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Anderson-like localization in disordered LN photonic crystal slab cavities Juan Pablo Vasco, and Stephen Hughes ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00967 • Publication Date (Web): 15 Jan 2018 Downloaded from http://pubs.acs.org on January 16, 2018
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ACS Photonics
Anderson-like lo alization in disordered
LN
photoni rystal slab avities
Juan Pablo Vas o∗ and Stephen Hughes∗ Department of Physi s, Engineering Physi s and Astronomy, Queen's University, Kingston, Ontario, Canada, K7L 3N6
E-mail: jpv queensu. a; shughesqueensu. a
Abstra t We present a detailed theoreti al study of the ee ts of stru tural disorder on LN photoni rystal slab avities, ranging from short to long length s ales (N =3-35 avity lengths), using a fully three-dimensional Blo h mode expansion te hnique. We ompute the opti al density of states (DOS), quality fa tors and ee tive mode volumes of the
avity modes, with and without disorder, and ompare with the lo alized modes of the orresponding disordered photoni rystal waveguide. We demonstrate how the quality fa tors and ee tive mode volumes saturate at a spe i avity length and be ome bounded by the orresponding values of the Anderson modes appearing in the disordered waveguide. By means of the intensity u tuation riterion, we observe Anderson-like lo alization for avity lengths larger than around L31, and show that the eld on nement in the disordered LN avities is mainly determined by the lo al
hara teristi s of the stru tural disorder as long as the on nement region is far enough from the avity mirrors and the ee tive mode lo alization length is mu h smaller than the avity length; under this regime, the disordered avity system be omes insensitive to hanges in the avity boundaries and a good agreement with the intensity u tuation
riterion is found for lo alization. Surprisingly, we nd that the Anderson-like lo alized 1
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modes do not appear as new disorder-indu ed resonan es in the main spe tral region of the LN avity modes, and, moreover, the disordered DOS enhan ement is largest for the disordered waveguide system with the same length. These results are fundamentally interesting for appli ations su h as lasing and avity-QED, and provide new insights into the role of the boundary ondition (e.g., open versus mirrors) on nite-size slowlight waveguides. They also point out the lear failure of using models based on the
avity boundaries/mirrors and a single slow-light Blo h mode to des ribe avity systems with large N, whi h has been ommon pra tise.
Keywords Anderson lo alization, Disordered photoni s,
LN
avities, Photoni rystals, Disordered
waveguides For more than a de ade, unavoidable imperfe tions arising in the fabri ation pro ess of photoni rystal slabs (PCSs) have been shown to have a major in uen e on the opti al performan e of PCS-based stru tures, and an a
urate onsideration of their ee ts on the opti al properties of PCS devi es has been the fo us of intense resear h. Most of these studies have fo used on smaller PCS avities non-linear media, intrinsi disorder.
811 12
14
and extended PCS waveguides, in both linear
57
and
where su h imperfe tions are usually understood as a small amount of
Disorder in PCS waveguides is known to be quite ri h, and has the apa-
bility of indu ing phase transitions from extended to lo alized states;
13
in addition, multiple
oherent ba k-s attering phenomena leads to the spontaneous formation of random avities and the system enters into a simple 1D-like Anderson lo alization regime.
1416
Su h phenom-
ena are intensi ed near to the band-edges of the guided modes, i.e., in the slow-light regime, where the ba k-s attering losses in reases approximately as ity).
17
1/vg2
(with
vg
the group velo -
Interestingly, these unavoidable problems reated by random imperfe tions in PCS
waveguides have re ently motivated novel approa hes in whi h disorder is required in order to the system works into the desired regime; for instan e, re ent and important examples are
2
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the enhan ement of photoni transport in disordered superlatti es, high-Q avity modes
19,20
(where
Q is the
with appli ations in avity-QED,
21,22
18
random formation of
quality fa tor), disordered light-matter intera tion
and open transmission hannels in strongly s attering
media to ontrol light-matter intera tions.
2325
On the other hand, ee ts of disorder-indu ed
s attering on PCS avities have shown to ae t mostly the opti al quality fa tor of the avity modes and indu e small u tuations (in omparison to the frequen ies.
4
Q u tuations) on their resonant
These aforementioned studies, in the ase of avities, have been mainly limited
to small avities (su h as the
L3) whose
ee tive sizes are no more than a few periods of the
underlying photoni latti e, leading to avity resonan es whi h fall far from the W 1 waveguide (a omplete row of missing holes) band-edge, i.e., the orresponding system without the
avity mirrors. Nevertheless, re ent experiments in slow-light PC lasers show lear eviden e that the ee ts of disorder on large LN PCS avities (with N the number of missing holes along the waveguide latti e) are not trivial and poorly misunderstood;
26
the lower-frequen y
avity modes, whi h ontribute dominantly to the lasing phenomenon, have resonan es that approa h the orresponding waveguide band-edge for in reasing avity length, leading to a more sensitive response of those modes to unintentional stru tural imperfe tions.
