The efficiency of evanescent coupling between a silica optical fiber taper
and a silicon photonic crystal waveguide is studied. A high reflectivity mirror
on the end of the photonic crystal waveguide is used to recollect, in the
backwards propagating fiber mode, the optical power that is initially coupled
into the photonic crystal waveguide. An outcoupled power in the backward
propagating fiber mode of 88% of the input power is measured, corresponding to
a lower bound on the coupler efficiency of 94%

Barclay

Kartik Srinivasan

Knight

Kuang

Matthew Borselli

Notomi

Oskar Painter

Paul E. Barclay

English

arXiv

Crossref

The efficiency of evanescent coupling between a silica optical fiber taper and a silicon photonic crystal waveguide is studied. A high-reflectivity mirror on the end of the photonic crystal waveguide is used to recollect, in the backward-propagating fiber mode, the optical power that is initially coupled into the photonic crystal waveguide. An outcoupled power in the backward-propagating fiber mode of 88% of the input power is measured, corresponding to a lower bound on the coupler efficiency of 94%

Barclay, Paul E.

Srinivasan, Kartik

Borselli, Matthew

Painter, Oskar

Caltech Authors - Main

April 1, 2004 / Vol. 29, No. 7 / OPTICS LETTERS 697Efficient input and output fiber coupling
to a photonic crystal waveguide
Paul E. Barclay, Kartik Srinivasan, Matthew Borselli, and Oskar Painter
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125Received November 3, 2003
The efficiency of evanescent coupling between a silica optical fiber taper and a silicon photonic crystal wave-
guide is studied. A high-ref lectivity mirror on the end of the photonic crystal waveguide is used to recollect,
in the backward-propagating fiber mode, the optical power that is initially coupled into the photonic crystal
waveguide. An outcoupled power in the backward-propagating fiber mode of 88% of the input power is mea-
sured, corresponding to a lower bound on the coupler efficiency of 94%. © 2004 Optical Society of America
OCIS codes: 060.1810, 130.3120.Evanescent coupling between optical fiber tapers1 and
photonic crystal waveguides2,3 (PCWGs) allows highly
efficient and mode-selective interfacing between fiber
optics and planar photonic crystal (PC) devices. Cir-
cumventing the intrinsic spatial and refractive-index
mismatch of fiber to PCWG end-fire coupling,4 this
coupling scheme takes advantage of the strong disper-
sion and undercut geometry inherent to PC membrane
structures to allow efficient power transfer between
modes of a PCWG and a fiber taper. The coherent
interaction over the length of the coupling region
between phase-matched modes of each waveguide
manifests in nearly complete resonant power transfer
between waveguides, and varying the position of the
fiber taper makes it possible to tune the coupling
strength and phase-matching frequency, allowing the
dispersive and spatial properties of PCWG modes to
be eff iciently probed.5
In previous work2,5 the coupling efficiency of
this technique was studied by analyzing the signal
transmitted through the f iber taper as a function
of wavelength and fiber taper position. Although
transmission measurements are suff icient to infer the
efficiency of this coupling scheme, a measurement
of power coupled into and then back out of a PCWG
is of interest since it allows the lower bound of the
input–output coupling efficiency to be directly es-
tablished. By studying light coupled into a PCWG,
ref lected by an end mirror at the PCWG termination,
and then coupled into the backward-propagating fiber
mode, we show in this Letter that this coupling scheme
has near-unity eff iciency.
Figure 1(a) shows an illustration of the fiber taper
evanescent coupling scheme. A fiber taper, formed
by simultaneously heating and stretching a standard
single-mode fiber until it has a minimum diameter of
roughly 1 mm, is positioned above and parallel to an
undercut planar PCWG. The small diameter of the
fiber taper allows the evanescent tail of its fundamen-
tal mode to interact with the PCWG, and the under-
