of 7
ARTICLE
Received 7 Mar 2016
|
Accepted 11 Nov 2016
|
Published 9 Jan 2017
Coherent ultra-violet to near-infrared generation in
silica ridge waveguides
Dong Yoon Oh
1,
*, Ki Youl Yang
1,
*, Connor Fredrick
2,
*, Gabriel Ycas
2
, Scott A. Diddams
2
& Kerry J. Vahala
1
Short duration, intense pulses of light can experience dramatic spectral broadening when
propagating through lengths of optical fibre. This continuum generation process is caused by
a combination of nonlinear optical effects including the formation of dispersive waves. Optical
analogues of Cherenkov radiation, these waves allow a pulse to radiate power into a distant
spectral region. In this work, efficient and coherent dispersive wave generation of visible to
ultraviolet light is demonstrated in silica waveguides on a silicon chip. Unlike fibre broadeners,
the arrays provide a wide range of emission wavelength choices on a single, compact chip.
This new capability is used to simplify offset frequency measurements of a mode-locked
frequency comb. The arrays can also enable mode-locked lasers to attain unprecedented
tunable spectral reach for spectroscopy, bioimaging, tomography and metrology.
DOI: 10.1038/ncomms13922
OPEN
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
2
Time and Frequency Division,
National Institute of Standards and Technology, Boulder, Colorado 80305, USA. * These authors contributed equally to this work. Correspondence and
requests for materials should be addressed to K.J.V. (email: vahala@caltech.edu).
NATURE COMMUNICATIONS
| 8:13922 | DOI: 10.1038/ncomms13922 | www.nature.com/naturecommunications
1
C
ontinuum generation in microstructured optical fibre
1
gained importance with the generation of broad spectra
2
for application in optical frequency combs. There, a mode-
locked laser is broadened as a precurser to
f
-2
f
measurement of
the comb offset frequency
3
and self-referencing
4
. The advent of
frequency microcombs
5
and most recently the demonstration of
femtosecond pulse generation in microcavities
6–9
has focused
attention on techniques for supercontinuum generation on a chip.
Beginning with studies of Raman and four-wave-mixing effects in
monolithic waveguides
10
, there has been steady progress towards
supercontinuum generation using a variety of on-chip waveguide
materials
11–17
. Low pulse energies for efficient broadening have
been demonstrated on account of large nonlinear coefficients
combined with nano-scale waveguide cross sections
14,15
, and
devices operating in the mid-infrared
18,19
are possible. Moreover,
self-referencing has been achieved using a silicon nitride based,
monolithic waveguide on silicon
20
.
However, while there has been remarkable progress on
visible
21,22
and ultraviolet band
23–25
continuum generation in
microstructured optical fibres, as well as harmonic ultraviolet
and deep ultraviolet generation in gas-filled fibres
26,27
, there
have been no reports of chip-based ultraviolet generation.
Furthermore, widely tunable emission in a single device is a
feature of monolithic arrays that has no parallel in optical fibres.
Arrays enable precisely targeted wavelength emission for optimal
self-referencing. As shown here, their compactness also eliminates
the need for delay lines in these systems. Finally, this class of
devices can be combined with microcombs and compact mode-
locked lasers to provide user-designed coherent, short pulse light
for optical clocks
28
, laser cooling
29
, quantum manipulation of
atoms and ions
30
, and bioimaging
31
.
After describing the silicon ridge waveguide array fabrication
process, measurements of dispersive-wave generation are pre-
sented that include a study of the phase matching wavelength
dependence upon the waveguide geometry. Application of the
waveguide array to optimized self-referencing of a mode-locked
Yb fibre laser is then presented. Here, the dispersive wave phase
matching wavelength is precisely matched to the 2
f
frequency so
as to provide maximum signal-to-noise in the frequency-comb
offset-frequency measurement. Also, the approach eliminates the
delay line used in the standard self-referencing method. Finally,
the continuum spectra produced by the waveguide arrays are
analysed and modelled in detail.
Results
Silica ridge waveguide arrays
. Continuum generation in wave-
guides results from a combination of nonlinear processes
10,32
.
Self-phase modulation in combination with second-order
anomalous dispersion induces temporal compression of an
input pulse. As the pulse spectrally broadens, higher-order
dispersion and Raman interactions become important and the
pulse undergoes soliton fission. The resulting series of
fundamental solitons experiences Raman self-frequency shifting
towards longer wavelengths. If the soliton spectrum has
significant overlap with spectral regions that feature normal
dispersion, then it can radiate energy into a dispersive wave that
phase matches to the soliton phase
33
. The dispersive wave, which
can also be understood in terms of an analogy to Cherenkov
radiation
34
, provides a powerful way to both engineer the spectral
extent of the resulting optical continuum and to also spectrally
concentrate optical power in new bands
22,35
. The waveguides
demonstrated here apply lithographic control to engineer the
generation of coherent ultraviolet to visible dispersive-wave
radiation.
The waveguide geometry is shown in Fig. 1a,b and features a
silica ridge design that is air-clad on three sides. The air-cladding
enables a high level of optical confinement to both increase the
optical nonlinearity and shift the wavelength for zero group
velocity dispersion towards visible wavelengths. The fabrication
of the waveguides is an adaptation of the process used for
dispersion engineering in high-
Q
resonators
36
. The process flow
is presented in Fig. 2. More detail on the fabrication process is
provided in Supplementary Note 1. The waveguides support
both transverse electric (TE) and transverse magnetic (TM)
polarizations. The TM-polarization provides dispersive wave
generation at shorter wavelengths.
Waveguide arrays of varying mode area (0.76–2.22
m
m
2
) were
fabricated on the silicon chip. The effective mode areas are
determined by input of waveguide cross sections (measured in a
scanning electron microscope, SEM) into a finite-element-
method (FEM) solver. The wavelength dependence of the
second-order dispersion is also calculated using the FEM solver.
The Sellmeier equation was used to include the wavelength
dependence of the silica refractive index
37
. While bulk silica
features anomalous dispersion only for wavelengths beyond
1,270 nm, the geometrical dispersion introduced by the strong
optical confinement of the ridge waveguide enables anomalous
dispersion at much shorter wavelengths. In particular, the zero
crossing of the dispersion (
l
ZDW
) can be engineered to occur over
a wide range of wavelengths from 557 to 731 nm as is apparent in
Fig. 1c,d. Importantly, these lie well below the pumping
wavelengths of 830 and 1,064 nm that we employ.
The phase matching condition for the dispersive wave
generation satisfies the following equation
38
:
bo
DW
ðÞ
bo
p


