1 | 2 | 0.5 | .0046 | .0046 | .0045 | .0046 | .0086 | .0861 | .0009 | .0167 | .0099 | .0096 | .0082 | .0115 | .0055 | .0213 | .0144 | .0141 | .0168 | .0976
1.5 | 0.5 | 3 | .0046 | .0045 | .0046 | .0046 | .0085 | .0872 | .0009 | .0045 | .0028 | .0027 | .0023 | .0056 | .0055 | .0090 | .0073 | .0073 | .0108 | .0928
1.5 | 1 | 1.5 | .0046 | .0045 | .0045 | .0046 | .0086 | .0870 | .0009 | .0081 | .0050 | .0047 | .0041 | .0110 | .0055 | .0127 | .0095 | .0093 | .0127 | .0981
1.5 | 2 | 0.75 | .0046 | .0046 | .0045 | .0047 | .0089 | .0871 | .0009 | .0199 | .0119 | .0114 | .0098 | .0183 | .0055 | .0246 | .0164 | .0160 | .0187 | .1055
2 | 0.5 | 4 | .0047 | .0047 | .0047 | .0048 | .0087 | .0870 | .0009 | .0044 | .0027 | .0026 | .0022 | .0056 | .0056 | .0090 | .0074 | .0074 | .0109 | .0926
2 | 1 | 2 | .0047 | .0047 | .0047 | .0047 | .0087 | .0875 | .0009 | .0081 | .0049 | .0047 | .0040 | .0122 | .0056 | .0128 | .0096 | .0094 | .0128 | .0997
2 | 2 | 1 | .0047 | .0048 | .0047 | .0048 | .0090 | .0871 | .0009 | .0214 | .0127 | .0121 | .0106 | .0243 | .0056 | .0261 | .0174 | .0169 | .0195 | .1113
4 | 0.5 | 8 | .0047 | .0047 | .0047 | .0048 | .0087 | .0866 | .0009 | .0044 | .0027 | .0026 | .0022 | .0057 | .0056 | .0090 | .0074 | .0074 | .0109 | .0923
4 | 1 | 4 | .0047 | .0047 | .0047 | .0048 | .0087 | .0876 | .0009 | .0083 | .0050 | .0049 | .0042 | .0135 | .0056 | .0129 | .0097 | .0096 | .0129 | .1010
4 | 2 | 2 | .0047 | .0047 | .0047 | .0048 | .0090 | .0889 | .0009 | .0225 | .0136 | .0131 | .0112 | .0390 | .0056 | .0272 | .0182 | .0179 | .0202 | .1280
50th Quantile
|  |  | $ABias^{2}$ | $Avar$ | $AIMSE$
$\sigma_{X_{2}}$ | $\sigma_{U_{2}}$ | Ratio | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
1 | 0.5 | 2 | .0046 | .0046 | .0046 | .0047 | .0086 | .0867 | .0008 | .0037 | .0024 | .0023 | .0020 | .0044 | .0055 | .0083 | .0070 | .0070 | .0105 | .0911
1 | 1 | 1 | .0046 | .0047 | .0046 | .0047 | .0087 | .0867 | .0008 | .0065 | .0041 | .0039 | .0034 | .0069 | .0055 | .0112 | .0088 | .0086 | .0121 | .0936
1 | 2 | 0.5 | .0046 | .0047 | .0046 | .0047 | .0088 | .0868 | .0008 | .0131 | .0082 | .0077 | .0067 | .0094 | .0055 | .0178 | .0128 | .0125 | .0156 | .0962
1.5 | 0.5 | 3 | .0047 | .0047 | .0047 | .0047 | .0086 | .0864 | .0008 | .0037 | .0023 | .0022 | .0019 | .0046 | .0055 | .0083 | .0070 | .0070 | .0105 | .0910
1.5 | 1 | 1.5 | .0047 | .0047 | .0047 | .0048 | .0087 | .0862 | .0008 | .0067 | .0043 | .0041 | .0035 | .0086 | .0055 | .0114 | .0089 | .0089 | .0122 | .0948
1.5 | 2 | 0.75 | .0047 | .0047 | .0047 | .0048 | .0088 | .0869 | .0008 | .0160 | .0097 | .0093 | .0081 | .0149 | .0055 | .0207 | .0144 | .0140 | .0168 | .1018
2 | 0.5 | 4 | .0047 | .0047 | .0047 | .0047 | .0086 | .0867 | .0008 | .0037 | .0023 | .0022 | .0019 | .0047 | .0055 | .0083 | .0070 | .0070 | .0105 | .0914
2 | 1 | 2 | .0047 | .0047 | .0047 | .0048 | .0087 | .0865 | .0008 | .0067 | .0043 | .0041 | .0036 | .0095 | .0055 | .0114 | .0090 | .0089 | .0123 | .0960
2 | 2 | 1 | .0047 | .0047 | .0046 | .0048 | .0088 | .0864 | .0008 | .0168 | .0103 | .0099 | .0085 | .0205 | .0055 | .0215 | .0149 | .0146 | .0173 | .1069
4 | 0.5 | 8 | .0047 | .0047 | .0047 | .0048 | .0086 | .0866 | .0008 | .0036 | .0023 | .0022 | .0019 | .0048 | .0055 | .0083 | .0070 | .0070 | .0105 | .0914
4 | 1 | 4 | .0047 | .0047 | .0047 | .0048 | .0088 | .0862 | .0008 | .0066 | .0042 | .0040 | .0035 | .0110 | .0055 | .0113 | .0089 | .0089 | .0123 | .0972
4 | 2 | 2 | .0047 | .0047 | .0047 | .0048 | .0089 | .0861 | .0008 | .0181 | .0115 | .0111 | .0095 | .0315 | .0055 | .0228 | .0162 | .0159 | .0184 | .1176
75th Quantile
|  |  | $ABias^{2}$ | $Avar$ | $AIMSE$
$\sigma_{X_{2}}$ | $\sigma_{U_{2}}$ | Ratio | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
1 | 0.5 | 2 | .0044 | .0044 | .0043 | .0044 | .0084 | .0859 | .0011 | .0045 | .0029 | .0027 | .0024 | .0054 | .0054 | .0089 | .0072 | .0072 | .0108 | .0914
1 | 1 | 1 | .0044 | .0043 | .0043 | .0044 | .0086 | .0862 | .0011 | .0077 | .0049 | .0047 | .0040 | .0085 | .0054 | .0121 | .0092 | .0091 | .0126 | .0947
1 | 2 | 0.5 | .0044 | .0043 | .0042 | .0044 | .0087 | .0867 | .0011 | .0171 | .0103 | .0099 | .0085 | .0122 | .0054 | .0213 | .0145 | .0143 | .0173 | .0989
1.5 | 0.5 | 3 | .0046 | .0045 | .0045 | .0046 | .0085 | .0864 | .0010 | .0045 | .0028 | .0026 | .0023 | .0057 | .0055 | .0090 | .0073 | .0072 | .0108 | .0921
1.5 | 1 | 1.5 | .0046 | .0046 | .0045 | .0046 | .0087 | .0874 | .0010 | .0077 | .0048 | .0046 | .0040 | .0107 | .0055 | .0123 | .0093 | .0092 | .0127 | .0981
1.5 | 2 | 0.75 | .0046 | .0045 | .0044 | .0046 | .0089 | .0882 | .0010 | .0198 | .0119 | .0114 | .0099 | .0186 | .0055 | .0244 | .0163 | .0160 | .0188 | .1068
2 | 0.5 | 4 | .0046 | .0046 | .0046 | .0047 | .0086 | .0865 | .0009 | .0044 | .0027 | .0026 | .0023 | .0059 | .0056 | .0090 | .0073 | .0073 | .0108 | .0924
2 | 1 | 2 | .0046 | .0046 | .0046 | .0047 | .0088 | .0883 | .0009 | .0077 | .0047 | .0045 | .0039 | .0121 | .0056 | .0123 | .0093 | .0092 | .0127 | .1004
2 | 2 | 1 | .0046 | .0046 | .0046 | .0048 | .0091 | .0904 | .0009 | .0209 | .0128 | .0121 | .0106 | .0247 | .0056 | .0255 | .0174 | .0169 | .0197 | .1151
4 | 0.5 | 8 | .0047 | .0047 | .0047 | .0047 | .0086 | .0862 | .0009 | .0044 | .0027 | .0026 | .0022 | .0060 | .0056 | .0090 | .0074 | .0073 | .0109 | .0922
4 | 1 | 4 | .0047 | .0047 | .0047 | .0048 | .0089 | .0868 | .0009 | .0077 | .0047 | .0045 | .0039 | .0137 | .0056 | .0124 | .0094 | .0093 | .0128 | .1005
4 | 2 | 2 | .0047 | .0048 | .0047 | .0049 | .0091 | .0879 | .0009 | .0217 | .0134 | .0127 | .0111 | .0384 | .0056 | .0265 | .0181 | .0176 | .0202 | .1263
95th Quantile
|  |  | $ABias^{2}$ | $Avar$ | $AIMSE$
$\sigma_{X_{2}}$ | $\sigma_{U_{2}}$ | Ratio | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
1 | 0.5 | 2 | .0034 | .0033 | .0033 | .0034 | .0073 | .0857 | .0027 | .0146 | .0076 | .0073 | .0063 | .0132 | .0061 | .0179 | .0109 | .0107 | .0136 | .0989
1 | 1 | 1 | .0034 | .0034 | .0033 | .0034 | .0073 | .0844 | .0027 | .0238 | .0123 | .0119 | .0102 | .0222 | .0061 | .0272 | .0157 | .0153 | .0175 | .1065
1 | 2 | 0.5 | .0034 | .0036 | .0033 | .0035 | .0078 | .0859 | .0027 | .0486 | .0249 | .0245 | .0206 | .0309 | .0061 | .0522 | .0283 | .0280 | .0284 | .1168
1.5 | 0.5 | 3 | .0037 | .0037 | .0037 | .0038 | .0077 | .0856 | .0026 | .0149 | .0076 | .0075 | .0063 | .0137 | .0063 | .0186 | .0113 | .0112 | .0140 | .0993
1.5 | 1 | 1.5 | .0037 | .0037 | .0037 | .0037 | .0076 | .0832 | .0026 | .0250 | .0128 | .0125 | .0106 | .0276 | .0063 | .0287 | .0165 | .0162 | .0182 | .1108
1.5 | 2 | 0.75 | .0037 | .0037 | .0035 | .0036 | .0078 | .0830 | .0026 | .0592 | .0303 | .0292 | .0250 | .0492 | .0063 | .0629 | .0337 | .0328 | .0328 | .1322
2 | 0.5 | 4 | .0037 | .0036 | .0036 | .0037 | .0077 | .0861 | .0026 | .0147 | .0077 | .0075 | .0063 | .0143 | .0063 | .0183 | .0113 | .0112 | .0140 | .1003
2 | 1 | 2 | .0037 | .0035 | .0035 | .0036 | .0075 | .0853 | .0026 | .0250 | .0129 | .0126 | .0107 | .0319 | .0063 | .0286 | .0165 | .0162 | .0182 | .1172
2 | 2 | 1 | .0037 | .0034 | .0034 | .0037 | .0078 | .0862 | .0026 | .0636 | .0318 | .0306 | .0262 | .0650 | .0063 | .0670 | .0352 | .0342 | .0340 | .1512
4 | 0.5 | 8 | .0036 | .0035 | .0035 | .0036 | .0074 | .0855 | .0026 | .0151 | .0080 | .0076 | .0066 | .0143 | .0062 | .0186 | .0115 | .0112 | .0140 | .0999
4 | 1 | 4 | .0036 | .0034 | .0034 | .0035 | .0074 | .0861 | .0026 | .0260 | .0131 | .0129 | .0109 | .0352 | .0062 | .0294 | .0165 | .0164 | .0183 | .1213
4 | 2 | 2 | .0036 | .0032 | .0032 | .0033 | .0075 | .0873 | .0026 | .0699 | .0354 | .0345 | .0293 | .1006 | .0062 | .0731 | .0386 | .0378 | .0368 | .1879

Table 14: The effect of $\beta_{2}$ on the performance of
$\widehat{\beta}_{1\tau}(t)$ obtained under the six methods: oracle, SIMEX,
FUI, FSMI, average, and naive considered for Simulation Study 5. The ABias2,
Avar, and AIMSE of the estimators for $\beta_{2}=(0.5,1.0,1.5,2.0,4.0)$;
$\tau=(0.25,0.5,0.75,0.95)$; with $n=500$, AR(1) error structure, $\rho=0.5$,
$\sigma_{X_{1}}(t)=3$, $U_{1}(t)\sim MVN(0,2.5^{2})$, $\sigma_{X_{2}}=0.5$,
$U_{2}\sim N(0,0.25^{2})$, $\sigma_{Z_{c}}=0.5$, and $p_{Z_{b}}=0.6$.