cut geometry of the PCWG prevents the fiber taper
from radiating into the PC substrate. A schematic
detailing the optical path within the fiber taper and
the PCWG is shown in Fig. 1(b). To spectrally char-
acterize the taper–PCWG coupler, an external-cavity
swept-wavelength source with a wavelength range of0146-9592/04/070697-03$15.00/01565–1625 nm was connected to the f iber taper input
by means of a 3-dB coupler. At the f iber taper out-
put a photodetector (DT ) was connected to measure the
transmitted power past the taper–PCWG coupling re-
gion. An additional photodetector (DR) was connected
to the second 3-dB coupler input to measure the light
ref lected by the PCWG.
The PCWGs used in this work were fabricated from
a silicon-on-insulator wafer. The airholes were pat-
terned in the 300-nm-thick silicon layer with standard
electron-beam lithography and dry etching techniques,
after which the suspended membrane structure was
created by selective removal of the 2-mm-thick un-
derlying silicon dioxide layer with a hydrof luoric
acid wet etch. To form the PCWGs a grade in hole
radius was introduced along the transverse direction
of a compressed square lattice of airholes,6 as shown
in Figs. 2(a) and 2(b). Coupling in this case occurs
between the fundamental linearly polarized fiber
Fig. 1. (a) Schematic of the coupling scheme. (b) Illus-
tration of the contradirectional coupling process and the
feedback within the PCWG caused by the ref lectivities r1,2
of the waveguide terminations.© 2004 Optical Society of America
698 OPTICS LETTERS / Vol. 29, No. 7 / April 1, 2004Fig. 2. (a) Waveguide geometry and finite-difference time-
domain calculated magnetic field profile (By ) of the TE-1
mode. (b) Scanning electron microscopy image of the high
(r1) ref lectivity waveguide termination. The PCWG has
a transverse lattice constant Lx  415 nm, a longitudi-
nal lattice constant Lz  536 nm, and length L  200Lz.
(c) Dispersion of the TE-1 PCWG mode. Also shown are
the mirror termination band edges for momentum along the
waveguide axis (zˆ). The corresponding partial bandgap of
the mirror region is indicated by the shaded region.
mode and the TE-1 PCWG mode, the field profile of
which is shown in Fig. 2(a). This PCWG mode has a
negative group velocity, resulting in contradirectional
coupling as depicted in Fig. 1, and was designed to
be integrated with recently demonstrated7 high-Q
cavities formed with a similar grading in hole radius.
In previous studies of evanescent coupling to such
PCWGs5 the waveguide termination consisted of a
2-mm region of undercut silicon, followed by an 8-mm
length of nonundercut silicon on insulator and a final
air interface. The interference from multiple ref lec-
tions at the three interfaces within the waveguide ter-
mination resulted in a highly wavelength-dependent
ref lection coeff icient, making quantitative analysis of
signals ref lected by the PCWG difficult. Here, the
waveguide terminations were engineered to have ei-
ther high or low broadband ref lectivities. On one end
of the PCWG, shown in Fig. 2(b), a PC mirror with high
modal ref lectivity (r1) for the TE-1 PCWG mode was
used. This high-ref lectivity end mirror was formed
by removal of the lattice compression of the waveguide
while maintaining the transverse grade in hole radius.
The change in lattice compression results in the TE-1
PCWG mode lying within the partial bandgap6 of the
high-ref lectivity end mirror section [Fig. 2(c)] while
the transverse grade in hole radius reduces diffraction
loss within the mirror. On the opposite end of the
PCWG a poor ref lector with low ref lectivity (r2) was
realized by removal (over several lattice constants) of
the transverse grade in hole radius while keeping the
central waveguide hole radius fixed, resulting in a
loss of transverse waveguiding within the end mirror
section and allowing the TE-1 PCWG mode to diffract
into the unguided bulk modes of the PC.
In the absence of ref lections caused by the PCWG
termination (i.e., r1, 2  0) the f iber–PCWG junction
can be characterized by"
a1F L
a2PC0
#