1
v
g

o
DW

o
p


1
2
g
P
p
¼
0
ð
1
Þ
where
b
(
o
DW
) and
b
(
o
p
) are the propagation constants at the
dispersive wave frequency (
o
DW
) and the pump frequency (
o
p
),
respectively, and
v
g
is the group velocity at the pump frequency.
P
p
is the peak power of the pulse when the dispersive wave
is generated.
g
is the nonlinearity of the waveguide at the
pump frequency and is given by
g
¼
o
p
n
2
/(
cA
eff
) where
n
2
is the
Kerr coefficient and
A
eff
is the effective mode area. Defining
D
b
(
o
)
¼
b
(
o
)

b
(
o
p
)

(
o

o
p
)/
v
g
, the dispersive wave phase
matching condition is given by
D
b
(
o
DW
)
¼
g
P
p
/2 and is plotted
in Fig. 1d for an 830 nm pump. At this pump wavelength, the
phase matching wavelength can be engineered to vary from 310
to 576 nm. Discussion on the range of phase matching
wavelengths can be found in Supplementary Fig. 2.
Imaging dispersive waves in the array
. Photographs of pumped
waveguides on a single chip are combined in Fig. 1e. The laser
pulse in these images is provided by a mode-locked titanium-
sapphire laser (830 nm emission, 60 fs pulse width measured on
an autocorrelator, and 81 MHz repetition frequency) and is
coupled into the silica ridge waveguides using a

