B. Analytic investigation on the group velocity of the optical branch
B.2 Analytic investigation for the group velocity of optical branch
First, please recall Eq. (10), the equation of dispersion relation of the extended mass-spring system.
( )( ) ( )( ) ( )( )
24 2 /m /I coska 1 0.5a2 /I coska 1 2 4 /Im coska 1 0
w +éë a +b - - a + ùûw + ab - =
(B-1) Before further proceed, the dispersion relation is expressed in the simpler form as shown below.
( ) ( )
4 A k 2 B k 0
w + w + = (B-2)
where A k
( ) (
=2 a/m+b /I)(
coska- -1)
0.5a2(
a/I)(
coska+1 ,)
(B-3)( ) (
4 /)(
cos 1 .)
2B k = ab Im ka- (B-4)
Since the A k
( )
and B k( )
are the function of k , Eq. (B-2) can be considered as the quadratic equation for w2. Therefore, the solution of Eq. (B-2) can be easily obtained as( ) ( )
2( )
2 4
2
A k A k B k
w =- - - (B-5a)
( ) ( )
2( )
2 4
2
A k A k B k
w =- + - (B-5b)
The solution in Eq. (B-5a) is for the acoustic branch, and Eq. (B-5b) is for the optical branch. Note that the group velocity of the wave can be obtained as ¶w/¶k. From Eq. (B-2), derivative of the angular frequency w with respect to wavenumber k, the group velocity, can be obtained by differentiate Eq. (B- 2) with respect to k as shown below
( ) ( ) ( )
3 2
4 A k' 2A k B k' 0
k k
w w
w ¶ + w + w¶ + =
¶ ¶ (B-6) where A k'
( )
= -asinka×éë2(
a/m+b /I)
-0.5a2(
a/I)
ùû, (B-7)( ) ( )( )
' sin 8 / cos 1 .
B k = -a ka× ab Im ka- (B-8)
By re-arranging Eq. (B-6) for the derivative of w with respect to k , the equation for the slope of the dispersion curve can be achieved as shown below
( ) ( )
( )
2 2
' '
2 2
A k B k
k A k
w w
w w
é ù
- +
¶ = ë û
¶ éë + ùû
(B-9)
The equation of slope in the dispersion curve can be reduced further reminding that our interest is the optical branch. The w2 term in Eq. (B-5b) can be substituted by Eq. (B-9). As a result, the derivative of w2 is expressed as shown below
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
2 2
2 2
' ' ' '
2 4
2 4
A k B k A k B k
k A k A k B k A k A k B k
w w
w w w
é ù é ù
- + - +
¶ = ë û = ë û
¶ éë- + - + ùû -
(B-10)
The analytically expressed slope of the optical branch for the flexural wave is obtained. From Eq. (10), the condition of determining whether the optical branch has a positive branch or a negative branch.
Since Eq. (B-10) is the slope of the optical branch of the flexural wave, when Eq. (B-10) has a negative sign within the IBZ, the flexural wave has a negative group velocity in its optical branch. Therefore, determining the sign of Eq. (B-10) tells whether the optical branch has a negative, positive, or positive- negative group velocity.
To determine whether Eq. (B-10) has a negative or positive value, we focus on the numerator of the equation because the denominator of the equation always has a positive value due to the root term. Note that the root term in Eq. (B-10) has only a real number because the optical branch is basically a pass
47
band. Therefore, only the numerator is related to the sign of the group velocity of the optical branch.
The numerator in Eq. (B-10) can be extended as shown below
( )
2( ) { ( )
2( )
2( )( ) }
' ' sin 2 / / 0.5 / 8 / cos 1
A k w B k a ka a m b I a a I w ab Im ka
é ù é ù
-ë + û= × ë + - û + -
(B-11) Since our interest is the optical branch in IBZ where 0<ka<p , asinka has always positive value.
Therefore, Eq. (B-11) can be reduced as below
( )
2( )
2( )( )
2 a /m b/I 0.5a a/I w 8 ab/Im coska 1
é + - ù + -
ë û (B-12)
In similar, the coska-1 has always negative value in IBZ. So, the w2 terms in Eq. (B-12) determine the sign of the group velocity of the optical branch.
( )
2( )
2 a /m+b/I -0.5a a /I (B-13) From this, they can be split into two cases: one is the case of Eq. (B-13) has a negative value, which can be directly concluded that the group velocity of the optical branch has a negative value for IBZ.
The second case, where Eq. (B-13) has a positive value, has more possible cases. In this case, the term about angular frequency w2 and the term about wavenumber k in Eq. (B-12) determine the sign of the group velocity of the optical branch. It means that if the w2 term is bigger than the k term for the given frequency and wavenumber, the group velocity of the optical branch would be positive, and vice versa. Note that Eq. (B-12) can be considered as the dispersion relation equation by setting Eq. (B-12) to be zero.
( )
2( )
2( )( )
2 a/m b/I 0.5a a /I w 8 ab/Im coska 1 0
é + - ù + - =
ë û (B-14)
For the given wavenumber, if the optical branch of the flexural wave take place above the curve plotted by Eq. (B-14), the optical branch has a positive group velocity. It can be verified by plotting Eq. (B-14) as shown in Figure B-2(a).
