of 7
On
-chip two-octave supercontinuum generation
by enhancing self-steepening of optical pulses
Lin Zhang,
1,*
Yan Yan,
1
Yang Yue,
1
Qiang Lin,
2
Oskar Painter,
2
Raymond G.
Beausoleil,
3
and Alan E. Willner
1
1
Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA
2
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California
91125, USA
3
HP Laboratories, Palo Alto, California 94304, USA
*linzhang@usc.edu
Abstract:
Dramatic advances in supercontinuum generation have been
made recently using photonic crystal fibers, but it is quite challenging to
obtain an octave-spanning supercontinuum on a chip, partially because of
strong dispersion in high-index-contrast nonlinear integrated waveguides.
We show by simulation that extremely flat and low dispersion can be
achieved in silicon nitride slot waveguides over a wavelength band of 500
nm. Different from most of previously reported supercontinua that were
generated either by higher-order soliton fission in anomalous dispersion
regime or by self-phase modulation in normal dispersion regime, a two-
octave supercontinuum from 630 to 2650 nm (360 THz in total) can be
generated by enhancing self-steepening in pulse propagation in nearly zero
dispersion regime, when an optical shock as short as 3 fs is formed.
©2011 Optical Society of America
OCIS codes:
(130.3120) Integrated optics devices; (130.3060) Infrared; (190.4390) Nonlinear
optics, integrated optics; (190.7110) Ultrafast nonlinear optics; (320.2250) Femtosecond
phenomena; (350.4238) Nanophotonics and photonic crystals; (999.9999) Silicon photonics.
References and links
1. J. K. Ranka
, R. S. Windeler, and A. J. Stentz
, “
Visible continuum generation in air-silica microstructure optical
fibers with anomalous dispersion at 800 nm
,”
Opt. Lett.
25
(1),
25
27 (2000
).
2. J. M. Dudley
, G. Genty, and
S. Coen
, “
Supercontinuum generation in photonic crystal fiber
,”
Rev. Mod. Phys.
78
(4), 1135
1184 (2006
).
3. T. A. Birks
, W. J. Wadsworth, an
d P. St. J. Russell
, “
Supercontinuum generation in tapered fibers
,”
Opt. Lett.
25
(
19), 1415
1417 (2000
).
4.
G. Qin
, X.
Yan
, C. Kito
, M.
Liao
, C.
Chaudhari
, T.
Suzuki, and
Y. Ohishi
, “
Ultrabroadband supercontinuum
generation from ultraviolet to 6.28 μm in a
fluoride fiber
,”
Appl. Phys. Lett.
95
(
16), 161103 (2009
).
5. W. H. Reeves
, D. V. Skryabin
, F.
Biancalana
, J. C. Knight
, P. St. J. Russell
, F. G. Omenetto
, A.
Efimov, and A.
J. Taylor
, “
Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres
,”
Nature
424
(6948), 511
515 (2003
).
6. M. L. V. Tse
, P.
Horak
, F.
Poletti
, N. G. Broderick
, J. H. Price
, J. R. Hayes, and D. J. Richardson
,
Supercontinuum generation at 1.06 mum in holey fibers with dispersion flattened profiles
,”
Opt. Express
14
(
10), 4445
4451 (2006
).
7.
W.
-Q. Zhang
, S.
Afshar V, and T. M. Monro
, “
A genetic algorithm based approach to fiber design for high
coherence and large bandwidth supercontinuum generation
,”
Opt. Express
17
(21), 19311
19327 (2009
).
8.
J. Leuthold
, C. Koos, and
W.
Freude
, “
Nonlinear silicon photonics
,”
Nat. Photonics
4
(8), 535
544
(2010
).
9. B. J. Eggleton
, B.
Luther-Davies, and
K. Richardson
, “
Chalcogenide photonics
,”
Nat. Photonics
5
(3), 141
148
(2011
).
10. Ö. Boyraz
, P.
Koonath
, V.
Raghunathan, and
B. Jalali
, “
All optical switching and continuum generation in
silicon waveguides
,”
Opt. Express
12
(17), 4094
4102 (2004
).