60 objective
lens. The coupling efficiency is measured to be 25–35%. As the
laser pulse propagates along a given waveguide (left-to-right
in image), the infrared pulse is initially invisible. The pulse
undergoes temporal compression and spectral broadening
because of the self phase modulation and anomalous second-
order dispersion. As broadening occurs, portions of the pulse’s
spectrum becomes visible. Solition fission and dispersive wave
generation occur at the bright orange spot. Waveguide areas are
larger for upper waveguides in the photograph and decrease
towards the lower portion of the plot. Decreasing mode areas
shift the phase matching condition to shorter wavelengths so that
the dispersive wave generation transitions from the visible to
ultraviolet. For wavelengths below 400 nm the dispersive wave is
not visible in the photograph.
ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13922
2
NATURE COMMUNICATIONS
| 8:13922 | DOI: 10.1038/ncomms13922 | www.nature.com/naturecommunications
More details about the pulse propagation in the waveguide are
provided in Fig. 3. Here, calculated spectra are plotted versus
propagation length for pulse energies of 330 pJ (Fig. 3a) and
1,100 pJ (Fig. 3c). The pulse initially propagates as a higher-order
soliton that undergoes temporal compression and spectral
broadening until experiencing soliton fission and dispersive wave
generation at a fission length,
L
f
, that depends upon the pulse
energy (330 pJ:
L
f
¼
0.39 cm; 1,100 pJ:
L
f
¼
0.16 cm). As the input
pulse energy increases, the calculated fission length decreases.
This behaviour is observable in Fig. 3b where a composite of
photographs of the waveguide is shown at coupled pulse energies
from 330 to 1,100 pJ. Soliton fission is observable as the reddish-
orange emission. The observed soliton fission length agrees well
with a numerical simulation shown as the grey line.
The dispersive wave initially overlaps with the soliton pulse but
walks off from the soliton with continued propagation because of
group velocity mismatch (see Fig. 3d,e). Also, the dispersive wave
is temporally stretched as it propagates in the waveguide because
of the normal dispersion it experiences. Walk-off is a typical
feature of fibre broadeners and requires a delay line to spatially
overlap the octave wave (2
f
) and dispersive wave. However, as
shown in the next section, the waveguide length can be adjusted
to optimize overlap between 2f and dispersive-wave pulses.
Moreover, the array, itself, provides the user with the ability to
optimize the dispersive wave efficiency around a specific laser
source.
Walk-off-free dispersive-wave-enhanced octave generation
.
Beyond the generation of broadly tunable visible light, precise
dispersive-wave engineering in a monolithic waveguide can be
applied both to enhance the signal-to-noise ratio as well as
simplify the setup for detection of the carrier-envelope offset
frequency of a laser frequency comb
3
. As a demonstration,
f
-2
f
offset frequency generation of a Yb fibre laser comb is performed
using a silica ridge waveguide. The waveguide having a mode area
of 3.13
m
m
2
and length of 1.50 cm is dispersion engineered so as
to enhance dispersive wave formation at twice the frequency of
the Yb laser comb. The experimental setup is shown in Fig. 4a.
The Yb laser emits 90 fs pulses with centre wavelength of
1,064 nm (100 MHz repetition rate). Before waveguide coupling, a
halfwave plate rotates the Yb laser polarization so that 80% of the
total coupled pulse energy (2,300 pJ) is transmitted in the TE
mode and 20% is transmitted in the TM mode of the waveguide.
The TE wave forms a dispersive wave near 532 nm.
A photograph of the spectrum of the continuum is shown in
Fig. 4b and a spectral scan is provided in 4c. Both the TE and TM
waves are subsequently coupled to a potassium niobate (KNbO
3
)
crystal where the TM component is phase matched for second
545
529
509
497
485
472
462
446
432
416
395
366
331
322
1.5 cm
300
500
700
900
1,100
0
50
–50
0
–250
250
GVD (ps nm
–1
km
–1
)
Δ

