring diameter showing the output ring is smaller than the input ring. . . 75 Figure 5.17 Mode solution in blue for higher order mode matching to plasma waveguide with red
line showing the step profile approximation of the waveguide wall. . . 76 Figure 5.18 Bessel zone transverse profile in (a) and the line out of this profile with guided mode
profile in (b) . . . 76 Figure 5.19 Final Bessel-Gauss vortex beam imaged Bessel zone with line out in the x-dimension,
showing a mode size of 24 µm. . . 77 Figure 5.20 Plasma breakdown in air, a couple centimeters in length, seen by naked eye of the
Bessel-Gauss beam. . . 77 Figure 5.21 Probe geometry for Bessel-Gauss vortex plasma waveguide with translation to delay
the probe and matching pump and probe focusing lenses, also the plasma (shown in
maroon) is relay imaged with magnification to the CCD. . . 78 Figure 5.22 SHG of ring pump beam mixed with Gaussian probe beam showing concentric mixing. . 78
Figure 5.23 The probe when pump beam is unblocked in red showing a plasma distortion or dip in the intensity, characteristic of the nonlinear spatial profile of the plasma. The probe
beam when pump beam is blocked in blue. . . 79 Figure 5.24 Normalized intensity of input field (red dashed) and best mode solution (blue) versus
the radial coordinate. . . 80 Figure 5.25 Coupling coefficient versus Gaussian apodization parameter (1/e2radius) for m = 4
mode in blue, m = 3 mode in red and m = 5 mode in black. . . 81 Figure 5.26 Coupling coefficient versus wavelength for m = 4 mode in blue, m = 3 mode in red and
m = 5 mode in black. . . 81 Figure 5.27 Coupling coefficient versus angle for m = 4 mode in blue, m = 3 mode in red and
LIST OF TABLES
Table 1.1 Intensity range for common nonlinearities in W/cm2 . . . 2 Table 1.2 Dispersion compensation (second, third and fourth order) sign for material, a grating
pair and a prism pair . . . 8 Table 4.1 PF Matching Design 1 parameters with 110 lines/mm transmission gratings . . . 50 Table 4.2 PF Matching Design 2 parameters with 460 lines/mm transmission grating beamsplitters . . 51 Table 4.3 PF Matching Design 3 parameters with designed 200 lines/mm transmission grating
LIST OF ABBREVIATIONS
Chirped Pulse Amplification . . . CPA Simultaneous spatial and temporal focusing . . . SSTF Pulse Front Tilt . . . PFT Second Harmonic Generation . . . SHG Third Harmonic Generation . . . THG High Harmonic Generation . . . HHG Four Wave Mixing . . . FWM
ACKNOWLEDGMENTS
I would like to thank my advisor, Charles Durfee, for his optics insight and always having an idea to try out. We would like to acknowledge funding support from AFOSR under grants FA9550-10-1-0394 and FA9550-10-0561. Also, this thesis would not have been successful without the support of my husband, Scott.
CHAPTER 1
GENERAL INTRODUCTION
Development of laser amplifiers for ultrashort pulses has made possible experiments and applications that make use of a nonlinear response of the medium to the light. Ultrashort pulses are less than picoseconds (10 12 s) in duration and have a broad spectral bandwidth, as frequency and time are a Fourier conjugate pair ( ! t = 4 ln 2 ). In-lab examples of applications that take advantage of short pulse durations include micro-machining, laser eye surgery, and nonlinear conversion to other wavelengths, THz to high harmonic generation (HHG), etc. [1, 2]. Some outdoor examples are long distance propagation for remote sensing and guiding of discharges, e.g. lightning [3]. Propagation of ultrashort pulses occurs in a nonlinear regime and is complicated. Phenomena such as di↵raction, ionization and self-focusing, where the intensity dependent nonlinear index is higher on-axis creates a lens, inhibit propagation [4]. An alternate approach that utilizes nonlinearities over long distances is filamentation or creation of filaments that are chains of light that self-focus causing plasma generation that deself-focuses, until losses dominate the process [5]. With filaments and other nonlinearities, the beam can be controlled with linear optics where nonlinear propagation is suppressed up until the desired interaction target zone. The beam structure can also control how the pulse arrives at the target, i.e. through spatial variation of the temporal peak of the pulse. Another example of how to structure light is simultaneous spatial and temporal focusing (SSTF), or the control of spatial arrangement of frequency components [6]. SSTF is used in micro-machining and tissue ablation to create a temporal localization at the spatial focal volume for precise cutting [7, 8]. There are many spatial structures of beams that include Bessel beams, Bessel-Gauss beams, vortex beams, and radial/azimuthal polarizations [9]. Combining di↵erent spatial structures of beams is possible [10, 11]. Our work covers diagnostics to measure the structure of beams, controlled pulse front matching, generation of Bessel-Gauss and vortex beams and nonlinear production of plasma waveguides with vortex beams.
In this introduction, we will discuss a general background of nonlinear optics in Subsection 1, generation of ultrashort pulses in Subsection 2 and characterization of these pulses in Subsection 3.
1.1 Ultrafast and nonlinear optics background
Ultrashort pulses are characterized by examining an intensity spectrum and spectral phase that describes the spectral components of the pulse. When using ultrafast pulses, the intensities can reach 107W/cm2and higher. With light intensities this high, there is a nonlinear dielectric polarization response to the electric field of the light. The nonlinear response is found in the refractive index that comes from a material’s response to
an incident wave. The wave in this medium is then described as the sum of the incident and scattered waves. We study many nonlinear optics phenomenon in our laboratory. Nonlinear behavior is observed in frequency mixing processes such as second harmonic generation (SHG) or four wave mixing (FWM), nonlinear beam propagation in self-focusing and ionization creating filaments. In order to understand more about these nonlinear processes, we use those frequency mixing processes that lend themselves to interferometry and Fourier phase analysis to retrieve the intensity spectrum and spectral phase. The intensities of these nonlinear processes can be found in the following Table 1.1.