25th Quantile
---
| $ABias^{2}$ | $Avar$ | $AIMSE$
$\beta_{2}$ | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
0.5 | .0043 | .0043 | .0043 | .0044 | .0083 | .0861 | .0011 | .0036 | .0023 | .0023 | .0019 | .0037 | .0054 | .0080 | .0066 | .0066 | .0102 | .0898
1 | .0046 | .0045 | .0045 | .0046 | .0085 | .0872 | .0009 | .0044 | .0028 | .0027 | .0023 | .0053 | .0055 | .0089 | .0073 | .0073 | .0108 | .0925
1.5 | .0046 | .0045 | .0045 | .0046 | .0086 | .0876 | .0009 | .0062 | .0038 | .0036 | .0032 | .0083 | .0055 | .0107 | .0084 | .0083 | .0118 | .0959
2 | .0047 | .0047 | .0047 | .0047 | .0087 | .0875 | .0009 | .0081 | .0049 | .0047 | .0040 | .0122 | .0056 | .0128 | .0096 | .0094 | .0128 | .0997
4 | .0047 | .0047 | .0047 | .0048 | .0090 | .0889 | .0009 | .0225 | .0136 | .0131 | .0112 | .0390 | .0056 | .0272 | .0182 | .0179 | .0202 | .1280
50th Quantile
| $ABias^{2}$ | $Avar$ | $AIMSE$
$\beta_{2}$ | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
0.5 | .0045 | .0044 | .0044 | .0045 | .0083 | .0868 | .0009 | .0031 | .0020 | .0019 | .0017 | .0029 | .0054 | .0075 | .0065 | .0064 | .0100 | .0897
1 | .0046 | .0046 | .0046 | .0047 | .0086 | .0867 | .0008 | .0037 | .0024 | .0023 | .0020 | .0044 | .0055 | .0083 | .0070 | .0070 | .0105 | .0911
1.5 | .0047 | .0047 | .0047 | .0047 | .0087 | .0860 | .0008 | .0050 | .0031 | .0030 | .0026 | .0065 | .0055 | .0096 | .0078 | .0077 | .0112 | .0925
2 | .0047 | .0047 | .0047 | .0048 | .0087 | .0865 | .0008 | .0067 | .0043 | .0041 | .0036 | .0095 | .0055 | .0114 | .0090 | .0089 | .0123 | .0960
4 | .0047 | .0047 | .0047 | .0048 | .0089 | .0861 | .0008 | .0181 | .0115 | .0111 | .0095 | .0315 | .0055 | .0228 | .0162 | .0159 | .0184 | .1176
75th Quantile
| $ABias^{2}$ | $Avar$ | $AIMSE$
$\beta_{2}$ | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
0.5 | .0043 | .0043 | .0042 | .0043 | .0082 | .0854 | .0011 | .0037 | .0024 | .0023 | .0020 | .0038 | .0054 | .0079 | .0066 | .0066 | .0101 | .0892
1 | .0044 | .0044 | .0043 | .0044 | .0084 | .0859 | .0011 | .0045 | .0029 | .0027 | .0024 | .0054 | .0054 | .0089 | .0072 | .0072 | .0108 | .0914
1.5 | .0046 | .0045 | .0045 | .0046 | .0086 | .0875 | .0010 | .0059 | .0036 | .0035 | .0030 | .0082 | .0055 | .0104 | .0081 | .0081 | .0116 | .0957
2 | .0046 | .0046 | .0046 | .0047 | .0088 | .0883 | .0009 | .0077 | .0047 | .0045 | .0039 | .0121 | .0056 | .0123 | .0093 | .0092 | .0127 | .1004
4 | .0047 | .0048 | .0047 | .0049 | .0091 | .0879 | .0009 | .0217 | .0134 | .0127 | .0111 | .0384 | .0056 | .0265 | .0181 | .0176 | .0202 | .1263
95th Quantile
| $ABias^{2}$ | $Avar$ | $AIMSE$
$\beta_{2}$ | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive | Oracle | SIMEX | FUI | FSMI | Ave | Naive
0.5 | .0030 | .0029 | .0029 | .0030 | .0068 | .0835 | .0029 | .0121 | .0065 | .0063 | .0054 | .0090 | .0059 | .0150 | .0094 | .0092 | .0122 | .0925
1 | .0034 | .0033 | .0033 | .0034 | .0073 | .0857 | .0027 | .0146 | .0076 | .0073 | .0063 | .0132 | .0061 | .0179 | .0109 | .0107 | .0136 | .0989
1.5 | .0037 | .0037 | .0037 | .0038 | .0077 | .0838 | .0026 | .0193 | .0099 | .0095 | .0082 | .0205 | .0063 | .0230 | .0136 | .0133 | .0159 | .1044
2 | .0037 | .0035 | .0035 | .0036 | .0075 | .0853 | .0026 | .0250 | .0129 | .0126 | .0107 | .0319 | .0063 | .0286 | .0165 | .0162 | .0182 | .1172
4 | .0036 | .0032 | .0032 | .0033 | .0075 | .0873 | .0026 | .0699 | .0354 | .0345 | .0293 | .1006 | .0062 | .0731 | .0386 | .0378 | .0368 | .1879


11institutetext: Lincoln Institute of Agri-food Technology, University of
Lincoln, UK, 11email<EMAIL_ADDRESS>
22institutetext: School of Computer Science, University of Lincoln, UK,
22email<EMAIL_ADDRESS>
33institutetext: University of Surrey, UK, 33email<EMAIL_ADDRESS>

# Enabling Tactile Feedback for Robotic Strawberry Handling using AST Skin

Vishnu Rajendran S 11 Kiyanoush Nazari 22 Simon Parsons 11 Amir Ghalamzan E 33

###### Abstract

Acoustic Soft Tactile (AST) skin is a novel sensing technology which derives
tactile information from the modulation of acoustic waves travelling through
the skin’s embedded acoustic channels. A generalisable data-driven calibration
model maps the acoustic modulations to the corresponding tactile information
in the form of contact forces with their contact locations and contact
geometries. AST skin technology has been highlighted for its easy
customisation. As a case study, this paper discusses the possibility of using
AST skin on a custom-built robotic end effector finger for strawberry
handling. The paper delves into the design, prototyping, and calibration
method to sensorise the end effector finger with AST skin. A real-time force-
controlled gripping experiment is conducted with the sensorised finger to
handle strawberries by their peduncle. The finger could successfully grip the
strawberry peduncle by maintaining a preset force of 2 N with a maximum Mean
Absolute Error (MAE) of 0.31 N over multiple peduncle diameters and strawberry
weight classes. Moreover, this study sets confidence in the usability of AST
skin in generating real-time tactile feedback for robot manipulation tasks.

###### Keywords:

Soft Tactile skin Acoustics Grip-force control.

##  1 Introduction

Figure 1: Demonstration of AST sensing technology to generate tactile force
feedback for a robot manipulation task: The AST Skin is attached to a custom
end effector finger (the AST finger), and the force feedback from the AST
finger is used in performing force-controlled handling of strawberries by
gripping their peduncle. This finger assembly is mounted on an SMC gripper for
validation purposes, but later, it will be attached to a custom-built
strawberry harvesting end-effector

Soft tactile sensors are generally used in manipulation tasks that involve
handling soft deformable objects. These sensors are typically characterised by
soft deformable skin whose deformation is converted to tactile feedback as
tactile readings using an integrated transduction mechanism. There has been
continuous progress in the development of soft tactile sensors using various
skin materials and transduction mechanisms [16, 23].

Various robotic applications call for customisable soft tactile sensors. This
is especially important to generate tactile feedback from robot counterparts
for the effective execution of manipulation tasks. The customisation can be in
the form of sensor shape and size, material and sensing specifications (such
as sensitivity, measurement range) to fit the requirement. Since the
conventional soft-tactile sensors have their transduction mechanism integrated
with the sensing skin, customisation is not straightforward. Usually, the
transduction mechanism uses electric principles (e.g., resistive [27],
capacitive [9], piezoelectric [19], magnetic [15, 4], impedance [26]) non-
electric principles (e.g., camera-based [24, 6, 8, 2, 18], fluid-based [7]) or
their combinations [11]. When it comes to sensors using electric principles,
they have closely knit circuitry embedded beneath the sensing surface and it
is the camera with its accessories for camera-based sensors. Similarly, fluid
pressure-based sensors have intricate fluid lines integral to the sensing
skin. Hence, customising the skin requires considerable effort to also
customise the integrated transduction mechanism elements. However, the
possibility of keeping the sensing skin and transduction elements modular
offers a better scope for customisation. This is possible when the
transduction mechanism uses propagating mediums (such as vision/light, fluids
or acoustics) to sense the disturbances on the skin by analysing their
respective modulations. While using vision, the camera and its accessories
result in a bulky form factor of the sensor [25]. Hence, using a camera
provides limitations in minimising the form factor of the whole sensor.
Moreover, fluid-based methods are known for their delayed measurement
responses [5]. In this context, acoustics has a promising scope. It only
requires minimal hardware components, typically a speaker and a microphone
[10, 22]. Most importantly, it can potentially derive diverse tactile
information such as force, contact location, temperature and contact material
nature [22].

With the modularity concept for customisation and harnessing the capabilities
of acoustics, a low-cost novel tactile sensing technology, namely, Acoustic
Soft Tactile skin (AST Skin), has been developed [13]. This sensing technology
keeps the soft sensing skin and transduction mechanism elements modular. The
sensing skin only needs hollow acoustic channels beneath the sensing surface.
A speaker and microphone unit form the transduction elements that need to be
connected to the acoustic channel. A reference acoustic wave emitted by a
speaker propagates through these channels and returns to the microphone. When
external forces act on the sensing surface, they deform the channel; hence,
the modulation of the acoustic waves varies. This variation is used to measure
tactile interactions. Moreover, AST skin technology uses a generalisable data-
driven machine-learning model for its calibration. Such a calibration model
can account for the variation of the sensing skin’s form factor or material
change that may arise from customisation, which is complex to establish
through an analytical model. AST skin technology has been proven to measure
normal forces, 2D contact locations, and contact surface geometries [13, 21].
Moreover, the skin’s resilience to external sound disturbances is also
validated.

In previous work, various skin configurations and their impact on tactile
measurements are studied in detail [13]. This paper evaluates the
customisability of AST skin by integrating it on a custom end effector finger
(the AST finger) and tests its sensing capabilities (refer to figure 1). This
finger has been developed as a retrofit to a strawberry harvesting robotic
end-effector. The AST finger is expected to grip the strawberry peduncle and
provide tactile feedback about the peduncle grip status when the robot’s
vision unit cannot confirm the grip due to occlusions. Moreover, the feedback
from the AST finger facilitates effective grip force control for the
strawberry handling involved in the harvesting cycle. This paper does not
discuss the details of the harvesting end effector mechanism; instead, the AST
finger is tested using a general purpose two jaw gripping end effector. The
proposed AST finger has a reduced size factor compared to other sensorised
gripper fingers available for strawberry handling [20]. The size of the
fingers impacts its usability in handling strawberries grown in clusters.
Moreover, gripping the strawberry body while harvesting can lead to fruit
bruising [1], reducing its shelf life. Thus, the AST finger aims to grip the
peduncle rather than the fruit body.

The remainder of the paper outlines the methodology to develop the AST finger
and its sensing performance during the real-time strawberry peduncle gripping
trials.

##  2 Methods

This section discusses the methodology for the end-to-end development of the
AST finger. Initially, it outlines a study to determine the maximum gripping
force that can be applied to the strawberry peduncle. This maximum grip force
value helps to decide the force range for which the AST skin needs to be
calibrated. Later in this section, the design of the AST finger, prototyping
details, and the adopted data-driven calibration approach are detailed.

###  2.1 Study on Gripping Force

During harvesting, the robot’s end effector fingers should grip the peduncle
before the cutting and continue holding it until the detached strawberry is
placed in the onboard storage (e.g., punnet). Applying the proper force is
essential to prevent the strawberry from slipping off the fingers during this
manipulation cycle. There are two reasons for slippage: applying insufficient
force or crushing the peduncle due to excessive force.

To determine the safe range of force that can be applied to the peduncle, a
compression study was conducted with peduncle samples from two breeds of
strawberries [14]. The study found that the safe limit of gripping force can
be approximated to 10 N. Hence, the AST finger is calibrated for a range of 0
to 10 N.

###  2.2 AST Finger Design

The AST finger comprises of a back plate on which the AST skin is attached
(refer to figure 2(a)). The back plate is 3D printed using PLA material, while
the AST skin is made by moulding. The AST skin is moulded in two halves
separately and then joined together. Figure 2(b) shows the mould design. The
two moulds are 3D printed, and a Silicone rubber compound with a 10 A shore
hardness value (PlatSil Gel 10) is poured into the moulds. After the curing
period, the two halves of skin are joined together to form the whole skin.
Later, the AST skin is attached to the back plate. The same silicone material
is used as the adhesive to join the skin halves and to attach it to the back
plate. The acoustic channel is given a 3 mm diameter and is at 1 mm below the
sensing surface of the skin so that a gentle touch can affect the channel
geometry. The AST finger has two ports to connect the modular acoustic
hardware components (refer to figure 2(a)). In this prototype, a regular
headphone speaker and microphone are connected to the port via a flexible tube
as a test case. In the future, a miniature speaker-microphone will be used.