(mm
–1
)
Wavelength (nm)
(nm)
a
bc
e
d
2.0
3.0
4.0
700
600
1.0
1.5
2.0
550
650
Ridge width (
μ
m)
Mode area(
μ
m
2
)
λ
ZDW
(nm)
2.5
3.5
4.5
1.5
750
Figure 1 | Phase matching condition and direct observation of dispersive wave generation in silica ridge waveguides.
(
a
,
b
) Scanning electron
microscope images of an array of silica ridge waveguides on a silicon chip. Silicon pillars support silica layers containing waveguides. The red box i
n
a
contains a silica waveguide whose cross section is shown in
b
; the cross section shows the calculated mode profile of the TM mode at wavelength 830 nm
superimposed. Scale bar, 100
m
m for (
a
) and 1
m
m for (
b
). (
c
) Calculated mode area and zero dispersion wavelength (
l
ZDW
) are plotted versus the ridge
base width. (
d
) Calculated group velocity dispersion (GVD, dashed lines) and phase-matching parameter
D
b
(solid lines) for dispersive wave generation in
TM polarization are plotted versus wavelength. Blue, red and yellow solid and dashed lines correspond to mode areas of 0.83, 1.03, 1.69
m
m
2
respectively.
The phase matching condition
D
b
¼
1
2
g
P
p
(830 nm pump) is satisfied at the intersections of the coloured solid lines and the black dashed line. The points
with error bars are measured dispersion values obtained from sets of ten scans of spectral fringes measured using a Mach-Zehnder interferometer
(see Methods). (
e
) Ultraviolet–visible dispersive wave generation in a silicon chip containing an array of waveguides with varying mode area (322 nm
emission has area of 0.83
m
m
2
and 545 nm emission has area of 2.09
m
m
2
). Image of multiple photographs of scattered light taken from above the 1.5 cm
long chip in which infrared pulses are launched at the left side of each waveguide. Pulse energies are set to the threshold pulse energy for dispersive w
ave
generation. The colour emission at the left side of the image is dispersive wave emission that has been reflected at the far right facet of the chip. The in
itial
spectral broadening of the input pulse can be seen as the orange-red emission that gradually shifts diagonally upward right. The visible scattered li
ght
shown in the figure is a very small amount of the total generated. The vast majority of the light is forward propagating and collected into a multimode fibr
e
(not shown).
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13922
ARTICLE
NATURE COMMUNICATIONS
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3
harmonic generation. The phase-matching condition generates
the second-harmonic beam that is rotated by 90
°
relative to the
TM fundamental so that the second-harmonic polarization is
aligned with the TE polarized dispersive wave. The measured
doubled spectrum and the dispersive-wave spectrum are shown in
Fig. 4d. After spectral filtering around 532 nm the two waves are
mixed on a photodiode for offset frequency (
f
ceo
) beatnote
generation. In this final step, the enhanced dispersive wave comb
lines increase the beatnote strength. The electrical spectrum of the
beatnotes at
f
ceo,1
¼
25.8 MHz and
f
ceo,2
¼
f
rep