Table 1.1: Intensity range for common nonlinearities in W/cm2 Nonlinear process Intensity (W/cm2) Harmonic conversion > 107 108 Nonlinear beam propagation > 3⇥ 1010
Ionization > 1013
Elaser > Eatom > 1016 Relativistic e↵ects > 1018
The nonlinear processes resulting from high intensities can be described macroscopically with Maxwell’s equations, specifically with the electric displacement field eD containing the polarization vector eP ,
e
D = ✏0E + ee P eP = ✏0 E eeD = ✏0E + ✏e 0 E = ✏e 0(1 + ) eE = ✏0E ✏ = ne 2, (1.1) where ✏0 is the permittivity of free-space and ✏ is the permittivity of the material. These multi-photon processes are expressed through the nonlinear polarization power series in field strength, and are a function of nonlinear optical susceptibilities, = n2+ 1,
˜
P (t) = ✏0[ (1)E(t) +˜ (2)E˜2(t) + (3)E˜3(t) + ...]⌘ ˜P1(t) + ˜P2(t) + ˜P3(t) + ... (1.2) The di↵erent orders of susceptibility are commonly referred to as di↵erent ordered nonlinearities and are functions of the refractive index, n, that is the ratio of the velocity in a medium to free-space and is therefore a function of the permittivity and permeability:
n = c v = ✓ ✏ µ ✏0µ0 ◆1/2 > 1 in medium. (1.3)
We are interested in intensities that can be calculated from the electric field by taking the modulus squared, I(t) =| ˜E|2. The most common intensity dependent nonlinearities we encounter in our laboratory are described more below.
1.1.1 Harmonic conversion (frequency mixing)
Nonlinear processes can either be direct harmonic generation from a single input beam or mixing photons from one or more beams that overlap in a nonlinear medium.
A commonly used nonlinear optical process is harmonic conversion, specifically frequency doubling or second-harmonic generation (SHG) to create shorter wavelengths [12]. SHG is a second order nonlinearity that is described by the addition of 2 photons of frequency ! that emit a photon at the second harmonic frequency 2!. Our fundamental frequency is the near infrared (IR) wavelength of 800 nm and the SH frequency emits in the visible spectrum at a wavelength of half the fundamental, 400 nm. The electric field can be described by
E(2!)⇠ P(2)(2!) = (2)E(!)E(!). (1.4)
The same process can be extended to a third order nonlinearity, third harmonic generation (THG), in which adding together 3 photons gives a third harmonic frequency 3!, at one third the fundamental wavelength at 266 nm. Both the SHG and THG wave vector diagrams can be seen in Figure 1.1, where the ground state is shown in solid black and the excited state is shown as a dotted black line.
Figure 1.1: Wave-vector diagrams for SHG showing two frequencies (red) adding to emit a second harmonic frequency (blue) and THG showing three frequencies (red) adding to emit a third harmonic frequency (purple), the solid line is the ground state and the dashed line is the excited state
These frequency mixing processes are usually achieved by overlapping the frequencies in a nonlinear medium such as a birefringent crystal. Birefringence is the polarization and direction dependence of the nonlinear index, and makes possible the phase matching condition. The phase matching condition for harmonics is4k = nk1 kn and is equivalent to matching the phase velocities. This expression is a vector expression that projects vectors onto the direction of the output beam with a cosine factor. Phase matching is most commonly achieved using a birefringent nonlinear medium with one polarization of light seeing the ordinary axis of the material (no) and the other polarization seeing the extraordinary axis of the material (ne).
1.1.2 Nonlinear beam propagation
The nonlinear processes above involve generation of a signal at a new frequency. It is also possible to have frequency degenerate mixing. For example degenerate four-wave mixing on a single beam is described by the polarization term (3)E2E⇤. This leads to intensity dependent phase shifts. Intensity dependent phase shifts can also be expressed in the optical Kerr e↵ect which is a third order nonlinearity that induces a change in the refractive index with intensity when traveling in a medium. This index change follows n(I) = n0+ n2I, where n0 is the linear refractive index, n2 is the nonlinear refractive index and I is the intensity. The Kerr e↵ect is commonly used to mode lock lasers. Nonlinear e↵ects that stem from the intensity-dependent refractive index are self-phase modulation (SPM), self-focusing and n2accumulation. SPM comes from the temporal dependence of the phase shift whereas self-focusing comes from the transverse spatial dependence of the phase shift.
The nonlinear phase shift or B-integral of the traveling wave can be written as
NL(r, t) = kn2I(r, t)z (1.5)
where k is the wave vector, n2 is the nonlinear refractive index, I(r, t) is the intensity of the traveling wave and z is the thickness of the material. The time-dependent nonlinear phase shift leads to frequency shifts described by the instantaneous frequency as long as the amplitude of the pulse varies slowly compared to the optical period according Chapter 7.5.1 in [12]. These frequency shifts are described by !(t) =d NL(t)/dt and add new frequencies to the pulse that result in a broader bandwidth pulse as seen in the right side of the panel in Figure 1.2.