11. L. Yin
, Q.
Lin, and G. P. Agrawal
, “
Soliton fission and supercontinuum generation in silicon waveguides
,”
Opt.
Lett.
32
(4),
391
393 (2007
).
12. I.-W.
Hsieh
, X.
Chen
, X.
Liu
, J. I. Dadap
, N. C. Panoiu
, C.-Y. Chou
, F.
Xia
, W. M. Green
, Y. A. Vlasov, and R.
M.
Osgood
, “
Supercontinuum generation in silicon photonic wires
,”
Opt. Express
15
(
23), 15242
15249 (2007
).
13. M. R. E. Lamont
, B.
Luther-Davies,
D. Y. Choi
, S.
Madden, and B. J. Eggleton
, “
Supercontinuum generation in
dispersion engineered highly nonlinear (γ = 10 /W/m) As
2
S
3
) chalcogenide planar waveguide
,”
Opt. Express
16
(
19), 14938
14944 (2008
).
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11584
14. D. Duchesne
, M.
Peccianti
, M. R. E. Lamont
, M.
Ferrera
, L. Razzari
, F.
Légaré
, R.
Morandot
ti,
S. Chu
, B. E.
Little, and D. J. Moss
, “
Supercontinuum generation in a high index doped silica glass spiral waveguide
,”
Opt.
Express
18
(2), 923
930 (2010
).
15. L. Zhang
, Y.
Yue
, R. G. Beausoleil, and A. E. Willner
, “
Flattened dispersion in silicon slot waveguides
,”
Opt.
Express
18
(19), 20529
20534 (2010
).
16. A. M. Heidt
, “
Pulse preserving flat-top supercontinuum generation in all-normal dispersion photonic crystal
fibers
,”
J. Opt. Soc. Am. B
27
(3), 550
559
(2010
).
17. G. Genty
, P.
Kinsler
, B.
Kibler
, an
d J. M. Dudley
, “
Nonlinear envelope equation modeling of sub-cycle
dynamics and harmonic generation in nonlinear waveguides
,”
Opt. Express
15
(9), 5382
5387 (2007
).
18. S. Wabnitz and V. V. Kozlov
, “
Harmonic and supercontinuum generation in quadratic and cubic nonlinear
optical media
,”
J. Opt. Soc. Am. B
27
(9), 1707
1711 (2010
).
19. G. Genty
, B.
Kibler
, P. Kinsler, and J. M. Dudley
, “
Harmonic extended supercontinuum generation and carrier
envelope phase dependent spectral broadening in silica nanowires
,”
Opt. Express
16
(
15), 10886
10893 (2008
).
20. P. Kinsler
, “
Optical pulse propagation with minimal approximations
,”
Phys. Rev. A
81
(1), 013819 (2010
).
21. J. C. A. Tyrrell
, P.
Kinsler, and G. H. C. New
, “
Pseudospectral spatial-domain: a new method for nonlinear pulse
propagation in the few-cycle regime with arbitrary dispersion
,”
J. Mod. Opt.
52
(7), 973
986 (2005
).
22. G. P. Agrawal
,
Nonlinear Fiber Optics
, 3rd ed. (Academic, 2001).
23. J. S. Levy
, A. Gondarenko
, M. A. Foster
, A. C. Turner-Foster
, A. L. Gaeta
, and
M.
Lipson
, “
CMOS-compatible
multiple-wavelength oscillator for on-chip optical interconnects
,”
Nat. Photonics
4
(1),
37
40 (2010
).
24. K. Ikeda
, R. E. Saperstein
, N.
Alic, and
Y. Fainman
, “
Thermal and Kerr nonlinear properties of plasma-deposited
silicon nitride/ silicon dioxide waveguides
,”
Opt. Express
16
(17), 12987
12994 (2008
).
25. S. Afshar V and T. M. Monro
, “
A full vectorial model for pulse propagation in emerging waveguides with
subwavelength structures part I: Kerr nonlinearity
,”
Opt. Express
17
(4), 2298
2318 (2009
).
26. Y. Wang
, R. Yue
, H.
