モルフォ蝶

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モルフォ蝶

2010/1/28
宇宙理論セミナー
Structural color
家篤史
2010年1月27日水曜日
1
2006年6月8日
光の散乱と青空
2010年1月27日水曜日
2
2007年1月31日
自然と偏光
青空の観察 (2/2)
荒川区南千住
隅田川土手にて
1月27日15:50頃
N
W
2010年1月27日水曜日
E
3
そして今回、
「自然と物理」シリーズ第３回
生物の色彩と光学効果
∼モルフォ蝶の羽はなぜ青い？∼
2010年1月27日水曜日
4
紹介論文
Structural color of Morpho butterflies
G.S. Smith
Am. J. Phys. 77 (2009) 1010
2010年1月27日水曜日
5
モルフォ蝶
Wikipediaより
生息地：北アメリカ南部∼南アメリカ
種類：80種ほど
分類：タテハチョウ科・モルフォチョウ亜科・
モルフォチョウ族・モルフォチョウ属
特長：体にくらべて非常に大きな翅をもち、表面は金属光
沢をもつ。この光沢はほとんどの種類で青色に発色
する（鮮やかな翅の色を持つのは雄）。
2010年1月27日水曜日
6
Morpho Rhetenor
表(背中)
裏(腹)
http://www.pref.mie.jp/Haku/Hp/Osusume/morho%20rhetenor.htm
2010年1月27日水曜日
7
様々なモルフォ蝶（雄）
1 (2008) 076401
e 8. Typical male Morpho butterflies (Courtesy of The Museum of Nature and Human Activities, Hyogo, Ja
S. Kinoshita, et al. Rep. Prog. Phys. 71 (2008) 076401
2010年1月27日水曜日
8
モルフォ蝶の翅の色(1)
鮮やかな青色に隠された秘密
100 !m
100 !m
トルエンに浸すと（翅の左部分）、
• モルフォ蝶の翅から青色が消える
5 cm
Direction of Ridge
5 cm
Direction of Ridge
(a)
モルフォ蝶
(a)
(a)
(a)
• ベニイロタテハ蝶の翅の色は
Ridges
Ridges
同じまま
Lamellae
Lamellae
22 cm
cm
(b)
(b)
ベニイロタテハ蝶
（学名は、Cymothoe sangaris。中央アフリカに生息）
Fig. 1. !a" Photograph of the dorsal side of a male Morpho rhetenor butterFig.
!a"Photograph
Photograph
of the
dorsal
of a male
Morpho
rhetenor
butterfly.1.!b"
of the
dorsal
side side
of a male
Cymothoe
sangaris
butterfly.
2010年1月27日水曜日
fly.For
!b"both
Photograph
of thethedorsal
of aright-hand
male Cymothoe
butterfly.
photographs,
wingsside
on the
side aresangaris
in their natural
9
モルフォ蝶の翅の色(2)
evaluated to obtain an expression for the scattered light. Numerical results obtained with this expression are presented in
Sec.
IV,an and
theyforshow
the observed
evaluated to
obtain
expression
the scattered
light. Nu- blue color and iridesmerical results
obtained
with thisfor
expression
presented in In Sec. V a simplified
cence
observed
these are
butterflies.
Sec. IV, and
they show
observed
blueiscolor
and to
iridesversion
ofthethe
model
used
provide insight into the
cence observed for these butterflies. In Sec. V a simplified
thattoproduce
theseinto
effects.
version ofmechanisms
the model is used
provide insight
the
mechanisms that produce these effects.
II.FOR
MODEL
FOR A RIDGE
II. MODEL
A RIDGE
漂白すると、
(a)
モルフォ蝶の翅
(a)
Two main features stand out in the transmission and scanmainoffeatures
standtheout
in the
ning electron Two
micrographs
Morpho scales:
nearly
par- transmission and scanelectron
of Morpho
allel, long,ning
thin ridges,
and micrographs
the regularly spaced
lamellae scales: the nearly par25
within a ridge.
al. ridges,
deduced from
allel,Kinoshita
long, etthin
and their
the mearegularly spaced lamellae
surements that the ridges, although similar, are offset
25 ranwithinanda this
ridge.
Kinoshita
from their meadomly in height,
variation
is sufficientettoal.
make deduced
the
collective surements
scattering fromthat
the ridges
incoherent.
This inco- similar, are offset ranthe ridges,
although
herence implies
thatin
theheight,
scatteringand
from this
a scale
can be chardomly
variation
is sufficient to make the
acterized by the scattering from a single ridge.
scattering
the ridges
We will collective
adopt the simple
model for from
a ridge shown
in Fig. incoherent. This inco5. It contains
the major
featuresthat
for scattering
commonfrom
to
herence
implies
the scattering
a scale can be charmany Morpho
butterflies.
this scattering
model there are
N identical
acterized
byInthe
from
a single ridge.
lamellae, each a slab of width w and height h. The lamellae
will
the g.simple
model for a ridge shown in Fig.
are separated We
by air
gapsadopt
of height
In the longitudinal
direction !y",
lamellae are the
inclined
at thefeatures
angle ! tofor
the scattering common to
5. the
It contains
major
plane of the
scale, which
makes
the structure periodic
this
many
Morpho
butterflies.
In this inmodel
there are N identical
direction with period l = !h + g" / sin !. A unit cell of the perilamellae,
each
a frame
slab !dashed
of width
odic structure
is enclosed
by the
lines"winand
Fig. height h. The lamellae
areassume
separated
by air gaps
of height
5!a". We will
that the randomness
also applies
in the g. In the longitudinal
longitudinal
direction so!y",
that the
the scattering
from
can at the angle ! to the
direction
lamellae
area ridge
inclined
be characterized by the scattering from a single unit cell. The
plane
ofsections
the scale,
which
makes
theFig.structure periodic in this
andN!b"
slabs whose
cross
are shown
in dark
gray in
5!a" are thedirection
structures that
weperiod
will analyze
in +
theg"remainder
with
l = !h
/ sin !. A unit cell of the peri-10
• モルフォ蝶の翅は青色のまま
• ベニイロタテハ蝶の翅は脱色
して色あせる
(b)
3. Photographs of wings after bleaching: !a" Morpho rhetenor and !b"
mothoe sangaris.
(b)
ベニイロタテハ蝶の翅
ht by the Morpho butterfly scale in a way that is approprifor use in courses on electromagnetism and optics. This
alysis
the after
majorbleaching:
features of !a"
the Morpho
scatteredrhetenor
light. In
graphspredicts
of wings
me ways, it combines components of the earlier apngaris.
2010年1月27日水曜日
26
青い翅の秘密
• トルエン：翅の表皮とほぼ同じ屈折率（n~1.56）
index-matching liquid
• 漂白しても色あせない
化学的発色（色素沈着）によるものではない
モルフォ蝶の翅の色は、表皮の微細構造から来る
光学特性に起因している（散乱・干渉）
⇨ 構造色 (structural color)
2010年1月27日水曜日
11
構造色研究の歴史
1665 Hooke
光学顕微鏡による孔雀や鴨の羽の観察
1917 Rayleigh 電磁気理論にもとづく構造発色の研究
1924-27 Mason 色彩と微視的構造の関係（干渉の影響）
モルフォ蝶の翅の表面構造を予言
電子顕微鏡の開発（1931年）により、構造発色の
物理的メカニズムが徐々に解明
2010年1月27日水曜日
12
表面構造
鱗粉
光学顕微鏡で拡大
100 !m
71 (2008)
076401
Phys.
71 (2008)
076401
(a)
5 cm
Direction of Ridge
(c) (c)
S Kinoshita et
et al
al
(e)(e)
(a)
Direction of Ridge
(a)
Ridgeの電顕写真
Ridges
2010年1月27日水曜日
(b)
(d)(d)
(f)(f)
13
表面構造のまとめ
Direction of Ridge
(a)
模式図
Ridges
ラメラ(lamellae)と呼ばれる膜が何
層にも積み重なった棚構造(ridges)
Lamellae
（注・実際はもっと不規則な構造）
モルフォ蝶の翅
幾層ものラメラで散乱（屈折・
反射）された光が
s
Supporting
Structure
Plane of Scale
1 !m
2010年1月27日水曜日
多層膜干渉
を起こした結果、青色に見える
14
多層膜干渉
S Kinoshita et al
多層膜からの散乱光がお互い干渉する
ことで、特徴的な光学特性（角度依存
性・波長依存性）が現れる
gure 3. Schematic illustration of multilayer interference.
c.f. 薄膜干渉
hys. 71 (2008) 076401
Reflectivity
S Kinoshita et al
m should be satisfied because of the restriction of the
ess. In particular, the relations with m = 1 and 1m" = 1
(a)
(c)
d=0.1µm
pond to the lowest-order case, where the optical path
薄膜の表裏から反射された光が干渉すること
0.8
s, defined as the length multiplied by the refractive index,
67.5°
and B layers are equal to each other. Land called0.6
this case
で、波長により強め合ったり弱め合ったりする
eal multilayer [25]. On the other hand, if the thickness
of
0.4
45°
layer does not satisfy the soap-bubble relation, while the
22.5°
0.2
f the A and B layers satisfies equation (5), the reflection
at
0°
–B interface works destructively and the peak reflectivity
2010年1月27日水曜日
0.1
例: CD、シャボン玉
1
15
Unit Cell
Lamellae
Lamellae
1
2
3
g
h
Unit Cell
h
x
g
1
2
3
モデル
i
ETM
Ai
N
モルフォ蝶の構造発色を理解するモデル
w
s
N
w
s
Effecd
Medi
l
dn
• 鱗粉表面に対して鉛直に光が入射、ridge中の各ラメラで散乱
l
Plane of Scale
(a)
!!rc −Cuti1
（比誘電率）
!rm = 1 +
• 各ラメラは鱗粉表面に対して傾斜角 γ で等間隔に整列 !h + g
Plane of Scale
(a)
h
Fig.
6. Mo
Effec
the ridge.
(幅 w, 長さ l, 厚さ h, 間隔 g Medi
)
!
Lamellae
i
ETE
Incident
Plane Wave
The semi-infinite lay
x
h g
Fig. 6. Model used
for dis
est
tive permittivity.
We
Incident
large
x
E
the ridge.
E Wave
region, t
Plane
#
P param
with effective
E
E! !r!
E
! "
r
z ties that exist in the n
y
x
ylarge distance from t
Air
lar with
and
region, this field
is g
O am unequal
Top Lamella
P
v
Effective #
lae," etc. These
irreg
− j !#
Medium
!