Figure 1.2: An input Gaussian pulse shape (red) with a nonlinear frequency shift adds new frequencies to give the self-phase modulation broadened spectrum on the right in blue
When an intense laser pulse travels through a medium, the beam itself is altered to come to a focus within the material . This occurs because the intensity of the laser pulse is higher in the center of the beam inducing a higher refractive index change than on the edges of the beam as seen in Figure 1.3. This lensing e↵ect takes place when the beam’s power greatly exceeds the critical power, Pcr ⇡ 20/8n0n2. Self-focusing
Figure 1.3: An input Gaussian going through a material with an intensity dependent refractive index (red) causes self-focusing or collapse that curves the wavefront like a beam through a lens
causes higher intensities in the focal volume which can lead to ionization. 1.1.3 Ionization
With high beam intensity, propagation of the pulse can induce optical damage. For ultrafast pulses, the optical damage can be from avalanche breakdown (109 1012W/cm2), multi-photon ionization (1012 1016W/cm2) or direct field ionization (> 1016W/cm2). Avalanche ionization occurs from a small number of free electrons in the material accelerating to higher energies once they interact with the beam. These higher energy electrons move around and ionize other atoms creating more accelerating electrons. The increased thermal energy of these electrons can lead to cracking or melting of the material. Multi-photon ionization happens with multiple photons, for example two-photon ionization occurs in the focal volume of the incident laser beam, and therefore ionizes the atoms in a specific space. Direct field ionization results from the field strength being so large as to rip electrons from the atomic nucleus. This occurs when the laser field strength exceeds the atomic field strength, Eat = e/4⇡✏0a20, or at a laser intensity of Iat = n✏0cE
2
at/2. Above these intensities, relativistic e↵ects dominate, which lead to nonlinearities in the atomic response.
We need to amplify ultrashort pulses to explore these nonlinear phenomena which requires use of a chirped pulse amplification (CPA) system to avoid damaging the amplifier gain medium. In this thesis, our CPA system and focusing conditions reach intensities up to 1016W/cm2 at the highest, so our experiments utilize the harmonic generation, nonlinear pulse propagation, and ionization regimes. We want to further explore the photon-electron interactions in these regimes.
1.2 Chirped Pulse Amplification
Ultrafast lasers were made possible with a mode-locking mechanism that locks di↵erent modes of the laser in phase to generate periodic broad-bandwidth pulses at the output coupler at the end of an oscillator resonator. Specifically, Ti:Sapphire oscillators, like the one in our laboratory, use Kerr lens mode-locking, as mentioned in Section 1.1.2, that takes advantage of nonlinear focusing to have smaller modes in the gain
medium and therefore a higher gain than the continuous wave regime. Ti:Sapphire oscillators also have a prism-pair for dispersion compensation to generate the broad-bandwidth pulse with central wavelength of 800 nm. A more detailed discussion of dispersion compensation can be found in Section 1.2. The ultrafast pulse train from an oscillator only outputs nanoJoule (nJ) energy pulses, which does not reach the high intensity regime desired for the many applications discussed. Therefore, a chirped pulse amplification (CPA) system is needed. The CPA system can be described by the schematic in Figure 1.4, where pulses generated in the oscillator are stretched in time using dispersive elements, then the overlapping bandwidth in the gain medium is amplified and finally compressed using the opposite sign dispersion elements to give a femtosecond pulse train with milliJoule (mJ) energies. For our high-power experiments, a traditional CPA system is used with two amplifier stages. Using the pulse train output from the oscillator, the pulses then pass through a stretcher based on o↵-axis cylindrical mirrors and gratings. This allows for aberration control up to 4th order dispersion on the pulse train. Stretching the pulses allows for amplification using a regenerative amplifier pumped with a Photonics Industries (Nd:YLF, 20 W CW, 532 nm) pump into the gain medium, allowing up to 1 mJ (before compression) output energy. Compressing the pulse with a double-passed grating compressor allows for pulse durations on the order of 45 fs, with an output energy of 0.5 mJ per pulse. Further amplification with a multi-pass amplifier gives higher output energies up to 10 mJ (before compression). Typical double-passed grating compressors have an efficiency on the order of 50%, yielding around 5 mJ output energy pulses at 60 fs pulse durations with 4th-order limited dispersion.
Figure 1.4: A chirped pulse amplification (CPA) system process schematic showing temporal pulse shape after each system component, where the colors represent spectral bandwidth. A short pulse is generated in the oscillator, spectrally dispersed in the stretcher, amplified for more intensity and dispersion compensated for in the compressor giving a short pulse with all spectral components overlapped. Figure adapted from Fig 1.3 of [13].
Knowing how the above nonlinearities propagate through the CPA system will give us the means to mitigate them. One technique to lower peak intensity is to spectrally chirp out the pulse or spatially spread the spectral components of the pulse, i.e. dispersion. Spectrally chirped out pulses are the basis of the chirped pulse amplification (CPA) system described above, and is achieved by going through material, a pulse stretcher or pulse compressor. Dispersion comes from a frequency-dependent phase velocity. A one-dimensional plane wave, E(z, t) = A exp[i(!t kz)], propagates with a phase velocity of ⌫p =!/k=c/n( ),
where ! is the angular frequency, k is the wavenumber, c is the speed of light and n( ) is the refractive index of the material. In most transparent materials, the refractive index changes with wavenumber and therefore frequency or wavelength and can be described by the Sellmeier equations to determine the dispersion of light in the particular medium. An increase in the refractive index with frequency is the normal dispersion regime and a decrease with frequency is the anomalous dispersion regime. Most transparent materials, such as air and glass that we mainly use, have normal dispersion. The phase velocity’s dependence on wavelength or frequency is commonly referred to as chromatic dispersion and causes refraction angles at optical surfaces to be frequency dependent which is commonly referred to as angular dispersion.
The phase velocity di↵ers from the group velocity since a wave packet or superpositions of plane waves at di↵erent frequencies represents an actual displacement of z giving finite time values. Any modulation of the wave in a medium, such as dispersion, will propagate with the group velocity,
⌫g= d! dk = vp 1 k n dn dk , (1.6)
where vg is the group velocity, vp is the phase velocity, k is the wave vector and n is the refractive index. In vacuum, where the medium has a constant refractive index, the angular frequency is proportional to the wavenumber, ! = c k, and this linear dispersion relationship means the phase velocity and group velocity are the same.