"
t k 0
k t0
#"
a1F 0
a2PCL
#
, (1)where k and k0 are coupling coefficients; t and t0 are
transmission coefficients; and a1F z and a
2
PCz are the
amplitudes of the forward-propagating fundamental
fiber taper mode and the backward-propagating TE-1
PCWG mode, respectively. The coupling region ex-
tends along the z axis, with z  0 corresponding to the
input and z  L to the output of the coupler. As illus-
trated in Fig. 1(b), nonzero r1, 2 introduce feedback into
the system. This allows input light coupled from the
forward-propagating fundamental f iber taper mode
into the TE-1 PCWG mode to be partially ref lected
by the PCWG termination and subsequently coupled
to the backward-propagating fundamental fiber taper
mode. In the presence of this feedback within the
PCWG, the normalized ref lected and transmitted
powers in the fiber taper are given by8
T  ja1F Lj2 
Ç
t 1
kk0t0r1r2
1 2 r1t0r2t0
Ç2
, (2)
R  ja2F 0j2 
Ç
kk0r1
1 2 r1t0r2t0
Ç2
, (3)
for a1F 0  1 and a
2
PCL 0. By measuring T and R
and considering Eqs. (2) and (3), we can determine the
efficiency of the f iber–PCWG coupling as measured by
jkk0j.
In the measurements presented here a fiber taper
with 2-mm diameter was used to probe the PCWG of
Fig. 2(b). Figures 3(a) and 3(b) show T and R as a
function of wavelength when the taper is aligned with
the PCWG at a height of g  0.20 mm and indicate
that the phase-matched wavelength is l  1598 6
5 nm. As described in Ref. 5, coupling to the guided
TE-1 mode of the PCWG was confirmed by study of
the coupling dependence upon polarization, lateral
taper displacement, and fiber taper diameter. Fig-
ure 3(c) shows T and R at the resonant wavelength
of the PCWG nearest the phase-matching condition
[the minima and maxima in T l and Rl] as a
function of taper height g above the PCWG. Included
in Fig. 3(c) is the off-resonant (away from phase
matching) transmission, Toff , through the fiber. A
maximum normalized ref lected power Rmax  0.88
was measured at a height of g  0.25 mm, where
the corresponding transmission was T , 0.01. As
can be seen in the wavelength scans of Fig. 3(a),
at this taper height the Fabry–Perot resonances
caused by multiple ref lections from the end mirrors
of the PCWG are suppressed for wavelengths within
the coupler bandwidth due to strong coupling to the
fiber taper. Ignoring for the moment the effects of
multiple ref lections in Eq. (3), the maximum optical
power coupling efficiency is then jkk0j  pRmaxjr1j,
where the square-root dependence upon Rmax is a
result of the light passing through the coupler twice in
returning to the backward-propagating fiber mode.
Assuming the high-ref lectivity mirror is perfect (unity
ref lection), for the measured Rmax  0.88 this implies
94% coupling of optical power from taper to PC wave-
guide (and vice versa).
By further comparison of T g and Rg with
the model given above, this time including feedback
April 1, 2004 / Vol. 29, No. 7 / OPTICS LETTERS 699Fig. 3. (a) Ref lection and (b) transmission of the fiber
taper as a function of wavelength for a taper height of
0.20mm. Both signals were normalized to the taper trans-
mission with the PCWG absent. (c) Measured taper trans-
mission minimum (l 1600 nm), ref lection maximum (l
1600 nm), and off-resonant transmission (l  1565 nm) as
a function of taper height. Also shown are f its to the data,
and the resulting predicted coupler efficiency, jkj2. Asym-