Han, and
X. Liao
, “
Raman study of structural order of a-SiN
x
:H and its change upon
thermal annealing
,”
J. Non-Cryst. Solids
291
(1-2), 107
112
(2001
).
27. A. L. Gaeta
, “
Catastrophic collapse of ultrashort pulses
,”
Phys. Rev. Lett.
84
(16), 3582
3585 (2000
).
28. N. Aközbek
, A.
Iwasaki
, A.
Becker
, M.
Scalora
, S. L. Chin, and C. M. Bowden
, “
Third-harmonic generation and
self-channeling in air using high-power femtosecond laser pulses
,”
Phys. Rev. Lett.
89
(
14), 143901 (2002
).
29. V. Roppo
, M.
Centini
, C. Sibilia
, M.
Bertolotti
, D.
de Ceglia
, M.
Scalora
, N. Akozbek
, M. J. Bloemer
, J. W.
Haus
, O. G. Kosareva, and V. P. Kandidov
, “
Role of phase matching in pulsed second-harmonic generation:
walk-off and phase-locked twin pulses in negative-index media
,”
Phys. Rev. A
76
(3), 033829 (2007
).
1. Introduction
One basic building block in nonlinear optics is a supercontinuum generator, which has
experienced a revolutionary development after its realization using photonic crystal fibers
(PCFs) [1,2]. Supercontinua of a few octaves in width have been reported [3,4] for numerous
applications such as frequency metrology, optical coherence tomography, pulse compression,
microscopy and spectroscopy, telecommunication, and sensing. A key figure of merit is the
width of a supercontinuum, which is greatly affected by the spectral profile of the dispersion
in a nonlinear medium. The success of PCF-based supercontinuum generation is partially
attributed to advanced dispersion engineering allowed by design freedom of the 2D lattice in
the fiber cladding [2,5]. Generally, the dispersion engineering is aimed at desirable zero-
dispersion wavelengths and low dispersion over a wide spectral band. Flat dispersion of ± 2
ps/(nm·km) over a 1000-
nm
-wide wavelength range was proposed [6,7
].
Highly nonlinear integrated waveguides and photonic wires with high index contrast have
generated much excitement in recent years [8,9], forming the backbone of compact devices in
a photonic-integrated-circuit platform. However, to the best of our knowledge, demonstrated
on-chip supercontinua have a spectral range of ~400 nm [
10
14], far less than one octave,
which is partially because of insufficient capability to engineer the dispersion of nonlinear
waveguides. Recently, the dispersion profile of a silicon waveguide was made 20 times flatter
by introducing a nano-scale slot structure [
15
], but this is still not sufficient to support more
than one octave spectral broadening of femtosecond optical pulses.
Here, we propose a silicon nitride slot waveguide, which exhibits further improvement in
dispersion flatness by 30 times, compared with that reported in Ref [
15
]. In our simulation, a
two-octave supercontinuum can be obtained on a chip by enhancing pulse self-steepening and
forming an optical shock as short as 3 fs. The supercontinuum can be „transferred‟ to third
-
harmonic spectral range by phase-locked pulse trapping effect. On-chip supercontinuum
generation is believed to be a key enabler for building portable imaging, sensing, and
frequency-metrology-based positioning systems. We expect that the advanced dispersion
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11585
engineering technique in integrated waveguides opens the door to combine ultrafast optics and
nano-photonics and to apply ultra-wideband optical information technologies ubiquitously.
2. Flat all-normal dispersion in silicon nitride slot waveguide
The proposed silicon nitride (Si
3
N
4
) slot waveguide is shown in Fig. 1. A horizontal silica slot
is formed between two Si
3
N
4
layers. The substrate is 2-
μm
-thick SiO
2
. The waveguide
parameters are: width
W
= 980 nm, upper height
H
u
= 497 nm, lower height
H
l
= 880 nm, and
slot height
H
s
= 120.5 nm. Material dispersions of Si
3
N
4
and SiO
2
are considered. The
waveguide has a single fundamental mode at the vertical polarization beyond 1800 nm.