As a mating finger for the AST finger to grip the strawberry peduncle, a dummy
finger has been 3D printed with PLA material. To actuate both fingers for the
gripping trials, they are attached to an SMC gripper as shown in figure 1 and
5(a).

(a)

(b)

Figure 2: (a). AST finger with its mating dummy finger, (b). Skin mould: The
two semi-cylindrical projections in each mould help to create the skin’s
cylindrical cavity while joining two halves of the cured silicone

###  2.3 Skin Calibration

As mentioned in Section 1, the AST skin uses a data-driven calibration model.
In here, the AST finger is only calibrated to measure contact forces. So, the
data set used for the calibration only involves the Fast Fourier Transform
(FFT) of the modulated acoustic wave and the corresponding force that causes
the modulation. To derive this dataset, a robot-based calibration setup is
used to apply known forces on the sensing surface, and the corresponding FFT
of the acoustic wave modulation is recorded against the applied force. The
robot wrist is fitted with an axial load cell-peg assembly, and the peg has a
cylindrical profile with a diameter of 1.5 mm (refer to the figure 3, right).
This cylindrical profile simulates the average diameter of the strawberry
peduncle considered for studying the gripping force limit. During actual
deployment on the strawberry harvesting end effector, the finger will be
calibrated with a diverse data set generated with different diameter pegs
covering the possible peduncle diameters and orientations ranges.

Figure 3: AST finger calibration: The calibrated area of the AST finger
(left), the calibration set up consisting of 6 DOF UFactory xArm, axial load
cell with peg attached to the robot arm wrist (middle), the calibration peg
with cylindrical profiled edge to simulate the shape of strawberry peduncle
(right)

As a test case, only a 14 mm section of the skin is calibrated, and this
section was divided into seven subsections at a 2 mm gap, as shown in the
figure 3 (left). The robot drives the peg vertically downwards at subsections
with an equal increment of 0.5 mm from the skin surface until the force value
reaches 10 N. During this, the modulated FFT data is recorded against the
force values. The resulting data set is used to train the regression machine
learning model to predict force from FFT data. To select the best-suited
regression model, MATLAB regression learner is used. Different regression
models are trained with a data partition of 90:10 ratio and 10% cross-
validation folds (refer to table 1). The Exponential Gaussian Process
regression model is selected based on its lower validation error (RMSE: 0.27)
and used as the calibration model for the skin.

This calibration model is tested with the 10% test data, and the predictions
obtained are presented in the table 2. About 91.1% of predictions are made
within $\pm 0.5$ N tolerances, while it could make 99 % of predictions at $\pm
1.0$ N tolerances. Moreover, the mean absolute error of the predictions is
0.16 N with a standard deviation of 0.22 N.

Table 1: Comparison of Regression models using MATLAB Regression Learner Regression Models | Validation Error (RMSE)
---|---
Linear Regression |
Linear | 0.91
Interactions Linear | 0.56
Robust | 0.97
Step-wise Linear | 0.56
Regression Trees |
Fine Tree | 0.42
Medium Tree | 0.43
Coarse Tree | 0.46
Support Vector Machines |
Linear | 0.95
Quadratic | 12
Cubic | 101.93
Fine Gaussian | 0.43
Medium Gaussian | 0.49
Coarse Gaussian | 0.59
Gaussian Process |
Rational Quadratic | 0.28
Squared Exponential | 0.40
Matern 5/2 | 0.34
Exponential | 0.27
Ensemble of Trees |
Boosted Trees | 0.50
Bagged Trees | 0.37
Neural Networks |
Narrow Neural | 0.46
Medium Neural | 0.43
Wide Neural Network | 0.41
Bi-layered Neural Network | 0.44
Tri-layered Neural Network | 0.42
Table 2: Prediction performance of the calibration model | Absolute
---
Error (N)
| Percentage
---
Predictions (%)
$\pm 0.5$ | 91.13924
$\pm 1.0$ | 99.27667
$\pm 1.5$ | 99.81917
$\pm 2.0$ | 100

##  3 Performance Evaluation of AST Finger

This section evaluates the usability of the AST finger’s feedback for force-
controlled gripping of the strawberry peduncle.

For this, a series of trials are conducted where the AST finger grips
strawberries by their peduncles for a simple pick-and-drop sequence. The AST
and dummy finger are connected to an SMC LEZH gripping end effector using an
extension adaptor, as depicted in figure 5(a). The end effector with the
fingers is then attached to a Franka Emika robotic arm. This SMC gripper
increases the gripping force by closing the fingers in adjustable grip widths.

For the gripping trials, five strawberries are collected with 35-60 mm
peduncle left on them (refer to figure 4). These strawberries have an average
weight and peduncle diameter of 0.141 N and 1.73 mm, respectively (refer to
table 3). A gripping force of 4 N is assumed for gripping these strawberries
during the pick-and-drop manipulation cycle. With this gripping force, a mass
of 91 grams (approximately 1 N) can be manipulated when the manipulator
acceleration is kept below 1.0 m/s2 and when the gripping surface of the
finger has a friction coefficient of 0.5 (refer to the equation 1 and with S
as 2). Generally, the friction coefficient of silicone material is closer to
1, but it is assumed to be 0.5 here, providing an additional safety factor.
Since the AST finger assembly forms a 2-jaw finger configuration, the total
gripping force of 4 N can be considered shared between the two fingers. Hence,
2 N is the target force to be read from the AST finger to complete the
gripping action before the pick-and-drop traverse.

$F_{g}=\frac{m.(g+a).S}{\mu}$ (1)

where; ’Fg’ is the net gripping force (N), ’m’ is the mass to be handled (Kg),
’g’ is the acceleration due to gravity (m/s2), ’$\mu$’ is the coefficient of
friction, and ’S’ is factor of safety [17].

Figure 4: Strawberry samples selected for the gripping trials Table 3: Features of strawberries used for the gripping trials Sample | Weight (N) | Peduncle Diameter (mm)
---|---|---
1 | 0.084 | 1.24
2 | 0.111 | 1.38
3 | 0.155 | 1.88
4 | 0.176 | 1.90
5 | 0.181 | 2.29
Mean | 0.141 | 1.73

###  3.1 Gripper Force Controller

Tactile feedback-based reactive grip force control is commonly used for grip
stabilisation in robotic manipulation [3]. Here, the real-time force feedback
($f_{m}$) from the AST finger is leveraged for such a grip force controller.
As such, the controller is set to stop gripping by settling the measured grip
force ($f_{m}$) within a safety zone close to the desired grip force
$[f_{d}-\epsilon,f_{d}+\epsilon]$ (refer to equation 2). As described earlier,
the desired grip force ($f_{d}$) is 2 N. For this control implementation, the
grip width of the SMC gripper ($g_{t}$) is changed with a step size of
$\sigma_{h}$ to close the AST finger assembly. Therefore, the grip width
command at each time step reads as follows:

$g_{t+1}=\begin{cases}g_{t}-\sigma_{h}&\text{if }f_{m}<f_{d}-\epsilon,\\\
g_{t}&\text{if }f_{d}-\epsilon<f_{m}<f_{d}+\epsilon,\\\
g_{t}+\sigma_{h}&f_{m}>f_{d}+\epsilon.\end{cases}$ (2)

The grasp width change’s step size ($\sigma_{h}$) is tuned based on the target
grip force to avoid overshooting in grip force changes or significant control
lags. In our experimental trials, the hyper-parameters are defined as
$\sigma_{h}=1$ mm and $\epsilon=0.1$ N.

###  3.2 Experimental Trials

The strawberries are placed individually in the picking location on the
workbench with the peduncle upright (refer to figure 5(b)). At the start of
the trial, the fingers will be positioned so that the peduncle is between the
AST finger (calibrated area) and the dummy finger. Later, the end effector
controller is triggered manually, and the fingers start closing with equal
increments ($\sigma_{h}$ = 1 mm) until the gripping force value reaches the
target gripping force ($f_{d}$=2N). Afterwards, the robot arm moves in a fixed
trajectory towards the dropping point, where the strawberry is released into a
punnet. The experiment scenario is shown in the figure 5(b).

(a)

(b)

Figure 5:  (a). AST finger assembly connected to the SMC gripper for gripping
trials, (b). Experimental setup for the gripping trials: Point A is the
picking point, B is a via point between A and C, and C is the dropping point

This gripping trial is repeated five times for each strawberry sample, and the
respective force readings from the AST finger ($f_{m}$) are logged in real-
time to evaluate the performance. The upcoming section will discuss the
analysis of the force readings.

##  4 Results and Discussions

A sample of the force readings recorded during a gripping trial is shown in
the figure 6. Three sections can be visualised from the force profile. In
section S1, the peduncle comes in contact with the fingers. After S1, contact
force increases as the fingers continue closing/gripping until the set limit
($f_{d}$) of 2 N is reached (in section S2). Later, the end effector travels
from the picking to the dropping location (from point A to C via B), and force
readings during this traverse are between S2 and S3 in figure 6. In section
S3, the grip force suddenly drops due to the opening of fingers to release the
strawberry into the punnet.

From this force profile, the readings ($f_{m}$) between S2 and S3 signify how
close the grip force settles around the desired grip force ($f_{d}$= 2N). The
table 4 presents the Mean Absolute Error (MAE), which quantifies the variation
of $f_{m}$ from $f_{d}$. It has been studied that the maximum MAE is about
0.31 N, which is negligible for this task as it doesn’t damage any of the
peduncles. It is to be noted that this variation in sensor reading can be for
two reasons: (i). the effect of swinging the strawberry during the traverse
from picking to dropping location, or (ii). the inherent calibration error of
the AST skin, as discussed in section 2.3.

The results of the above study provide confidence in the usability of AST
skin-embedded fingers in force-controlled gripping tasks using its feedback.

Figure 6: AST readings ($f_{m}$) during a gripping sequence  Table 4: Mean Absolute Error (MAE) recorded for each gripping trial against the set grip force value of 2 N Sample | Trials |  Average MAE
---|---|---
1 | 2 | 3 | 4 | 5
1 | 0.079 | 0.167 | 0.143 | 0.102 | 0.089 | 0.116
2 | 0.082 | 0.684 | 0.560 | 0.047 | 0.182 | 0.311
3 | 0.219 | 0.307 | 0.401 | 0.178 | 0.358 | 0.293
4 | 0.238 | 0.202 | 0.095 | 0.050 | 0.222 | 0.161
5 | 0.106 | 0.059 | 0.142 | 0.031 | 0.115 | 0.090

##  5 Conclusion

This paper tests the usability of Acoustic Soft Tactile (AST) skin as a
customisable tactile skin for enabling tactile feedback for robot manipulation
tasks. This has been evaluated by building and testing an end effector finger
with AST skin to perform force-controlled strawberry gripping. The AST finger
facilitated the end effector to handle the strawberry by gripping its peduncle
with a set target force of 2 N with a maximum mean absolute error of 0.31 N.
As a future work, the AST fingers will be integrated into the custom-built end
effector and tested in the field by mounting it on a strawberry harvesting
robot [12]. Moreover, the entire skin area will be calibrated to measure the
gripping force with the description of the peduncle location on the skin
surface. This contact localisation approach helps confirming a successful grip
at the target locked by the robot’s integrated vision system.