Ad-
ditional lo alization ee ts, onne ted with ideas of Sajeev John's work on lo alization in light-s attering media,
15
are onsequently expe ted, however, the role of the Anderson phe-
nomenon in su h large avities is not well understood yet, sin e the non-disordered mode is bounded by the avity mirrors (the end fa et regions of the waveguide); therefore, this is not a truly extended mode that ould be ome a lo alized state when disorder is introdu ed in the system. In the present paper, by making a dire t omparison with the disordered waveguide system (namely, without the avity mirrors), we present an intuitive and detailed explanation of how to approa h the Anderson-like lo alization phenomenon in these longer-length LN PCS avities, whi h, to the best of our knowledge, has not been addressed in previous works. For our full ve torial 3D al ulations, we use the Blo h mode expansion method (BME)
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27
and
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a photon Green fun tion formalism
28
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to study the disordered LN PCS avities; in parti ular,
we onsider random u tuations of the in-plane hole positions, with Gaussian probability, as the main disorder ontribution to the system, and the orresponding standard variation
σ
of the Gaussian distribution as the main disorder parameter. We show in Figure 1 disorder realizations of the L7 avity for
σ = 0.01a, σ = 0.02a
and
σ = 0.05a,
i.e., 1%, 2% and 5%
of the PC latti e parameter, respe tively. The non-disordered (ideal) holes are represented by lled dark-gray ir les, while the disordered ones are represented by transparent red
ir les. By onsidering
σ = 0.01a,
i.e., a typi al value between typi al amounts of intrinsi
and deliberate disorder, we ompute the opti al density of states (DOS) to hara terize the spe tral properties of the ordered and disordered avities. We nd that additional resonan es, namely, dierent from the ones omputed in the non-disordered system, do not appear due to disorder in the avity mode spe tral region, and the orresponding DOS enhan ement in the waveguide system (i.e., with no avity mirrors) is found to be similar to the ones of the largest LN avities and larger than the ones al ulated for the small avity lengths. We also show that the avity mode
Q
fa tors naturally saturate at a spe i avity length
rather than be ome maximized as previously suggested in Ref. 26, and the saturation value is bounded by the Anderson modes of the orresponding disordered W 1 waveguide.
In
addition, we identify an equivalent behavior for the ee tive mode volumes, whi h leads to a redu tion of the lo alization length of the avity modes; therefore, we nd that in general two important ee ts must be taken into a
ount in disordered LN avities: disorder-indu ed losses and disorder-indu ed lo alization.
By means of the intensity u tuation riterion,
ommonly employed in disordered photoni s,
21,22,29,30
we assess the Anderson lo alization
phenomenon in the disordered slab avities and observe Anderson-like lo alization for avity lengths equal or larger than L31. Moreover, in order to understand the role of the avity boundary onditions on this phenomenon, we also systemati ally study the properties of the
avity's fundamental mode for dierent avity lengths (ranging from L3 to L35) and ompare with the fundamental disorder-indu ed mode of the disordered W 1 waveguide system (whi h
4
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Figure 1: Top view of a disordered L7 avity for disorder magnitudes of 1%, 2% and 5% of the PC latti e parameter. The non-disorder (ideal) pro le is represented by lled dark-gray
ir les, while the disordered one is represented by transparent red ir les.
has ompletely dierent boundary onditions). We also show that, in presen e of disorder and under ertain onditions, the eld on nement inside the avity region is mainly determined by the lo al hara teristi s of the stru tural disorder and the avity mirrors do not play an important role in the disordered photoni mode; under this regime, the light on nement displays similar behavior to the ones seen in disordered waveguides, thus suggesting Anderson-like lo alization in disordered
avities with large N. Importantly, sin e the modal properties be ome insensitive to the avity boundary ondition, this learly leads to the breakdown of the single slow-light mode approximation, previously employed to study PCS avities, regime.
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26,31
in the disordered long- avity
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The rest of our paper is organized as follows. In Se . we brie y review the BME method and establish the main parameters of the system. In Se . , results for the avity resonan es, opti al quality fa tor
Q, ee tive
mode volume
V
and DOS are presented for non-disordered
LN PCS avities. In Se . , we introdu e the model of disorder, ompute the disordered DOS,
the varian e of the normalized intensity distribution, as well as the disordered avity mode
Q
fa tors and the orresponding
V.
Finally, in Se . , we study the role of the boundary
ondition in the mode lo alization phenomena and ompare with results of Se . . The main
on lusions of the work are presented in Se . .
Theoreti al method Large size disordered PCS an be e iently des ribed by the Blo h Mode Expansion method (BME) under the low loss regime.
Hβ (r),
eld,
stru ture,
27,28
To use the BME approa h, the disordered magneti
is expanded in the magneti eld Blo h modes of the non-disordered (ideal)
Hkn (r),
with expansion oe ients
Hβ (r) =
X
Uβ (k, n):
Uβ (k, n)Hkn (r),
(1)
k,n
where
k and n are the wave
ve tor and band index, respe tively, in the rst Brillouin zone of
the non-disordered stru ture. Assuming linear, isotropi , non-magneti , transparent (lossless), and non-dispersive materials, the time-independent Maxwell equations take the form of the following eigenvalue problem when the expansion of Eq. (1) is onsidered:
X
Vkn,k′n′
k,n
2 ωβ2 ωkn + 2 δkk′ ,nn′ Uβ (k, n) = 2 Uβ (k′ , n′ ), c c
(2)
η(r) [∇ × Hkn (r)] · [∇ × H∗k′ n′ (r)] dr.