metry in the fiber taper loss about the coupling region was
taken into account by repeating the measurements with the
direction of propagation through the taper and the orien-
tation of the PCWG sample reversed; the geometric mean
of the values obtained from the two orientations takes any
asymmetry into account.
within the PCWG, we find that the high-ref lectivity
PCWG end mirror is imperfect and that Rmax is in
fact limited by mirror ref lectivity, not the efficiency
of the coupling junction. For this comparison we
take the elements of the coupling matrix to satisfy
the relations jtj2 1 jkj2  1, k 0  k, and t0  t
of an ideal (lossless) coupling junction.8 For the
phase-matched contradirectional coupling considered
here, the dependence of k on g can be approximated
by jkgj  tanhk, 0 exp2gg0L,9 where k, 0 and
g0 are constants. Substituting these relations into
Eqs. (2) and (3), we can fit T and R to the experi-
mental data with r1, 2, k, 0, and g0 as free parame-
ters.10 Waveguide loss can also be included in the
model; however, it is found to be small compared withthe mirror loss and will be studied in detail elsewhere.
Figure 3(c) shows the fits to the data for g $ 0.20 mm;
for g , 0.20 mm nonresonant scattering loss is no
longer negligible and the ideal coupling junction
model breaks down. Note that this scattering loss
is observed to affect the ref lected signal at a smaller
taper–PCWG gap than for the transmitted signal; Toff
begins to decrease for g # 0.30 mm, whereas R begins
to decrease for g # 0.20 mm. This indicates that,
within the coupler bandwidth, 1 2 Toff is an overesti-
mate of the amount of power scattered into radiation
modes. From the fits in Fig. 3(c) the PCWG end mir-
ror ref lectivities are estimated to be jr1j2  0.90 and
jr2j2  0.52, and the optical power coupling efficiency
(jkj2) occurring at Rmax (g  0.25 mm) is approxi-
mately 97%.
In conclusion, these results indicate that this cou-
pling scheme allows highly eff icient input and output
coupling between photonic crystal devices and fiber ta-
pers and is promising for eff icient probing of future in-
tegrated planar PC devices.
K. Srinivasan thanks the Hertz Foundation and M.
Borselli thanks the Moore Foundation for f inancial
support. P. Barclay’s e-mail address is pbarclay@
caltech.edu.
References
1. J. Knight, G. Cheung, F. Jacques, and T. Birks, Opt.
Lett. 22, 1129 (1997).
2. P. E. Barclay, K. Srinivasan, M. Borselli, and O.
Painter, IEE Electron. Lett. 39, 842 (2003).
3. W. Kuang, C. Kim, A. Stapleton, and J. O’Brien, Opt.
Lett. 27, 1604 (2002).
4. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C.
Takahashi, and I. Yokohama, IEEE J. Quantum Elec-
tron. 38, 736 (2002).
5. P. E. Barclay, K. Srinivasan, M. Borselli, and O.
Painter, arXiv.org e-Print archive, quant-ph/0308070,
August 12, 2003, http://arXiv.org/quant-ph/0308070.
6. P. E. Barclay, K. Srinivasan, and O. Painter, J. Opt.
Soc. Am. B 20, 2274 (2003).
7. K. Srinivasan, P. E. Barclay, M. Borselli, and O.
Painter, arXiv.org e-Print archive, quant-ph/0309190,
September 25, 2003, http://arXiv.org/quant-ph/
0309190.
8. H. A. Haus, Waves and Fields in Optoelectronics, Pren-
tice-Hall Series in Solid State Physical Electronics
(Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 3,
pp. 59–71.
9. P. Yeh and H. F. Taylor, Appl. Opt. 19, 2848 (1980).
10. We take r1t0r2t0jr1t0r2t0j  1, corresponding to a reso-
nant condition within the PCWG.