E !r!" =
e
d
r
"
in
which
4
$
r
z
are correlated multi
= rr̂ loca
y
k of!" the− l1
mc
therefore
ignore
mu
= lamellae
!
O
Top Lamella
4$r
#
Cuticle
h
各ラメラからの散乱光の総和を計算
field within the cuti
E! !r!
uETMi
Unit Cell
i
i
TE
1
2
3
TE
i
i
sr
TM
TM
N
0
sr
w
s
l
2
0
鱗粉表面
Plane of Scale
2010年1月27日水曜日
n
rc
sr
cm
16
Orientation of the ridge with respect to the coordinate system !x , y , z". The
large
distancefrom
from
the
ridge,ininthe
theFraunhofer
Fraunhoferororfar-zo
far-z
x
scattered field islarge
to bedistance
computed
at
point
P.ridge,
the
x
cmfollowing
30,31
large distance
fromdistance
the ridge, from
inregion,
the Fraunhofer
orin
far-zone
large
thethis
ridge,
the
Fraunhofer
or
far-zone
30,31
field
is
given
by
the
integral:
III.
ANALYSIS
OF
THE
SCATTE
30,31
region,
this
field
is
given
by
the
following
integral:
30,31
region,#thisregion,
field is given
by
the
following
integral:
Effective
P field
this
is given by the following integral:
$$$
$$
Radiation
formula
$
$$
$$
# the ridge, in thePFraunhofer or far-zone
Pe distance from
w
30,31
−
j
!
#
Medium
P
0
The
incident
light
produces
a
polar
on, this field is given−byj !the
following integral: ! sr
−jk
r
jk0r̂·r
!!dV
−
j
!
#
#
!
0
0
0
E
!r
"
=
e
!1
−
r̂r̂·"
J
!r
"e
sr
−jk
r
jk
r̂·r
sr
−jk
r
jk
r̂·r
!
!
!
!
OF
THE
!dVe! 0 !1 − r̂r̂·" LIGHT
! !r!" = III.
!!b"!r
0 !1 − r̂r̂·" E
0 SCATTERED
b!!"e 0 !!dV! ! x
!
!
E
eANALYSIS
J
"e
!
r
!
"
!r
=
J
!r
b
− j!#0 V−jk r 4$
4$
r unit volume" P
r$!r sr
" − j !#
!
V
4
jk
r̂·r
!
for
the
bound
cha
per
r
!
0 !1 − r̂r̂·"
0V !dV
0
sr
−jk
r
jk
r̂·r
E
!r
"
=
e
J
!r
"e
!
!
!
!
!
!
b
!1 − r̂r̂·"
J! b!r!!"e 0 light
dV! produces a polarization
rE! !r!" = 4$r e 0Fig.
The
incident
!dipole
moment
2can
2 6. Model used
4
$
r
be
expressed
as
an
equivalent
vo
for
estimating
the
electric
field
within
the
nth
lamella
of
V
V
y
2
!
"
−
1"
k
!
"
−
1"
k
rc
fi
0
rc
0"jk0r̂·r−
−jk0r
!!dV
−jk
r
jk
r̂·r
y per unit
!
!
1"
k
!
!
!
0
0
!
rc
=
e
!1
−
r̂r̂·"
E
!r
"e
,
!
!
! 0 bound charge
=
e
−
r̂r̂·"
E
!r
"e
−jk
r !1of
jk0r̂·r!! dV!
!
!
for
the
the
lamellae
that
volume"
P
2
the
ridge.
!
0
入射光がラメラ部に当たると分極を起こす
=
e
!1
−
r̂r̂·"
E
!r
"e
dV! ,u
!
!
!"rc − 1" −jk r 4$r
k0O
2
V
4
$
r
jk
r̂·r
!
!
!
!
!
O
! !r!!!""e 0− as
V
=
j
!
P
=
j
!
"
!
"
−! 1"E ,
J
=
e 0 !1
− r̂r̂·"
E
dV!an
, equivalent
4
$
r
1"
k
can
be
expressed
volume
current
density,
b
0
rc
V
rc
0
−jk
r
jk
r̂·r
!
4$r
s
=V
e 0 !1 − r̂r̂·"!4"
E! !r!!"e 0 dV! ,
電流
$$
$$
$$ #
$$$$
!
c
! b = j! P
! 4=$jr!"0!"!4"
!
V
"
=
n
=
2.43−
j0.19
is
the
in
which
−
1"E
,
!3"
J
;
rc the lamellae, r! rc
in which r!! locates the source point within
(分極電場の変動)
n
locates
the
source
point
within
the
lamella
in
which
r
!
!
the the
source
point
within
the
lamellae,
r
which r!!=locates
!
rr̂ locates
field
point
P
in
Fig.
5!b",
and
V
is
the
volume
script
r"
permittivity
offar-zone
cuticle,
and
!4"
the
source
point relative
within
the
lamellae
in
which
r!ridge,
! locates
#
large
distance
from
the
in
the
Fraunhofer
or
"
=
n
=
2.43−
j0.19
is
the
complex
!subin
which
29V is the vold
locates of
the the
field lamellae.
point P in Fig.
5!b",
and
thelocates
volume
rcV =isrr̂
the
field
point
P
in
Fig.
5!b",
and
After
introducing
the
dimensions
of
the
30,31
= rr̂islocates
the
field
point
P in
Fig.
5!b",
and
V
is
the
volum
This
cur
field
within
the
lamellae.
region,
this
field
given
by
the
following
integral:
he lamellae.
After
introducing
the
dimensions
of
the
!
この分極電場をもとに散乱光が作られる:
lamellae and the
appropriate
coordinates,
Eq.
!4"
becomes
of
the
lamellae.
After
introducing
the
dimensions
of
t
is
the
total
electric
script
r"
permittivity
of
cuticle,
and
E
of
the
lamellae.
After
introducing
the
dimensions
of
locates
the
source
point
within
the
lamellae,
r
in
which
r
!! Eq. !4" becomes 29tered electromagnetic field that !is thet
ellae and the
coordinates,
P appropriatefield
lamellae
andthe
theappropriate
appropriate
coordinates,
Eq.!4"
!4"become
becomr
This
current
produces
the
scatwithin
the
lamellae.
2
lamellae
and
coordinates,
Eq.
!"rc
− 1" −jk the
krr̂
=
locates
field
point
P
in
Fig.
5!b",
and
V
is
the
volume
2
0
#
sr
r− j !
!
0
!
"
−
1"
k
十分遠方で
0
rc !r
e
sr electromagnetic
−jk0rfield2 that is the!light we
jk0r̂·r
!" e=−jk0r tered
!!dV
u
observe.
At a
!
! sr!r!" = 0 E
E
E
!r
"
=
e
!1
−
r̂r̂·"
J
!r
"e
!
!
!
!
4$r lamellae. After introducing
b dimensions
of the
the
of No.
the11, N
2k01013
!
"
−
1"
r 成り立つ表式
4$r
Am.
J.
Phys.,
Vol.
77,
rc
sr
−jk
r
!
"
−
1"
k
4$r ! Esr! !r!" = 0 rc
−jk
eV0r0
N
$ $ $$
$$
$ $
!
$
$
%%$$ $$ $$
$ $
! %$
w/2
x2E !r
lamellae
coordinates,
Eq. !4" becomes
e
!" =
0
x2 and 0the appropriate
4
$
r
1013
Am.
Phys.,
77,
! 4!r!$
%%
!1 −Vol.
r̂r̂·"E
"r 11, November 2009ラメラ中の電場
2 !J.
!No.
)rse % %
!1 −kr̂r̂·"E
!r
"
!
!
!
"
−
1"
N
rc
0
n=1
z
=−w/2
0!
w/2
y
=−l
x
=x
x2 jk0r̂·r!!
2
!
!
!
n=1
z
=−w/2
y
=−l
x
=x
of
!
!
! 1 = k ! " − 11"e −jk0r!1
N − r̂r̂·"
0 E
w/2
x
!r! 2"e
dV! , !
rc
0
sr
−jk
r
!b"
!!0transverse
!1− −r̂r̂·"E
r̂r̂·"E
!r
!"!"
jk
!x
&'=
cos
+y &!4
&rsin '+ze
&0"%
jk0!x! sin
& cos
'+y
sin!!r
&sin
sin
+z!'cos
"sin
!, cos%
$
!
the
dimensions:
Left
side,
"
E
!
dx
dy
dz
%
e
!1
!r
dx
dy
dz
,
%
e
!
!
!
V
!
!
!
!
!
he
he dimensions: Left side, transverse
n=1 z!=−w/2 y !=−l x!=x1
4
$
r
oss
section.
Note
that
the
lamellae
of
n=1
z!=−w/2
y !=−l x!=x1
ss section. Note that the lamellae of
!5"
!5"
!4"
respecttotothe
theplane
planeofofthe
thescale.
scale.!b"!b"
jk
!x
sin
&
cos
'
+y
sin
&
sin
'
+z
cos
&
"
espect
!
!
!
N
0
N
w/2
w/2 and %0jk
e0!x! sinx2& cos '+y! sin & sin '+z! cos &"dxdx
!dy
!dz
!
othe
thecoordinate
coordinate
system
!x
,
y
,
z".
The
x
=
−&!n
−
1"!h
+
g"
+
h
+
y
sin
(
'
/
cos
(
dy
dz
,
%
e
!
!
!
1
, y , z".− The
with system
x1 !x
= −&!n
1"!h + g" + h + y sin (' / cos (
and
!
nt
P.
−&!n
−
1"!h
+
g"
+
y
sin
(
'
/
cos
(
.
%
!1
−
r̂r̂·"E
!r
! !"
locates
the
source
point
within
the
lamellae,
r!
in
which
r
,t transverse
P.
x2 = −&!n − 1"!h + g" + y sin! (' / cos (.
!
olamellae
proceed with
the
calculation,
we
must
know
the
electric
n=1 z!=−w/2 y !=−l x!=x
of
1
To proceed
with
the calculation,
wepoint
must know
electric
= rr̂
locates
the field
P inthe
Fig.
5!b",
nt2010年1月27日水曜日
and V is the volume
17
ラメラ中の電場
• １つのridgeのみに着目（他のridgesの影響は無視）
• 計算の単純化のために、
仮定
u
x
i.
i
ETE
i
ETM
Air
!
y
入射光が第n層のラメラに到達するまでを、
!!rc − 1"wh
!rm を持った有効媒質での屈折とし
=1+
.
比誘電率 !h + g"s
て表現（左図）
The semi-infinite layer below the lamella also h
!!rc − 1"wh
v
Effective
比誘電率
tive
permittivity.
We
model
these two
!
!