(3)
with disordered matrix elements
Vkn,k′n′ =
Z
s.cell
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The integral of Eq. (3) is omputed in the total super ell of the disordered photoni stru ture and
η(r)
is de ned as the dieren e between the disordered and the non-disordered pro les:
η(r) =
1 1 − . ǫ′ (r) ǫ(r)
(4)
The matrix elements of Eq. (3) are onveniently omputed from the guided mode expansion (GME) approximation,
32
where the Blo h modes of the non-disordered stru ture are
expanded in the eigenmodes of the ee tive homogeneous slab. In addition, we employ the so- alled photoni golden rule probability
Γβ
33
to estimate the out-of-plane losses, in whi h the transition
from a disordered mode
|Hβ i to
a radiative mode
|Hrad i (above
of the slab) is omputed and weighted with the radiative density of states
Γβ = π
2 X ˆ ′ |Hβ i ρrad , hHrad|Θ
the light line
ρrad ,
so that
(5)
rad
where
ˆ ′ = ∇ × 1 ∇×, Θ ǫ′ (r)
(6)
is the Maxwell operator asso iated to the disordered pro le. frequen y,
The imaginary part of the
Ωβ = Γβ /(2ωβ ), obtained through the golden rule in Eq. (5), and its real part, ωβ ,
determine the opti al quality fa tor of the avity mode,
Qβ = ωβ /(2Ωβ ).
On e the magneti
eld of the system is obtained from Eq. (2), the ele tri eld of the disordered mode is
omputed via
Eβ (r) =
ic ∇ × Hβ (r), ωβ ǫ′ (r)
(7)
whi h is then normalized through
Z
ǫ′ (r)|Eβ (r)|2 dr = 1.
s.cell
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(8)
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The ee tive mode volume an be de ned as
Vβ =
where
r0
1 , ǫ′ (r0 )|Eβ (r0 )|2
(9)
is usually taken at the antinode position of the ele tri eld peak.
One advantage of the BME method (whi h is a full 3D approa h for slabs) is that it is
apable of des ribing a large volume super ell using the basis of a small super ell or unit
ell. Su h big super ells an be asso iated to a system with any degree of disorder, or even to a non-disordered one. In parti ular, we employ the basis of the W 1 waveguide, de ned in a one-period (x dire tion) super ell, to des ribe the LN avity system de ned in a super ell of 70 periods (x dire tion). Both systems have the same size along the is
√ 9 3a.
We have veri ed that this super ell of dimensions
√ 70a × 9 3a
y
dire tion whi h
is large enough to
a hieve the onvergen e of the avity modes and to avoid non-physi al behavior oming from residual oupling between avities at neighboring super ells (see Supporting Information for details). Using the GME approa h, we solve the
√ a×9 3a W 1 waveguide, the basis of our LN
avity system, by onsidering the same parameters of Ref. 26, i.e., refra tive index (InP), latti e parameter
a = 438
nm, slab thi kness
d = 250
nm and hole radii
n = 3.17
r = 0.25a.
The proje ted band stru ture is shown in Figure 2. Here we have employed 437 plane waves and 1 guided TE mode in the GME basis. Figure 2 determines the basis to be used into the BME approa h, in parti ular, we onsider
l
bands and 70
k
points, uniformly distributed
within the rst Brillouin zone, to solve the eigenvalue problem of Eq. (2). It is well known that the BME method is quite suitable to optimize large disordered stru tures
27
and provides
a fast onvergen e for the eigenvalues of Eq. (2), nevertheless, the onvergen e of the quality fa tor as a fun tion of the number of bands has been shown to be slow in PCS avities.
4
We have arried out onvergen e tests with the non-disordered and disordered avities and found that
l = 200
is enough to a
urately des ribe the system with a maximum overall
error of 0.05% and 10% for the resonant frequen ies and quality fa tors, respe tively, with
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Figure 2: The GME proje ted band stru ture of the
√ a × 9 3a
W 1 PCS waveguide; 200
bands (bla k urves) are onsidered in the BME approa h.
respe t to the onvergent values (see Supporting Information for details on the onvergen e test and the orresponding FDTD omparison).
Non-disordered
LN
avities
In order to understand the ee ts of stru tural disorder on large LN PCS avities, we rst study the non-disordered or ideal latti e system employing the BME approa h, where the
avity pro le takes the pla e of the disordered one in Eq. (4). Figure 3 shows the omputed results. The avity resonan es as a fun tion of the avity length are displayed in Figure 3a, where one learly sees a frequen y redshift for in reasing avity lengths. Interestingly, the fundamental mode frequen y tends to the fundamental W 1 band-edge, whi h is predi ted at 189.8 THz from our GME al ulations; su h trend an be understood given the geometry of the system, i.e., large avities resemble the waveguide system, and onsequently, in the limit of
N → ∞ the fundamental avity resonan e has to be exa tly 189.8 THz.