f
ceo,1
¼
73.8 MHz
are shown in Fig. 4e. Both signals have a signal-to-noise ratio
4
34 dB at an electrical resolution bandwidth of 300 kHz.
This establishes the coherent nature of the dispersive wave and
is more than sufficient for subsequent self-referenced servo
control of the comb.
Beyond the enhanced signal-to-noise provided by the ability to
select an optimal waveguide in the array for matching to the 2
f
frequency, there is a key simplification enabled by colinear
generation of dispersive and second harmonic waves. The overall
group velocity dispersion between the TE dispersive wave and the
TM wave at 1,064 nm in both the waveguide array and the
KNbO
3
crystal is such that both the dispersive wave and the
second-harmonic pulses emerge from the doubling crystal with a
high degree of spatial overlap (that is, walk-off-free). As a result, a
major simplification is possible by eliminating the traditional step
of path-length balancing using a Michelson interferometer.
Instead, the two pulses (the dispersive wave and the second-
harmonic pulse) can be directly coupled to the photodetector
39
.
Spectral measurements
. To further probe the behaviour of the
dispersive wave generation process, spectral measurements were
performed at varying waveguide cross sectional areas. For these
measurements, the titanium sapphire laser was used (see earlier
discussion) and the experimental setup included an attenuator
to vary the input pulse energy, as well as a half-wave plate to
control polarization. The light output from the waveguides is
endfire-coupled to a multimode fibre for the measurement on a
spectrometer. To cover the entire spectral range of interest two
spectrometers were used: a Yokogawa (AQ6370D, 600–1,700 nm)
and an Ocean Optics (HR4000, 200–900 nm).
g
ab
cd
ef
Figure 2 | Ridge waveguide array microfabrication process.
(
a
)
Photolithography on thermal silica layer. Mask width is denoted by the
double-arrow line. (
b
) HF wet-etching to define silica ridge. (
c
) Additional
oxide layer is grown by thermal oxidation. (
d
) Supporting structure is
patterned by photolithography. (
e
) HF wet-etching creates striped openings
in the silica layer. (
f
) Isotropic etching of the silicon (XeF
2
) is performed to
undercut the silica layer. (
g
) Rendering of final ridge waveguide array
structure. Dependence of ridge dimension on mask width is shown in
Supplementary Fig. 1.
300
700
1,100
Wavelength (nm)
1,100
1,100
700
300
700
300
–1
Delay (ps)
Wavelength (nm)
Wavelength (nm)
Pulse energy (pJ)
400
600
800
1,000
Waveguide position (cm)
Wavelength (nm)
0.4
0.3
0.2
0.1
0
300
700
1,100
330 pJ
0
–50
Power (dB)
1,100 pJ
0.4
0.3
0.2
0.1
0
Waveguide position (cm)
0.4
0.3
0.2
0.1
0
Waveguide position (cm)
01
0.16 cm
0.47 cm
–1
Delay (ps)
01
abcd
e
Figure 3 | Numerical simulation of pulse propagation in the waveguide and comparision with measurement.
(
a
) Calculated continuum spectra as a
function of waveguide position at a coupled pulse energy of 330 pJ. The pulse is launched in the TM mode of the waveguide with mode area 0.76
m
m
2
.
The colour bar (inset) applies to (
a
,
c
e
). (
b
) Top-view photographs of scattered light from the surface of the waveguide. The photographs were taken at
pulse energies ranging from 330 to 1,100 pJ (left to right side). As indicated by the dashed line, the pulse travels in the waveguide from the bottom to th
e
top of the image. Dispersive wave generation occurs in the ultraviolet and is therefore not visible in the image. The grey line superimposed on the
photographs is the length at which dispersive wave generation occurs as predicted by the simulation. The spectral breathing of the input pulse result
sina
periodically visible orange-red emission that correlates with the calculation. (
c
) Calculated continuum spectra as a function of waveguide position at a pulse
energy of 1,100 pJ. (
d
,
e
) Calculated spectrograms of the optical pulse propagating in the waveguide corresponding to (
c
). The spectrogram in
d
is at
waveguide position 0.16 cm where dispersive wave is generated. The spectrogram in
e
is at the waveguide output.
ARTICLE
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Measured supercontinuum spectra are shown in Fig. 5a for TM
and TE polarized pulses launched into waveguides of varying
mode area in the waveguide array shown in Fig. 1e. Conversion of
pump light in the near infrared to visible and ultraviolet
wavelength dispersive waves is apparent in each spectrum.
A summary of measured dispersive wave emission wavelengths
l
DW
versus mode area is provided in Fig. 5b. Controlled tuning of
the dispersive wave from 310 to 576 nm with an average 8 nm
interval is demonstrated in the TM mode and from 475 to 613 nm
in the TE mode. Also included in Fig. 5b is the predicted
dispersive wave emission wavelength using the phase matching
condition (equation (1)). The measurement result agrees well
with the calculation.
To investigate the conversion efficiency of the input pulse
from the near-infrared into ultraviolet–visible wavelengths
(300–650 nm), a ultraviolet fused silica aspheric lens (transmis-
sion 200–2,000 nm) was used to collect the light exiting the
waveguides. The optical power in the ultraviolet–visible wave-
lengths was then filtered out with a bandpass filter and measured
by a thermopile power metre (nearly uniform spectral response).
The power was calibrated using the wavelength-dependent
transmission curves of the bandpass filter and the measured
supercontinuum spectra. The scatter plot of measured conversion