## References

  * [1] Aliasgarian, S., Ghassemzadeh, H.R., Moghaddam, M., Ghaffari, H., et al.: Mechanical damage of strawberry during harvest and postharvest operations. World Applied Sciences Journal 22(7), 969–974 (2013)
  * [2] Chen, Z., Zhang, S., Luo, S., Sun, F., Fang, B.: Tacchi: A pluggable and low computational cost elastomer deformation simulator for optical tactile sensors. IEEE Robotics and Automation Letters 8(3), 1239–1246 (2023). https://doi.org/10.1109/LRA.2023.3237042
  * [3] Deng, Z., Jonetzko, Y., Zhang, L., Zhang, J.: Grasping force control of multi-fingered robotic hands through tactile sensing for object stabilization. Sensors 20(4), 1050 (2020)
  * [4] Diguet, G., Froemel, J., Muroyama, M., Ohtaka, K.: Tactile sensing using magnetic foam. Polymers 14(4), 834 (2022)
  * [5] Fujiwara, E., de Oliveira Rosa, L.: Agar-based soft tactile transducer with embedded optical fiber specklegram sensor. Results in Optics 10, 100345 (2023)
  * [6] Gomes, D.F., Luo, S.: Geltip tactile sensor for dexterous manipulation in clutter. In: Tactile Sensing, Skill Learning, and Robotic Dexterous Manipulation, pp. 3–21. Elsevier (2022)
  * [7] Gong, D., He, R., Yu, J., Zuo, G.: A pneumatic tactile sensor for co-operative robots. Sensors 17(11), 2592 (2017)
  * [8] Lambeta, M., Chou, P.W., Tian, S., Yang, B., Maloon, B., Most, V.R., Stroud, D., Santos, R., Byagowi, A., Kammerer, G., et al.: Digit: A novel design for a low-cost compact high-resolution tactile sensor with application to in-hand manipulation. IEEE Robotics and Automation Letters 5(3), 3838–3845 (2020)
  * [9] Li, Q., Ullah, Z., Li, W., Guo, Y., Xu, J., Wang, R., Zeng, Q., Chen, M., Liu, C., Liu, L.: Wide-range strain sensors based on highly transparent and supremely stretchable graphene/ag-nanowires hybrid structures. Small 12(36), 5058–5065 (2016)
  * [10] Ono, M., Shizuki, B., Tanaka, J.: Sensing touch force using active acoustic sensing. In: Proceedings of the Ninth International Conference on Tangible, Embedded, and Embodied Interaction. pp. 355–358 (2015)
  * [11] Park, K., Yuk, H., Yang, M., Cho, J., Lee, H., Kim, J.: A biomimetic elastomeric robot skin using electrical impedance and acoustic tomography for tactile sensing. Science Robotics 7(67), eabm7187 (2022)
  * [12] Parsa, S., Debnath, B., Khan, M.A., E, A.G.: Modular autonomous strawberry picking robotic system. Journal of Field Robotics (2023)
  * [13] Rajendran, V., Mandil, W., Parsons, S., E, A.G.: Acoustic soft tactile skin (ast skin). arXiv preprint arXiv:2303.17355 (2023)
  * [14] Rajendran, V., Parsa, S., Parsons, S., E, A.G.: Peduncle gripping and cutting force for strawberry harvesting robotic end-effector design. In: 2022 4th International Conference on Control and Robotics (ICCR). pp. 59–64 (2022). https://doi.org/10.1109/ICCR55715.2022.10053882
  * [15] Rehan, M., Saleem, M.M., Tiwana, M.I., Shakoor, R.I., Cheung, R.: A soft multi-axis high force range magnetic tactile sensor for force feedback in robotic surgical systems. Sensors 22(9), 3500 (2022)
  * [16] Roberts, P., Zadan, M., Majidi, C.: Soft tactile sensing skins for robotics. Current Robotics Reports 2, 343–354 (2021)
  * [17] for Robots, I.P.: Calculation of gripping force. https://en.iprworldwide.com/calculation-of-gripping-force/ ([Accessed 20052022])
  * [18] Sferrazza, C., D’Andrea, R.: Design, motivation and evaluation of a full-resolution optical tactile sensor. Sensors 19(4), 928 (2019)
  * [19] Song, K., Kim, S.H., Jin, S., Kim, S., Lee, S., Kim, J.S., Park, J.M., Cha, Y.: Pneumatic actuator and flexible piezoelectric sensor for soft virtual reality glove system. Scientific reports 9(1), 8988 (2019)
  * [20] Visentin, F., Castellini, F., Muradore, R.: A soft, sensorized gripper for delicate harvesting of small fruits. Computers and Electronics in Agriculture 213, 108202 (2023)
  * [21] Vishnu, R.S., Parsons, S., E., A.G.: Single and bi-layered 2-d acoustic soft tactile skin. In: 2024 IEEE 7th International Conference on Soft Robotics (RoboSoft). pp. 133–138 (2024). https://doi.org/10.1109/RoboSoft60065.2024.10522056
  * [22] Wall, V., Zöller, G., Brock, O.: Passive and active acoustic sensing for soft pneumatic actuators. The International Journal of Robotics Research 42(3), 108–122 (2023)
  * [23] Wang, C., Liu, C., Shang, F., Niu, S., Ke, L., Zhang, N., Ma, B., Li, R., Sun, X., Zhang, S.: Tactile sensing technology in bionic skin: A review. Biosensors and Bioelectronics 220, 114882 (2023)
  * [24] Ward-Cherrier, B., Pestell, N., Cramphorn, L., Winstone, B., Giannaccini, M.E., Rossiter, J., Lepora, N.F.: The tactip family: Soft optical tactile sensors with 3d-printed biomimetic morphologies. Soft robotics 5(2), 216–227 (2018)
  * [25] Wei, Y., Xu, Q.: An overview of micro-force sensing techniques. Sensors and Actuators A: Physical 234, 359–374 (2015)
  * [26] Wu, H., Zheng, B., Wang, H., Ye, J.: New flexible tactile sensor based on electrical impedance tomography. Micromachines 13(2), 185 (2022)
  * [27] Zimmer, J., Hellebrekers, T., Asfour, T., Majidi, C., Kroemer, O.: Predicting grasp success with a soft sensing skin and shape-memory actuated gripper. In: 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). pp. 7120–7127. IEEE (2019)


* (11) M. König, S. Wiedmann, C. Brüene, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, S.-C. Zhang, ”Quantum Spin Hall Insulator State in HgTe Quantum Wells”, Science 318, 766 (2007).
  * (12) M. He, H. Sun, and Q. L. He, ”Topological insulator: Spintronics and quantum computations”, Front. Phys. 14, 43401 (2019).
  * (13) A. Miyake, ”Quantum Computation on the Edge of a Symmetry-Protected Topological Order”, Phys. Rev. Lett. 105, 040501 (2010).
  * (14) D. Pesin and A. H. MacDonald, ”Spintronics and pseudospintronics in graphene and topological insulators”, Nat. Mater. 11, 409 (2012).
  * (15) A. Stern and N. H. Lindner, ”Topological quantum computation–from basic concepts to first experiments”, Science 339, 1179 (2013).
  * (16) R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, P. Fendley, C. Nayak, Y. Oreg, A. Stern, E Berg, K. Shtengel, and M. P. A. Fisher, ”Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure”, Phys. Rev. X 4, 011036 (2014).
  * (17) R. W. Bomantara and J. Gong, ”Quantum computation via Floquet topological edge modes”, Phys. Rev. B 98, 165421 (2018).
  * (18) A. Kitaev, ”Periodic table for topological insulators and superconductors”, AIP Conference Proceedings 1134, 22 (2009).
  * (19) A. Altland and M. R. Zirnbauer, ”Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures”, Phys. Rev. B 55, 1142 (1997).
  * (20) M. Z. Hasan and C. L. Kane, ”Colloquium: Topological insulators”, Rev. Mod. Phys. 82, 3045 (2010).
  * (21) B. A. Bernevig, ”Topological Insulators and Topological Superconductors”, (Princeton University Press, 2013).
  * (22) R. D. King-Smith and D. Vanderbilt, ”Theory of polarization of crystalline solids”, Phys. Rev. B 47, 1651(R) (1993).
  * (23) D. Vanderbilt and R. D. King-Smith, ”Electric polarization as a bulk quantity and its relation to surface charge”, Phys. Rev. B 48, 4442 (1993).
  * (24) N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, ”Maximally localized Wannier functions: Theory and applications”, Rev. Mod. Phys. 84, 1419 (2012).
  * (25) D. Vanderbilt, ”Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators”, (Cambridge University Press, 2018).
  * (26) J.-H. Park, G. Yang, J. Klinovaja, P. Stano, and D. Loss, ”Fractional boundary charges in quantum dot arrays with density modulation” Phys. Rev. B 94, 075416 (2016).
  * (27) M. Thakurathi, J. Klinovaja, and D. Loss, ”From fractional boundary charges to quantized Hall conductance”, Phys. Rev. B 98, 245404 (2018).
  * (28) M. Pletyukhov D. M. Kennes, J. Klinovaja, D. Loss, and H. Schoeller, ”Surface charge theorem and topological constraints for edge states: Analytical study of one-dimensional nearest-neighbor tight-binding models”, Phys. Rev. B 101, 165304 (2020).
  * (29) M. Pletyukhov, D. M. Kennes, J. Klinovaja, D. Loss, and H. Schoeller, ”Topological invariants to characterize universality of boundary charge in one-dimensional insulators beyond symmetry constraints”, Phys. Rev. B 101, 161106(R) (2020).
  * (30) M. Pletyukhov, D. M. Kennes, K. Piasotski, J. Klinovaja, D. Loss, and H. Schoeller, ”Rational boundary charge in one-dimensional systems with interaction and disorder”, Phys. Rev. Research 2, 033345 (2020).
  * (31) C. Weber, K. Piasotski, M. Pletyukhov, J. Klinovaja, D. Loss, H. Schoeller, and D. M. Kennes, ”Universality of Boundary Charge Fluctuations”, Phys. Rev. Lett. 126, 016803 (2021).
  * (32) K. Piasotski, M. Pletyukhov, C. S. Weber, J. Klinovaja, D. M. Kennes, and H. Schoeller, ”Universality of Abelian and non-Abelian Wannier functions in generalized one-dimensional Aubry-André-Harper models”, Phys. Rev. Research 3, 033167 (2021).
  * (33) N. Müller, K. Piasotski, D. M. Kennes, H. Schoeller, and M. Pletyukhov, ”Universal properties of boundary and interface charges in multichannel one-dimensional models without symmetry constraints”, Phys. Rev. B 104, 125447 (2021).
  * (34) S. Miles, D. M. Kennes, H. Schoeller, and M. Pletyukhov, ”Universal properties of boundary and interface charges in continuum models of one-dimensional insulators”, Phys. Rev. B 104, 155409 (2021).
  * (35) P. G. Harper, ”Single Band Motion of Conduction Electrons in a Uniform Magnetic Field”, Proc. Phys. Soc. A 68, 874 (1955).
  * (36) S. Aubry and G. André, ”Analyticity breaking and Anderson localization in incommensurate lattices”, Ann. Israel Phys. Soc. 3, 133 (1980).
  * (37) W. Kohn, ”Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms”, Phys. Rev. Lett. 76, 3168 (1996).
  * (38) E. Prodan and W. Kohn, ”Nearsightedness of electronic matter”, PNAS 102, 11635 (2005).
  * (39) R. Jackiw and C. Rebbi, ”Solitons with fermion number $1/2$”, Phys. Rev. D 13, 3398 (1976).
  * (40) W.-P. Su, J. R. Schrieffer, and A. J. Heeger, ”Solitons in Polyacetylene”, Phys. Rev. Lett. 42, 1698 (1979); ”Soliton excitations in polyacetylene”, Phys. Rev. B 22, 2099 (1980).
  * (41) M. J. Rice and E.J. Mele, ”Elementary Excitations of a Linearly Conjugated Diatomic Polymer”, Phys. Rev. Lett. 49, 1455 (1982).
  * (42) A. J. Heeger, S. Kivelson, J.R. Schrieffer, and W.-P. Su, ”Solitons in conducting polymers”, Rev. Mod. Phys. 60, 781 (1988).
  * (43) J. Goldstone and F. Wilczek, ”Fractional Quantum Numbers on Solitons”, Phys. Rev. Lett. 47, 986 (1981).
  * (44) I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, and M. Wimmer, ”Spin-orbit interaction in InSb nanowires”, Phys. Rev. B 91, 201413(R) (2015).
  * (45) C. Kloeffel, M. Trif, and D. Loss, ”Strong spin-orbit interaction and helical hole states in Ge/Si nanowires”, Phys. Rev. B 84, 195314 (2011).
  * (46) D. Rainis, L. Trifunovic, J. Klinovaja, and D. Loss, ”Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions”, Phys. Rev. B 87, 024515 (2013).
  * (47) J. Klinovaja and D. Loss, ”Composite Majorana fermion wave functions in nanowires”, Phys. Rev. B 86, 085408 (2012).
  * (48) F. Maier, J. Klinovaja, and D. Loss, ”Majorana fermions in Ge/Si hole nanowires”, Phys. Rev. B 90, 195421 (2014).
  * (49) J. Klinovaja, P. Stano, and D. Loss, ”Transition from Fractional to Majorana Fermions in Rashba Nanowires”, Phys. Rev. Lett. 109, 236801 (2012).
  * (50) D. Liang and X. P.A. Gao, ”InAs Nanowire Transistors as Gas Sensor and the Response Mechanism”, Nano Lett. 12, 3263 (2012).
  * (51) T. Nautiyal, T. H. Rho, and K. S. Kim, ”Nanowires for spintronics: A study of transition-metal elements of groups $8–10$”, Phys. Rev. B 69, 193404 (2004).
  * (52) A. Iorio, M. Rocci, L. Bours, M. Carrega, V. Zannier, L. Sorba, S. Roddaro, F. Giazotto, and E. Strambini, ”Vectorial Control of the Spin–Orbit Interaction in Suspended InAs Nanowires”, Nano Lett. 19, 652 (2018).
  * (53) F. Wilczek and A. Zee, ”Appearance of Gauge Structure in Simple Dynamical Systems”, Phys. Rev. Lett. 52, 2111 (1984).
  * (54) G. Juzeliūnas, J. Ruseckas, A. Jacob, L. Santos, and P. Öhberg, ”Double and Negative Reflection of Cold Atoms in Non-Abelian Gauge Potentials”, Phys. Rev. Lett. 100, 200405 (2008).
  * (55) J. Ruseckas, G. Juzeliūnas, P. Öhberg, and M. Fleischhauer, ”Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States”, Phys. Rev. Lett. 95, 010404 (2005).
  * (56) M. Burrello and A. Trombettoni, ”Non-Abelian Anyons from Degenerate Landau Levels of Ultracold Atoms in Artificial Gauge Potentials”, Phys. Rev. Lett. 105, 125304 (2010).
  * (57) I. I. Satija, D. C. Dakin, and C. W. Clark, ”Metal-Insulator Transition Revisited for Cold Atoms in Non-Abelian Gauge Potentials”, Phys. Rev. Lett. 97, 216401 (2006).
  * (58) V. Pietilä and M. Möttönen, ”Non-Abelian Magnetic Monopole in a Bose-Einstein Condensate”, Phys. Rev. Lett. 102, 080403 (2009).
  * (59) Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, ”Synthetic magnetic fields for ultracold neutral atoms”, Nature 462, 628 (2009).
  * (60) P. San-Jose, J. González, and F. Guinea, ”Non-Abelian Gauge Potentials in Graphene Bilayers”, Phys. Rev. Lett. 108, 216802 (2012).
  * (61) L.-J. Yin, J.-B. Qiao, Wei-Jie Zuo, Wen-Tian Li, and Lin He, ”Experimental evidence for non-Abelian gauge potentials in twisted graphene bilayers”, Phys. Rev. B 92, 081406(R) (2015).
  * (62) S. K. Ghosh, P. Vyasanakere, and V. B. Shenoy, ”Trapped fermions in a synthetic non-Abelian gauge field”, Phys. Rev. A 84, 053629 (2011).
  * (63) H. Tercas, H. Flayac, D. D. Solnyshkov, and G. Malpuech, ”Non-Abelian Gauge Fields in Photonic Cavities and Photonic Superfluids”, Phys. Rev. Lett. 112, 066402 (2014).
  * (64) N. Marzari and D. Vanderbilt, ”Maximally localized generalized Wannier functions for composite energy bands”, Phys. Rev. B 56, 12847 (1997).
  * (65) A. Komnik and S. Heinze, ”Analytical results for the Green’s functions of lattice fermions”, Phys. Rev. B 96, 155103 (2017).
  * (66) A. Zazunov, R. Egger, M. Alvarado, and A. Levy Yeyati, ”Josephson effect in multiterminal topological junctions”, Phys. Rev. B 96, 024516 (2017).
  * (67) A. Zazunov, A. Iks, M. Alvarado, A. Levy Yeyati, and R. Egger, ”Josephson effect in junctions of conventional and topological superconductors”, Beilstein J. Nanotechnol. 9, 1659 (2018).
  * (68) L. Arrachea, G. S. Lozano, and A. A. Aligia, ”Thermal transport in one-dimensional spin heterostructures”, Phys. Rev. B 80, 014425 (2009).
  * (69) Y. Peng, Y. Bao, and F. von Oppen, ”Boundary Green functions of topological insulators and superconductors”, Phys. Rev. B 95, 235143 (2017).
  * (70) A. Zazunov, R. Egger, and A. Levy Yeyati, ”Low-energy theory of transport in Majorana wire junctions”, Phys. Rev. B 94, 014502 (2016).
  * (71) M. Alvarado, A. Iks, A. Zazunov, R. Egger, and A. Levy Yeyati, ”Boundary Green’s function approach for spinful single-channel and multichannel Majorana nanowires”, Phys. Rev. B 101, 094511 (2020).
  * (72) E. W. Cheney and D. R. Kincaid, ”Numerical Mathematics and Computing”, (Brooks/Cole Publishing Co., USA, 2007).