Figure 2 shows the flat dispersion of quasi-TM (vertically polarized) mode in the slot
waveguide in terms of dispersion coefficient β
2
= d
2
β(ω)/dω
2
, where β(ω) is propagation
constant. The avera
ge β
2
and Δβ
2
are 0.0137 and 0.00195 ps
2
/m, respectively, over a 610-
nm
-
wide band from 1210 to 1820 nm. Compared to a dispersion-flattened silicon slot waveguide
[15
], which has β
2
of ± 0.024 ps
2
/m from 1565 to 2100 nm, the proposed slot waveguide
shows 30 times flatter dispersion. We intentionally modify the slot waveguide to obtain
normal dispersion at all wavelengths because the supercontinuum generated in all-normal
dispersion regime typically would have good spectral coherence [2,16]. The dispersion
flattening results from an anti-crossing effect [
15
]. The strip mode at short wavelength
becomes more like a slot mode as wavelength increases, which causes a slightly negative
waveguide dispersion [
15
] to balance the convex dispersion in strip waveguides without a slot
structure. The negative dispersion is made well-matched to the convex dispersion when Si
3
N
4
and SiO
2
are adopted. Thus we obtain a much flatter dispersion than where using Si/SiO
2
slot
waveguides. One can tailor dispersion value and slope for various nonlinear applications by
changing waveguide structural parameters with similar trends presented in Ref. [
15
].
Fig. 1. Silicon nitride slot waveguide for dispersion flattening and supercontinuum generation.
A horizontal silica slot is between two silicon nitride layers.
Fig. 2. Flattened all normal dispersion for supercontinuum generation in the slot waveguide.
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11586
3. Supercontinuum generation by enhanced nonlinear self-steepening effect
We use a generalized nonlinear envelope equation (GNEE) [17,18], with third harmonic
generation considered, to model supercontinuum generation. It has been confirmed that the
simulation of even sub-cycle pulse propagations using this envelope equation is quite accurate
[17
20
], which is in excellent quantitative agreement with numerical integration of accurate
Maxwell's equations [
21
]. We use split-step Fourier method [
22
] to solve the following
GNEE:
2
()
(
)
[
( )
( )]
2!
m
m
m
Kerr
Raman
m
m
i
i
A
N
A
N
A
zm
t

 
where
0
0
2
23
2'
22
1
( )
1
|
|
and
3
1
( )
1
(
')(|
|
) ' .
3
it
Kerr
e
Kerr
t
it
Raman
R
R
Raman
i
N
A
i
A
A
A e
t
i
N
A
i
A h t t
A
A e
dt
t



 






 




We denote A = A(z,t) as the complex amplitude of an optical pulse. Its Fourier transform is
( , ) (1/ 2 )
( , ) exp(
)
A z
A z t
i t dt


. The terms |A|
2
A and (1/3)A
3
exp(2iω
0
t) describe the
self-phase modulation and third-
harmonic generation, respectively. α is the propagation loss,
and β
m
is the
m
th-
order dispersion coefficient. The nonlinear coefficient γ
e
of the waveguide is
obtained using a full-vectorial model. Th
e shock times τ
Kerr
and τ
Raman
for Kerr and Raman
nonlinearities are calculated with consideration on wavelength-dependent effective mode area
and nonlinear index n
2
. h
R
(t) is the Raman response function.
The propagation loss is set to be 1 dB/cm, which is achievable [
23
], and nonlinear loss
induced by two-photon absorption is ignored [
23
]. Wavelength dependence of the loss is not
considered, since the propagation loss is low at almost all wavelengths of interest. A large
group birefringence, which is 0.053 at 2200-nm wavelength, causes a quick walk-off between
the two polarizations with a characteristic length of walk-
off equal to ~350 μm, much less
than the propagation distance for supercontinuum generation. Thus, the polarization coupling
in pulse propagation is ignored. We obtain high modal field overlap between intra-pulse
spectral components, which is >75% between 800-nm and 2200-nm wavelengths. All order
dispersion terms are included as shown in Ref [
11
]. The nonlinear index n
2
for Si
3
N
4
is
measured at 1550 nm [
24
]. We assume n
2
is independent of wavelength for Si
3
N
4
and SiO
2
since their bandgap wavelengths are below 245 nm. The Kerr nonlinear coefficient γ
e
is
computed versus wavelength using the full-vectorial model [
25
], with γ
e
= 0.49 /(m·W) at
2200 nm. The Raman scattering properties are given in Ref [
26
]. We take the wavelength-
dependence of the nonlinearity into account by correcting the shock terms. The Kerr shock
time is 2.94 fs at 2200 nm, and the Raman shock time is 2.68 fs. In our simulations, the total
length of time window is 50 ps, and the whole bandwidth in the frequency domain is 3000
THz. More details in our model will be given in future publications.