−
1"wh
"
!rm = 1 +
. regions a
rc
Medium
!rm = 1 +
.
!6"
+ g"s of the rando
dn
（体積平均）
with
effective
parameters!hbecause
!h + g"s
ties that existThe
in semi-infinite
the natural scale.
The ridges
are
layer
below
the
lamel
mc
ii.
lar with unequal
and offset heights,
bent
tops,
rore
ラメラ内での多重反射は無視
tive
permittivity.
We
model
these
two
#
Cuticle
layer
below
the
lamella
also
has
this
effech
−
1"wh
ラメラ The!!semi-infinite
rc
lae, etc. These
also make
it unlike
withirregularities
effective parameters
because
of th
ラメラ中の電場
cm
!
=
1
+
.
!6"
tive
permittivity.
We
model
these
twoexist
regions
as single
layer
rm
are
correlated
multiple
reflections
within
thes
ties
that
in
the
natural
scale.
The
ri
!
!
−
1"wh
Effective
rc
+
g"s
with!!h
effective
of
the random
Medium
=
1
+ parameters
. because
!6"
lar ＝入射波＋第１反射波
with
unequal
and offsetirregulariheights,
bent
therefore
ignore
multiple
reflections
and consi
rm
!hthe
+ g"s
lae,
etc.
These
irregularities
also
make
ties
that
exist
in
natural
scale.
The
ridges
are
often
irregufield
within
the
cuticle
layer
to
be
the
sum
of
18
6. 2010年1月27日水曜日
Model used for estimating the electric field within the nth lamella of
am
two rays in Fig. 6. Notice that there are two r
coordinate systems shown in Fig. 6, the one !x , y
associated with the overall structure in Fig. 5, and
one !u , v , w" that is aligned with the lamella. The la
former rotated clockwise in the x , y plane by the a
The electric field within the lamella is easily dete
the !u , v , w" coordinate system using standard resu
reflection and refraction of a plane wave at a plana
between two materials !the Fresnel equations".32 T
&tion
transmission
coefficients:
and transmission
coefficients for an interface
regions 1 and 2 are
Fresnel’s formulae
Reflection
#!r1cos #1 − #!r2cos #2
R TE =
,
12
#!r1cos #1 + #!r2cos #2
比誘電率
�r1
ξ1
#!r2cos #1 − #!r1cos #2
R TM =
,
12
#!r2cos #1 + #!r1cos #2
T TE =
�r2
12
ξ2
2010年1月27日水曜日
T TM =
12
2#!r1cos #1
#!r1cos #1 + #!r2cos #2
2#!r2cos #1
,
#!r2cos #1 + #!r1cos #2
,
in which #1 is the angle of incidence in region 1, an
angle of refraction in region 2. Then the electric19fi
sr
&
R−sin!ŷ"ecos!
'.(a)
sr
2
ˆ k2%!+x! cos!−"1"wl!h/cos
ˆ
!that
"
+
%
"%
!
s easily
determined
in
cm
!r
"
=
−
'
"
"
E
!
one
!x
,
y
,
z"
is
k
!r
"
=
−
'
!
!
−
1"wl!h/cos
""
E
jk
$d
cos
−
%
"+y
−
%
"%
cm
!
rceasily
srin
2
TE
0
c n
rc
TE
0
ˆ
egrations
are
of
exponential
functions".
This
process
is
latter
is
the
!
−
$x̂
sin!
"
+
%
"
&
e
ic
field
within
the
lamella
is
determined
Plane
of Scaleand !12"" "
To complete the analysis,
we
substitute
Eqs.
!11"
k
!r
"
=
−
'
!
!
−
1"wl!h/cos
E
!
#
rc
TE
0
andard
results
for
the
cos
#
−
!
cos
#
−jkc$!dn+2h"cos
%+x
cos!"+%"+y sin!
"+%analysis,
"% −jk
!12"
g.
5,
and
a
second
To
complete
the
we sub
1
r2
2tedious
−jk
r
TM
!12"
&
R
e
'.
aightforward
but
and
gives
0r
0
into
Eq.
!5"
and
perform
the
indicated
integrations
!all
of
the
system
using
standard
results
for
the
e
!ecoordinate
cos
#
"
.
angle
e
,
(a) −jk0ir E
r2
2
Incident
cm
atThe
a planar
interface
i
−
ŷ
cos!
"
+
%
"%
la.
latter
is
the
TE
TE
#
into
Eq.
!5"
and
perform
the
indica
e
sin
T
E
'C
!', )
&T
TE
TE
,
Plane Wave
sin
T
E
'
C
!
'
,
)
"
&T
cos
#
+
cos
#
integrations
are
of
exponential
functions".
This
process
is
32 !
A
TE
i
A
TE
1
r2
2
termined
in
d
refraction
of
a
plane
wave
at
a
planar
interface
Thethe
reflecuations".
am
mc
TET TEEam mc
4exponential
('!12"
rCA!', )" fu
To
complete
analysis,
we
substitute
Eqs.
!11"
and
!12"
sin
&T
4
(
r
!by
cos
#
−jk
$!d
+2h"cos
%
+x
cos!
"
+
%
"+y
sin!
"
+
%
"%
TE
"
.
the
angle
2
integrations
are
of
32
r2! sr
2
c
n
straightforward
but
tedious
and
gives
E
ˆ
the""integrations
analysis,
we
substitute
!11"
and
!12"
mc
RrcTM−complete
e1"wl!h/cos
'.the am Eqs.
4
(
r
!r!!5"
" =the
−
'k&
!To
Einterface
an
between
The
reflecmaterials
!the
equations".
sults
for
Fig. 6.
nto
Eq.
and
perform
the
indicated
!all
of
Incident
TE
0!Fresnel
cm
E
N
x
asily
determined
in
N
ridg
straightforward
but
tedious
andthegiv
#
Plane
Wave
into
Eq.
!5"
and
perform
the
indicated
integrations
!all
of
the
cos
#
−
!
cos
#
ntegrations
are
of
exponential
functions".
This
process
is
nsmission
coefficients
for
an
interface
between
N
2ar interface
1
r1
2
rsr
Toe−jk
complete
the
analysis,
we substitute
Eqs. e!11"
and$−k!12"
−j!km cos
cos#%"dn
2cos
0−j!k
P
c
ˆ
!12"
!
cos
$
−k
%
"d
dard
results
for
the
&
$C
!'
,
k
!r
"
=
−
'
!
!
−
1"wl!h/cos
"
"
E
m
c
n
!
(
B
!
cos
#
i
rc
integrations
are
of
exponential
functions".
This
process
is
TE
0
&
e
$C
!
'
,
)
"
E
r1
2
traightforward
but
tedious
and
gives
−j!km cos $−k!all
%"d"the
B indicated integrations
d
2
are
#
TET
TEE into Eq.sin
!5"
and
perform
the
'
C
!
'
,
)
"
&T
c2cos of
n$C ! ' ,r ) "
refleccos
#
+
!
cos
#
A
n=1
&
e
TE
,
sr
2tThe
1
r1
2
a planar interface
B
z
ˆ
am straightforward
mc
!
−jk0rgives E
n=1
large d
4
(
r
x
but
tedious
and
k
!r
"
=
−
'
!
!
−
1"wl!h/cos
!
e
rc
integrations
are
of
exponential
functions".
This
process
is
32
TE
0
!ce
cos
#
To
complete
the
analysis,
we
substitute
Eqs.
!11"
and
!12"
i
n=1
y
region
r1 between
2 #reflecThe
ons".
TET !7"
TEE
sin
'
C
!
'
,
)
"
&T
sr
2
TE
ˆ
A
+
#
P
!Eq.
!!7"
#!r!1!5"
!'r2Nkcos
O
TopR
LamellaC C! ' , ) "%,
"−= −and
!!rc#−+
"
"
E
r1cos
2 1"wl!h/cos
straightforward
but
and
gives
am tedious
mc TE 4 (
−jk0r
into
perform
the
indicated
integrations
!all
of
the
TE
r
TE
0
cm
#
C
!
'
,
)
"%,
!13"
R
!7"
C
2 !r1cosbetween
#1
interface
e
,
!s
TEC C! ' , ) "%,"
sr cm
2
i
E
+
R
−j!k
cos
$
−k
cos
%
"d
ˆ
r
!
m
c
n
, #exponential
#
−jk
are
of
This
process
iscm
sin
&T TET TEETE
k
!r
"
'
!
!
−
1"wl!h/cos
"
"
E
かなり煩雑な式になる：
!
&
e
$C
!
'
,
)
"
N
z
0r= −functions".
!integrations
cos
#
+
!
cos
rc
B
TE
0
e
#
r1
1
r2
2
am mc
1 # + #! cos #
i
srsin 'C !'
2 " −j!k cos $−k cos %"d
cos
4(r
ˆ
!
1
r2
2
TE
TE
T
E
,
)
&T
y
n=1but tedious
straightforward
and
gives
,
k
!r
"
=
−
!
!
−
1"wl!h/cos
"
"
E
!
m
c
n
A
TE
& (0 e rc −jk0r
$CB!', )
TE
1" 2
am mc
sr
4
(
r
O
e
Top
Lamella
!
!r2cos #2
N
!r
"
=
k
!
!
−
1"wl!h/cos ""
E
1
i
!
n=1
! rc
TM
0
−jk
r
sr
2
TE
TE
#
0
TE
T ETE "i sr" e sin !13"
'
C1A!'2, )#"!rc
&T
!
C
!
'
,
)
"%,
R
!r22cos
!
cos
#
N
!r
"
=
k
!
!
−
1"wl!h/cos
E
C
!
#E!!srr2#cos
2
1 −+
r1
2
rc
−j!km"cos
$−kc cos
ˆ
TM
0
#1− 'cmk!7"
am
mc
!
4
(
r
!r
"
=
!
!
−
1"wl!h/cos
"
"
!r
"
=
k
!
!
−
1"wl!h/cos
"
E
TE
TE
!
!
sin
T
E
'
C
!
'
,
)
"
&T
&
e
#
,
rc cos
TE
0 −j!k
TM
A0 rc
CCmc
!', )
"%,
!13"−jk r
TE
!
$−kc cos %"dn + R TE
,
in wh
rc
m
am
#
&
e
$C
!
'
,
)
"
(b)
4
(
r
#
cm
!
0
(
B
!
cos
#
+
!
cos
#
rc
n=1i e
#r21 #1 +1#!r1cos
= rr̂ lo
r1 #2 2
−jk0r
N
2cos
−jk
r
n=1
e
of "the
C A! ' , )
&T TMT TM−jk
ETM
0
, &T
i
N
e
r
1 TE
am
mc 0
i 'CA!−j!k
sin
T TE&T
ETETMT TM&
', )mC"cos!'$,!7"
−kc cos %"dn !
lamell
(
r
e
sr #
2
(C 4
TE
!he
cos
e
$C
!