The more general spectral chirp can be seen in the spectral phase:
(!) = ~k· ~r = 2⇡n l =2⇡P, (1.7)
where is the wavelength, n is the refractive index, l is the length of the medium and P is the optical path length traveled by an individual wavelength. A Taylor series expansion of the spectral phase yields the di↵erent dispersion terms. This expansion is
(!) = (!0) + (! !0) ✓d d! ◆ !0 + 1 2!(! !0) 2✓d2 d!2 ◆ !0 + 1 3!(! !0) 3✓d3 d!3 ◆ !0 + 1 4!(! !0) 4 ✓ d4 d!4 ◆ !0 + ... (1.8)
where the group delay (GD) is 1= ⇣
d d!
⌘ !0
, the second order dispersion (SOD) or group delay dispersion (GDD) is 2=
⇣d2
d!2 ⌘
!0
, the third order dispersion (TOD) is 3= ⇣d3
d!3 ⌘
!0
and the fourth order dispersion (FOD) is 4 =
⇣d4
d!4 ⌘
!0
. The di↵erent orders of dispersion can be seen in Figure 1.5, where the original Gaussian pulse is in red and the e↵ect from the ordered dispersion times, first order (a) through fourth order (d) respectively, are in blue. First order is a shift in delay, second order increases the pulse duration, third order shows a non-symmetric shift in delay and increase in pulse duration in and fourth order shows an
increased pulse duration symmetrically and decreased peak intensity. (a)-400 -200 200 400t 0.2 0.4 0.6 0.8 1.0 IntensityHtL (b)-400 -200 200 400t 0.2 0.4 0.6 0.8 1.0 IntensityHtL (c)-1000 -500 500 1000t 0.2 0.4 0.6 0.8 1.0 IntensityHtL (d)-1000 -500 500 1000t 0.2 0.4 0.6 0.8 1.0 IntensityHtL
Figure 1.5: Results of dispersion orders (blue) on a Gaussian temporal pulse profile (red). First order dispersion showing linear temporal shift in (a), second order dispersion showing increase in pulse duration in (b), third order dispersion showing non-symmetric shift and increase in pulse duration in (c) and fourth order dispersion showing increased pulse duration symmetrically and decreased intensity in (d).
For example, materials such as glass impart a positive GDD and TOD and gratings impart a negative GDD and positive TOD. Refer to Table 1.2 for more information on the sign of the di↵erent orders of dispersion. Important to note is that grating dispersion compensates for the material second and fourth order dispersion the pulse accumulates through refractive optics. This is due to short wavelengths (blue) di↵racting less and refracting more while longer wavelengths (red) di↵ract more and refract less.
Table 1.2: Dispersion compensation (second, third and fourth order) sign for material, a grating pair and a prism pair
Device SOD/GDD (fs2) TOD (fs3) FOD (fs4)
Material + +
+/-Grating Pair - +
-Prism Pair - -
-It is important that whichever dispersion order is imparted to the beam traveling through the optical components of the stretcher and amplifier are compensated for in the grating compressor. Ideally we would
want a frequency-independent spectral phase, yet in our actual CPA system, our best compressor alignment is usually transform limited in fourth-order, i.e. the minimum spectral width possible for a given pulse duration, minimizing GD, GDD, and TOD.
Di↵erent dispersion elements can be used for pulse compression but most common is a di↵raction grating pair [14]. The common design for the grating compressor is shown in Figure 1.6, where a dispersive beam comes in and di↵racts o↵ the first grating and then hits the second grating, which is aligned parallel to the first, which collimates the beam with a larger spectral spread in space. The collimated beam hits a roof mirror that retro-reflects the beam back through the gratings at a vertical o↵set.
Figure 1.6: Double pass grating compressor design with parallel, equal groove density gratings and a roof mirror to retro-reflect the beam back at a vertical o↵set allowing recombination of all spectral components back at the first grating
The grating equation utilizes Snell’s law and is expressed by n2sin ✓2 n1sin ✓1=m
d , (1.9)
where ✓1is the incident angle, ✓2is the di↵racted angle, m is the di↵racted order and d is the groove density. The grating pair can be thought of as a tilted window. The spectral phase of a tilted window can be expressed by
w(!) = k0Lw(n2cos ✓2 n1cos ✓1), (1.10) where k0=!/c and Lw is the window thickness [15]. This can then be combined with the grating equation to give the spectral phase of the grating
gr(!) = k0Lgr (n22 ✓ m d n1sin ✓1 ◆2!1/2 ! k0Lgr 1 ✓ d sin ✓1 ◆2!1/2 , (1.11) where the right hand side of Eqn. 1.11 is the reduced form of grwhen n1= n2= m = 1. Alignment of the grating pair is key to obtaining transform limited dispersion pulses.