A chirp-free hyperbolic secant pulse spectrally centered at 2200-nm wavelength, with a
full width at half-maximum (FWHM) of 120 fs and a peak power of 6 kW (pulse energy of
0.8 nJ), is launched into the proposed Si
3
N
4
slot waveguide. Figure 3(a) shows that, along the
waveguide, the pulse generates significantly blue-shifted spectral components down to 800
nm wavelength mainly due to self-steepening [22,27]. In our case, the flattened dispersion
reduces walk-off of newly generated spectral components, which facilitates the formation of
an optical shock at the pulse falling edge that is as short as 3 fs (see Fig. 3(b)), as it travels 5
mm. The optical shock induces so much spectral blue shift that it reaches a short-wavelength
region where third harmonics are generated. The blue part of the spectrum becomes stable at a
propagation distance of 15 mm and shows a small power fluctuation of 3 dB over a 754-
nm
-
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11587
wide wavelength range from 847 to 1601 nm at 20 mm. The high-power part of the spectrum
is red-shifted and extended to 2650 nm, due to self-phase modulation and Raman self-
frequency shift [
22
]. The supercontinuum is formed from 630 to 2650 nm (that is, 360 THz in
total) at
35 dB, covering a two-octave bandwidth.
Fig. 3. Two-octave supercontinuum generation in the dispersion-flattened slot waveguide. (a)
The input 120-fs pulse is centered at 2200-nm wavelength. A supercontinuum is generated
from 630 to 2650 nm mainly due to self-steepening of the pulse. (b) In time domain, an optical
shock as short as 3 fs (see the inset) is formed, as the pulse travels 5 mm.
Spectral evolution along the waveguide length is illustrated in Fig. 4. A few nonlinear
interactive processes responsible for the formation of the supercontinuum can be seen. First,
self-steepening of the optical pulse, associated with intensity-dependent group velocity [
22
],
causes a sharp falling edge of the pulse. On the other hand, self-phase modulation produce
s
blue-shifted spectral components at the falling edge, which walk-off very little relative to the
edge, due to the low dispersion. Together with the self-steepening effect, these high-frequency
components in turn help form a shaper edge, resulting in bluer shifts. Therefore, the flat and
low dispersion triggers this positive feedback mechanism for optical shock formation and
spectral broadening, which follows from A to B as shown in Fig. 4. Such a steep pulse edge
transfers energy to a frequency range near 370 THz, ~230 THz away from the pulse carrier
frequency. Second, tracking from B to C in Fig. 4, we note that, with accumulated dispersion,
the falling edge becomes less steep, and newly generated blue-shifted frequencies are closer to
the carrier, which improves the spectral flatness of the supercontinuum. Another effect of the
dispersion is that the blue-shifted components walk away from the steep edge and overlap
with the pulse tail, forming a beating pattern as shown in Fig. 3(b). Third, the pulse waveform
in Fig. 3(b) has a high-power 'shoulder' at the beginning of its falling edge before the optical
shock, which generates blue-shifted frequencies near the carrier following from A to D in Fig.
4. Fourth, third-harmonic generation occurs at 408 THz, and some frequency-resolved fringes
are observed mainly due to cross-phase modulation by the input pulse [
22
]. The third-
harmonic pulse sees a larger group delay and escapes from the envelope of the input pulse.
This is why the third-harmonic spectrum becomes stable after a distance of ~1 mm.