The fundamental
avity mode is then the one whi h is expe ted to be more sensitive to disorder ee ts for larger
avity lengths; the strong ba k-s attering o
urring lose to the band-edge frequen y in the presen e of disorder leads to additional lo alization ee ts oming from the interplay between order and disorder into the Sajeev John lo alization phenomenon.
15
These results are also
in very good agreement with previous experimental measurements in InP LN avities, with
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Figure 3: (a) Cavity frequen ies of the rst 7 modes as a fun tion of the avity length, for the ideal avities (no disorder). Quality fa tors (b) and ee tive mode volumes ( ) of the rst 4 avity modes as a fun tion of the avity length.
N ranging from 2 to 20;
26
the modal urves
M 1, M 2, M 3, M 4
and
M5
have exa tly the
same behavior. We also show, in Figure 3b, the opti al out-of-plane quality fa tors, of the rst 4 avity modes, omputed with Eq. (5) as a fun tion of the avity length. The quality fa tors of the modes in rease for in reasing length and those modes whose trend is to fall below the
W1
in reasing of a nite
Q
light line for very large avity length are expe ted to have an unbounded fast
Q;
and those ones whi h tend to fall above the light line are expe ted to have
in the waveguide limit. Evidently, the fundamental avity mode,
M 1,
displays
the largest quality fa tor (from all the omputed modes) given its the trend to approa h the fundamental
W1
band-edge for in reasing avity lengths. These BME
Q
fa tors are in
very good agreement with re ent FDTD al ulations in non-disordered LN avities ranging from
L5
to
L15 34
(see also Supporting Information). Correspondingly, we have omputed
the ee tive mode volumes of these avity modes and these are shown in Figure 3 . The mode volumes, whi h are proportional to the lo alization length of the avity modes, display an approximately linear in reasing for in reasing avity length, re overing the in nite mode volume value (or lo alization length) in the non-disordered waveguide regime. With the aim of studying in more detail the spe tral properties of the LN avities we also analyze the DOS of the system, whi h an be omputed using the photoni Green fun tion formalism. The transverse Green's fun tion of the system is expanded in the basis
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of photoni normal modes
35
X ω 2 E∗β (r′ )Eβ (r) ← → G (r, r′, ω) ≈ , 2 2 − ω ω ˜ β β where
ω ˜ β = ωβ − iΩβ
is the omplex frequen y of the mode
β
(10)
and the ele tri eld is subje t
to the normalization ondition of Eq. (8). Sin e we are dealing with high
Q resonators, in the
present formulation we negle t the quasinormal mode aspe ts of the avity modes, therefore, the Green's fun tion of Eq. (10) and the mode volume de nition in Eq. (9) onverge and are expe ted to be an ex ellent approximation.
36,37
The lo al DOS of the system is de ned as
n h← io → 6 , ρ(r, ω) = Im Tr G (r, r, ω) πω
(11)
where the operation Tr[ ℄ represents the tra e over the three orthogonal spatial dire tions. By integrating Eq. (11) over all spa e (total DOS), it an be shown that, in units of Pur ell fa tor (whi h gives the enhan ement fa tor of a dipole emitter), i.e., over the bulk DOS
√ ω ǫ/(π 2 c3 ),
the DOS of the system an be written as:
28
) ( X 1 6πc3 DOS(ω) = , Im ωǫ3/2 ω ˜ β2 − ω 2 β where
ǫ
(12)
is the bulk/slab diele tri onstant (asso iated to InP in our ase). Figure 4 shows
the omputed DOS in a log-s ale for the stru ture of the
W1
L7, L11, L15, L21
and
L29
avities, and the band
is shown in bottom where the shaded gray regions en lose the TE-like
photoni band gap of the
W1
PC. The frequen y region of interest, 189.5 to 200 THz [same
of Figure 3(a)℄, is shaded in light-blue.
As expe ted from results of Figure 3, the DOS
intensity in reases, with de easing peak-width, for in reasing avity length at the avity mode frequen ies (at resonan e with the avity mode
β
the DOS is proportional to
Qβ ),
and the frequen y spa ing between the peaks de reases as N in reases. Sin e smaller LN
avities lead to smaller average refra tive index in the PC slab (more holes), the photoni
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Figure 4: The DOS in Pur ell fa tor units (see text) for the non-disordered L7, L11, L15, L21, L29 avities. The proje ted band stru ture of the non-disordered W 1 system is shown
in the bottom, with the TE-like photoni band gap en losed by the shaded gray regions, and the frequen y region of interest (189.5 to 200 THz), where the main avity mode appear, has been highlighted in the light-blue shaded area.
band-edges of the avity stru tures, en losing the photoni band gap, are slightly blue-shifted (with respe t to the
W1
ase) as the avity length de reases. In the shaded blue region, we
found a total of 3, 4, 6, 8 and 11 avity resonan es of the
L7, L11, L15, L21 and L29 avities,
respe tively.
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Disordered
LN
avities
Planar PCS are always subje t to a small degree of unintentional imperfe tions, oming from the fabri ation pro ess, whi h an be understood as intrinsi disorder in the diele tri latti e.