# Spatiotemporal control of laser intensity through cross-phase modulation

T.T. Simpson<EMAIL_ADDRESS>Laboratory for Laser Energetics,
University of Rochester, Rochester, NY 14623, USA  D. Ramsey  Laboratory for
Laser Energetics, University of Rochester, Rochester, NY 14623, USA  P. Franke
Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14623,
USA  K. Weichman  Laboratory for Laser Energetics, University of Rochester,
Rochester, NY 14623, USA  M.V. Ambat  Laboratory for Laser Energetics,
University of Rochester, Rochester, NY 14623, USA  D. Turnbull  Laboratory for
Laser Energetics, University of Rochester, Rochester, NY 14623, USA  D.H.
Froula  Laboratory for Laser Energetics, University of Rochester, Rochester,
NY 14623, USA  J.P. Palastro<EMAIL_ADDRESS>Laboratory for Laser
Energetics, University of Rochester, Rochester, NY 14623, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Spatiotemporal pulse shaping provides control over the trajectory and range of
an intensity peak. While this control can enhance laser-based applications,
the optical configurations required for shaping the pulse can constrain the
transverse or temporal profile, duration, or orbital angular momentum (OAM).
Here we present a novel technique for spatiotemporal control that mitigates
these constraints by using a “stencil” pulse to spatiotemporally structure a
second, primary pulse through cross-phase modulation (XPM) in a Kerr lens. The
temporally shaped stencil pulse induces a time-dependent focusing phase within
the primary pulse. This technique, the “flying focus X,” allows the primary
pulse to have any profile or OAM, expanding the flexibility of spatiotemporal
pulse shaping for laser-based applications. As an example, simulations show
that the flying focus X can deliver an arbitrary-velocity, variable-duration
intensity peak with OAM over distances much longer than a Rayleigh range.

##  I Introduction

Spatiotemporal pulse shaping offers a new paradigm for controlling laser
intensity. By exploiting space-time correlations in the amplitude or phase of
a laser pulse, a number of recent techniques have created arbitrary-velocity
intensity peaks that remain nearly propagation invariant over distances much
longer than a Rayleigh range Kondakci and Abouraddy (2017); Sainte-Marie  _et
al._ (2017); Froula  _et al._ (2018); Kondakci and Abouraddy (2019); Li and
Kawanaka (2020); Palastro  _et al._ (2020); Caizergues  _et al._ (2020); Jolly
_et al._ (2020). These features promise to revolutionize a wide range of
laser-based applications, including high-power amplifiers Turnbull  _et al._
(2018), radiation sources Howard  _et al._ (2019); Franke  _et al._ (2021);
Ramsey  _et al._ (2021), and compact accelerators Debus  _et al._ (2019);
Palastro  _et al._ (2020); Caizergues  _et al._ (2020); Ramsey  _et al._
(2020); Palastro  _et al._ (2021). Nevertheless, each of the existing
techniques requires an optical configuration that constrains properties of the
intensity peak, such as its transverse or temporal profile, duration, or
orbital angular momentum (OAM). As an example, the original, “chromatic”
flying focus uses a chirp and a chromatic lens to control the time and
location at which each temporal slice of a laser pulse comes to focus,
respectively Froula  _et al._ (2018); Sainte-Marie  _et al._ (2017). While
this technique offers some flexibility to shape the transverse profile of the
far-field, the chromatic aberration and chirp can place a lower bound on the
duration of the intensity peak that is much larger than the transform-limited
duration Sainte-Marie  _et al._ (2017); Palastro  _et al._ (2018); Froula  _et
al._ (2019). An alternative technique, the “achromatic” flying focus, employs
the spherical aberration of an axiparabola Smartsev  _et al._ (2019); Oubrerie
_et al._ (2021) to focus different annuli in the near-field to different axial
locations in the far-field and an echelon to adjust their relative timing
Palastro  _et al._ (2020). Here, the intensity peak can have a near transform-
limited duration, but the flattop transverse profile required in the near-
field and the spherical aberration of the axiparabola fully determine the far-
field profile.

Both the chromatic and achromatic flying focus techniques use linear optical
elements to structure a pulse with space-time correlations, but nonlinear
processes, such as self-phase modulation, can also produce space-time
correlations Marburger (1975); Shen (1975); Brodeur  _et al._ (1997); Couairon
and Mysyrowicz (2007); Simpson  _et al._ (2020). Self-phase modulation refers
to the phase accrued by a laser pulse as it traverses a nonlinear medium and
modifies the refractive index. In a Kerr nonlinear medium, the change in the
refractive index ($\delta n$) is proportional to the intensity ($I$): $\delta
n=n_{2}I$, where $n_{2}$ is the nonlinear refractive index and linear
polarization is assumed throughout. When $n_{2}>0$, the phase velocity is
reduced in regions of high intensity, creating a wavefront curvature that can
counteract diffraction, i.e., the pulse self-focuses Kelley (1965); Marburger
(1975); Shen (1975); Sun  _et al._ (1987). At sufficiently high power, this
curvature can even overcome diffraction, resulting in the transverse collapse
of the pulse until the intensity reaches a threshold for activating an arrest
mechanism Fibich (1996); Bang  _et al._ (2002); Kosareva  _et al._ (2011);
Couairon and Mysyrowicz (2007). The recently described “self-flying focus”
combines this effect with temporal pulse shaping to control the trajectory of
an intensity peak in a Kerr medium Simpson  _et al._ (2020). This approach can
deliver an arbitrary velocity intensity peak over distances comparable to the
focal length, but nonlinear evolution of the transverse and temporal profiles
during self-focusing collapse and arrest significantly complicate any attempt
at far-field shaping. Further, the arrest mechanism, e.g., ionization, can
irreversibly damage the medium.

Cross-phase modulation (XPM) refers to the phase that one laser pulse induces
onto another as they copropagate through a nonlinear medium Agrawal (2000);
Boyd (2019). For two pulses ($P$ and $S$ for specificity) overlapped in a Kerr
medium, the change in refractive index experienced by pulse $P$ ($\delta
n_{P}$) will depend on both its intensity ($I_{P}$) and that of pulse $S$
($I_{S}$): $\delta n_{P}=n_{2}(I_{P}+\alpha I_{S})$, where $\alpha$ depends on
their relative polarizations. As a consequence, the phase velocity of pulse
$P$ can be lower in regions where either pulse has a high intensity. If
$I_{S}\gg I_{P}$, the intensity profile of pulse $S$ will dictate the local
phase velocity and resulting wavefront curvature acquired by pulse $P$. That
is, a high-intensity pulse can induce the focusing of a colocated low-
intensity pulse McKinstrie and Russell (1988); Agrawal (1990); Hafizi  _et
al._ (2015).

Here we describe a novel technique for spatiotemporal control: the “flying
focus X,” which combines temporal pulse shaping with XPM to produce an
ultrashort, arbitrary velocity intensity peak, with or without OAM, over
distances far greater than a Rayleigh range. Specifically, a temporally
shaped, high-intensity “stencil” pulse induces the time-dependent focusing of
a second, primary pulse through XPM in a Kerr lens [Fig. 1]. The minimum and
maximum intensity of the stencil pulse ($S$) set the focal range of the
primary pulse ($P$), while its duration sets the velocity of the resulting
intensity peak. Use of a stencil pulse mitigates the constraints on the
primary pulse, allowing for spatiotemporal control independent of properties
such as the far-field duration, transverse profile, or OAM. In effect, these
constraints are offloaded onto the stencil pulse. As a result, the flying
focus X provides unprecedented flexibility for structuring the far-field
properties of the intensity peak, promising to further enable or enhance a
wide range of laser-based applications.

Figure 1: Schematic of the flying focus X. A high-intensity stencil pulse with
a flattop transverse and shaped temporal profile co-propagates with a low-
intensity primary pulse. The stencil induces a time-dependent focusing phase
on the primary in a parabolic Kerr lens. A filter removes the stencil, while
an ideal lens provides additional focusing for the primary. The resulting
focus travels over a distance $L_{f}$ at a velocity $v_{f}$, both of which can
be tuned by shaping the temporal profile of the stencil. A stencil pulse with
an intensity that ramps up in time can produce either negative (left) or
positive superluminal (right) focal velocities, depending on its duration
($\tau_{S}$).

##  II The Flying Focus X

Figure 1 illustrates how temporal pulse shaping and XPM can be used to control
the trajectory and range of an intensity peak. A stencil pulse and a primary
pulse, overlapped in space and time, co-propagate along the z axis and enter a
convex, parabolic Kerr lens. The two pulses have distinct wavelengths or
polarization vectors and $I_{S}\gg I_{P}$. The stencil pulse has a flattop
transverse profile, which, in combination with the parabolic shape of the Kerr
lens, ensures that the primary pulse acquires an aberration-free wavefront
curvature. The instantaneous intensity of the stencil dictates the time
dependence of that curvature. Specifically, the primary pulse exits the Kerr
lens with the phase

$\phi_{K}(r,\tau)=-\frac{\alpha k_{0}r^{2}n_{2}I_{S}(\tau)}{2R},$ (1)

where $\tau=t-z/c$ denotes time within the primary pulse, $k_{0}$ its vacuum
wavenumber, and $R$ the Kerr lens radius of curvature (see Appendix I). The
phase ($\phi_{K}$) is equivalent to that applied by an ideal lens, but with a
time-dependent focal length: $f_{K}(\tau)=R/\alpha n_{2}I_{s}(\tau)$. After
the Kerr lens, a wavelength or polarization filter removes the stencil pulse.
The primary pulse continues through a second, linear lens such that its total
phase ($\phi_{P}$) becomes $\phi_{P}=\phi_{K}+\phi_{l}$, where
$\phi_{l}=-k_{0}r^{2}/2f_{l}$ and $f_{l}$ is the linear lens focal length.