Efficient input and output fiber coupling to a photonic crystal waveguide

April 1, 2004 / Vol. 29, No. 7 / OPTICS LETTERS 697Efficient input and output fiber coupling
to a photonic crystal waveguide
Paul E. Barclay, Kartik Srinivasan, Matthew Borselli, and Oskar Painter
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125Received November 3, 2003
The efficiency of evanescent coupling between a silica optical fiber taper and a silicon photonic crystal wave-
guide is studied. A high-ref lectivity mirror on the end of the photonic crystal waveguide is used to recollect,
in the backward-propagating fiber mode, the optical power that is initially coupled into the photonic crystal
waveguide. An outcoupled power in the backward-propagating fiber mode of 88% of the input power is mea-
sured, corresponding to a lower bound on the coupler efficiency of 94%. © 2004 Optical Society of America
OCIS codes: 060.1810, 130.3120.Evanescent coupling between optical fiber tapers1 and
photonic crystal waveguides2,3 (PCWGs) allows highly
efficient and mode-selective interfacing between fiber
optics and planar photonic crystal (PC) devices. Cir-
cumventing the intrinsic spatial and refractive-index
mismatch of fiber to PCWG end-fire coupling,4 this
coupling scheme takes advantage of the strong disper-
sion and undercut geometry inherent to PC membrane
structures to allow efficient power transfer between
modes of a PCWG and a fiber taper. The coherent
interaction over the length of the coupling region
between phase-matched modes of each waveguide
manifests in nearly complete resonant power transfer
between waveguides, and varying the position of the
fiber taper makes it possible to tune the coupling
strength and phase-matching frequency, allowing the
dispersive and spatial properties of PCWG modes to
be eff iciently probed.5
In previous work2,5 the coupling efficiency of
this technique was studied by analyzing the signal
transmitted through the f iber taper as a function
of wavelength and fiber taper position. Although
transmission measurements are suff icient to infer the
efficiency of this coupling scheme, a measurement
of power coupled into and then back out of a PCWG
is of interest since it allows the lower bound of the
input–output coupling efficiency to be directly es-
tablished. By studying light coupled into a PCWG,
ref lected by an end mirror at the PCWG termination,
and then coupled into the backward-propagating fiber
mode, we show in this Letter that this coupling scheme
has near-unity eff iciency.
Figure 1(a) shows an illustration of the fiber taper
evanescent coupling scheme. A fiber taper, formed
by simultaneously heating and stretching a standard
single-mode fiber until it has a minimum diameter of
roughly 1 mm, is positioned above and parallel to an
undercut planar PCWG. The small diameter of the
fiber taper allows the evanescent tail of its fundamen-
tal mode to interact with the PCWG, and the under-
cut geometry of the PCWG prevents the fiber taper
from radiating into the PC substrate. A schematic
detailing the optical path within the fiber taper and
the PCWG is shown in Fig. 1(b). To spectrally char-
acterize the taper–PCWG coupler, an external-cavity
swept-wavelength source with a wavelength range of0146-9592/04/070697-03$15.00/01565–1625 nm was connected to the f iber taper input
by means of a 3-dB coupler. At the f iber taper out-
put a photodetector (DT ) was connected to measure the
transmitted power past the taper–PCWG coupling re-