1.3 Characterization of ultrashort pulses in space and time
Alignment of CPA systems is very important to the ultrashort pulse characteristics needed for our ex-periments. So just as important as the experiments themselves is the initial characterization of the output pulse in both the space and time domains to have a clean reference pulse free of distortion. Most common are nonlinear techniques such as the autocorrelation to retrieve the spectral phase mentioned in the previous subsection. The expression for the intensity autocorrelation is as follows
A(⌧ ) = ˆ +1
1 |I(t) + I(t
⌧ )|2dt, (1.12)
where I is the intensity and ⌧ is the delay. The autocorrelation gives a qualitative measure of the pulse duration but the shape of the pulse has to be known or assumed to get a meaningful number. If the autocorrelation signal is spectrally resolved, there is enough information to retrieve the intensity and phase with methods like frequency resolved optical gating (FROG) [16] and single-shot second-harmonic generation FROG, commonly referred to as SHG FROG or GRENOUILLE [17]. A schematic for the scanning FROG can be seen in Figure 1.7, where re-imaging of the SH crystal could be done by inserting a lens between the SH crystal and spectrometer. These techniques take advantage of an autocorrelation or interference between
Figure 1.7: Scanning FROG schematic showing beamsplitter (BS) to have a translation stage delay (mirror pair) in one arm and then recombining in a focusing lens in to the SH crystal where the SH mixing signal is collected in a spectrometer. Figure adapted from [16].
two copies of a pulse in a nonlinear medium with time delay between the pulses to vary the nonlinear signal versus delay. Crossing pulses allows encoding of the time delay in position. The result is described in the schematic in Figure 1.8(a), where the time sweep across the overlapped part of the pulses is represented by ⌧ (x) = (2x sin ✓)/c. The resulting interference pattern is shown in Figure 1.8(b), where the fringes smear out (less fringe contrast) outside the overlapped area.
(a) (b)
Figure 1.8: Two wave vectors (k1, k2) crossed with half crossing angle ✓, beam waist radius x and resulting delay between the pulses ⌧ in (a) and the resulting interference pattern of these two crossed pulses showing the fringe smearing outside overlap area in (b)
The crossing angle of these pulses introduces a relative pulse front tilt between the pulses, which can be thought of as one side of the pulse arriving before the other edge and can be calculated from the group delay of the spectral phase and will be shown in detail in Chapter 4. Also, these techniques can be utilized in a single-shot arrangement, as in GRENOUILLE, by using cylindrical lens instead of a spherical lens to map the wavelength to a vertical camera position, where di↵erent spatial positions correspond to di↵erent time delays.
The spectral phase and its derivatives can be extracted from many established techniques such as use of a spectral phase interferometer for direct electric field reconstruction of ultrashort optical pulses (SPIDER) [18], frequency resolved optical gating (FROG) [16], or dispersion scans done with a SHG crystal in our laboratory as in Figure 1.9, where the axial position of the SH crystal is varied and the integrated SH spectrum is collected in a fiber spectrometer. The SH position for each wavelength beamlet is then extracted from the data collected [19].
Figure 1.9: Dispersion scan done by scanning SH crystal through focus in z that maps the axial position to second order phase that allows dispersion information to be gathered for each frequency beamlet to align the single-pass compressor with fourth-order limited dispersion. Figure from conference presentation [20].
1.3.1 Spectral Interferometry
Spectral interferometry can be used to characterize the pulse with linear optics. The simple, one-dimensional case of spectral interference can retrieve the spectral phase of material dispersion. More
compli-cated spectral phases can be spatially and spectrally characterized with an imaging spectrometer. We image the pulse crossing plane to an entrance slit (10 µm) of an imaging spectrometer that images the slit through the spectrometer with magnification to the camera. This imaging provides a spatial reference so the pulse’s spatial information can also be resolved [21].
Since time and frequency are a conjugate Fourier pair, we use Fourier analysis on the temporal delay that causes spectral interference to extract information about the spectral phase. A forward traveling electromagnetic (EM) wave has the form E(x, t) = exp[i(kx !t)] where a backward traveling EM wave is the complex conjugate. When two waves overlap in space and time the result is a superposition of the wave. This can be seen mathematically as the absolute value squared of the two fields, which simplifies to
I(x, y) = a21+ a22+ a1a2e 2i ⇣ x!sin✓x c + y!sin✓y c ⌘ + a1a2e2i ⇣ x!sin✓x c + y!sin✓y c ⌘ (1.13) where ✓xis the crossing angle between the wavefronts in the x-dimension and ✓yis the crossing angle between the wavefronts in the y-dimension. The sum of the waves gives the DC terms shown as the amplitudes squared and the product of the waves gives the AC terms or complex interference terms. When two waves are in phase with each other they constructively interfere and when they are out of phase they destructively interfere and combinations of constructive and destructive interference give interference patterns that we then analyze to understand the wavefronts that make them.
Spectrometers are common for measuring the intensity of light over a spectral range. Therefore, they can be useful in measuring the interference pattern or variation of intensity of short temporal pulses or wide spectral bandwidths. For example, a fiber spectrometer can measure intensity versus frequency to show spatial interference from an angle between the wavefronts. And an imaging spectrometer with entrance slit can show spectral and spatial interference at the same time.
The interferograms collected are then processed using Fourier analysis as described in Figure 1.10. A Fourier transform is done in the frequency domain to give the pulse front tilt. And since the interference is in either of the AC peaks we can select one of them and recenter to zero in position and time, then Fourier transform back to frequency space. A more detailed explanation of spectral interferometry is given in Chapter 3. Selecting a specific frequency of the inverse Fourier transform field gives the phase information shown in Figure 1.11(A) and a linear fit of the phase gives the slope and therefore, the local angle between the beams in (B).
In this thesis, we will discuss more details of pulse front tilt and spatio-temporal distortion of spatial chirp in Chapter 2. Chapter 3 will focus on the shearing interferometer we developed to characterize the spatial chirp and divergence of our di↵erent compressor arrangements (submitted to Optics Letters). An experimental apparatus utilizing PFT matching for many applications is covered in Chapter 4. Extensions
Figure 1.10: Fourier analysis block diagram for phase retrieval, starting with a spatial-spectral interferogram we Fourier transform in the frequency domain to give the spatio-temporal domain that shows the DC (amplitude) and two AC (phase) peaks. Selection of an AC peak or sidelobe and re-centering to x = 0, T = 0. We then Fourier transform back to the spatial-spectral domain that yields the phase information of the interferogram.