'
,
)
"
!
'
,
)
"%,
+
R
E
)
"
!
i
mcin
r1
−j!k
cos&T
%"dnTM
C !', )"
B T!TM
A m cos $−kc!13"
region
is2ethe
!r!"2of
= incidence
kam
!C!!rc
−)1"wl!h/cos
""#1(
Eangle
TM&
4!(srrmc1, and
TE
C
'
,
"%,
+
R
2
TM
0
C
E
cm
#
am
$C
'
,
)
"
A
4
(
r
TM
B mc N
2
!
cos
#
#
cm
!r
"
=
k
!
!
−
1"wl!h/cos
"
"
E
!
r1
1
n=1
am
!s
!
rc
TM
0
E
4
(
r
tion
in region rc
2. NThen ,the electric field
within
he
#
n=1
!
f incidence#in region 1, and
#
is
the
rc
−j!km cos $−kc cos %"dn ˆ
2
(b)
N
&
e
!7" !r2!7"
iscos #1 +
III. ANALYSIS
OF THE SCATTERED LIGHT &$ ' c
!ahin
cos
#
(
−j!k
cose$−jk
−kc0rcos
%
"d
N
r1
2
m
n
TE
C
!cos
',C)
)!"d'""%,
!13"
+ Rcos+$$C
−jk0r
e
ion 2. Then &
the
electric
within
C
TE
i field−j!k
B
−k
%
C
,
)
"%,
!13"
R
1
Fig.
5.
!a"
Model
for
the
ridge
showing
the
dimensions:
Left
side, transverse
ˆ
e
n=1
1
The
incident
light
produces
a
polarization
!dipole moment
m
c
n
cm, ) "cm
TM
TM
sr
2
i
C
&T
T
E
!
'
sr
2
cos
&
e
&$
'
'
sin!
"
−
%
−j!k
cos
$
−k
cos
%
"d
ˆ
!
A
!
cross
section;
right
side,
longitudinal
cross
section.
Note
that
the
lamellae
TM
m = P! for the
c kbound
n−
TMT TME
i ETM!r
C
!
'
,
)
"
−jkmdn=1
cos
$rc −jk1"wl!h/cos
!u+dn"cos % "&T
charge
of1"wl!h/c
theof
lamellae'
thats
per
unit
volume"
"
=
k
!
!
"
!r
"
!
!
E
!
!
cos
&
e
&$
'
am
mc
A
n
c
TM
rc
0
TM
0
TE
TE
(
with
respect
to
the
plane
of
the
scale.
!b"
the
ridge
are
tilted
at
the
angle
ETET T e #!
$e n=14(r
am
mc
can be expressed#
as an equivalent volume current density,
4(
r
am
mc
Orientation of the
ridge
with
respect
to
the
system !x , y , z". The
!coordinate
ˆ
#
rc
n=1
rc− 1"E!"
region
and
is1 TE
theC !', )"%,
−
)
"
−
)
cos!
−
%
−
)
"%C
!
'
2 1,!r2
cos#+2 #R
!
!
B
scattered
field
is
to
be
computed
at
point
P.
=
j
!
P
=
j
!
"
!
"
,
!3"
J
!13"
N!u+d %C
dn−jk
cos
$ nwithin
jk
% cv −jk
N
!u+d
+2h"cos
sin 0%r 1
m
c cm n"cos,−jk
E+
c
ee−jk
electric
field
1
$e
−jk0!subr
TE
R
e
%e
ŵ,
!8"
ˆ
in which " = #n = 2.43− j0.19 is the complex relative
e
sr
2
sr
2
#
−
)
"
−
)
cos!
"
−
%
−
)
"%C
!
'
,
)
"
−j!k
cos
$
−k
cos
%
"d
e
i
!
!
ˆ
−j!k
cos
$
−k
cos
%
"d
with
ˆ $r"cos!
ˆpermittivity
ˆ−"script
B" −
m
c=!r
n,&
!c r2cos
!r1TM
cos
#2E
cm # 1 + &
ˆ
!%isi −
msin!
c % −"
n&$
!r
"
k
!
!
−
1"wl!h/cos
"
"
=
k
!
!
−
1"wl!h/cos
"
E
!
!
the
total
electric
ofsin!
cuticle,
and
E
cos
eT TM
&$
'
'
)
"
)
"
−
%
−
)
"%C
!
'
,
)
"
C
&T
ETM
!
'
)
"
cos
'
"
+
)
"
−
)
−
cos
e
'
'
sin!
"
−
%
rc
rc
TM
0
0
(
B
A
III.
ANALYSIS
OF
THE
SCATTERED
LIGHT
TMThis
C
T TM
ETM
= −&
current
produces thex2scatfield within the &T
lamellae.
am
mc TM 4 ( r #! #!
!8"
u+dn+2h"cos % −jk
sin
%
v
rc
amthat ismc
n=1
rc n=1
tered electromagnetic field
the light we observe.
c
Toap
%e
ŵ,
!8"
The
incident
light
produces
a
polarization
!dipole
moment4 ( rAt
ˆ
ˆ
1
!u+d
"cos
%
TMC C! ' , ) "', ˆ
cos
' sin!1,"and
+ % #−2)is" the
−−jk
) per
cos!
"
+
%
−
)
"%R
field w
sc the1n! angle
of
incidence
region
ˆ
sr
2 − $'in
!
N
−jk
r
for
the
bound
charge
of
the
lamellae
that
unit
volume"
P
cos
'
'
sin!
"
+
%
−
)
"
−
)
cos!
"an
−
$
r 0
i !" =
cm
−jk
d−n cos
$
!r
k
!
!
1"wl!h/cos
"
"
E
use
e
0
m
ˆ
rc
N
TM
0
TM
TM
ei" −can
−
EinTMregion
T− )
T
ˆe−j!k
expressed
as
volume current density,
−
)
"'
−, )
)"within
cos!
% −be )
"%C
',"an)equivalent
"
#
B,!)
action
2.
Then
electric
field
cos"$the
−kc%cos
%)
"d"%C
"
−
)
cos!
−
−
!
i
shown
ˆ
TM
TM
m
n
!
C
&T
T
E
!
'
B
am
mc
cos
&
&$
'
'
sin!
"
−
%
rc e
A
TM
TM
TMC
TM
C
&T
T
E
!
'
,
)
"
−
)
"%R
!
'
,
)
"',
!14"
kcv #
sin!%
(
$−kccontai
cos
! ="%R
! ,! ' , ) "', −j!km cos!3"
C am am
mc
rcŵ,
TM with
=−
j!)
P
j!"0!"TM
1"E
J!rbA
!8"
4
(
rc −C
cm
mc
&
e
C
the
coefficients
lla
is
4"(+r % − )" −# )ˆ cos!
n=1
ness h
ˆˆ cos ' sin!
cm " + %
−jk0r
ion
1,
and
#
is
the
−jk
d
cos
$
2010年1月27日水曜日
20
'
−
$
2
m
n
ˆ
TMthe
e %
jk !u+d "cos
andTM# is
i
(
結果の表式
TE
i
TM
i
TE
(
i
TM
(
(
(
(
(
Fig. 5. !a" Model for the ridge showing the dimensions: Left side, transverse
cross section; right side, longitudinal cross section. Note that the lamellae of
the ridge are tilted at the angle with respect to the plane of the scale. !b"
Orientation of the ridge with respect to the coordinate system !x , y , z". The
scattered field is to be computed at point P.
(
0
b
(
rc
rc
29
1013
Am. J. Phys., Vol. 77, No. 11, November 2009
(
(
&
e TM
&$' cos ' sin!" − ˆ%
ˆ
!
ˆ
T
=
,)
−
)
"
−
cos!
"
−
%
−
)
"%C
!
'
,
)
"
E
TM
n=1
ˆ cos!" − % −− )
cos
'
sin!
"
+
%
−
)
"
−
)
cos!
"
+
%
$'"%C
−
)
"%R
C
!
'
,
)
"',
!1
N
B
−
)
"
−
)
!
'
,
)
"
C
1
n=1
12
#!r2−jkcos
#
cm
−jkm
dn −jk
cos $v sin %B 1
!u+d%n"cosE%i−jkT
− )"%R TMCC!', )
!u+d
+2h"cos
%
#
+
!
cos
#
#
TM
TM
c
n
c
u+d
e
"e jk=nc"cos
−
T
1
r1
2
−j!km cos $%e
−kc cos %"dn ŵ,
2
!
cos
#
ˆ
+#R TEeTM
!8"
i
cm
r2 $
1
!
mdn cos
cos
&
e
&$
'
'
sin!
"
−
%
am
mc
TM
TM
(
TM
e
!r
"
=
−
E
T
T
E
!
ˆ
ˆ
−
)
"%R
C
!
'
,
)
"',
!14"
TM
ˆ cos!" + %
!rccm − $n=1
TM
)+cos!
"12%−am=)"#mc
cos!
−"%C
% coefficients
−B#
)'"%C
!'$,'), "cos ' sin!" + % − )" − )
ˆT
B−
'ˆ cos ' sin!" +TM,n
% − )" −−cm
)ˆ#)cos!
"rcC−−"
)
−!with
%cos
−")
!
,
)
"
!
the
#1 + !ofr1cos
#2
#
is
incidence
in region
and #2 is the
in
which
r2the angle
with the1,coefficients
sin c%v sin %ŵ,
1
jk
!u+d
"cos
%
ev −jk
!8"
c
n
ŵ,
!8"
&$!sin1 %−û )+"%R
cosTM
"e
v̂the
jk"
% "−)
ˆ+cos
−ˆ2.+)
"%R"TM
, )"', field within
C
)
!14"
c!u+d
n"cos
ˆ
ˆ
ˆcm%cos!
C! '
C!'electric
'
'
sin!
+
%
−
)
cos!
+C
%
−
$
&$!sin
%
û
cos
%
"e
v̂
coefficients
angle
of
refraction
in
region
Then
the
− i)" −with
)
",−jk
− "',
%
−
)
"%C
!
'
,
)
"
cos
'
'
sin!
"
+
%
−
)
"
−
)
cos!
"
%
−
$
cm
d
cos
$
B
in region
1,Cand
issinc$!k
in which #1 is theCangle
" =incidence
sinc$!k0w/2"cos
'%sinc&$k
sin
%the w/2"co
A!', )of
c#2
e mn
!r!" = −
ETMT TMT TMTM
!