gion. An additional photodetector (DR) was connected
to the second 3-dB coupler input to measure the light
ref lected by the PCWG.
The PCWGs used in this work were fabricated from
a silicon-on-insulator wafer. The airholes were pat-
terned in the 300-nm-thick silicon layer with standard
electron-beam lithography and dry etching techniques,
after which the suspended membrane structure was
created by selective removal of the 2-mm-thick un-
derlying silicon dioxide layer with a hydrof luoric
acid wet etch. To form the PCWGs a grade in hole
radius was introduced along the transverse direction
of a compressed square lattice of airholes,6 as shown
in Figs. 2(a) and 2(b). Coupling in this case occurs
between the fundamental linearly polarized fiber
Fig. 1. (a) Schematic of the coupling scheme. (b) Illus-
tration of the contradirectional coupling process and the
feedback within the PCWG caused by the ref lectivities r1,2
of the waveguide terminations.© 2004 Optical Society of America
698 OPTICS LETTERS / Vol. 29, No. 7 / April 1, 2004Fig. 2. (a) Waveguide geometry and finite-difference time-
domain calculated magnetic field profile (By ) of the TE-1
mode. (b) Scanning electron microscopy image of the high
(r1) ref lectivity waveguide termination. The PCWG has
a transverse lattice constant Lx  415 nm, a longitudi-
nal lattice constant Lz  536 nm, and length L  200Lz.
(c) Dispersion of the TE-1 PCWG mode. Also shown are
the mirror termination band edges for momentum along the
waveguide axis (ẑ). The corresponding partial bandgap of
the mirror region is indicated by the shaded region.
mode and the TE-1 PCWG mode, the field profile of
which is shown in Fig. 2(a). This PCWG mode has a
negative group velocity, resulting in contradirectional
coupling as depicted in Fig. 1, and was designed to
be integrated with recently demonstrated7 high-Q
cavities formed with a similar grading in hole radius.
In previous studies of evanescent coupling to such
PCWGs5 the waveguide termination consisted of a
2-mm region of undercut silicon, followed by an 8-mm
length of nonundercut silicon on insulator and a final
air interface. The interference from multiple ref lec-
tions at the three interfaces within the waveguide ter-
mination resulted in a highly wavelength-dependent
ref lection coeff icient, making quantitative analysis of
signals ref lected by the PCWG difficult. Here, the
waveguide terminations were engineered to have ei-
ther high or low broadband ref lectivities. On one end
of the PCWG, shown in Fig. 2(b), a PC mirror with high
modal ref lectivity (r1) for the TE-1 PCWG mode was
used. This high-ref lectivity end mirror was formed
by removal of the lattice compression of the waveguide
while maintaining the transverse grade in hole radius.
The change in lattice compression results in the TE-1
PCWG mode lying within the partial bandgap6 of the
high-ref lectivity end mirror section [Fig. 2(c)] while
the transverse grade in hole radius reduces diffraction
loss within the mirror. On the opposite end of the
PCWG a poor ref lector with low ref lectivity (r2) was
realized by removal (over several lattice constants) of
the transverse grade in hole radius while keeping the
central waveguide hole radius fixed, resulting in a
loss of transverse waveguiding within the end mirror
section and allowing the TE-1 PCWG mode to diffract
into the unguided bulk modes of the PC.
In the absence of ref lections caused by the PCWG
termination (i.e., r1, 2  0) the f iber–PCWG junction
can be characterized by"
a1F L
a2PC0
#