Figure 1.11: With the phase information in the spatial-spectral domain, we can extract Fourier analysis phase slope as in (a) and the retrieved local angle as in (b)
of PFT matching to the Bessel, Bessel-Gauss and vortex beam geometries is discussed in Chapter 5. And utilizing a combination of Bessel-Gauss and vortex beams allows for a plasma waveguide design that is discussed in Chapter 5 as well. Conclusions and further work can be found at the end of each chapter to keep similar material grouped together.
1.4 References Cited
[1] G Mourou. The ultrahigh-peak-power laser : present and future. Applied Physics B: Lasers and Optics, 65:205–211, 1997.
[2] S Backus, C G Durfee, M M Murnane, and H C Kapteyn. High power ultrafast lasers. Review of Scientific Instruments, 69:1207–1223 ST – High power ultrafast lasers, 1998. ISSN 0034-6748.
[3] A Couairon and A Mysyrowicz. Femtosecond filamentation in transparent media. Physics Reports, 441: 47–189, March 2007. ISSN 03701573. doi: 10.1016/j.physrep.2006.12.005. URL http://linkinghub. elsevier.com/retrieve/pii/S037015730700021X.
[4] J´erˆome Kasparian and Jean-Pierre Wolf. Physics and applications of atmospheric nonlinear optics and filamentation. Optics express, 16(1):466–93, January 2008. ISSN 1094-4087. URL http://www.ncbi. nlm.nih.gov/pubmed/18521179.
[5] S L Chin, S A Hosseini, W Liu, Q Luo, F Th´eberge, N Ak¨ozbek, A Becker, V P Kandidov, O G Kosareva, and H Schroeder. The propagation of powerful femtosecond laser pulses in optical media : physics , applications , and new challenges. Can. J. Phys., 83:863–905, 2005. doi: 10.1139/P05-048. [6] Charles G Durfee, Michael Greco, Erica Block, Dawn Vitek, and Je↵ A Squier. Intuitive analysis of
space-time focusing with double-ABCD calculation. Optics Express, 20(13):14244–59, June 2012. ISSN 1094-4087. URL http://www.ncbi.nlm.nih.gov/pubmed/22714487.
[7] Dawn N Vitek, Erica Block, Yves Bellouard, Daniel E Adams, Sterling Backus, David Kleinfeld, Charles G Durfee, and Je↵rey A Squier. Spatio-temporally focused femtosecond laser pulses for nonre-ciprocal writing in optically transparent materials. Optics Express, 18(24):24673–24678, 2010.
[8] Erica Block, Michael Greco, Dawn Vitek, Omid Masihzadeh, David A Ammar, Malik Y Kahook, Naresh Mandava, Charles Durfee, and Je↵ Squier. Simultaneous spatial and temporal focusing for tissue ab-lation. Biomedical Optics Express, 4(6):831–41, June 2013. ISSN 2156-7085. doi: 10.1364/BOE. 4.000831. URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3675863&tool= pmcentrez&rendertype=abstract.
[9] Jill Morris. Studies of novel beam shapes and applications to optical manipulation. PhD thesis, 2010. [10] M Clerici, D Faccio, E Rubino, a Lotti, a Couairon, and P Di Trapani. Space-time focusing of Bessel-like
pulses. Optics letters, 35(19):3267–9, October 2010. ISSN 1539-4794. URL http://www.ncbi.nlm.nih. gov/pubmed/20890355.
[11] Matteo Clerici. Ultra-short laser pulse spatio-temporal shaping. PhD thesis, 2010. [12] Robert W. Boyd. Nonlinear Optics Third Edition. 2008.
[13] Marin Iliev. Cross-polarized wave generation (XPW) for ultrafast laser pulse characterization and in-tensity contrast enhancement. PhD thesis, Colorado School of Mines, 2014.
[14] E. Treacy. Optical pulse compression with di↵raction gratings. Ieee Journal of Quantum Electronics, 5 (9):454–458, 1969.
[15] Charles G Durfee, Je↵ a Squier, and Steve Kane. A modular approach to the analytic calculation of spectral phase for grisms and other refractive/di↵ractive structures. Optics express, 16(22):18004–16, October 2008. ISSN 1094-4087. URL http://www.ncbi.nlm.nih.gov/pubmed/18958079.
[16] Rick Trebino. Frequency-resolved optical gating: the measurement of ultrashort laser pulses. Klewer, 2002.
[17] Selcuk Akturk, Mark Kimmel, Patrick O Shea, and Rick Trebino. Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE. Optics Express, 11(5):491–501, 2003.
[18] C Iaconis and I a Walmsley. Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses. Optics letters, 23(10):792–4, May 1998. ISSN 0146-9592. URL http://www. ncbi.nlm.nih.gov/pubmed/18087344.
[19] Michael Greco and Charles G. Durfee. In-situ dispersion scan (D-scan) pulse characterization. 2015. [20] Amanda K. Meier, Michael Greco, and Charles G. Durfee. Characterization techniques for aligning
spatio-temporal focused ultrafast pulses. 2014.
[21] Daniel E Adams, Thomas a Planchon, Alexander Hrin, Je↵ a Squier, and Charles G Durfee. Character-ization of coupled nonlinear spatiospectral phase following an ultrafast self-focusing interaction. Optics letters, 34(9):1294–1296, 2009. ISSN 0146-9592. doi: 10.1364/OL.34.001294.
CHAPTER 2
SPATIAL CHIRP INTRODUCTION
As the general introduction explained, ultrashort pulses lend themselves to nonlinearities that distort the pulse shape in space and time. Characterization of the amplitude and phase of pulses that have a pulse shape that does not depend on position within the beam has been developed. What is more difficult is the characterization of these pulses when space and time are coupled within the pulse.