'
,
)
"
=
A
0
+ !sin
%û − ˆcosam
%v̂"R
mc
the
nth
lamella
is
#
TM
angle
of
refraction
in
region
2.
Then
the
electric
field
within
TM
!
+
!sin
%
û
−
cos
%
"R
v̂
−
)
"%R
C
!
'
,
)
"',
!14"
TM
cm + % − ) " −−)
"%R " +C%
, )"',
!14"
rc − $' cos 'Csin!
C cm
C! '
with
the coefficients
cos!
, )" = sinc$!kˆ0)
w/2"cos
sin %the coefficients
+ k0 sin ' sin!
" − )"%l/!2 cos "+"'k0 sin ' sin!"
cm
with
kos
cm ' %sinc&$k
A!'"
c
md$n cos $
the
nth
lamella
is
−jkc!u+dn+2h"cos % −jkcv jk
sin!u+d
%
% −jkc!u+d
−jkmdn cos $ jkc!u+dn"cos %
+2h"cos % −jkicv sin %
c , n"cos &e
&e
%e
! TE,n
n!9"
&$!sin
%
û
+
cos
%
"e
TE,T TEe
v̂
%e
!9"$e
!r
"
=
E
T
E
!
TMC C! ' , ) "', + k sin ' sin!
−
)
"%R
!14"
TE
"
−
)
"%l/!2
cos
"
"'
&
exp!j&$k
%"cos
+ k%0 sin ' sin!
"exp!j&$kc sin
0
&
%sinc&$k
sin % !
CA!', )" = sinc$!kcm0w/2"cos
am mc−jk d cos $ cjk sin
with
thec coefficients
with 'the
coefficients
i
!u+d
'%sinc&$kc sin %
CTA!TE'e, )"m=nsinc$!k
n
TE
0cw/2"cos
!r
"
=
E
T
$e
E
!
!9"
TE,n
!u+d
"cos
%
TE
dknc"cos
%
#
n
#
#
TM
km!sin
= !%rmûk−
and%the
beam −jk
mc !u+d
"Rfollowing
v̂with
% −jk
k−c =)"%l/!2
!rcexp!j&$k
k0relations
,cos
km ="c"'!sin
k%0, +and
the
following
rck0, +
0, cos
&
k
sin
'
sin!
"
rm
−)
"%l/!2becos
""'",
!15"
c relations
n+2h"cos
cv sin %ŵ,
− )"%l/!2
cos
TE
+
k
sin
'
sin!
"
0
+
R
e
%e
!8"
cm
各係数の表式：
0
with
the
coefficients
+ k%
sin
sin!" − )"%l/!2 !8"
cos ""'
ngles:
%
−jk'
)" = sinc$!k
%sinc&$k
!'angles:
,C
)A"!=',sinc$!k
'+%sinc&$k
sin n%+2h"cos
tweenCthe
cm'−jk
0 %e
cv sin %ŵ,
0w/2"cos
c sin
Asin
0w/2"cos
cc!u+d
TE
R
e
relations
be−jk
!u+d
+2h"cos
%
−jk
%
v
M
"%l/!2
cos" ""'",!9"
!15a"
cm
&e c & nexp!j&$kc%esin c% + −k0)
,sin
' sin!
m
#!10"
&"exp!j&$k
sin
%B!+',kcos!
sin
'+sin!
"cos!" −
#!rcCsin
" =#!sinc$!k
%sinc&$k
sin
k%0%=sin
sin!
)C
"%l/!2
"sinc&$k
"'cos
+
sin$"'. −sin!
"B!−',))cos
"%l/!2
"'
= #!rcsin
!k'rm0 sin
!10"
sin'"
c
0
sin
$
.
A!',%)=
0w/2"cos
c+
)
"
=
sinc&$k
C
C
!
!
,
"
"
=
sinc#$k
#
$
"
"
=
cos!
"
−
%
"
rm
c
Cc
c
1
"%l/!2
cos
"
"'",
!15a"
!9"
#c%!v,rcsink0%,, km = #−!+rm)k k!9"
i
−jk
d
cos
$
, and
the" −following
relations
be!&
m cos
n " "'",
1 the
TM
TM
0sin
'
sin!
)
"%l/!2
cos
"
"'
e
!r
"
=
−
E
T
T
E
!
−
)
"%l/!2
&
exp!j&$k
sin
%
+
k
sin
'
sin!
"
i air-effective
−jk
d
cos
$
exp!j&$k
sin
%
+
k
sin
'
sin!
"
0
sin ' cos
cos )
+"
kexp!
!10"
TM,n
The
subscripts
am,
mc,
and
cm
refer
to
meTM
c
0
!
m
n
= sinc&$k
cos!
"
−
%
"
CB!',to)"the
c
0
sin
'
cos
)
%h/!2
cos!"cos
"' &
+
k
0%h/!2
pts
am,
mc,
and
cm
refer
air-effective
meTMam
TMmc
−
k
sin
e
!r
"
=
−
E
T
T
E
c
0
!
#
0
TM,n
TM
angles:
CC!medium-cuticle,
!, "" = sinc#$k
cos!
#!!cuticle-effective
+rc$" c amcos!mc# +meC
!
!
,
"
"
=
sinc#$k
$"
#
c
dium,
effective
and
C
owing
relations
bective
medium-cuticle,
and
cuticle-effective
mefollowing
relations
be&
exp!j&$k
sin
%
+
k
sin
'
sin!
"
rc
"%l/!2
cos "cos
"'",
!15a"
" −t
− j&$k
c " − %" + 0k sin−')cos
− )%h/!2
"%l/!2
cos
""'"'",
with
c cos!
" = sinc&$kc cos!
CB!', )me"Eq.
&
exp!
"jk −!u+d
%%exp$−
" !15a"
+"cos
k0 %sinj2k
' cos
)+%$d
− j&$kc cos!
effective
0 in Fig. 6.)For
n
dium
interfaces
use
in
!5"
we
express
Eqs.
!d
h"cos
$
c
n
#
#
aces
in
Fig.
6.
For
use
in
Eq.
!5"
we
express
Eqs.
c
n
−
k
sin
!
cos
"
%h/!2
cos
#
"&
370
nm
jk
!u+d
"cos
%
=
!
sin
%
=
!
sin
$
.
!10"
−
k
sin
!
cos
"
%h/!2
cos
#
"&
C
!
!
,
"
"
=
sinc#$k
cos!
#
+
$
"
0
&$!sin
%
û
+
cos
%
"e
v̂
n " − %"
" = %sinc&$k
C, %
rc me- − )rm
C
c
&$!sin
+, )cos
v̂"e c c cos!
ffective
B0!,û'
"%l/!2
cos
")"'",
!15a"
+ h/2%/cos
"!15
'",
!8"
and
!9"
in
terms
of
the
coordinates
!x
y
z",
sin
'
cos
%h/!2
cos
"
"'
&
exp!
+
k
400
nm
j&$kc cos!" − %" + k0 sin ' cos )%$d+
in terms of the coordinates
!x , y−, z",
0
n h/2%/cos "'", %exp!j#$k cos!
C
#
+
$
"
−
k
express
Eqs.
370
nm $%
%exp$−
j2k
!d
+
h"cos
$
%
!10"
c
C
!
!
,
"
"
=
sinc#$k
cos!
#
+
$
"
−
k
sin
!
cos
"
%h/!2
cos
#
"&
c
n
0
!
'
,
)
"
=
sinc&$k
cos!
"
−
%
"
C
%exp$−
j2k
!d
+
h"cos
!10"
C
c
430
nm
!', )" = sinc&$k
cos!
"
−
%
"
C0Bair-effective
sin
'
cos
)
%h/!2
cos
"
"'
&
exp!
+
k
TM
B the
c mecripts am, mc,− and
cm
refer
to
TM
c
n
+
!sin
%
û
−
cos
%
"R
v̂
+ !sin %û − cos %v̂"R0 400 nm
c
" − %" ++kAm.
sin
'
cos
)
%$d
j&$kc cos!1014
o
"
'",
!15b"cm cm
0h/2%/cos
n
J.
Phys.,
Vol.
77,
No.
11,
November
2009
460
nm
2
"
=
20
!
'
,
)
"
=
sinc&$k
cos!
"
−
%
"
C
ective
medium-cuticle,
and
cuticle-effective
%exp!j#$k
# "+"'+2h"cos
$cos
" −exp!
k0"'
sin
!+
cos
"−%$d
%k− sin
h/2%/cos
#&",)%$d10
BPhys.,
kmesincme! cos
"%h/!2
cos #+"&j2k
0"++
Am.
J.the
Vol.−77,
11,
November
2009
Glenn
S."Smith
%exp$−
+meh"cos
$c% cos!
cos
)
%h/!2
cos
&
k0 sin
430
nm
he
0 No.
n
%exp!j#$k
cos!
#
$
"
!
cos
%
c!d
nk'0
sin
'
cos
)
%h/!2
"
&
exp!
+
−jk
!u+d
%
−jk
sin
%
v
cos!
"
%
k
sin
'
cos
−
j&$k
to air-effective
air-effective
c
0
−jk
!u+d
+2h"cos
%
−jk
sin
%
v
cc
n n
cc c ,
0 490
&e
%e %e460
!9"nm !9" n
+6.h/2%/cos
"'",
!15b"
&e
,
rfaces
in
Fig.
For
use
in
Eq.
!5"
we
express
Eqs.
nm
'j2k
cos
cos$"
&cos!
exp!
+
kmed cuticle-effective
520 nm
0 sinme%exp$−
!d)n %h/!2
+ h"cos
%−"'j&$k
+" %−h/2%/cos
#cos
&",
!15c"
%$d
9uticle-effective
Glenn
S.
Smith
1014
%exp!j#$k
#
+
$
k
sin
!
"
%
c
n
cos!
"
−
"
+
k
sin
'
cos
)
%$d
with
+
h/2%/cos
#
&",
!15c"
%$d
c
0
c
0
n
cos!
"
−
%
"
+
k
sin
'
cos
)
%$d
−
j&$k
490
nm
+
h/2%/cos
"
'",
9"
inwe
terms
of the
coordinates !x , y , z",
n0
c
n
!5"
express
Eqs.
#!!rc1014
##!!rmk0k, and
45
Eq. !5" we express
Eqs.
with
k
=
k
,
k
=
the
following
relations
be#
c
0
m
cos!
"
−
%
"
+
k
sin
'
cos
)
%$d
−
j&$k
520
nm
Glenn
S.