The problem of modelling disorder in PCS has been addressed in the literature
by onsidering simple hole-size or hole-position random u tuations,
5,38
as well as more
sophisti ated models where a random- orrelated surfa e roughness is introdu ed in the holes surfa es.
39,40
Here, we onsider random u tuations of the hole positions as the main disorder
ontribution.
Su h a model of disorder has shown to be a
urate for understanding the
ee ts of unintentional stru tural disorder in PCS waveguides, u tuations in the hole sizes.
δ
and is similar to random
28
Starting from the non- u tuated position u tuations
41
(0)
(0)
(xm , ym ) of the m-th hole, we onsider random
with Gaussian probability within the BME super ell:
(0) (xm , ym ) = (x(0) m + δx, ym + δy),
where
(xm , ym )
(13)
is the u tuated position. We adopt the standard deviation of the Gaussian
probability distribution
σ = σx = σy
as our disorder parameter. In the present work, we
set a medium quantity of stru tural disorder
σ = 0.01a,
i.e., 1% of the latti e parameter,
orresponding to a disorder magnitude whi h is between typi al amounts of intrinsi and deliberate disorder values that have been used previously.
38
In all the disordered ases that
will be shown below, we have onsidered 20 independent statisti al realizations of the disordered system, and we have veri ed that this number of instan es is enough to des ribe the main physi s of the disordered PCS stru ture (see Supporting Information for details). In 5a, we show the averaged DOS over 20 statisti al realizations of the disordered system for the L7, L11, L15, L21, L29 avities and the W 1 system. The proje ted band stru ture of the non-disordered W 1 waveguide is also shown inside the redu ed Brillouin zone and, as in Figure 4, the shaded gray regions en lose the TE-like photoni band gap of the waveguide.
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Figure 5: (a) Averaged DOS in Pur ell fa tor units (see text) for the L7, L11, L15, L21, L29 avities and the orresponding W 1 waveguide.
We have onsidered 20 independent
statisti al instan es and the amount of intrinsi disorder is
σ = 0.01a.
(b) The DOS, in
units of Pur ell fa tor, is shown for six independent statisti al realizations of the disordered L21 avity with
σ = 0.01a.
The proje ted band stru ture of the non-disordered W 1 system
is shown in bottom of both (a) and (b), the frequen y region of interest ( avity modes) has been highlighted in the light-blue shaded area and the regions en losing the band gap of the non-disordered waveguide are shaded in gray.
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The
hDOSi
displays sets of sharp resonan es inside the photoni band gap of the system
orresponding to the avity modes; the number of resonan es in reases as the avity length in reases re overing the Lifshitz tail in the waveguide limit, at both fundamental and se ond waveguide band-edges.
Interestingly, the Pur ell enhan ement determined by the dis rete
resonan es in the W 1 waveguide, inside the shaded region (dominant avity mode spe tral region), is similar to the L29 ase and larger than those of the smaller LN ases, suggesting that the random avity modes, spontaneously indu ed by disorder in the waveguide, are as good or even better than designed/engineered avity modes of the perfe t latti e. From Figure 5a, it is possible to pi ture the overall trend of the system but it is di ult to
on lude if new modes are appearing in the sets of sharp resonan es due to disorder, whi h would be, at rst insight, asso iated to Anderson-like lo alized modes inside the avities. To identify if su h new modes are in fa t appearing inside the avities, we show in Figure 5b six independent statisti al instan es of DOS for the L21 avity. Here, individual sharp peaks,
orresponding to the system resonan es, an easily be resolved. From this gure, it is possible to see that the number of peaks in the shaded region is the same of the orresponding non-disordered DOS of Figure 4, i.e, 8 resonant peaks, and it is always the same for all disordered realizations of the avity. We have also seen the same behavior for all disordered instan es of the avities studied in this work. Therefore, we an on lude that no new modes appear in the LN avities due to disorder (at least not in the avity modes' spe tral region). This result is also valid for larger amounts of disorder as long as the avity resonan es are not mixed with the indu ed Anderson modes at the band edges of the PC band gap (see Supporting Information for further details). As in the ase of the non-disordered systems, the band-edges en losing the photoni band gap of the disordered avities, are also slightly blue-shifted for de reasing avity length. Taking into a
ount the fa t that no new resonan es are indu ed by disorder in the light-blue shaded area of Figure 5(b), we have omputed the averaged frequen ies of the avity modes as a fun tion of the avity length and the results are shown in Figure 6, where the standard deviation is represented by the error bars. The
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200 198
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M3 M5 M6 M7 M2 M4
196 194 M1 192 190 W1 5
10
15
20
25
Cavity Length (N)
30
35
Figure 6: Average frequen ies of the rst 7 avity modes as a fun tion of the avity length. The horizontal dashed line orresponds to the Anderson modes of the disordered W 1 waveguide (ensemble average). The standard deviation is represented by the error bars. We have used 20 independent statisti al instan es with an amount of intrinsi disorder of
σ = 0.01a.
averaged frequen y spe trum of the disordered system is slightly blue-shifted, in omparison to Figure 3a, but all the fun tional features are equivalent to the non-disordered ase. In parti ular, we note larger u tuations for smaller avities, whi h is in agreement with the approximate s aling of the disorder-indu ed resonan e shifts with the inverse mode volume.
