We report on the design and performance of an echelon-based single shot visible/near-infrared spectrometer with adequate sensitivity to measure the nonlinear optical and terahertz Kerr effects in neat molecular liquids at room temperature. Useful molecular information spanning tens of picoseconds can be measured in just a few milliseconds, and the signal-to-noise performance scales favorably with respect to the standard stage scan technique. These results demonstrate the viability of stage-free nonlinear Kerr effect measurements and provide a route for improvements to the speed of future multidimensional Kerr effect studies

Blake, Geoffrey A.

Katayama, Ikufumi

Mead, Griffin

Takeda, Jun

Caltech Authors - Main

Supplementary Materials for “An Echelon-based Single Shot Optical and Terahertz Kerr
Effect Spectrometer"
Authors: Griffin Mead, Ikufumi Katayama, Jun Takeda, Geoffrey A. Blake∗
∗Corresponding author: gab@gps.caltech.edu
Contents of Supplementary Materials:
A. Echelon Imaging
B. Data Processing
C. Diiodomethane OKE Response
D. Solvent OKE Responses: Stage vs. Echelon
E. Calculation of the Heterodyne and Homodyne Components in the Single Shot OKE Response
F. Quadratic Scaling of OKE Signal
1
A. Echelon Imaging
Properly imaging the echelon surface onto the camera array is critical for accurate measure-
ments. The following example demonstrates how the echelon image is propagated along the opti-
cal path, and extensively uses the imaging equation
1
f x
=
1
Ox
+
1
Ix
(1)
The echelon surface plane is the first object O1 in the imaging pathway (Fig. 1 (a)). Light
passing through the first lens L1 (focal length f1 = 200 mm) produces the first image I1 at a
distance provided by the imaging equation. This image acts as the second object O2 for the second
lens L2. The O2 distance from L2 must account for the 140 mm separation between L1 and L2.
The remainder of the imaging pathway follows from this same procedure of propagating images
along the optical path.
As the light passes through each lens, the images formed will change size. While the spheri-
cal lenses L1 and L2 uniformly change the horizontal and vertical dimensions of the image, the
cylindrical lenses L3 and L4 are used to adjust the image magnification along a single axis. The
L3 cylindrical lens forms image I3, and de-magnifies the horizontal axis of the echelon image.
Similarly, the L4 cylindrical lens de-magnifies the vertical axis. The initial and final dimensions
of the echelon object and images are shown in Fig 1 (b). Only a central slice of the final image is
acquired by the active sub-array on the camera, as shown (c).
2
FIG. 1. (a) The location of the echelon image is followed through the optical path between the echelon
surface and the camera array. (b) The relative size of the original echelon object (O1) and the images
formed after the two cylindrical lenses L3 and L4 (I3 and I4 respectively). (c) A schematic comparison of
the size of final image I4 and the active sub-array of the camera used to record the OKE/TKE data. The
dimensions of I4 were approximately 14.3 mm wide by 5.1 mm high. The camera subarray used to record
data was 16.64 mm wide by 0.52 mm high.
3
B. Data Processing
The fast frame rate camera can produce large quantities of data quickly; for example, 10 sec-
onds of data from the 1280×10 pixel sub-array with 2×8 binning and 1 kHz acquisition rate pro-
duces several hundred megabytes of data. Two data storage and analysis approaches were tested,
using ascii (‘.asc’) and raw (‘.dat’) data file formats. In Andor Solis, writing .dat files to disk was
found to be around 10-20× quicker than storing the data in the .asc format.