efficiency (and average dispersive wave power) plotted versus
dispersive wave peak wavelength for both polarizations is shown
in Fig. 5c. For this measurement, conversion efficiency is defined
as the dispersive wave power at the waveguide output divided by
the total collected spectral power at the waveguide output. The
spectral extent of the dispersive wave was defined by selecting
wavelengths where the power spectral density of the dispersive
wave had fallen to 5% of the maximum value.
Using the Yb laser emission at 1,064 nm, it was also possible to
investigate even deeper ultraviolet dispersive wave generation
below 300 nm. This observation is consistent with the
phase-matching condition for dispersive wave generation
(Supplementary Fig. 2). Figure 5d shows supercontinuum spectra
measured in the TM mode of two waveguides having cross-
sectional areas of 1.09 and 1.12
m
m
2
using a coupled pulse energy
of approximately 2,000 pJ. The spectra feature multiple peaks
because of soliton breathing and subsequent dispersive wave
emission. The shortest wavelength edge of the spectrum is nearly
265 nm for the case of the waveguide with mode area 1.09
m
m
2
.
Discussion
A silicon chip based waveguide array has been applied to generate
ultraviolet to visible light by conversion of an input pulse into a
dispersive wave. The dispersive wave emission wavelength is
precisely tuned from ultraviolet to visible by lithographic control
of the waveguide dimensions. Generation of ultraviolet emission
as short as 265 nm has been demonstrated. The measured and
predicted dispersive wave emission wavelengths are in excellent
agreement. Arrays-on-a-chip containing hundreds of waveguides
are easily fabricated and provide ready access to a range of
emission wavelengths using a single pump laser. This chip-based
tuning control allows for optimization of emission for spectro-
scopy and metrology using a single device. As a demonstration,
offset frequency generation in a Yb mode-locked laser frequency
comb was demonstrated by designing a waveguide to generate a
dispersive wave that is optimally matched to the second harmonic
of the original 1
m
m comb. Significantly, this demonstration also
confirms the high coherence of the dispersive wave generated by
these waveguides. Moreover, the ability to tailor the chip length to
provide walk-off-free self-referencing was demonstrated. Mono-
lithic waveguide arrays for wide-band coherent optical generation
(up to two octaves) provide a new capability for integration with
other optical elements on a chip and can also find application in
other areas including bioimaging.
Methods
Dispersion characterization
.
To verify that the waveguide exhibits anomalous
dispersion near the value predicted by the FEM simulations in Fig. 1d, the group
velocity dispersion is characterized using a Mach-Zehnder interferometer having
one arm with an adjustable delay
40
. In the dispersion measurement, a probe pulse
is attenuated to a low enough pulse energy (
o
30 pJ) so as to prevent significant
λ
/2
SiO
2
waveguide
KNbO
3
crystal
Photodiode
ESA
Bandpass filter
Yb fibre laser
1,064 nm
a
b
Frequency (MHz)
Wavelen
g
th (nm)
0
–20
–40
–60
–80
0
Power (dBm)
20
40
60
80
100
500
900
1,300
1,700
Power (20 dB per div)
700
1,100
1,500
frequency
doubling
–60
–50
–40
–30
–70
520
530
540
550
Power (dBm)
DW
SH
Wavelen
g
th (nm)
f
ceo,1
f
ceo,2
f
rep
cde
Figure 4 | Application of dispersive wave engineering to self-referencing a Yb fibre laser frequency comb.
(
a
) Experimental setup for measuring
f
ceo
of
the Yb fibre laser. The ridge waveguide has a mode area 3.13
m
m
2
and length 1.50 cm. (
b
) Photograph of the emitted light dispersed through a prism and
reflected on a white screen. Scale bar, 5 cm. (
c
) Measured optical spectrum of the collimated beam at the output facet of the waveguide. The coupled pulse
energy is 2,300 pJ and the input spectrum is shown in black. (
d
) Spectra of the dispersive wave (DW, blue) and second harmonic light (SH, black) filtered
by a bandpass filter. (
e
) Radio-frequency spectrum measured with an electrical spectrum analyser (ESA) shows the pulse repetition rate
f
rep
and the carrier-
envelope-offset beat frequency
f
ceo,1
and
f
ceo,2
. The resolution bandwidth (RBW) is 300 kHz.
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nonlinear phase shifts in the waveguide. The pulse is split using a beamsplitter and
propagates into two arms of a Mach-Zehnder interferometer. One pulse is coupled
to the waveguide and the other one propagates through the adjustable delay line in
the air. The light emitted from the waveguide is collimated and the pulses from the
two arms are combined using a second beamsplitter. The combined beams are then
focused into a single mode fibre and sent to a spectrometer. The time delay
between the two pulses is calculated using the Fourier-transform of the measured
spectrum which shows spectral fringes
41
. By tuning the laser wavelength from 770
to 810 nm, the dispersion of the waveguide is then measured from the observed
wavelength-dependence of the group delay in the waveguide arm. The measured
dispersion is plotted in Fig. 1d and agrees well with the FEM simulation. The error
bars in Fig. 1d are derived from the standard deviation of group delay from
10 spectral measurement at each wavelength.
Numerical simulations
.
Numerical simulations of pulse propagation in the
waveguides were performed using a generalized nonlinear Schro
̈
dinger
equation. To model the frequency dependence of the nonlinear response,
the following equation is used
42
.
@
~
A
@
z
¼
i
bo
ðÞ
bo
p