Smith
with
k
=
k
,
k
=
,
and
the
following
relations
bec
n"%'",
%exp!j#$k
$0" −n k+0+h/2%/cos
sin
! cos
with
c #"
rc 0
rm 0 !15c" !15b"
c cos!# +%$d
h/2%/cos
xes, y !x
, z",
sin!x"
+with
h/2%/cos
"'",m
!15b"
, y , z",
tween
the&",
angles:
!, D
tween
the
angles:
.
sinc!x"
=
+
h/2%/cos
"
'",
!15b"
Am. J. Phys., Vol.
77,
No.
11,
November
2009
Glenn
S.
Smith
ただし、
%$dwith
sin!x" !15c" sin!x"
n + h/2%/cos #&",
x !16"
#!rcsin
#
sinc!x" = sin "sinc!x"
=. #
%
=
!
sin
$
.
!10" !16" C
ber
2009
Glenn
S.
Smith
1014
rm
.
=
#
ovember
2009
Glenn
S.
Smith
1014
!rcsin %x = !rmsin $ .
!10"
sin x" = 1014
with
sin!x"Glenn S. Smith
our
calculations,
we will
be interest
. The subscripts am, mc, andFor
!16"
sinc!x" =
cm
refer
to
the
air-effective
mesin!x"
x ourThe
For
calculations,
weam,
will mc,
be interested
in
of the
scattered
field,
readily
subscripts
and
cm
toirradiance
the
air-effective
Foreffective
our !16"
calculations,
we
will refer
bethe
interested
in which
the irradiance
dium,
medium-cuticle,
and
cuticle-effective
me-is me.
sinc!x" =
0.01
of
the
scattered
field,
which
is
readily
obtained
from
Eqs.
N,
h,
w,
g,
γ,
s
x
パラメーター
!13"
and
dium,
effective
and
cuticle-effective
me- 1.0
theinterested
scattered
which
readily
obtained
Eqs.
dium
inmedium-cuticle,
Fig.
6. For
use
inis !14",
Eq.
!5"
we
expressfrom
Eqs.
For our calculations, we
willofinterfaces
be
infield,
the
irradiance
0.01
0.1
!13" and !14",
sr
!8"
and
!9"
terms
of the
coordinates
!x , ysr!5"
, z", we express Eqs.
!13"
andin!14",
of the
scattered
field, dium
which
isirradiance
readily
obtained
sr Eq.
interfaces
in
Fig.
6.from
ForIEqs.
use
in
sr
(a)
or our calculations,
we will
be interested
in
the
I
+
I
=
I
sr
sr
TE
TM
!13" which
and !14",
Isr obtained
+ ITM
=!8"
ITEand
が決まれば、散乱光は
λ,
θ,
Φ
srterms
sr の関数として与えられる。
f the scattered
field,
is readily
from
Eqs.
sr
!9"
in
of
the
coordinates
!x
, y(a)
, z",
I = ITE + ITM
sr
sr
sr
sr
13" and !14",
sr
!2009
!
1014
sr Am. J. Phys.,
sr Vol. 77, No. 11,=November
r̂
·
$Re!S
"
+
Re!S
"%
I
+
I
=
I
!
!
c,TE
c,TM
TE
TM
= r̂ · $Re!S " + Re!S sr "%
2010年1月27日水曜日
21
sr
TE
am
mc
結果の表式（続き）
c,TE
c,TM
CC!!, "" = sinc#$kc cos!# + $"
the lamellae are −
tilted
at the
angle"#%h/!2
= 10° cos
to the
of the
k0 sin
! cos
#"&base
%exp$−
j2k
!d
+
c
n
scale,
this!orientation
of the
beam
− k0sosin
cos "%h/!2
cos
#"&is what we would expect for a specular
reflection
lamellae.
patterns c cos!#
%exp$−
j2kfrom
+ h"cos
$%The
%exp!j#$k
c!dnthe
具体的計算
%exp$− j2kc!dn + h"cos $%
# + $half
" −space
k0 sin
"%
+ h/2%/cos
#
%$d!ncos
c cos!
Fig. 9. The total time-averaged power%exp!j#$k
scattered into
the upper
I. Representative
dimensions
%exp!j#$k
# + $for
" −a Morpho
k0 singround
! cosscale.
"%
versus wavelength. Table
c cos!
with
#&",
!15
%$dn + h/2%/cos
Quantity
Description
Value
!15c"
%$dn + h/2%/cos #&",
sin!x"
with
N
Number of lamellaesinc!x" =
8.
moves away from
the Thickness
absenceofoflamellae
scattered light x 65 nm
h ! = 90°. Note
sin!x"
!16
w sinc!x"
of lamellae show that 400 nm
at angles near grazing,
! = 0=° , 180°.. Width
Observations
For our calculations, we will be
x
g brown at such angles;
Spacing between
lamellae
155 nm
the scale appears
the of
color
is
due
to
Fig.
the
scattered
field,
which
is
#
Tilt
angle
of
lamellae
10°
angle
pigmentation not
structural
scattering.we will!13"
For
our
calculations,
be
interested
in
the
irradianc
and !14", 700 nm
inclu
s
Spacing
between
ridges
Figure 8 is of
a different
plot
for
showing
the
iridescence:
the scattered field, which is readily
from chan
Eq
sr obtained
sr
sr
I = ITEof+"ITM
The scattered irradiance
is
plotted
in
relief
as
a
function
!13" and !14",
with
モデル
sin!x"
パラメーター
.
sinc!x" =
!16"
x
For our calculations, we will be interested in the irradiance
andthe
! for
the plane
# = 20°.
The
spectrum
for
the scattered
of
scattered
which
is
readily
obtained
from
sr Eqs.
sr
1015 field,
Am.
J.
Phys.,
Vol.
77,
No.
11,
November
2009
!
!
sr
sr
sr
=
r̂
·
$Re!S
"
+
Re!S
c,TM
light clearly
shifts to
!from blue to vio- c,TE
I shorter
ITM
= field:
ITE + wavelengths
!13"
and
Irradiance
of !14",
scattered
let" as the angle of observation approaches grazing. 1
sr
sr
sr
sr
sr 2
sr 2
sr
!
!
!
!
=
r̂
·
$Re!S
"
+
Re!S
"%
=
!'E
'
+
'E
+ Ithe
= ITE of
A Imeasure
total
time-averaged
power
scattered
at
a
c,TE
c,TM
TE
TM' ",
TM
2&0
given wavelength can be obtained by numerically integrating
sr
sr
1
!
!
sr
2
=
r̂
·
$Re!S
"
+
Re!S
"%! sr hemisphere,
!17" for the irradiance
i
i
! TEthe
TotalEq.
power:
c,TE
c,TM
!
!
'
",
!17
= over
!'E
'2 +upper
'E
with
'E
'
=
'E
TM
TE
TM'.
%& & '&
$/2
2$2 & 0
$
sr
2
sr
sr
2
2
#Psr=
$ = 1 !'E
+
I
!r,
!
,
#
"r
sin !d!d# .
! TE'! i+ 'E! TM
i
'
",
!17"
!
IV.
COLOR
AND
IRIDESCEN
with
'E
'
=
'E
'.
#=0
#TE
=3$/2 TM
!=0
2&0
2010年1月27日水曜日
22
me species. For example, the values for the
lamellae h of the Morpho rhetenor range
m.20,23 For our calculations, we will use the
imensions for a Morpho ground scale given
Fig. 7. Patterns for the scattered irradiance from a ridge. Each curve is for a
different wavelength. !a" Plane ! = ) / 2. !b" Plane " = 2# = 20°. Note that the
scale for Isr is logarithmic.
角度と波長依存性
on of the scattered
radiation in space is conin Fig. 7!b" are “cuts” through this beam, " = 2# = 20°. The
370 nm
yed using graphs
akin to the polar patterns
400nm
nm
o
scattering is most intense at the wavelengths normally asso370
2
"
=
20
0
d to described400
the
430
nm radiation from antennas.
nm
o
◦ with blue and violet light !( = 400, 430, 460, and 490
ciated
2
"
=
20
460
nm
に固定
Fig. 7, the radial
distance0 from the originθ at= 90
430 nm
nm"
and
small outside of this range !( = 370 and 520 nm".
490 nm
sr
460
nm
that
le is proportional
to the irradiance I at 45
The plots in Fig. 7!b" clearly show iridescence; the domi520 nm
490
nm
at the radial scale is logarithmic and that45
two!, Deg.nant wavelength !color" shifts with the angle of observation
played. There 520
arenmsix curves, each for a dif!. These results are consistent with observations; the color
!, Deg.
gth within the range of 370 nm' (
changes from blue to blue-violet as the angle of observation
◦
curves are normalized so that the maximum
(θ > 90 )
up is one.33 The inset in the left-hand side of
90
ws the relative orientation
for
the
incident
0.01
0.1
sr 1.0
mellae of the ridge.
90
I
0.01
0.1 of srthe1.0
! = ) / 2, shown in Fig. 7!a",
(a) the beam
tilted at about 20° to◦ the normal. RecallI that
◦
(a)
ϕ =at 2γ
= 20# = 10°
に固定
tilted
the angle
to the base of the
ϕ
=
20
1.0
ientation of the beam is what
we would ex90
#, Deg.
lar reflection from the lamellae. The patterns
青・紫が強く発色
青から青紫へ
45
90
#, Deg.
135
45 for a Morpho ground scale.
ive dimensions
Description
Number of lamellae
0
Thickness of lamellae 0.01
0 Width of lamellae
2010年1月27日水曜日
0.01
Spacing between lamellae
(b)
I
sr
135
Value
0.0
V
90
400
B
120
G
500
8
Y
O
150
180
R
$, nm
600
#, Deg.
0.165 nmsr 1.0
180
400 nm
I
180
23
0.1
155 nm 1.0 Fig. 8. The scattered irradiance as a function of the wavelength and the
lae, are included in the model described in Sec. II. The sca
tering of light !a plane wave" by this structure is determine
in Sec. III. The scattered field is written as an integral !ra
diation integral" over the total field within the lamellae. The
an approximation for the field in the lamellae is obtaine
using the theory for reflection and refraction of a plane wav
by a planar layered medium, with appropriate simplificatio
based on the randomness in the structure. The approximat
field is inserted into the radiation integral, and the integral
実験との比較
Scatter
(F )
Incident
total power
(normalized)
"
!
Lamella
(a)
Incident
9. The total time-averaged power scattered into the upper half space
Fig. 4. Measured reflectance versus wavelength for a scale or a portion of
us wavelength.
wing of a Morpho rhetenor butterfly. Results are for normal incidence e
cept those of Plattner which are for 10° from normal.