Figure B-2. The Eq. (B-14) is plotted on the case of (a) the optical branch having a positive group velocity and (b) having positive-negative group velocity.
Eq. (B-14) is plotted as a dotted line in Figure B-2. In addition, the optical branch of the flexural wave is also plotted in Figure B-2 as a solid line. As expected above, in the case of the optical branch having the positive group velocity, the optical branch takes place above the dotted line.
If the optical branch meets the dispersion curve of Eq. (14), the optical branch starts to have negative group velocity as shown in Figure B-2(b). It means that the optical branch has both positive group velocity and negative group velocity in IBZ. Once the optical branch encounters the dotted line by Eq.
(B-14), the optical branch keeps having a negative group velocity. Therefore, there are two possibilities:
The case where the optical branch keeps taking place above the dotted line and has a positive group velocity, and the case where the optical branch meets the dotted line and has a positive-negative group velocity. These two cases can be easily distinguished by observing the frequency at k=p /a. Let’s say the frequency of the optical branch at k=p /a is wk=p/a and the frequency of Eq. (B-14) at
/
k=p a is w*. These two frequencies can be obtained by substituting k=p /a in Eq. (B-5b) and Eq. (B-14).
( )
2
/ max 4 / , 4 /
k p a m I
w = = a b B-15
49
( )
( ) ( )
*2
2
16 /
2 / / 0.5 /
mI
m I a I
w ab
a b a
= + - B-16
Consequently, if Eq. (B-15) is higher than Eq. (B-16), the optical branch has a positive group velocity and if Eq. (B-15) is lower than Eq. (B-16), the optical branch has a positive-negative group velocity.
In summary, there are three cases for the optical branch: having a negative, positive, and positive- negative group velocity. The negative group velocity can be achieved when Eq. (B-14) has a negative value, so it can be written as
( )
2( )
2 a /m+b/I -0.5a a/I <0 B-17 Therefore, the condition for negative group velocity can be written as
2
4
I a
m b
+a < B-18 Then if Eq. 14 has a positive value, there are two cases. One is the pure positive group velocity where Eq. (B-15) is higher than Eq. (B-16), and the other is the positive-negative group velocity where Eq.
(B-15) is lower than Eq. (B-16). They can be written as
( ) ( )
( )
2( )
16 /
max 4 / , 4 /
2 / / 0.5 /
m I mI
m I a I
a b ab
a b a
> + - B-19
( ) ( )
( )
2( )
16 /
max 4 / , 4 /
2 / / 0.5 /
m I mI
m I a I
a b ab
a b a
< + - B-20
If a/m>b/I Eq. (B-19) and Eq. (B-20) can be simplified as
2
4 a m I a b- >
B-21
2
4 a m I
a -b < B-22
If a/m<b/I Eq. (B-21) and Eq. (B-22) can be simplified as
2
4 a I m b a- >
B-23
2
4 a I m b a- <
B-24
The Eq. (B-21) and Eq. (B-23), Eq. (B-22) and Eq. (B-24) can be merged as shown below.
2
4 a m I
a -b > B-25
2
4 a m I
a -b < B-26
Consequently, the condition for positive group velocity can be expressed as
2 2
and
4 4
I a I a
m m
b b
a a
+ > - > B-28
,and the condition for positive-negative group velocity can be expressed as
2 2
and
4 4
I a I a
m m
b b
a a
+ > - < B-29
Finally, these conditions for judging the group velocity of the optical branch are obtained.
Interestingly, the conditions, Eq. (B-18), Eq. (B-28), and Eq. (B-29), are expressed in terms of the ratio of rotational inertia and mass /I m , the ratio of bending stiffness and shear stiffness b a/ , and the periodicity. Using this fact, the condition obtained above can be visualized by drawing the map which has two axes of the inertia ratio /I m and stiffness ratio b a/ . It can be drawn as shown below,
51
Figure B-3. The condition for the sign of group velocity in optical branch with respect to the inertia ratio and stiffness ratio.
First, the region highlighted with a blue color indicates the place where the stiffness ratio and inertia ratio satisfy Eq. (B-18), the condition for negative group velocity. The red region indicates the place where the ratios satisfy Eq. (B-28), the condition for positive group velocity. Lastly, the purple region indicates the place where the ratios satisfy Eq. (B-29). As shown in Figure B-3, if the sum of the stiffness ratio and inertia ratio is less than the a2/ 4, the metamaterial has a negative group velocity. If the sum of the stiffness ratio and inertia ratio exceeds a2/ 4 , the metamaterial has a positive or positive- negative group velocity. Moreover, if the difference between the two ratios is smaller than the a2/ 4, the positive-negative group velocity would be achieved. Otherwise, if the difference is bigger than the
2/ 4
a , the pure positive group velocity would be achieved. From this visualized parametric map, one may easily find the group velocity of the optical branch of the interested metamaterial and how unit cell design is changed to achieve desired group velocity.