3
These results show then that the avity frequen ies, apart from small u tuations around an average value, are not fundamentally ae ted when sizable amounts of disorder are present in the LN avity system. Dierent from the avity resonan e properties, the quality fa tors and ee tive mode volumes display mu h more dramati behaviors in the presen e of disorder. Figure 7a shows the averaged averaged
Q
as a fun tion of the avity length for the rst 4 avity modes, and the
Q values of the Anderson modes in the W 1 system are represented by the horizontal
dashed line. The asso iated standard deviations are represented by the error bars and the
orresponding non-disorder results (3) are shown in thin dashed lines.
When disorder is
onsidered, the quality fa tors of the avity modes are redu ed (as expe ted), and those modes whi h are loser to the band-edge of the waveguide, i.e., in the slow-light regime, are mu h more sensitive to disorder ee ts, as is the ase of the fundamental avity mode M 1 (red
urve), whi h de reases from
∼ 938×104 (see Figure 3b) to ∼ 27×104 for the L35 avity, and
the se ond mode M 2 (blue urve) de reases from
16
∼ 225 × 104
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∼ 27 × 104
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for the same avity. By omparing Figure 7a with Figure 3b (or with the thin dashed lines in luded in Figure 7a) we see that su h a redu tion is less strong for those modes whi h are not lose to the waveguide band-edge, i.e., far from the slow-light regime. Moreover, we have not identi ed any spe i avity length at whi h the quality fa tor is maximized when disorder is onsidered, whi h was previously suggested in Ref. 26 as a possible explanation of the threshold minimization in slow-light PC lasers (when the avity length was in reased). In ontrast, we have found that the quality fa tor in reases for in reasing avity length and eventually saturates, with small os illations, below the averaged of the orresponding waveguide system.
Q
of the Anderson modes
Sin e the mean quality fa tor of su h Anderson
modes depends on the degree of disorder, the avity length at whi h the mean also depends on the spe i value of roughly bounded by the average
Q
σ,
implying that the
Q
Q
saturates
performan e of the avities is
performan e of the W 1 Anderson modes. The averaged
mode volumes are shown in Figure 7b, where it is demonstrated that
V
is redu ed in large
avities, with respe t to the non-disordered ase (see thin dashed lines or 3 ), in the presen e of disorder, and, as for the ase of the average
V
Q,
the volume of the avities also remains bounded by
of the Anderson-like modes in the W 1 waveguide.
The disorder-indu ed
lo alization suggested by Figure 7b is learly seen in Figure 7 , where the intensity pro le
|D(r)|2
[with
σ = 0.01a in
D(r) = ǫ0 ǫ(r)E(r)℄
is shown for the rst two avity modes with
σ = 0
and
the upper and bottom panels, respe tively. The non-disordered and disordered
pro les are orrespondingly represented by thi k white ir les.
In Figure 7 , the eld is
on entrated in a smaller region within the avity, implying a redu tion of the ee tive lo alization length of the mode.
Thus, as remarked earlier, we on lude that stru tural
disorder in LN avities auses disorder-indu ed losses and disorder-indu ed lo alization. In order to assess the quasi-1D Anderson lo alization phenomenon in the disordered LN
avity system, a lear broadening of the intensity distribution has to be seen with
respe t to the Rayleigh distribution.
29
Given the proportionality between the verti al emit-
ted intensity with the lo al DOS through the out-of-plane radiative de ay,
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we em-
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Figure 7: Averaged (a) quality fa tor and (b) mode volumes as a fun tion of the avity length. The standard deviation is represented by the error bars and the thin dashed lines
orrespond to the results without onsidering disorder. We have onsidered 20 independent statisti al instan es and the amount of intrinsi disorder is
σ = 0.01a.
The horizontal dashed
lines are for the Anderson modes of the disordered W 1 waveguide (ensemble average) with the error bars representing the orresponding standard deviation. ( ) Intensity pro les in 2 the enter of the slab |D(x, y, z = 0)| of the rst two avity modes for the L21 avity; we
have used
σ = 0.0
and
σ = 0.01a.
The non-disordered (ideal) and disordered PC pro le are
orrespondingly represented by the thi k white ir les.
ploy the lo al DOS u tuations to obtain the intensity u tuations, i.e., Var (I/ hIi) Var [ρ(r, ω)/ hρ(r, ω)i].
42,43
=
The varian e of the normalized intensity distribution, Var(I/ hIi),
is then omputed by employing the lo al DOS of all disordered realizations along the avity dire tion,
ρ(x, y = 0, z = 0, ω),
averaged over the desired frequen y range.
30
We show in
Figure 8, the varian e of the intensity u tuations in the averaged frequen y region of the fundamental avity mode (usually the most important for pra ti al appli ations) and the
orresponding value obtained for the disordered
W 1 system,
5.9, whi h is in good agreement
with previous experimental measurements in similar disordered
W 1 waveguides. 21
By assum-
ing that the intensity u tuations follows the same statisti s of the transmission u tuations, the system falls into the Anderson lo alization regime if Var(I/ hIi) ex eeds the riti al value 7/3,
21,29
i.e., the blue hain- he k line in 8. Consequently, we observe Anderson-like hara -
teristi s for avity lengths equal or larger than
18
L31
(see inset of the gure).