Upon exiting the linear lens, each time slice within the primary pulse will
have a focal length determined by its total wavefront curvature:

$f(\tau)=f_{l}\Bigl{[}1+\frac{\alpha f_{l}n_{2}I_{S}(\tau)}{R}\Bigr{]}^{-1}.$
(2)

A time slice that was overlapped with a low stencil intensity in the Kerr lens
will focus farther from the optical assembly and closer to the linear lens
focal point; a time slice that was overlapped with a high stencil intensity
will focus closer to the optical assembly and farther from the linear lens
focal point. Over the duration of the primary pulse, the focal point will move
within a range ($L_{f}$) defined by the minimum and maximum intensity of the
stencil pulse:

$\frac{L_{f}}{f_{l}}=\frac{1}{1+\frac{\alpha
f_{l}n_{2}}{R}I_{S,\mathrm{min}}}-\frac{1}{1+\frac{\alpha
f_{l}n_{2}}{R}I_{S,\mathrm{max}}}.$ (3)

For typical parameters, the linear lens has a smaller focal length (greater
focusing power) than the Kerr lens, i.e., $f_{l}\ll f_{K}$. Nevertheless, the
Kerr lens can still create a focal range that is a significant fraction of the
linear lens focal length: $L_{f}\approx(\alpha
f_{l}^{2}n_{2}/R)(I_{S,\mathrm{max}}-I_{S,\mathrm{min}})$.

By appropriately shaping the temporal profile of the stencil pulse, the focal
point can be made to move through the focal region with a desired trajectory.
Each time slice within the primary pulse will focus at a time
$t_{f}=\tau+f(\tau)/c$. A general focal trajectory, $f(t_{f})$, can be found
by iterating to solve for $f(\tau)$. For a focal point moving at a constant
velocity ($v_{f}$), $f(\tau)=f_{0}+u\tau$, where $f_{0}$ is the focal point of
the initial time slice and $u=cv_{f}/(c-v_{f})$ is the reduced velocity. Using
this expression for $f(\tau)$, one can invert Eq. (2) to find the required
temporal profile of the stencil pulse:

$I_{S}(\tau)=\frac{R}{\alpha
f_{l}n_{2}}\Bigl{(}\frac{f_{l}}{f_{0}+u\tau}-1\Bigr{)}.$ (4)

For either a superluminal ($v_{f}>c$) or backwards ($v_{f}<0$) focal velocity,
$u<0$ and the stencil intensity must ramp up in time [Fig. 2(a)]. For a
subluminal focal velocity ($0<v_{f}<c$), $u>0$ and the stencil intensity must
ramp down in time. The total duration of the stencil and primary pulses
($\tau_{S}$) depends only on the focal range and the reduced velocity, i.e.,
$\tau_{S}=L_{f}/|u|$. Because the focal range ($L_{f}$) is typically much
smaller than the linear lens focal length ($f_{l}$), the stencil profile can
often be approximated as a simple linear ramp: $I_{S}(\tau)\approx(R/\alpha
f_{0}n_{2})(1-f_{0}/f_{l}-u\tau/f_{0})$.

From the stencil pulse duration, the focal velocity can be expressed in terms
of physical parameters of the flying focus X: $v_{f}/c=(1\mp
c\tau_{S}/L_{f})^{-1}$, where the minus and plus signs correspond to ramp-up
($u<0$) and ramp-down ($u>0$) pulses, respectively [Fig. 2(b)]. This
illustrates two points. First, for a desired focal range, the focal velocity
can be tuned by adjusting the stencil pulse duration. Second, the transition
from a superluminal to backwards travelling focus occurs when the pulse
duration of a ramp-up pulse ($u<0$) becomes longer than the focal range (i.e.,
$\tau_{S}>L_{f}/c$).

Figure 2: Design space for a flying focus X pulse. (a) The intensity profile
for a ramp-up ($u<0$) stencil pulse as a function of time. Starting at
$\tau=0$ represents a pulse with $f_{0}=f_{l}$; starting at a time
$\tau=\Delta\tau$ represents a pulse with
$f_{0}=f_{l}-L_{f}\Delta\tau/\tau_{S}$. The intensity profile for a ramp-down
pulse ($u>0$) corresponds to flipping the horizontal axis and appropriately
shifting in time for a particular $f_{0}$. (b) The focal velocity as a
function of the stencil pulse duration and focal range.

The primary pulse can have any transverse profile. Because the combined Kerr
and linear lenses create an ideal parabolic wavefront, a natural choice is a
Laguerre-Gaussian (LG) mode, which, in addition to allowing for OAM, has well-
known expressions for its spot size, radius of curvature, and Gouy phase. With
this choice, the spot size for each time slice within the primary pulse is
given by $w_{P}(\tau)=w_{0}[(1-z/f(\tau))^{2}+(z/Z_{0})^{2}]^{1/2}$, where
$w_{0}$ is the spot size in the near-field ($z=0$) and
$Z_{0}=k_{0}w_{0}^{2}/2$. From the spot size and conservation of power in each
time slice of the primary pulse, one can derive the spatiotemporal profile of
the intensity peak

$I_{f}(z,t)=\Bigl{[}1+\frac{(t-z/v_{f}+f_{0}/u)^{2}}{\tau_{f}^{2}}\Bigr{]}^{-1}\frac{w_{0}^{2}}{w_{f}^{2}}I_{P0}(\tau),$
(5)

where $I_{P0}(\tau)$ is the intensity profile of the primary pulse in the
near-field,

$\tau_{f}=\frac{|c-v_{f}|}{|cv_{f}|}Z_{R}$ (6)

is the duration of the intensity peak, $w_{f}=2f_{l}/k_{0}w_{0}$ is its spot
size, $Z_{R}=k_{0}w_{f}^{2}/2$ is the Rayleigh range, and $L_{f}\ll f_{l}$ has
been assumed. Note that each LG mode can have its peak intensity at a
different radius; Eq. (5) is the temporal intensity profile at that radius.

The duration of the intensity peak ($\tau_{f}$) can be interpreted as the rate
at which adjacent time slices within the primary pulse come in and out of
focus, i.e., $Z_{R}/|u|$. The duration can be tuned by adjusting the focal
geometry: stronger (weaker) focusing shortens (lengthens) the Rayleigh range,
which decreases (increases) the duration. In general, the intensity peak can
have an ultrashort duration that is much smaller than that of the primary
pulse. The time-dependent refractive index experienced by the primary pulse in
the Kerr lens provides the bandwidth necessary to support the ultrashort
duration of the intensity peak in the far-field. The total bandwidth acquired
by the primary pulse can be calculated using the linear ramp approximation for
the stencil profile and the Kerr lens phase (Eq. (1)):
$\Delta\omega=2\partial_{\tau}\phi_{K}|_{r=w_{0}}$, or
$\Delta\omega=2|u|/Z_{R}\propto\tau_{f}^{-1}$, consistent with Eq. (6). Note
that the required bandwidth increases rapidly as $v_{f}$ approaches $c$. As an
example, with $v_{f}=1.01\,c$ and $Z_{R}=300\,\mathrm{\mu m}$,
$\tau_{f}=10\,\mathrm{fs}$.

In order for the intensity peak to have a constant maximum throughout the
focal region, the primary pulse must have a flattop intensity profile in time
with a duration equal to that of the stencil pulse ($\tau_{S}$). By shaping
the temporal intensity profile of the primary pulse ($I_{P0}$), the intensity
peak can also exhibit more exotic patterns. For instance, modulating the
temporal profile of the primary pulse will create an oscillating intensity
distribution along the optical axis, similar to Ref. Palastro  _et al._
(2018).

Designing a flying focus X pulse for an experiment or application follows a
straightforward procedure. Several laser-based applications require (1)
sustaining a high intensity over an extended distance and (2) that the
velocity of the intensity peak conform to some process Hebling  _et al._
(2002); D’Amico  _et al._ (2007); Kim  _et al._ (2008); Rundquist  _et al._
(1998); Durfee  _et al._ (1999); Popmintchev  _et al._ (2012); Durfee III and
Milchberg (1993); Milchberg  _et al._ (1996); Tajima and Dawson (1979); Esarey
_et al._ (2009); Malkin  _et al._ (1999); Trines  _et al._ (2011); Wilks  _et
al._ (1988); Dias  _et al._ (1997). The first step is then to specify the
focal range ($L_{f}$) and the focal velocity ($v_{f}$). Next, one would set
the focal geometry ($f_{l}$) based on either a desired far-field spot size
($w_{f}$) or intensity peak duration ($\tau_{f}$). As demonstrated by Eq. (6),
these two properties are interdependent. However, for a fixed $\tau_{f}$, a
larger root-mean-square spot size can be achieved by using higher-order
Laguerre-Gaussian modes. Finally, one would select a Kerr lens material with a
large $n_{2}$, low absorption and weak nonlinear dispersion at the primary and
stencil wavelengths, and the smallest possible radius of curvature. At optical
frequencies, for instance, ZnS is an excellent candidate Sheik-Bahae  _et al._
(1991). With all other parameters specified, the duration and profile of the
stencil pulse can be calculated from $\tau_{S}=(L_{f}/c)|c/v_{f}-1|$ [Fig
2(b)] and Eq. (2), respectively. Note that while the previous discussion
considered a positive Kerr lens, the flying focus X can also be designed with
a negative Kerr lens (see Appendix II).

Table 1: Near-field parameters for the two simulated flying focus X designs. Note that only the properties of the stencil pulse change, illustrating the versatility of a single optical configuration to accommodate different focal velocities and transverse profiles. For both examples, $\lambda_{0}=2\pi/k_{0}=1.054\,\mu\text{m}$, $L_{f}=1\,\text{cm}=33\,Z_{R}$, and $w_{f}=10\,\mu\text{m}$. ZnS was used for the Kerr lens and the stencil and primary pulses had parallel polarizations, such that $\alpha=2$ Sheik-Bahae  _et al._ (1991); Agrawal (2000).  | $v_{f}=1.01\,c$ | $v_{f}=-c$
---|---|---
$\ell$ | 0 | 1
$\tau_{S}$ (ps) | $0.33$ | $67$
$f_{l}$ (cm) | 60 | 60
$w_{0}$ (cm) | 2.0 | 2.0
$n_{2}$ (cm2/W) | $8.4\times 10^{-15}$ | $8.4\times 10^{-15}$
$R$ (cm) | 1 | 1
$I_{S,\mathrm{max}}$ (W/cm2) | $1.7\times 10^{10}$ | $1.7\times 10^{10}$

##  III Simulation Results

To demonstrate the versatility of the flying focus X, simulations (see
Appendix III for details) were conducted for two focal velocities, each with
relevance to laser-based applications (Table I). The transverse structure of
the primary pulse in each case was an LGℓ,p mode, where $\ell$ and $p$ denote
the OAM and radial quantum numbers, respectively. The first example features a
superluminal ($v_{f}=1.01c$) intensity peak with $p=\ell=0$, i.e., a standard
transverse Gaussian profile, ideal for compact accelerators (Fig. 3) Esarey
_et al._ (2009); Palastro  _et al._ (2020, 2021). The second example features
a backwards ($v_{f}=-c$) travelling intensity peak with $p=0$ and $\ell=1$,
i.e., a “donut” mode, ideal for exotic high-power amplification and photon
acceleration schemes (Fig. 4) Vieira  _et al._ (2016); Franke  _et al._
(2021).

Figure 3: Simulation results for an LG0,0 flying focus with $v_{f}=1.01c$,
demonstrating velocity control, an ultrashort duration, and near propagation
invariance through the focal region. (a) The temporal profile of the stencil
and primary pulses at the entrance of the Kerr lens. (b) The transverse
electric field of the primary pulse at the entrance of the Kerr lens. (c) The
on-axis intensity of the primary pulse in the far-field and a lineout of its
temporal profile. (d) The fluence of the primary pulse in the far-field. All
values are normalized to their respective maxima. Figure 4: Simulation results
for an LG0,1 flying focus with $v_{f}=-c$, demonstrating velocity control, a
short duration, and near propagation invariance through the focal region. (a)
The temporal profile of the stencil and primary pulses at the entrance of the
Kerr lens. (b) The transverse electric field of the primary pulse at the
entrance of the Kerr lens. (c) The intensity of the primary pulse in the far-
field at the radial location of its maximum intensity ($r=w_{f}/\sqrt{2}$) and
a lineout of its temporal profile. (d) The fluence of the primary pulse in the
far-field. All values are normalized to their respective maxima.