Python scripts were used for processing the camera data, and a similar time savings using the
.dat file format was evident. With an equivalent series of processing steps, .dat files were analyzed
>100× faster than .asc files. The Python script used for .dat file processing is shown below:
## b i n _ r e a d e r . py
i m p o r t numpy as np
i m p o r t m a t p l o t l i b . p y p l o t a s p l t
i m p o r t t ime
f i l e = ’DIM_10K_ . da t ’ # f i l e name
w = 1280 # a r r a y wid th i n p i x e l s , 2x b i n
h = 10 # a r r a y h e i g h t i n p i x e l s , 8x b i n
num = 10000 # number o f images a c q u i r e d
# i n i t . a r r a y s f o r even and odd image d a t a
odd = np . z e r o s (w)
even = np . z e r o s (w)
# s t a r t t i m e r
t 0 = t ime . t ime ( )
# r e a d i n f i l e w i th a p p r o p r i a t e e n c o d i n g
b = np . f r o m f i l e ( f i l e , d t y p e = np . i n t 3 2 )
t 1 = t ime . t ime ( )
c = np . r e s h a p e ( b , ( num , h , w) )
4
# sum a l o n g t h e a r r a y h e i g h t
d = np . sum ( c , a x i s =1)
# coadd images
f o r i i n r a n g e ( num ) :
i f i % 2 == 0 :
even += d [ i , : ]
e l s e :
odd += d [ i , : ]
# pe r fo rm s u b t r a c t i o n , n o r m a l i z a t i o n
i f np . max ( even ) > np . max ( odd ) :
d i f f = ( even−odd ) / odd
e l s e :
d i f f = ( odd−even ) / even
t 2 = t ime . t ime ( )
p r i n t ’ Read : ’ , np . round ( t1−t0 , 3 )
p r i n t ’ P r o c e s s : ’ , np . round ( t2−t1 , 3 )
# save p r o c e s s e d d a t a t o t x t f i l e
np . s a v e t x t ( ’% s _ d i f f . t x t ’ % f i l e , d i f f )
# p l o t r e s u l t s
p l t . s u b p l o t ( 2 , 1 , 1 )
p l t . p l o t ( even , l a b e l = ’ Even ’ )
p l t . p l o t ( odd , l a b e l = ’Odd ’ )
p l t . x l a b e l ( ’ P i x e l s (2 x b i n ) ’ )
p l t . y l a b e l ( ’ # p h o t o n s ( c o u n t s ) ’ )
p l t . l e g e n d ( f rameon = F a l s e )
5
p l t . s u b p l o t ( 2 , 1 , 2 )
p l t . s e m i l o g y ( d i f f , l a b e l = ’ Di f f ’ )
p l t . x l a b e l ( ’ P i x e l s (2 x b i n ) ’ )
p l t . y l a b e l ( ’ S i g n a l ( a . u . ) ’ )
p l t . l e g e n d ( f rameon = F a l s e )
p l t . show ( )
p l t . c l o s e ( )
6
C. Diiodomethane OKE Response
The OKE response of diiodomethane (CH2I2) was measured using the single shot apparatus,
with similar results to bromoform achieved.
CH2I2
−2 0 2 4 6 8 10
Time (ps)
0.0
0.2
0.4
0.6
0.8
1.0
S
ig
na
l (
a.
u.
)
(a)
CH2I2
0 2 4 6 8 10
Freq. (THz)
0.0
0.2
0.4
0.6
0.8
1.0
FF
T 
M
ag
. (
a.
u.
)
(b)
ν4
FIG. 2. (a) Diiodomethane OKE response measured using the echelon technique (N=10,000 shots). The
oscillatory molecular coherences are shown in the inset, along with the orientational response fit. (b) Fourier
transform of the fit residual reveals the ν4 Raman mode at 3.65 THz.
7
D. Solvent OKE Responses: Stage vs. Echelon
The six solvents measured using the single shot echelon and stage scan techniques are shown
below in Fig 3, with the difference (same scale, offset -0.2) between the two data sets shown to
demonstrate the degree of agreement between the two techniques. Some small diffraction arti-
facts and irregularities are present in the electronic response regime, which is defined as the few
hundred femtosecond region around the signal maximum. The longer picosecond scale molecular
orientational responses were found to agree well between the two techniques.
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
0.0
0.5
1.0
S
ig
na
l (
a.
u.
)
Acetonitrile
Ech
Stg
Diff
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
0.0
0.5
1.0
Acetone
Ech
Stg
Diff