実験データと同じように青色にピーク...でもなぜ？
"
2010年1月27日水曜日
ves
away from ! = 90°. Note the absence of scattered1012
light
Am. J. Phys., Vol. 77, No. 11, November 2009
24
cos !
i
!ETE,n!r!" = ETE
T TET TEe−jk
$
am
dn cos "
cos !
なぜ青くなる？
n
+ R TEe−jkc!u+dn+
cm
E! TM,n!r!" = −
モデルをさらに単純化：
(c)TE
Rcm = 0, Im[�rm ] = 0
mc
1
#!rc
i
ETM
T TM
am
&$!sin %û + co
g. 10. Schematic drawings showing the elements involved in the scatter◦ tilted
g from the ridge !simplified model". !a" Scattering from
a
single,
θ = 90
mella. !b" Scattering from an array of N equally spaced points. !c" Detail
r the scattering from the nth point.
TE成分のみに着目、
+ !sin %û − cos
の場合を考える
−jkc!u+dn+2h"
&e
with kc = #!rck0, km = #!rmk0,
tween the angles:
sr
k0 h < 1
ITE
& k40 sinc2()kc cos!' − (" + k0 cos #*h/!2 cos '"+
１
sin
% = #!rmsi
sin " = #!大体
rc
) sinc2(k0l)sin ' + sin!' − #"*/!2 cos '"+
単一ラメラ面による散乱
The
subscripts
am,
mc,
and
c
2
N
1
dium, 多層ラメラからの
effective medium-cut
−j)km cos!'−*"+k0 cos #*dn/cos '
e
.
!19"
)
dium interfaces in Fig. 6. For
N n=1
散乱光の干渉
!8" and !9" in terms of the co
here are four factors in Eq. !19" that depend on the
waveこれら２つの寄与が重要
4
ngth: k , the arguments of the two sinc functions, and the
レイリー散乱の特性
,
2010年1月27日水曜日
,
25
constructive or destructive interference that occurs for the
scattering
from
lamellae
is included
in term.
this term.
scattering
from
thethe
lamellae
is included
in this
antenna
analysis
describes
the scattering
InInantenna
analysis
the the
termterm
that that
describes
the scattering
fromthethe
single
lamella
is called
the “element
factor”
from
single
lamella
is called
the “element
factor”
F, andF, and
theterm
term
that
describes
scattering
the array
of points
the
that
describes
the the
scattering
fromfrom
the array
of points
37
isiscalled
thethe
“array
factor”
A.37A.
After
introducing
this notacalled
“array
factor”
After
introducing
this nota38
tion
evaluating
the the
geometric
series,
Eq. !19"
tionand
and
evaluating
geometric
series,
Eq. becomes
!19" becomes38
なぜ青くなる？
sr sr
4 4 2 2
ITE
&
k
#k0#k
l$sin
# + sin!# − %"%/!2 cos #"&
I &0ksinc
sinc
0l$sin # + sin!# − %"%/!2 cos #"&
TE
''
0
o
0
45
!, Deg.
2
|F|
2
|A|
λ =#425nm
= 425 nm
( (
2
sin#N$k
cos!
#
−
$
"
+
k
cos
%
%!h+g"/!2
cos
#
"&
m
0+ k cos % %!h+g"/!2 cos # "&
sin#N$k
cos!
#
−
$
"
m
0
''
N sin#$km cos!# − $" + k0 cos %%!h+g"/!2 cos #"&
2
N sin#$km cos!# − $" + k0 cos %%!h+g"/!2 cos #"&
2
2
= k40)F!
%
")
)A!
%
")
. 2
!20"
4
2
= k0)F!%") )A!%") .
!20"
In Fig. 11 the element and array factors from Eq. !20" are
In Fig.as11polar
the element
array460,
factors
fromnm.
Eq.The
!20" are
graphed
plots for and
! = 425,
and 495
=#460nm
= 460 nm
2 460, 2and 495 nm.λ The
graphed
as
polar
plots
for
!
=
425,
regions in which the two patterns, )F) and )A) , overlap are
regions
inそれぞれの寄与を波長毎
which
thethe
twooverlap
patterns,
)F)2 and
)A)2, overlap
shaded
gray.
Note that
increases
on going
from are
! shaded
= 425 nm
$Fig.Note
11!a"%
to the
! = 460
nm $Fig.
11!b"%on
andgoing
then from
gray.
that
overlap
increases
にわけてプロット
decreases
for
!
=
49511!a"%
nm $Fig.
sequence
shows
! = 425 nm $Fig.
to !11!c"%.
= 460 This
nm $Fig.
11!b"%
and then
◦
that
the
product
of
these
functions
is11!c"%.
significant
over ashows
θ =nm
90 $Fig.
decreases for !（ = 495
This only
sequence
の時）
band
shorter of
wavelengths
in the visible
spectrum.
thatofthetheproduct
these functions
is significant
only over a
To examine
this point
further, we in
consider
the angles
at
band
of the shorter
wavelengths
the visible
spectrum.
which the maxima %m and first zeros %0 of these functions λ = 495nm
= 495 nm
To examine this point further, we consider the angles # at
occur,
λ~460nm で重なりが最大
which the maxima %m and first zeros %0 of these functions
−1
%
=
2
#
,
%
=
#
+
sin
$sin # ( !!/l"cos #%
!21"
occur,
m,F
0,F
2010年1月27日水曜日
2" = 20
0.0
0.5
I
sr
1.0
90
(a)
o
0
2" = 20
45
!, Deg.
0.0
0.5
I
sr
1.0
90
(b)
0
2" = 20o
45
!, Deg.
0.0
0.5
I
sr
1.0
90
(c)
26
%m,A = cos−1#$!/!h + g"%cos # − *)rmcos!$ − #"&,
!22a"
= cos−1#$!/!h + g"%!1 * 1/N"cos # − *)rmcos!$ − #"&.
!22b"
ピーク波長
se angles are displayed on the ! , % plane in Fig. 12.39
black dot shows the point at which the maxima of the
functions coincide !! + 460 nm, % + 20°", and the
◦ gray
λ ≈ 460nm, φ ≈ 20 で強度が最大
具体的な条件式：
90
!, Deg. ($ = %/2)
60
30
0
-30
330
2
|F|0
zero
zero
max
1.0
λpeak
2
0.5
�
�
�
��
�
√
γ
−1
× cos 2γ + �rm cos sin
−γ
√
�rm
|F|m
0.1
2
|F|0
2
zero
-60
300
-90
270
300
|A|m
2
|A|0
max
(h + g)
=
cos γ
φpeak = 2γ
2
zero
400
|A|0
#, nm
500
600
!!rc − 1"wh
!rm = 1 +
.
!h + g"s
The semi-infinite layer belo
ラメラの性質に依存するが、
tive permittivity. We mode
with effective parameters b
枚数には依らない
12. The location in the ! , % plane of the maxima and first zeros of the
2
2
ties that exist in the natural
ent
factor
)F)
and
the
array
factor
)A)
. Results are for the plane +
2010年1月27日水曜日
27
UV
V
B
G
Y
%m,A = cos−1#$!/!h + g"%cos # − *)rmcos!$ − #"&,
!22a"
= cos−1#$!/!h + g"%!1 * 1/N"cos # − *)rmcos!$ − #"&.
!22b"
ピーク波長
se angles are displayed on the ! , % plane in Fig. 12.39
√the point at which the maxima of the
black
dot
shows
(h + g)(1 + �rm )
functions coincide !! + 460 nm, % + 20°", and the
◦ gray
で強度が最大
λ ≈ λ460nm,
peak [nm] φ ≈ 20
460
460
90
450
450
!, Deg. ($ = %/2)
60
30
0
具体的な条件式：
2
|F|0
440
440
zero
zero
max
430
430
1.0
2
0.5
�
�
�
��
�
√
γ
−1
× cos 2γ + �rm cos sin
−γ
√
�rm
|F|m
0.1
2
|F|0
420
420
2
-30
330
|A|m
2
|A|0
�30
�20 zero �10
-10
-30
max
-60 -20
300
-90
270
300
λpeak
0|A|2
zero
0
10
10
20
20
φpeak = 2γ
30
30
γ(= φpeak /2) [deg]
400
#, nm
500
(h + g)
=
cos γ
600
!!rc − 1"wh
!rm = 1 +
.
!h + g"s
The semi-infinite layer belo
ラメラの性質に依存するが、
tive permittivity. We mode
with effective parameters b
枚数には依らない
12. The location in the ! , % plane of the maxima and first zeros of the
2
2
ties that exist in the natural
ent
factor
)F)
and
the
array
factor
)A)
. Results are for the plane +
2010年1月27日水曜日
28
UV
V
B
G
Y
ラメラの枚数と波長特性
sr
�P �
1.0
Total power (normalized)
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
赤：32枚
マゼンタ：16枚
黒：8枚
緑：4枚
シアン：2枚
400
400
450
450
500
500
550
550
600
600
650
650
700
700
[nm]
スペクトル幅は変わるがピーク波長はほぼ同じ
2010年1月27日水曜日
29
まとめ
モルフォ蝶の翅はなぜ青い？その物理的メカニズム
多層膜からの散乱光の干渉に強め合う
単一ラメラによる散乱パターンの強弱
これらの条件が一致する波長域が青色だった
さらにレイリー散乱が短波長側の波長特性を強調
2010年1月27日水曜日
30
人工的に構造色を作る
L 50
Jpn. J. Appl. Phys., Vol. 44, No. 1 (2005)
FIB-CVD（集束イオンビームによる化学的気相成長法）を用い
Incident angle
てモルフォ蝶の構造色を再現する３次元立体ナノ構造を作成
= 30°
2005)
DLCフィルムによる多層膜構造
(Diamond-Like Carbon)
Jpn. J. Appl. Phys., Vol. 44, No. 1 (2005)
Jpn. J. Appl. Phys., Vol. 44, No. 1 (2005)
Intensity(a.u.)
Intensity(a.u.)
L 50
= 20°
L
49 = 5°
Incident
angle
K. W ATANABE et al.
K. W ATANABE et al.
= 30°
L 49
フィルムの厚さ
~200nm
= 20°
= 5°
モルフォ蝶
wavelength(nm)
(a)
Incident angle
Intensity(a.u.)
Intensity(a.u.)
wavelength(nm)
(a)
= 5°
Θ
using
3-D CAD data. This result demonstrates that FIB-CVD
2010年1月27日水曜日
Fig. 3.
φ
= 30°°
= 20°
= 5°
擬似構造
Incident angle
Intensity(a.u.)