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Var(I/)
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5
3
M1 W1 7/3
4 3
2.5 2 27 29 31 33 35
2 1 0
10
5
20
15
25
Cavity Length (N)
30
35
Figure 8: Varian e of the normalized intensity distribution (the lo al DOS has been averaged in the frequen y region of the fundamental avity mode) as a fun tion of the avity length (red line). The orresponding value for the
W1
waveguide is 5.9 (horizontal bla k dashed
line) and the riti al value 7/3 is represented by the horizontal blue hain line.
Role of the avity boundary onditions The apparent saturation of
Q and V
in Figure 7a and Figure 7b suggests that when disorder
is onsidered, on average, there is a minimum avity length at whi h the boundary avity
ondition is not fundamentally relevant to the avity mode. In order to address the role of the avity boundaries on the mode on nement, we now systemati ally study the DOS and the eld pro les for several avity lengths using exa tly the same disorder realization, i.e., the same distribution of disordered holes.
Results of this analysis are shown in Figure 9.
The DOS asso iated to the fundamental mode is displayed in Figure 9a for the L5, L15, L21, L29, L35 avities and the fundamental disorder-indu ed mode of the disordered W 1
waveguide, i.e., the lower frequen y mode appearing in the diagonalization within the TElike band gap of the W 1 PC slab, whi h we have denoted as M 1-W 1.
The frequen y
dieren e between the fundamental modes of the dierent avities de reases for in reasing
avity length (as expe ted from Figure 6) and, as already seen in the averaged DOS of Figure 5a, the intensity of the DOS is largest for the mode M 1-W 1 and quite similar to the orresponding intensity for avity lengths larger than L21. We also show in Figure 9b, Figure 9 and Figure 9d the frequen y, quality fa tor and ee tive mode volume (in luding the lo alization length), respe tively, of the fundamental avity mode, M 1, as a fun tion of
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Figure 9: (a) Computed DOS of the fundamental avity mode M 1 for several avity lengths by onsidering exa tly the same disorder realization with
σ = 0.01a.
The fundamental
disorder-indu ed mode of the disordered W 1 waveguide, M 1-W 1, is also shown. (b) Frequen y of the fundamental avity mode as a fun tion of the avity length; as in (a), exa tly the same disorder realization is onsidered with
σ = 0.01a.
W 1 mode is represented by the horizontal dashed line.
The frequen y of the M 1-
( ) Quality fa tors of the modes
shown in panel (b). (d) Ee tive mode volumes (left) and lo alization length (right) of the modes shown in panel (b). expansion oe ient
Uβ (m)
Absolute value of the (e) real and (f ) imaginary part of the for
β =M 1,
in the L5, L15 and L35 avities, and
W 1, in the disordered W 1 waveguide, where
UM 1−W 1 (m) >
β =M 1-
m = (k, n)
is a global index su h that UM 1−W 1 (m+1). (g) Intensity pro les in the enter of the slab |D(x, y, z = 0)|2
of the fundamental mode in the orresponding W 1 and avity systems. The disordered pro le is represented by thi k white ir les.
the avity length by (again) onsidering exa tly the same disordered PC. The orresponding values of the fundamental disorder-indu ed mode M 1-W 1 are represented by the horizontal dashed lines. The lo alization length in Figure 9d is omputed using the inverse parti ipation number as des ribed in Refs. 27 and 28. From these results we an learly see that, for large length disordered avities, the avity fundamental mode and the M 1-W 1 mode are totally
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equivalent, and this hypothesis is further strengthened in Figure 9e and Figure 9f, where the absolute value of
Im{Uβ (m)}
and
Re{Uβ (m)},
i.e., the expansion oe ient in Eq. (1),
are respe tively shown for the fundamental mode of the L5, L15 and L35 avities, and the mode that
β =M 1-W 1
β = M1,
in the disordered waveguide [the oe ients are organized su h
UM 1−W 1 (m) > UM 1−W 1 (m + 1)
between the expansion oe ients of
given the global index
β =M 1-W 1
and
m = (k, n)℄;
β =M 1
the dieren e
modes be omes negligible
for large avities lengths, thus demonstrating the equivalen e between these two on ned states in the presen e of disorder.
The similarity between the fundamental avity modes
and the M 1-W 1 mode an also be understood from Figure 9g, where we have plotted their intensity pro les with the disordered pro le represented by thi k white ir les. For strongly lo alized Anderson modes, whi h is the ase of M 1-W 1, the avity boundary ondition does not signi antly ae t the mode distribution as long as the boundaries of the avity are far from the lo alization region, then the on nement in su h large avities an be mainly determined by the spe i lo al hara teristi s of the stru tural disorder than by the
avity mirrors. It is important to stress again that the lo alized mode must be positioned far from the avity boundaries and its lo alization length must be mu h smaller than the
avity length in order to establish this equivalen e; in su h a regime, the avity boundaries do not play an important role and the on nement is mainly determined by Andersonlike lo alization phenomena.