Figure 3 illustrates that the flying focus X can create an ultrashort-duration
intensity peak traveling near the vacuum speed of light over many Rayleigh
ranges. As described by Eq. (4), the intensity of the stencil pulse ramps up
linearly in time. The primary pulse has a super-Gaussian (SG20) temporal
profile to ensure a constant intensity in the far-field. Throughout the focal
region, spanning $z=59\,\text{cm}$ to $z=60\,\text{cm}$, the intensity peak
remains ultrashort, with a duration comparable to the value predicted by Eq.
(6), i.e., $\tau_{f}=10\,\text{fs}$. The intensity peak maintains its
transverse Gaussian profile with a diffraction-limited spot over the entire
focal range.

Figure 4 illustrates that the flying focus X can create an intensity peak with
OAM that counterpropagates with respect to its phase fronts over many Rayleigh
ranges. The stencil and primary pulses have the same temporal shape as in Fig.
3, but have much longer durations to accommodate the negative focal velocity
[Fig. 2(b)]. As predicted by Eq. (6), the negative focal velocity results in
the intensity peak having a longer effective duration, $\tau_{f}\approx
2\,\text{ps}$. The intensity peak maintains its OAM profile with a
diffraction-limited spot over the entire focal range.

##  IV Discussion

While the flying focus X offers additional flexibility to shape the far-field
profile of the intensity peak, it does present some trade-offs. Foremost, the
stencil pulse is filtered out and discarded after the Kerr lens, making the
design inherently inefficient. This is exacerbated by the fact that the
stencil pulse must be much more intense than the primary pulse to ensure that
the induced focusing is not disrupted by self-phase modulation (i.e.,
$I_{S}\gg I_{P}/2$). The inefficiency can be mitigated by setting
$I_{S,\text{min}}=0$, which minimizes the total energy of the stencil pulse,
but fixes the maximum focal location to $f=f_{l}$. The constraint on the
relative intensities can be eliminated if both the stencil and the primary
pulses have a flattop, transverse intensity profile. From the resulting Kerr
lens phase, i.e., $\phi_{K}(r,\tau)=-k_{0}n_{2}[\alpha
I_{S}(\tau)+I_{P}]r^{2}/2R$, the temporal profile of the stencil can be shaped
to accommodate SPM for any primary intensity, while still providing control
over the far-field trajectory. Specifically, $I_{S}(\tau)=\frac{R}{\alpha
f_{l}n_{2}}(\frac{f_{l}}{f_{0}+u\tau}-1)-\frac{1}{\alpha}I_{P}$. Note,
however, that this approach limits the far-field transverse profile to an Airy
pattern and precludes a primary pulse with OAM. To eliminate the constraint on
the relative intensities for a primary pulse with OAM, the shape of the Kerr
lens can be changed to compensate the transverse spatial profile of the OAM
mode.

In addition to its inefficiency, the flying focus X requires stencil and
primary pulses with low levels of phase and intensity noise. As with
conventional optics, the primary pulse must have a clean wavefront as it
enters the optical assembly to ensure near diffraction-limited focusing to the
far-field. While XPM is insensitive to phase noise, the induced focusing
depends sensitively on the transverse intensity profile of the stencil. A
noisy stencil intensity can imprint phase aberrations onto the primary pulse
that inhibit its focusing and modify the far-field properties. Further, for a
large stencil intensity or in a thick Kerr lens, the intensity noise can seed
filamentation Yablonovitch and Bloembergen (1972). To avoid unwanted,
nonlinear propagation of noisy pulses, the phase acquired from XPM in the Kerr
lens should be no more than a few $\pi$, i.e., $\phi_{X}=\alpha
k_{0}n_{2}I_{S}\Delta z\lesssim 3\pi$, where $\Delta z$ is the maximum
thickness of the lens Fibich  _et al._ (2005); Boyd (2019). In practice, this
amounts to limiting the focal range to a few Rayleigh ranges. The use of low-
noise pulses significantly relaxes this condition:
$w_{0}^{-2}n_{2}I_{S}(\Delta z)^{2}\ll 1$, i.e., the self-focusing collapse
distance must be much larger than the lens thickness Marburger (1975).
Together, the conditions to avoid self-phase modulation and filamentation
place an upper bound on far-field intensity of the flying focus X
($I_{f,\text{max}}$): for clean beams, $I_{f,\text{max}}\ll
R^{2}/8n_{2}w_{f}^{2}$ or $I_{f,\text{max}}\ll 1\times
10^{19}\,\mathrm{W/cm^{2}}$ for the parameters in Table I. Thus, the
requirements of a low stencil pulse intensity in the near-field do not
preclude a high intensity in the far-field.

##  V Summary and Conclusions

A new technique for spatiotemporal control, the flying focus X, can create an
ultrashort, arbitrary trajectory intensity peak, with any transverse profile
or OAM, over distances much greater than a Rayleigh range. The technique
combines cross-phase modulation in a Kerr lens with temporal pulse shaping. In
a Kerr lens, a shaped auxiliary, or “stencil,” pulse imprints a second,
primary pulse with a time-dependent focusing phase. In the far-field, the
intensity peak formed by the time-dependent focusing of the primary pulse
follows a trajectory dictated by the shape of the stencil. Depending on the
shape, the peak can move forward or backwards and at any velocity. The use of
a stencil pulse and XPM overcomes constraints inherent to the optical
configurations employed in previous techniques for spatiotemporal control.
Specifically, the flying focus X provides independent control over the shape
of the transverse profile, OAM, and duration of the intensity peak.
Ultimately, this additional flexibility could further enhance the utility of
spatiotemporally shaped pulses for laser-based applications.

## Acknowledgments

The authors would like to thank J.L. Shaw and K.L. Nguyen for fruitful
discussions. The work published here was supported by the U.S. Department of
Energy Office of Fusion Energy Sciences under contract no. DE-SC0016253, the
Department of Energy under cooperative agreement no. DE-NA0003856, the
University of Rochester, and the New York State Energy Research and
Development Authority.

This report was prepared as an account of work sponsored by an agency of the
U.S. Government. Neither the U.S. Government nor any agency thereof, nor any
of their employees, makes any warranty, express or implied, or assumes any
legal liability or responsibility for the accuracy, completeness, or
usefulness of any information, apparatus, product, or process disclosed, or
represents that its use would not infringe privately owned rights. Reference
herein to any specific commercial product, process, or service by trade name,
trademark, manufacturer, or otherwise does not necessarily constitute or imply
its endorsement, recommendation, or favoring by the U.S. Government or any
agency thereof.

## Disclosures

The authors declare no conflicts of interest.

## Appendix I: Kerr Lens Model

To determine the profile of the primary pulse after the Kerr lens, consider
the wave equation for the transverse electric field ($\text{E}_{P}$) in a
time-dependent, Kerr-nonlinear medium:

$[\partial_{t}^{2}n(\textbf{x},t)^{2}-c^{2}\nabla^{2}]\text{E}_{P}=0,$ (A1)

where $n(\textbf{x},t)=n_{0}(\textbf{x})+\alpha n_{2}(\textbf{x})I_{S}(\tau)$
is the total refractive index, $n_{0}$ is the linear refractive index,
$\tau=t-\int{n_{0}dz}/c$, and $I_{S}\gg I_{P}$ has been assumed. The spatial
dependence of $n_{0}(\textbf{x})$ and $n_{2}(\textbf{x})$ indicates that the
refractive index changes across the Kerr lens-vacuum interface: outside the
lens $n_{0}=1$ and $n_{2}=0$. Upon switching to the coordinates $z$ and
$\tau$, using $n_{0}\gg\alpha n_{2}I_{S}$, and noting that the lens is thin
enough to neglect diffraction, Eq. (A5) becomes

$[c\partial_{z}+\alpha\partial_{\tau}n_{2}I_{S}(\tau)]\text{E}_{P}=0.$ (A2)

This equation admits analytic solutions. For the special case of
$I_{S}(\tau)=(R/\alpha f_{0}n_{2})(1-f_{0}/f_{l}-u\tau/f_{0})$, valid when
$L_{f}\ll f_{l}$, the solution is $\text{E}_{P}=e^{\int\kappa
dz}\text{E}_{0}[\textbf{r},\psi(\textbf{x},\tau)]$, where
$\text{E}_{0}[\textbf{r},\tau]$ is the electric field of the primary pulse at
the entrance to the Kerr lens, $\kappa=Ru/cf_{0}^{2}$,
$\psi(\textbf{x},\tau)=e^{\int\kappa dz}(\tau-\gamma)+\gamma$, and
$\gamma=(1-f_{0}/f_{l})(f_{0}/u)$. The exponential factor captures the
temporal broadening (compression) and the corresponding decrease (increase) in
amplitude resulting from the time-dependent refractive index induced by the
ramp-up (down) stencil.

The boundaries of the Kerr lens are defined by its entrance plane at
$z=z_{0}<0$ and its exit surface: $s(r)=z_{1}-r^{2}/2R$, where
$z_{0}<z_{1}<0$. The thickness and maximum radius are then $d=z_{1}-z_{0}$ and
$r_{\text{max}}=(2Rd)^{1/2}$, respectively. For the Kerr lenses of interest,
$\kappa d\ll 1$. As a result, the exponential factor in $\psi$ can be Taylor
expanded. At the exit of the Kerr lens, i.e., $z=z_{1}$, $\int\kappa
dz=\kappa(d-r^{2}/2R)$ and

$\psi(r,\tau)=\tau+\frac{r^{2}}{2cf_{lK}}+\frac{\alpha
n_{2}I_{S}(\tau)r^{2}}{2cR}-[n_{lK}+\alpha n_{2}I_{S}(\tau)]\frac{d}{c},$ (A3)

where $f_{lK}=R/(n_{lK}-1)$ and $n_{lK}$ are the linear focal length and
refractive index of the Kerr lens, respectively. A small radius of curvature
enhances the nonlinear focusing, but also results in strong linear focusing
that can limit the focal range ($L_{f}$). To overcome this, a preliminary
linear lens is used to compensate the linear focal length of the Kerr lens
($f_{lK})$, while a subsequent, weaker linear lens provides additional
focusing. After the optical assembly, i.e., at $z=0$,

$\psi(r,\tau)=\tau+\frac{r^{2}}{2cf_{l}}+\frac{\alpha
n_{2}I_{S}(\tau)r^{2}}{2cR}-\alpha n_{2}I_{S}(\tau)\frac{d}{c},$ (A4)

where constant phase terms have been dropped. Equation (A8) along with
$\text{E}_{0}[\textbf{r},\tau]$ define the initial condition:
$\text{E}_{P}=e^{\kappa(d-r^{2}/2R)}\text{E}_{0}[\textbf{r},\psi(r,\tau)]$.

The electric field of the primary pulse at the entrance to the Kerr lens can
be expressed as a rapidly oscillating carrier modulating a slowly varying
envelope:
$\text{E}_{0}=\frac{1}{2}e^{-i\omega_{0}\tau}E_{0}(\textbf{r},\tau)+\text{c.c.}$,
where $\omega_{0}=ck_{0}$ is the central frequency of the pulse. Performing
the substitution $\tau\rightarrow\psi(r,\tau)$, in accordance with the initial
condition, illustrates that the lens phases acquired in the optical assembly
appear in the exponential prefactor, while the radial group delay and
dispersion appear in $E_{0}$. Inclusion of the radial group delay and
dispersion are critical for correctly modeling the duration of the intensity
peak.

## Appendix II: Defocusing Kerr Lens

A Kerr lens with a concave, quadratic shape acts as a defocusing lens. In this
case, the primary pulse exits the Kerr lens with the phase

$\phi_{K}(r,\tau)=\frac{\alpha k_{0}r^{2}n_{2}I_{S}(\tau)}{2R},$ (A5)

such that each time slice has a focal length

$f(\tau)=f_{l}\Bigl{[}1-\frac{\alpha f_{l}n_{2}I_{S}(\tau)}{R}\Bigr{]}^{-1}.$
(A6)

This exhibits a somewhat counterintuitive effect: cross-phase modulation in a
self-focusing medium (i.e., one with a positive $n_{2}$) can induce
defocusing. Here, a time slice that was overlapped with a low stencil
intensity in the Kerr lens will focus closer to the optical assembly and the
linear lens focal point, while a time slice that was overlapped with a high
stencil intensity will focus farther from the optical assembly and the linear
lens focal point.