"
t k 0
k t0
# "
a1F 0
a2PCL
#
, (1)where k and k0 are coupling coefficients; t and t0 are
transmission coefficients; and a1F z and a
2
PCz are the
amplitudes of the forward-propagating fundamental
fiber taper mode and the backward-propagating TE-1
PCWG mode, respectively. The coupling region ex-
tends along the z axis, with z  0 corresponding to the
input and z  L to the output of the coupler. As illus-
trated in Fig. 1(b), nonzero r1, 2 introduce feedback into
the system. This allows input light coupled from the
forward-propagating fundamental f iber taper mode
into the TE-1 PCWG mode to be partially ref lected
by the PCWG termination and subsequently coupled
to the backward-propagating fundamental fiber taper
mode. In the presence of this feedback within the
PCWG, the normalized ref lected and transmitted
powers in the fiber taper are given by8
T  ja1F Lj
2 
Ç
t 1
kk0t0r1r2
1 2 r1t0r2t0
Ç2
, (2)
R  ja2F 0j
2 
Ç
kk0r1
1 2 r1t0r2t0
Ç2
, (3)
for a1F 0  1 and a
2
PCL  0. By measuring T and R
and considering Eqs. (2) and (3), we can determine the
efficiency of the f iber–PCWG coupling as measured by
jkk0j.
In the measurements presented here a fiber taper
with 2-mm diameter was used to probe the PCWG of
Fig. 2(b). Figures 3(a) and 3(b) show T and R as a
function of wavelength when the taper is aligned with
the PCWG at a height of g  0.20 mm and indicate
that the phase-matched wavelength is l  1598 6
5 nm. As described in Ref. 5, coupling to the guided
TE-1 mode of the PCWG was confirmed by study of
the coupling dependence upon polarization, lateral
taper displacement, and fiber taper diameter. Fig-
ure 3(c) shows T and R at the resonant wavelength
of the PCWG nearest the phase-matching condition
[the minima and maxima in T l and Rl] as a
function of taper height g above the PCWG. Included
in Fig. 3(c) is the off-resonant (away from phase
matching) transmission, Toff , through the fiber. A
maximum normalized ref lected power Rmax  0.88
was measured at a height of g  0.25 mm, where
the corresponding transmission was T , 0.01. As
can be seen in the wavelength scans of Fig. 3(a),
at this taper height the Fabry–Perot resonances
caused by multiple ref lections from the end mirrors
of the PCWG are suppressed for wavelengths within
the coupler bandwidth due to strong coupling to the
fiber taper. Ignoring for the moment the effects of
multiple ref lections in Eq. (3), the maximum optical
power coupling efficiency is then jkk0j 
p
Rmaxjr1j,
where the square-root dependence upon Rmax is a
result of the light passing through the coupler twice in
returning to the backward-propagating fiber mode.
Assuming the high-ref lectivity mirror is perfect (unity
ref lection), for the measured Rmax  0.88 this implies
94% coupling of optical power from taper to PC wave-
guide (and vice versa).
By further comparison of T g and Rg with
the model given above, this time including feedback
April 1, 2004 / Vol. 29, No. 7 / OPTICS LETTERS 699Fig. 3. (a) Ref lection and (b) transmission of the fiber
taper as a function of wavelength for a taper height of
0.20 mm. Both signals were normalized to the taper trans-
mission with the PCWG absent. (c) Measured taper trans-
mission minimum (l  1600 nm), ref lection maximum (l 
1600 nm), and off-resonant transmission (l  1565 nm) as
a function of taper height. Also shown are f its to the data,
and the resulting predicted coupler efficiency, jkj2. Asym-
metry in the fiber taper loss about the coupling region was
taken into account by repeating the measurements with the
direction of propagation through the taper and the orien-
tation of the PCWG sample reversed; the geometric mean
of the values obtained from the two orientations takes any
asymmetry into account.
within the PCWG, we find that the high-ref lectivity
PCWG end mirror is imperfect and that Rmax is in
fact limited by mirror ref lectivity, not the efficiency
of the coupling junction. For this comparison we
take the elements of the coupling matrix to satisfy
the relations jtj2 1 jkj2  1, k 0  k, and t0  t
of an ideal (lossless) coupling junction.8 For the
phase-matched contradirectional coupling considered
here, the dependence of k on g can be approximated
by jkgj  tanhk, 0 exp2gg0L,9 where k, 0 and
g0 are constants. Substituting these relations into
Eqs. (2) and (3), we can fit T and R to the experi-
mental data with r1, 2, k, 0, and g0 as free parame-
ters.10 Waveguide loss can also be included in the
model; however, it is found to be small compared withthe mirror loss and will be studied in detail elsewhere.
Figure 3(c) shows the fits to the data for g $ 0.20 mm;
for g , 0.20 mm nonresonant scattering loss is no
longer negligible and the ideal coupling junction
model breaks down. Note that this scattering loss
is observed to affect the ref lected signal at a smaller
taper–PCWG gap than for the transmitted signal; Toff
begins to decrease for g # 0.30 mm, whereas R begins
to decrease for g # 0.20 mm. This indicates that,
within the coupler bandwidth, 1 2 Toff is an overesti-
mate of the amount of power scattered into radiation
modes. From the fits in Fig. 3(c) the PCWG end mir-
ror ref lectivities are estimated to be jr1j2  0.90 and
jr2j2  0.52, and the optical power coupling efficiency
(jkj2) occurring at Rmax (g  0.25 mm) is approxi-
mately 97%.
In conclusion, these results indicate that this cou-
pling scheme allows highly eff icient input and output
coupling between photonic crystal devices and fiber ta-
pers and is promising for eff icient probing of future in-
tegrated planar PC devices.
K. Srinivasan thanks the Hertz Foundation and M.
Borselli thanks the Moore Foundation for f inancial
support. P. Barclay’s e-mail address is pbarclay@
caltech.edu.
References
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arXiv.org e-Print archive, quant-ph/0308070,

arXiv.org e-Print archive, quant-ph/0309190,

Waves and Fields in Optoelectronics, Prentice-Hall Series in Solid State Physical Electronics (Prentice-Hall,

We take r1t0r2t0jr1t0r2t0j  1, corresponding to a resonant condition within the PCWG.

https://authors.library.caltech.edu/2710/1/BARol04.pdf

Efficient input and output fiber coupling to a photonic crystal
  waveguide

Efficient input and output fiber coupling to a photonic crystal waveguide

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