2.1 Spatial chirp
Ultrafast pulses are usually described in a temporal domain, yet the geometric or spatial dimension is also important. In the temporal domain, temporal distortions can exist. A common spatio-temporal distortion in grating compressors is spatial chirp that arises, for example, when the gratings are not parallel.
Spatial chirp is defined as an individual frequency component separated in a spatial plane transverse to the propagation direction (z) [1]. Treating each frequency component as its own beam results in individual beamlets [2]. There are two types of spatial chirp, angular and transverse (or lateral) spatial chirp. Transverse spatial chirp arises from angular dispersion from dispersive optics such as gratings or prisms. Angular spatial chirp arises from each frequency component coming in at a di↵erent angle to a specific z-plane [3]. Both transverse and angular spatial chirp can be seen in the experimental intensity profile in Figure 2.1, where the line of circles represents x(!) whose slopedx(!)/d!determines transverse chirp and the line of triangles represents !(x) whose sloped!(x)/dxdetermines angular chirp [1] .
The cartoon in Figure 2.2a, describes pure lateral chirp and is adjusted from spectral chirp from a di↵raction grating (black dashed lines). The beamlets are separated at the focus giving a longer pulse duration. Pure angular chirp is described in Figure 2.2b, where the input beam parameters matter, as seen from all the wavelength components overlapping out of a grating compressor giving a round focus or transform-limited pulses and therefore a strong intensity localization.
The last panel in both (a) and (b) of Figure 2.2 is in the Fourier transformed domain {x, t} from the intensity profile domain{x, !} shown in Figure 2.1. Pulse front tilt (PFT) describes the pulse in the {x, t} domain and exists when the leading edge of the pulse arrives before or earlier than the trailing edge of the pulse. PFT is intrinsically tied to both angular and lateral spatial chirp due to frequency and time being Fourier transform pairs [3].
clarify the meaning of this term. Specifically, we show that there are two different definitions of spa-tial chirp, which we refer to as ‘‘spaspa-tial dispersion’’ and ‘‘frequency gradient.’’ Which definition to use depends on the situation. For Gaussian beams and pulses, we find the relationship between these two parameters to be analogous to that between the parameters describing temporal chirp in the time and frequency domains.
We begin with the case where no spatial chirp is present, and the amplitude of the electric field at position x and frequencyx (defined as frequency offset from the center frequency of the beam) can be written in the form:
Eðx; xÞ ¼ ExðxÞExðxÞ;
where the spatial amplitude Ex(x) and the spectral amplitude Ex(x) are completely separate and potentially arbitrary functions of one variable.1
In the presence of spatial chirp (here we assume that it exists in one transverse spatial dimension x only), E(x,x) becomes an inseparable two-varia-ble function, where the spatial and spectral depen-dences are coupled. We can easily measure the spatio-spectral intensity profile of the spatially chirped beam by sending the beam into an imaging spectrometer with a two-dimensional camera on its output image plane, as depicted inFig. 2. Fields sampled at different points along the entrance slit of the spectrometer are spectrally resolved onto different rows of the camera image, resulting in a trace of intensity in the x–x domain. With linear spatial chirp, the spatio-spectral intensity profile will appear tilted.Fig. 3(a) shows a typical x–x
intensity plot of an experimental beam with spatial chirp.
Obviously, the degree of spatial chirp can be characterized by measuring the tilt of the x–x trace. However, there is a subtlety in this meas-urement, namely, that there are two intuitive, 1An equivalent representation of a general spatio-temporal
ultrashort-pulse beam is the space–time Wigner function[21], a four-dimensional real-valued distribution function, which car-ries the same information about the ultrashort-pulse beam as the complex spatio-temporal (or spectral) field expression. The various two-dimensional marginals of the space–time Wigner function are the expressions of pulse-beam intensity in these domains. Although the space–time Wigner function is a powerful tool in the study of ultrafast beams in space and time, we choose not to use it in our analysis, because this work only involves studying the beam intensity in space and frequency. For that purpose, the simpler spatio-spectral field expression is a more appropriate tool.
Fig. 2. Measuring spatial chirp using an imaging spectrometer.
Fig. 3. Measured spatio-temporal intensity profile of an experimental spatially chirped beam. The line of triangles indicates the x0(x) function, which determines frequency
gradient, and the line of circles indicates the x0(x) function,
which determines spatial dispersion.
X. Gu et al. / Optics Communications 242 (2004) 599–604 601
Figure 2.1: Spatio-spectral intensity profile from Figure 3 in [1] showing transverse chirp in the line of circles and angular chirp in the line of triangles
There are two limiting cases that result in a beam with PFT. For a beam with transverse spatial chirp, an overall spectral chirp leads to a variation of the group delay with frequency, which in turn is mapped to position in the beam. Therefore, PFT can be adjusted by adjusting spectral chirp as in a chirped pulse amplification (CPA) system’s stretcher or compressor. A second source of PFT is angular chirp that also leads to a second-order dispersion, so a pulse can be temporally focused with the right combination of spectral phase and angular chirp. Because of relative positioning of the beamlet focus and the frequency overlap, characterization of the divergence is also important which is first analyzed with the shearing interferometer design explained in Chapter 3.