1
v
g
o

o
p



ao
ðÞ
2

~
Az
;
o
ðÞ
þ
i

go
ðÞF

Az
;
T
ðÞ
Z
1
1
RT
0
ðÞ

Az
;
T

T
0
ðÞ
jj
2
d
T
0

ð
2
Þ
Here,
~
Az
;
o
ðÞ
is the complex spectral envelope of the pulse at waveguide position
z
,

Az
;
T
ðÞ
is defined as
F

1
f
~
Az
;
o
ðÞ
=
A
1
=
4
eff
o
ðÞg
,
a
(
o
) is the linear loss and
R
(T) is
the Raman response function.

go
ðÞ
is the parameter related to the nonlinear
response defined as:

go
ðÞ¼
n
2
n
0
o
cn
eff
o
ðÞ
A
1
=
4
eff
o
ðÞ
ð
3
Þ
The frequency-dependent propagation constant (
b
(
o
)), effective mode index
(
n
eff
(
o
)) and mode area (
A
eff
(
o
)) of the waveguides were calculated using FEM
simulations and imported into the pulse propagation simulations.
Data availability
.
The data that support the plots within this paper and other
findings of this study are available from the corresponding author on reasonable
request.
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Acknowledgements
We gratefully acknowledge the Defense Advanced Research Projects Agency (DARPA)
under the PULSE (W31P4Q-14-1-0001) and QuASAR (W911NF-14-1-0284) programs,
the National Aeronautics and Space Administration (NASA) (KJV.JPLNASA-1-JPL.
1459106), the Kavli Nanoscience Institute, and the Institute for Quantum Information
and Matter, a National Science Foundation (NSF) Physics Frontiers Center
(PHY-1125565) with support of the Gordon and Betty Moore Foundation, NIST and the
National Science Foundation (AST-1310875).
Author contributions
Experiments were conceived by all authors. D.Y.O. and K.Y.Y performed modelling.
K.Y.Y. fabricated devices with assistance from D.Y.O. D.Y.O., C.F. and G.Y. performed
the measurement. Analysis of results was conducted by all authors. All authors
participated in writing the manuscript.
Additional information
Supplementary Information
accompanies this paper at http://www.nature.com/
naturecommunications
Competing financial interests:
The authors declare no competing financial interests.
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How to cite this article:
Oh, D. Y.
et al.
Coherent ultra-violet to near-infrared generation
in silica ridge waveguides.
Nat. Commun.
8,
13922 doi: 10.1038/ncomms13922 (2017).
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