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
0.0
0.5
1.0
S
ig
na
l (
a.
u.
)
Dimethyl  ulfoxide
Ech
Stg
Diff
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
0.0
0.5
1.0
Nitrobenzene
Ech
Stg
Diff
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
Time (p )
0.0
0.5
1.0
S
ig
na
l (
a.
u.
)
Benzene
Ech
Stg
Diff
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
Time (p )
0.0
0.5
1.0
Carbon di ulfide
Ech
Stg
Diff
FIG. 3. Comparison plots of the six solvents investigated by OKE, demonstrating the good agreement
between the two techniques across a range of response profiles. The echelon data consist of 10,000 laser
shots (10 seconds), the stage scan data 45,000 laser shots (45 seconds).
8
E. Calculation of the Heterodyne and Homodyne Components in the Single Shot OKE Response
The single shot OKE responses of the six solvents measured in this study were decomposed
into heterodyne and homodyne components using an algebraic solver written in Python. First, the
difference signal Y and background signal E2LO were extracted from the N=10,000 shot data set.
These two 1280×1 vectors were then used as inputs to a Python symbolic solver which found the
roots X to Eq. 4, where X = Esig. In Fig 4, the heterodyne and homodyne components are shown
for each solvent.
Y −X2 +2XELO = 0 (2)
−10 −5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
S
ig
na
l (
re
l.)
EmaxLO  = 4.73 
Emaxsig  = 0.11
Ace(on (r le
He(
Hom
−10 −5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
EmaxLO  = 2.8 
Emaxsig  = 0.18
Ace(one
He(
Hom
−10 −5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
S
 g
na
l (
re
l.)
EmaxLO  = 2.46 
Emaxsig  = 0.2
D me(hyl s)lfox de
He(
Hom
−10 −5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
EmaxLO  = 1.95 
Emaxsig  = 0.26
N (robenzene
He(
Hom
−10 −5 0 5 10
T me (ps)
0.0
0.2
0.4
0.6
0.8
1.0
S
 g
na
l (
re
l.)
EmaxLO  = 1.88 
Emaxsig  = 0.27
Benzene
He(
Hom
−10 −5 0 5 10
T me (ps)
0.0
0.2
0.4
0.6
0.8
1.0
EmaxLO  = 2.93 
Emaxsig  = 0.17
Carbon d s)lf de
He(
Hom
FIG. 4. The heterodyne and homodyne components of the single shot OKE responses of six solvents inves-
tigated in this study. The values in the upper left-hand corners specify the maximum relative magnitudes of
the calculated ELO and Esig fields, demonstrating the ELO Esig condition for linear heterodyne detection.
9
F. Quadratic Scaling of OKE Signal
To verify the nonlinear nature of the observed OKE signal, a series of stage scan measurements
were performed on CS2. Neutral density filters were used to attenuate the pump beam power. The
relative pump electric field strength of the measurements were calculated by taking the square root
of the fractional transmission through each neutral density filter used. Plotting the relative electric
field strength against the peak of the CS2 signal yielded a quadratic relationship, as required for a
nonlinear (i.e. ~E2pump dependent) signal.
0.1 0.2 0.3 0.4 0.5 0.6
E⃗ field (a.u.)
0
10
20
30
40
50
60
70
O
K
E
 S
ig
na
l (
a.
u.
)
Quadratic fit
Linear fit
Data
FIG. 5. The peak of the CS2 OKE signal is plotted as function of the relative pump electric field strength.
Data were taken using the stage scan technique, and neutral density filters were used to adjust the intensity
of the transmitted pump. A quadratic relationship best fits the data, in contrast to a linear fit, and confirms
the nonlinear origin of the OKE signal.
10


An echelon-based single shot optical and terahertz Kerr effect spectrometer

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An echelon-based single shot optical and terahertz Kerr effect spectrometer

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