In most situations, there is a combination of angular spatial chirp and spectral chirp (cause of lateral chirp) as demonstrated in Figure 2.3. An initial compressed input beam with no input spectral chirp shows that pure angular spatial chirp at focus leads to transverse spatial chirp away from the focus. Also, the PFT does not change direction as the beam goes through the focus when there is no input spectral chirp. The shortest pulse always has PFT in the same direction however because there is no overall spectral phase there. When there is input spectral chirp as in the under-compressed or over-compressed case in Figure 2.3, the PFT is a↵ected where the beam has transverse spatial chirp. The PFT that comes from the angular sweep controls the chirp. The colors on the pulse propagating at di↵erent z-positions show the intensity of the pulse not the wavelength. Also, the pulses are in{x, t}.
Figure 2.2: A schematic describing pure lateral chirp from spectral chirp is shown in (a). The spectral spread after the grating is focused yet the frequency components are not overlapped and therefore result in a larger pulse duration. A schematic describing pure angular chirp is shown in (b). All the frequency components with a strong angular dependence overlap at focus to shorten the pulse duration.
Figure 2.3: Combining spatial and spectral chirp results in the shown PFT in z. With a compressed pulse, the pulse duration is the shortest at z = 0 and PF direction does not change through the focus. With an under-compressed or over-compressed pulse, the pulse is shortest where there is no overall spectral phase and the angular chirp controls the chirp.
2.2 Optical arrangements utilizing spatio-temporal focusing
To achieve specific target delivery of energy, i.e. focusing the ultrashort pulse without experiencing any nonlinear e↵ects until the target plane, we use simultaneous spatial and temporal focusing (SSTF) [4]. Temporal focusing was first demonstrated in microscopy to give a large field of view without compromising axial resolution [5]. An SSTF focusing schematic is shown in Figure 2.4. The pulse is chirped out using a single-pass grating compressor so that at the focal plane of a lens or curved mirror the spectral components overlap at the beam waist. Aligning SSTF ensures equal beam waist and wavelength crossing planes.
Figure 2.4: A schematic showing simultaneous spatial and temporal focusing (SSTF). The wavelengths only overlap at the focus, which means there is only a short pulse at the focus and a longer pulse outside of it. The result is a large pulse front tilt from the angular chirp of the frequency components. Figure adapted from Block [6].
In contrast to SSTF, filamentation can be useful for mitigating nonlinear e↵ects. Filamentation comes from the nonlinear e↵ect of self-focusing, where long self-focusing channels can propagate over long distances. When the intensity is high enough, ionization of the surrounding molecules leads to plasma generation that causes defocusing. Ionization saturation leads to intensity clamping. So filamentation is a string of self-focusing, intensity clamping and defocusing events that repeat [7]. Knowing how these nonlinearities propagate allows mitigation of other nonlinearities.
Many broadband optics exist in our laboratory such as curved mirror telescopes that need astigmatism correction. Also important is aligning for flat wavefront in grating compressors, like the aforementioned SSTF compressor. Most simply, knife-edge and camera scans across the focus are used to examine individual spectral components in space to diagnose and align these optical systems. Yet these trial and error methods take table space with long focal length optics. A more compact and efficient way to diagnose these broadband optical systems is with interferometry, especially if phase information is important.
2.3 Shearing interferometer techniques
Important for collimation testing of ultrashort pulses are shearing interferometers that are used for overlapping two curved wavefronts which create interference that can be processed to give information on the exact wavefront curvature. This happens because the local crossing angle depends on the shear, as in sin[✓x] = xs/R, where ✓x is the local crossing angle, xs is the shear or half the distance between the wavefronts and R is the radius of curvature of the wavefront, shown in Figure 2.5. The processing of the interferograms is done with Fourier analysis mentioned in Chapter 1 and discussed further in Chapter 3.
Figure 2.5: Crossing wavefronts in solid black with each wavefront radius of curvature R, half the distance between the wavefronts xsand crossing angle between the beams ✓
There are many varieties of shearing interferometers, the most common being the shearing plate inter-ferometer as seen in Figure 2.6(a) that is simply a glass plate reflecting a wavefront o↵ the front and back surface giving a large temporal o↵set from the thickness of the glass making it good for monochromatic beams only [8] . A variation on this design is the air-wedge shearing interferometer as seen in Figure 2.6(b), that has two wedged glass plates, separated by an air gap, which gives 4 reflections [9]. The angle between reflections 1-2 and reflections 3-4 is set by the individual glass wedges. The angle between reflection 2 and 3 is variable by the air-wedge gap. This angular di↵erence gives more control over the spatial shear for moderately short time duration sources with less bandwidth. Best for short time duration sources is the Sagnac shearing interferometer as seen in Figure 2.6(c), that contains a beamsplitter and two mirrors to create a cyclic ring where one beam travels one direction around the ring while the other travels the opposite direction [10]. The three optics in this setup allow for adjusting shear in multiple ways, either by translating the beam splitter, one of the mirrors, or rotating both mirrors.
Since we are interested in characterizing wavefronts of ultrashort sources with a real-time diagnostic such as fringe rotation, just adding tilt in the other direction with the glass plate or wedge design would provide reference fringes for looking at rotation instead of null fringe but would not work well for our short pulse source. This is the reason for deciding to use a Sagnac interferometer design to get the interferograms
(a) (b) (c)
Figure 2.6: Di↵erent shearing interferometer designs with the shearing plate interferometer utilizing the o↵set of the front and back surfaces of the glass plate in (a), the air-wedge shearing interferometer that similarly uses the reflections of the front and back surface but of two glass wedges creating a variable angle in (b) and the Sagnac cyclic shearing interferometer with counter-rotating beams in (c). Figure adapted from [8].
for divergence analysis. Our design gives equal time delay when beams are in plane and a time delay is introduced with polarization optics creating interference in the spatial-spectral domain. This is further explained in Chapter 3. Interference in the spatial-spectral domain can also give more information about spatio-temporal e↵ects and a spatially inverted variation of this interferometer can be used to characterize spatial chirp.
2.4 References Cited
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