We show in Figure 10 the frequen y dieren e between the
fundamental avity mode and the fundamental disorder-indu ed mode of the disordered W 1 waveguide, i.e.,
∆f = fM 1 − fM 1−W 1 ,
as a fun tion of the avity length, ompared with the
photoni radiative linewidth of the M 1-W 1 mode (horizontal dashed line). As learly seen in the inset of the gure, the frequen y dieren e between the fundamental disordered avity modes and the Anderson mode of the disordered waveguide is blurred for avity lengths equal or larger than
L31,
suggesting that the fundamental mode of these disordered systems is not
signi antly sensitive to the boundary avity ondition for
N ≥ 31.
The sensitivity of the
system eigenvalues to the boundary onditions has been previously employed as a riterion
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0.003
∆f γM1-W1
3.2 0.002 2.4 0.001
1.6
0 23 25 27 29 31 33 35
0.8 0
Figure 10: Frequen y dieren e
5
10
15
20
25
Cavity Length (N)
∆f = fM 1 − fM 1−W 1
30
35
as a fun tion of the avity length. The
photoni radiative linewidth of the mode M 1-W 1 is represented by the horizontal dashed line.
for Anderson lo alization in semi ondu tors;
44
here, we also found a very good agreement
with the intensity u tuations riterion employed in disordered photoni s, thus reinfor ing our on lusions. As a onsequen e of these results, we highlight that models based on the avity boundaries and single slow-light Blo h modes to des ribe avity modes with large N, as it is the ase of the ee tive Fabry-Pérot resonator,
26,31
will be problemati for predi ting the behavior
of Anderson-like lo alized modes inside the disordered avity region. The main reason, as addressed above, is that the on ned state annot in general be understood as two ounterpropagating Blo h modes re e ting between the two avity mirrors, sin e the on ned mode does not ne essarily feel the avity boundaries when the system falls into the Andersonlike lo alization regime, i.e., when the slow-light Blo h modes are subje t to several strong ba k-re e tions originated in the lo al distribution of disordered holes.
Con lusions We have presented a detailed study of the ee ts of stru tural disorder on PCS LN avity modes by employing a fully 3D BME approa h and Green fun tion formalism. By onsidering in-plane
σ
magnitudes, whi h are between typi al amounts of intrinsi and deliberate
disorder, we found, on the one hand, that disorder indu es u tuations in the fundamental
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resonan e frequen ies and no new additional modes appear in the usual avity spe tral region of interest. However, disorder has a strong in uen e on the quality fa tor and mode volume of these modes, espe ially for longer length avities. We also studied the onset of lo alization modes for the avity stru tures, over a wide range of avity lengths and frequen ies; interestingly, our averaged DOS results suggest that the Anderson-modes of the disordered slow-light waveguides are as good or even better than non-optimized disordered LN avities in terms of enhan ing the total DOS, whi h opens new possibilities on designing
high quality disordered modes, e.g., by taking the disordered pro le as an improved design, that ould surpass the performan e of state-of-the-art engineered PCS avitieswhi h has been re ently demonstrated experimentally.
45
Furthermore, we have found that the mean
mode quality fa tors and respe tively mean ee tive mode volumes saturate for a spe i
avity length s ale (whi h depends on the disorder parameter), and they are bounded by the averages of the Anderson modes in the orresponding W 1 waveguide system. Next, we have shown that: (i) the quality fa tor is not maximized at a spe i length in disordered LN
avities as previously suggested in Ref. 26; (ii), apart from disorder-indu ed losses, disorderindu ed lo alization be omes riti ally important in longer length disordered LN avities; and (iii), by means of the intensity u tuation riterion we have observed Anderson-like lo alization for avity lengths equal or larger than
L31.
In order to further understand the role
of the avity boundary onditions on the avity mode on nement, we also systemati ally studied the ee ts of the avity mirrors on the fundamental avity mode (whi h is usually the most relevant one for pra ti al appli ations); we found that as long as the on nement region is far from the avity boundaries and the ee tive mode lo alization length is mu h smaller than the avity length, the photoni on nement is mainly determined by Andersonlike lo alization and the mirrors of the avity do not play an important role on the de nition of the resonant frequen y, quality fa tor and ee tive volume of the mode in system; this nding is signi ant as these are the key quantities to understand the light-matter oupling parameters in numerous appli ations su h as lasing, sensing, nonlinear opti s and avity-
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QED. Clearly, for signi antly long either
W1
LN
Page 24 of 30
avities, on e an expe t a similar performan e from
waveguides or the avities, wherein the physi s of the avity models an hange
signi antly.
A knowledgement This work was supported by the Natural S ien es and Engineering Resear h Coun il of Canada (NSERC) and Queen's University, Canada.
We gratefully a knowledge Jesper
Mørk and Lu a Sapienza for useful suggestions and dis ussions.
This resear h was en-
abled in part by omputational support provided by the Centre for Advan ed Computing (http:// a .queensu. a) and Compute Canada (www. ompute anada. a).
Supporting Information Available Further details regarding the numeri al al ulations presented in the manus ript are reported in the Supporting Information. This material is available free of harge via the Internet at
http://pubs.a s.org/.
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