For the same maximum stencil intensity, a defocusing Kerr lens can produce a
longer focal range than a focusing Kerr lens. The focal point of a defocusing
Kerr lens is bounded by $f=f_{l}$ and $f=\infty$, while that of a focusing
Kerr lens is bounded by $f=0$ and $f=f_{l}$. Specifically, the focal range for
a defocusing Kerr lens is given by

$\frac{L_{f}}{f_{l}}=\frac{1}{1-\frac{\alpha
f_{l}n_{2}}{R}I_{S,\mathrm{max}}}-\frac{1}{1-\frac{\alpha
f_{l}n_{2}}{R}I_{S,\mathrm{min}}},$ (A7)

where, in order for there to be a farthest focus, $I_{S,\mathrm{max}}<R/\alpha
f_{l}n_{2}$. Note, however, that the f-number, focal spot size, and on-axis
intensity of the primary pulse will vary substantially over long focal ranges
($L_{f}\lesssim f_{l}$).

To create an intensity peak that travels through the focal region at a
constant velocity ($v_{f}$), the intensity profile of the stencil pulse must
have the form

$I_{S}(\tau)=\frac{R}{\alpha
f_{l}n_{2}}\Bigl{(}1-\frac{f_{l}}{f_{0}+u\tau}\Bigr{)},$ (A8)

where $u=cv_{f}/(c-v_{f})$. In contrast to a focusing Kerr lens, superluminal
or backwards focal velocities ($u<0$) require a stencil intensity that ramps
down in time, while subluminal focal velocities ($u>0$) require a stencil
intensity that ramps up in time. For focal ranges much smaller than the linear
lens focal length ($L_{f}\ll f_{l}$), the stencil profile can be approximated
as a simple linear ramp: $I_{S}(\tau)\approx(R/\alpha
f_{0}n_{2})(f_{0}/f_{l}-1+u\tau/f_{0})$.

## Appendix III: Simulation Details

The simulations solved for the evolution of the primary pulse in three steps.
The first step used the analytic model presented in Appendix I to determine
the profile of the primary pulse after the optical assembly. The second step
used a frequency domain Fresnel integral, as described in the Appendix of Ref.
Palastro  _et al._ (2018), to propagate the envelope of the primary pulse to
the far-field. The final step used the modified paraxial wave equation to
propagate the primary pulse through the far-field. The results shown in Figs.
3 and 4 were obtained from this last step.

To solve for the evolution of the primary pulse after the optical assembly,
$\text{E}_{P}$ is written as a carrier modulating an envelope:
$\text{E}_{P}=\frac{1}{2}e^{-i\omega_{0}\tau}E_{P}(\textbf{r},z,\tau)+\text{c.c.}$,
with the initial condition
$E_{P}(\textbf{r},0,\tau)=e^{\kappa(d-r^{2}/2R)-i\omega_{0}(\psi-\tau)}E_{0}[\textbf{r},\psi(r,\tau)]$.
Propagation from the optical assembly to the far-field is calculated using the
frequency-domain Fresnel integral:

$\hat{E}_{P}(\textbf{r},z,\omega)=\frac{k_{\omega}}{2\pi
iz}\int\text{exp}\Bigl{[}i\frac{k_{\omega}}{2z}(\textbf{r}-\textbf{r}^{\prime})^{2}\Bigr{]}\hat{E}_{P}(\textbf{r}^{\prime},0,\omega)d\textbf{r}^{\prime},$
(A9)

where the caret represents a Fourier transform with respect to $\omega$, i.e.,
the conjugate variable to $\tau$, and $k_{\omega}=(\omega_{0}+\omega)/c$. This
allows for separate spatial resolution in the near and far-field, which
greatly reduces the number of grid points compared to numerically solving the
wave equation over the entire propagation distance Palastro  _et al._ (2018).
In the far-field, the envelope of the primary pulse is evolved using the
modified paraxial wave equation Zhu  _et al._ (2012):

$[2(i\omega_{0}-\partial_{\tau})\partial_{z}+c\nabla_{\perp}^{2}]E_{P}(\textbf{r},z,\tau)=0.$
(A10)

The mixed space-time derivative in Eq. (A10) ensures that effects such as
radial group delay and angular dispersion are modelled correctly—a requirement
for resolving the effective duration of the flying focus X. Note that after
the optical assembly $\tau=t-z/c$ consistent with propagation in vacuum.

The model described above does not account for the dispersion of $n_{2}$.
Simulations of a 20 fs pulse using the focal geometry described in Table I
predict that the dispersion of $n_{2}$ in ZnS increases the duration by
$\approx 10\,\text{fs}$.

## References

  * Kondakci and Abouraddy (2017) H. E. Kondakci and A. F. Abouraddy, Nature Photonics 11, 733 (2017).
  * Sainte-Marie  _et al._ (2017) A. Sainte-Marie, O. Gobert, and F. Quere, Optica 4, 1298 (2017).
  * Froula  _et al._ (2018) D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, Nature Photonics 12, 262 (2018).
  * Kondakci and Abouraddy (2019) H. E. Kondakci and A. F. Abouraddy, Nature Communications 10, 929 (2019).
  * Li and Kawanaka (2020) Z. Li and J. Kawanaka, Scientific Reports 10, 1 (2020).
  * Palastro  _et al._ (2020) J. P. Palastro, J. L. Shaw, P. Franke, D. Ramsey, T. T. Simpson, and D. H. Froula, Physical Review Letters 124, 134802 (2020).
  * Caizergues  _et al._ (2020) C. Caizergues, S. Smartsev, V. Malka, and C. Thaury, Nature Photonics 14, 475 (2020).
  * Jolly  _et al._ (2020) S. W. Jolly, O. Gobert, A. Jeandet, and F. Quéré, Opt. Express 28, 4888 (2020).
  * Turnbull  _et al._ (2018) D. Turnbull, S. Bucht, A. Davies, D. Haberberger, T. Kessler, J. Shaw, and D. Froula, Physical Review Letters 120, 024801 (2018).
  * Howard  _et al._ (2019) A. Howard, D. Turnbull, A. Davies, P. Franke, D. Froula, and J. Palastro, Physical Review Letters 123, 124801 (2019).
  * Franke  _et al._ (2021) P. Franke, D. Ramsey, T. T. Simpson, D. Turnbull, D. H. Froula, and J. P. Palastro, Physical Review A 104, 043520 (2021).
  * Ramsey  _et al._ (2021) D. Ramsey, B. Malaca, A. D. Piazza, M. F. P. Franke, D. H. Froula, M. Pardal, T. T. Simpson, J. Vieira, K. Weichman, and J. P. Palastro, (2021), arXiv:2108.04044 [physics.plasm-ph] .
  * Debus  _et al._ (2019) A. Debus, R. Pausch, A. Huebl, K. Steiniger, R. Widera, T. E. Cowan, U. Schramm, and M. Bussmann, Physical Review X 9, 031044 (2019).
  * Ramsey  _et al._ (2020) D. Ramsey, P. Franke, T. T. Simpson, D. H. Froula, and J. P. Palastro, Phyiscal Review E 102, 043207 (2020).
  * Palastro  _et al._ (2021) J. Palastro, B. Malaca, J. Vieira, D. Ramsey, T. Simpson, P. Franke, J. Shaw, and D. Froula, Physics of Plasmas 28, 013109 (2021).
  * Palastro  _et al._ (2018) J. P. Palastro, D. Turnbull, S.-W. Bahk, R. K. Follett, J. L. Shaw, D. Haberberger, J. Bromage, and D. H. Froula, Physical Review A 97, 033835 (2018).
  * Froula  _et al._ (2019) D. H. Froula, J. P. Palastro, D. Turnbull, A. Davies, L. Nguyen, A. Howard, D. Ramsey, P. Franke, S.-W. Bahk, I. A. Begishev, R. Boni, J. Bromage, S. Bucht, R. K. Follett, D. Haberberger, G. W. Jenkins, J. Katz, T. J. Kessler, J. L. Shaw, and J. Vieira, Physics of Plasmas 26, 032109 (2019), https://doi.org/10.1063/1.5086308 .
  * Smartsev  _et al._ (2019) S. Smartsev, C. Caizergues, K. Oubrerie, J. Gautier, J.-P. Goddet, A. Tafzi, K. T. Phuoc, V. Malka, and C. Thaury, Optics Letters 44, 3414 (2019).
  * Oubrerie  _et al._ (2021) K. Oubrerie, I. A. Andriyash, R. Lahaye, S. Smartsev, V. Malka, and C. Thaury, (2021), arXiv:2106.06802 [physics.optics] .
  * Marburger (1975) J. Marburger, Progress in Quantum Electronics 4, 35 (1975).
  * Shen (1975) Y. Shen, Progress in Quantum Electronics 4, 1 (1975).
  * Brodeur  _et al._ (1997) A. Brodeur, C. Chien, F. Ilkov, S. Chin, O. Kosareva, and V. Kandidov, Optics Letters 22, 304 (1997).
  * Couairon and Mysyrowicz (2007) A. Couairon and A. Mysyrowicz, Physics Reports 441, 47 (2007).
  * Simpson  _et al._ (2020) T. T. Simpson, D. Ramsey, P. Franke, N. Vafaei-Najafabadi, D. Turnbull, D. H. Froula, and J. P. Palastro, Optics Express 28, 38516 (2020).
  * Kelley (1965) P. Kelley, Physical Review Letters 15, 1005 (1965).
  * Sun  _et al._ (1987) G.-Z. Sun, E. Ott, Y. Lee, and P. Guzdar, The Physics of Fluids 30, 526 (1987).
  * Fibich (1996) G. Fibich, Physical Review Letters 76, 4356 (1996).
  * Bang  _et al._ (2002) O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Physical Review E 66, 046619 (2002).
  * Kosareva  _et al._ (2011) O. Kosareva, J.-F. Daigle, N. Panov, T. Wang, S. Hosseini, S. Yuan, G. Roy, V. Makarov, and S. L. Chin, Optics Letters 36, 1035 (2011).
  * Agrawal (2000) G. P. Agrawal,  _Nonlinear fiber optics_ (Springer, 2000).
  * Boyd (2019) R. W. Boyd,  _Nonlinear optics_ (Academic Press, 2019).
  * McKinstrie and Russell (1988) C. McKinstrie and D. Russell, Physical Review Letters 61, 2929 (1988).
  * Agrawal (1990) G. P. Agrawal, Physical Review Letters 64, 2487 (1990).
  * Hafizi  _et al._ (2015) B. Hafizi, J. P. Palastro, J. R. P. nano, D. F. Gordon, T. G. Jones, M. H. Helle, and D. Kaganovich, Optics Letters 40, 1556 (2015).
  * Hebling  _et al._ (2002) J. Hebling, G. Almasi, I. Z. Kozma, and J. Kuhl, Optics Express 10, 1161 (2002).
  * D’Amico  _et al._ (2007) C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, Physical Review Letters 98, 235002 (2007).
  * Kim  _et al._ (2008) K.-Y. Kim, A. Taylor, J. Glownia, and G. Rodriguez, Nature Photonics 2, 605 (2008).
  * Rundquist  _et al._ (1998) A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, Science 280, 1412 (1998).
  * Durfee  _et al._ (1999) C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, Physical Review Letters 83, 2187 (1999).
  * Popmintchev  _et al._ (2012) T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, Science 336, 1287 (2012), https://science.sciencemag.org/content/336/6086/1287.full.pdf .
  * Durfee III and Milchberg (1993) C. Durfee III and H. Milchberg, Physical Review Letters 71, 2409 (1993).
  * Milchberg  _et al._ (1996) H. Milchberg, T. Clark, C. Durfee III, T. Antonsen, and P. Mora, Physics of Plasmas 3, 2149 (1996).
  * Tajima and Dawson (1979) T. Tajima and J. M. Dawson, Physical Review Letters 43, 267 (1979).
  * Esarey  _et al._ (2009) E. Esarey, C. Schroeder, and W. Leemans, Reviews of Modern Physics 81, 1229 (2009).
  * Malkin  _et al._ (1999) V. Malkin, G. Shvets, and N. Fisch, Physical Review Letters 82, 4448 (1999).
  * Trines  _et al._ (2011) R. Trines, F. Fiuza, R. Bingham, R. Fonseca, L. Silva, R. Cairns, and P. Norreys, Nature Physics 7, 87 (2011).
  * Wilks  _et al._ (1988) S. Wilks, J. Dawson, and W. Mori, Physical Review Letters 61, 337 (1988).
  * Dias  _et al._ (1997) J. Dias, C. Stenz, N. Lopes, X. Badiche, F. Blasco, A. Dos Santos, L. O. e Silva, A. Mysyrowicz, A. Antonetti, and J. Mendonça, Physical Review Letters 78, 4773 (1997).
  * Sheik-Bahae  _et al._ (1991) M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, IEEE Journal of Quantum Electronics 27, 1296 (1991).
  * Vieira  _et al._ (2016) J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, Nature Communications 7, 10371 (2016).
  * Yablonovitch and Bloembergen (1972) E. Yablonovitch and N. Bloembergen, Physical Review Letters 29, 907 (1972).
  * Fibich  _et al._ (2005) G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, Optics Express 13, 5897 (2005).
  * Zhu  _et al._ (2012) W. Zhu, J. Palastro, and T. Antonsen, Physics of Plasmas 19, 033105 (2012).