Watanabe et al. (2005)
Iφ
Iθ
= 20°
Incident angle
Fig. 3. Morpho-butterfly-scale quasi-structure fabricated by FIB-CVD. (a)
SIM images of Morpho-butterfly-scale quasi-structure. (b) Optical microscope images of Morpho-butterfly-scale quasi-structure observed with a 5
to 45! incidence angle of white light.
Fig. 2. Inclined-view SIM images of Morpho-butterfly-scale quasi-structure.
= 30°°
= 30°
= 20°
wavelength(nm)
= 5°
Morpho-butterfly-scale quasi-structure
fabricated by FIB-CVD.
(a)
(b)
31
むという特性を
能工学）と呼ばれる新しい工学領域が
繊維
ォテックス」
モルフォ蝶の積層構造を再現
が付いており、その鱗片の複雑な構造が、あ
繊維が平行にな
80年代に始まり、注目を集めている。
る波長の光だけを反射させて鮮やかな青色を
を得た繊維である。
ぐに切れてしま
帝人が開発した構造発色繊維「モルフォ
アマゾン河流域を生息地とするモルフォ蝶
見せている。鱗粉の断面を電子顕微鏡で拡大
工学的応用
自然界に存在する色彩を表現することは繊
テックスはメタリックブルーに輝く羽を持つ「世界で
」は、モルフォ蝶の羽からヒント
維開発の研究テーマの一つである。このよ
鱗粉そのものには色素はない。モルフォ蝶の
るバイオミメティクス（生物模倣機
羽根には「鱗片」と呼ばれる粉のようなもの
能工学）と呼ばれる新しい工学領域が
が付いており、その鱗片の複雑な構造が、あ
80年代に始まり、注目を集めている。
繊維どうしの結
すると、タンパク質と空気が幾層にも重なっ
もっとも美しい蝶」と呼ばれている。羽根の
うに生物の持つすぐれた機能に学ぼうとす
を踏まなければ
た積層構造になっているのを確認できる。こ
程度の収縮率が
モルフォテックスの発色原理
あり、
「この構造発色を繊維で再現できれば、
図1
モルフォテックス® の発色原理のまま色の変化
図1
図 1 れは生物に見られる構造発色という現象で
モルフォテックスの発色原理
®
スに熱をかけた
る波長の光だけを反射させて鮮やかな青色を
モルフォテックス
• 帝人ファイバーが、日産と田中貴金属との共同研究の末、開発
帝人が開発した構造発色繊維「モルフォ
見せている。鱗粉の断面を電子顕微鏡で拡大
テックス」は、モルフォ蝶の羽からヒント
モルフォ蝶の積層構造を再現
を得た繊維である。
すると、タンパク質と空気が幾層にも重なっ
アマゾン河流域を生息地とするモルフォ蝶
た積層構造になっているのを確認できる。こ
自然界に存在する色彩を表現することは繊
はメタリックブルーに輝く羽を持つ「世界で
れは生物に見られる構造発色という現象で
維開発の研究テーマの一つである。このよ
もっとも美しい蝶」と呼ばれている。羽根の
あり、
「この構造発色を繊維で再現できれば、
うに生物の持つすぐれた機能に学ぼうとす
モルフォテックス
鱗粉そのものには色素はない。モルフォ蝶の
画期的な素材になる」と考え、95年、日産自
画期的な素材になる」と考え、95年、日産自
形に影響が出な
するための検討
動車株式会社と田中貴金属工業株式会社との
共同研究がスタートした。
るバイオミメティクス（生物模倣機
動車株式会社と田中貴金属工業株式会社との
羽根には「鱗片」と呼ばれる粉のようなもの
能工学）と呼ばれる新しい工学領域が
•
80年代に始まり、注目を集めている。
共同研究がスタートした。
が付いており、その鱗片の複雑な構造が、あ
帝人に与えられた課題は、モルフォ蝶と
る波長の光だけを反射させて鮮やかな青色を
温度で熱セット
れ以上分子運動
帝人に与えられた課題は、モルフォ蝶と
ることになる。
同じ輝きを繊維の上で再現すること。最も
した構造発色繊維（つまり、色落ちしない）
モルフォテックス
普通よりも高く
大きな課題はポリマーの選択と微細な積層
ラー上を走って
構造を実現するための装置設計であった。
ところが、これ
屈折率の異なるポリマーを交互に積層させて構造発色を実現
いくつかの方法を検討した結果、タンパク
帝人が開発した構造発色繊維「モルフォ
テックス」は、モルフォ蝶の羽からヒント
同じ輝きを繊維の上で再現すること。最も
見せている。鱗粉の断面を電子顕微鏡で拡大
モルフォテックスのパウダーが使われた化粧品
質と空気の代わりに、屈折率の異なるポリ
いくつかの方法を検討した結果、タンパク
れは生物に見られる構造発色という現象で
マーを交互に積層させることになった。高
モルフォテックス単糸断面
あり、
「この構造発色を繊維で再現できれば、
い干渉発色を可能にするためには、2 種の
質と空気の代わりに、屈折率の異なるポリ
マーを交互に積層させることになった。高
画期的な素材になる」と考え、95年、日産自
い干渉発色を可能にするためには、2 種の
ポリマーの屈折率差が大きいこと、また積
ポリマーの屈折率差が大きいこと、また積
層構造を安定させるためポリマーの親和性、
動車株式会社と田中貴金属工業株式会社との
共同研究がスタートした。
層構造を安定させるためポリマーの親和性、
帝人に与えられた課題は、モルフォ蝶と
適性粘度、繊維構造発現の類似性の条件を
同じ輝きを繊維の上で再現すること。最も
満たすことが必要である。最終的にはポリ
大きな課題はポリマーの選択と微細な積層
エステルとナイロンを交互に
61 層積み重ね
構造を実現するための装置設計であった。
た構造をもつ新素材「モルフォテックス
」が
モルフォテックスを使用したテキスタイル
モルフォテックスのパウダーが使われた化粧品
のパウダーが使われた楽器
モルフォテックス
テキスタイル
化粧品
モルフォテックスを使用したテキスタイル
TEIJIN LABORATORIES
適性粘度、繊維構造発現の類似性の条件を
満たすことが必要である。最終的にはポリ
エステルとナイロンを交互に 61 層積み重ね
誕生。
一層の厚みが 69ナノメートル。世界初
いくつかの方法を検討した結果、タンパク
た構造をもつ新素材「モルフォテックス」が
の構造発色繊維である。
質と空気の代わりに、屈折率の異なるポリ
誕生。一層の厚みが 69ナノメートル。世界初
マーを交互に積層させることになった。高
モルフォテックスのパウダーが使われた楽器
い干渉発色を可能にするためには、2 種の
楽器
の構造発色繊維である。
ポリマーの屈折率差が大きいこと、また積
層構造を安定させるためポリマーの親和性、
衣類以外に幅広い方面で利用
適性粘度、繊維構造発現の類似性の条件を
2010年1月27日水曜日
ことには成功し
大きな課題はポリマーの選択と微細な積層
すると、タンパク質と空気が幾層にも重なっ
モルフォテックス
のパウダーが使われた化粧品
構造を実現するための装置設計であった。
た積層構造になっているのを確認できる。こ
フォテックス
S
羽根には「鱗片」と呼ばれる粉のようなもの
満たすことが必要である。最終的にはポリ
１層69nm
32
Appendix
2010年1月27日水曜日
33
ing from the ridge !simplified model".
!a" Scattering from a single, ti
Incident
. The total time-averaged power scattered into the upper half space
lamella. !b" Scattering from an array of N equally spaced points. !c" De
wavelength.
the simple model for the
for the scattering from the nth point.
olor and iridescenceScattered
simi"
Scattered
Incident
pho butterflies. However,
(F )
(A )
es away from ! = 90°. Note the absence of scattered light
light,
!13",
gles nearnamely,
grazing, ! =Eqs.
0 ° , 180°.
Observations show
1
sr that 4
2
cale appears
brown at" such
ntly
complicated
thatangles;
the the color isITEdue&tok0 sinc ()kc cos!' − (" +2 k0 cos #*h/!2 cos '"+
mentation
not
!structural scattering.
2
or
these
characteristics
are
)
sinc
(k0l)sin 'Point
+ sin!' − #"*/!2 cos '"+
gure 8 is a different plot for showing the iridescence:
son
theirradiance
scattered
lightin relief
is as a function of "
scattered
is plotted
Scatterers
2
N
!
for
the
plane
#
=
20°.
The
spectrum
for
the
scattered
n
ous. Lamella
To gain some insight
1
−j)k
cos!
'
−
*
"+k
clearly shifts to shorter wavelengths !from blue to vio-)
m
0 cos #*dn/cos '
e
.
!1
eas athesimplified
model.
angle of observation approaches grazing.
N n=1
N
(a) plane
= $time-averaged
/ 2, ignore power scattered at a
the
measure
of the!total
(b)
ncle
wavelength
can
be
obtained
by
numerically
integrating
!R TE = 0" and assume
There
are
four
factors
in
Eq.
!19"
that
depend
on
the
wa
17" for
the
irradiance
over
the
upper
hemisphere,
cm
Incident
Planes of Constant
4
= 2.43". $The
irradiance
of
and
length: k0, the arguments of the two sinc functions,
2$
/2
$
Phase
Psr$ =
+
Isr!r, !, #"r2sum.
sin !d!The
d# . factor k4 = !2$ / ""4 is what gives rise to Rayle
0
#=0
#=3$/2
!=0
,
%& & '&
"
Scattered
(A )
!18"
November
re
9 is a plot 2009
of1#Psr$ versus wavelength, normalized to a
mum of one. As
2 expected from the patterns for the scatradiation in Fig. 7, it is mainly light at wavelengths in
Point range that is scattered from the ridge, with the
lue-violet
Scatterers
near
" = 450 nm.
It is interesting to compare these caln
ed results with the measurements shown in Fig. 4. For
ctual scale the dimensions of the ridges are distributed
N Thus, it is not difficult to imagine that
a range of values.
of (b)
curves like the one in Fig. 9, each for slightly differdimensions, could be Planes
superimposed
to produce results
of Constant
those shown in Fig. 4.
Phase
2010年1月27日水曜日
,
"
dn
1
dn cos (!#$%
cos !
1
$
n
(c)
Glenn
S. Smith
!
dn cos "
cos !
34
生物学的理由
そもそも、青い色素を持つ生物はほとんどいない
（青く見える魚のほとんどは構造発色）
青い色素胞を持つ魚
(cyanophores)
スポットマンダリン
（別名：サイケデリックフィッシュ）
（色素物質の化学特性はよくわ
かっていないらしい）
青色自体、自然界にはそう多く存在しない
2010年1月27日水曜日
警戒色？
35