Fig. 4. Spectral evolution in the slot waveguides. Low dispersion causes dramatic spectral
broadening and optical shock formation, from A to B. Then, accumulated dispersion makes the
pulse falling edge less steep and improves spectral flatness, from B to C. Self-phase modulation
produces blue-shifted frequencies near the carrier frequency, from A to D.
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11588
The dynamics of the self-steepening-induced supercontinuum generation, corresponding to
Fig. 4, can be intuitively represented using spectrograms generated by the cross-correlation
frequency-resolved optical gating (X-FROG) technique [2], in which an optical pulse is
characterized simultaneously in time and frequency domains. As explained above, the
fundamental pulse experiences dramatic self-steepening and spectral broadening in its
propagation from 0 to 4 mm. Due to frequency-dependent group delay, the blue part of the
edge walks off relative to the pulse, as seen at a distance of 10 mm, which forms a hockey-
stick-like pattern in the spectrogram shown in Fig. 5.
The third-harmonic pulse exhibits more complex dynamics, and its evolution is
significantly affected by group delay and pulse trapping induced by third harmonic generation
[28,29]. As shown in Fig. 5, the third-harmonic pulse is generated and cross-phase modulated
at the beginning of propagation. From 1 mm to 2 mm, the tail of the third-harmonic pulse,
which is after the steep edge of the fundamental pulse, walks away quickly, since its
frequency is not located in the dispersion-flattened spectral range. The rest part of the pulse
that coincides with the peak of the fundamental pulse is split into two parts. First, the low-
frequency part travels slowly, and after it arrives at the steep edge of the fundamental pulse, it
is blue-shifted due to cross-phase modulation and then escapes from the envelope of the
fundamental pulse. Second, the high-frequency part is trapped by the fundamental pulse due
to a nonlinear phase locking mechanism [
28
] and carries the dispersion property impressed by
the fundamental pulse [28,29], which is why its pattern in the spectrogram is also hockey-
stick-like, although stretched 3 times in the frequency domain. As seen in Fig. 5, such phase-
locked pulse trapping enables us to up-convert a 200-THz-wide supercontinuum that can be
2000 nm wide in wavelength, across a few-hundred-THz spectral region, to where a
supercontinuum cannot be efficiently formed with an optical pulse at a local frequency, with
flattened dispersion hardly achievable near material bandgap wavelength in practice.
Fig. 5. Spectrogram evolution of the pulse. Strong self-steepening and spectral broadening
occur in the pulse propagation from 0 to 4 mm. At 10 mm, the blue part of the pulse falling
edge walks off due to dispersion. This forms a hockey-stick-like pattern in the spectrogram. A
third-harmonic pulse trapped by the fundamental pulse due to a nonlinear phase locking also
has a hockey-stick-like spectrogram, though stretched 3 times in the frequency domain.
4. Discussion and conclusion
Different from most of previously investigated supercontinua that were generated mainly due
to either self-phase modulation in normal dispersion regime or high-order soliton fission and
dispersion wave generation in anomalous dispersion regime [2], the reported supercontinuum
features a highly asymmetric spectrum caused mainly by pulse self-steepening. Moreover,
using a silicon nitride waveguide, one can have high output power and extended spectral
range that are difficult to obtain in silicon waveguides.
In summary, we presented a dispersion tailoring technique that allows us to improve
dispersion flatness by 30 times in integrated high-index-contrast waveguides. Benefiting from
this, one can generate a two-octave supercontinuum on a chip by enhancing pulse self-
steepening, which paves the way for ultrafast and ultra-wideband applications on an integrated
nano-photonics platform. We believe that the on-chip supercontinuum generation would serve
as a key enabler for building portable imaging, sensing, and positioning systems.
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11589
Acknowledgments
The authors would thank Jacob Levy at Cornell University for helpful discussions. This work
is supported by HP Labs.
#146242
- $15.00
USD
Received
20
Apr
2011;
accepted
26
May
2011;
published
31
May
2011
(C)
2011
OSA
6 June
2011
/ Vol.
19,
No.
12
/ OPTICS
EXPRESS
11590