3. Orthogonal property of the optical modes and the coupling coefficients of incident beam to the optical
modes
We now show that the modes with different effective indices (propagation constants) are orthogonal. Suppose
two optical modes with the propagation constants 1 and  2 , respectively. The overlap
integral of the two optical modes can be calculated in the I, II, and III regions:
1 ( x) and 2 ( x) are

0
a



0
a

1 ( x)2 ( x)dx   1 ( x)2 ( x)dx   1 ( x)2 ( x)dx   1 ( x)2 ( x)dx ,
(14)
Using Eq. (1), one can express the overlap integral in the I region as:
1
d 21
 1 ( x)2 ( x)dx   k02n12  12 dx 2 2 ( x)dx ,
0
0
(15)
Applying integration by parts for two times and using Eq. (1), one gets the overlap integral in region I:

0

12 dx  
2 2
2
0 k n 
1
d1 0
1
d2 0
0 1
2

(
x
)
|


|

2

1 
 k02n12  12 12dx , (16), which can be
k02 n12  12
dx
k02 n12  12 dx
simplified as:

2
2
 12

0

12 dx  2 ( x)
d1 0
d
|  2 1 |0 ,
dx
dx
(17)
Similarly, the overlap integral of the two optical modes in regions II and III can be written as:

2
2
 12
   dx   ( x) ddx |

2
2
 12

a
1 2
0

a
2
12 dx  2 ( x)
1 a
0

d2 a
1 |0 ,
dx
d1  d2 
|a 
1 |a ,
dx
dx
(18)
(19)
Combining Eq. (17) through Eq. (19) and applying the boundary conditions of  and
d
continuous at the
dx
material interfaces, one gets:

2
2
 12



12 dx  0 ,
(20)
Eq. (20) indicates that two modes with different propagation constants (effective indices) are orthogonal.
Since the proof of the orthogonal property only involves the Helmholtz equations, the boundary conditions at
the material interfaces, and the assumption that the E field are zeros at -, and +-, one can argue that the guide
modes and radiation modes are also orthogonal.
The E field of the input light E0 can be written as the superposition of the guided modes and the radiation
modes:
E0 ( x)   aii ( x)   b  ( x)d ,
(21)
i
where,
i (x)
and
  (x) are the guided and radiation modes, respectively. ai and b are their coupling
coefficients. Since only the optical power of the guided modes is preserved after it travels a distance along the
waveguides, we focus on the coupling coefficients of the guided modes. According to the orthogonal property, if
one normalizes the guided mode profile i (x) , i.e.



i ( x) dx  1, the coupling coefficient ai can be written
2
as:

ai   E0 ( x)i ( x)dx ,
(22)

The power profile of each mode can be expressed as:
Pi ( x)  i ( x) ,
2
(23)
4. Optical power distribution in the coupled waveguides and its variation along the z-direction
Since only the optical power of the guided modes is preserved after it travels a distance along the waveguides,
the E field along the z-direction after a distance can be written as the superposition of the guided modes:
E ( x, z )   aii ( x)eii z ,
(24)
i
where,
ai is the coupling coefficient of the input beam to the ith-order mode as defined by Eq. (22), and  i is the
propagation constant of the corresponding mode. The integrated powers in the left, center and right waveguides at
1
different z-positions, i.e.
3
5
P1 ( z )   E ( x, z ) dx , P2 ( z )   E ( x, z ) dx , and P3 ( z )   E ( x, z ) dx are also
0
2
2
2
2
4
calculated using the BPM and the TMM methods. Figure 4(a), (b) (c) show the comparison of the results for
integrated powers in the left, center and right waveguides (i.e. P1 ( z) , P2 ( z ) and P3 ( z ) ), respectively. The
integrated powers
P1 ( z) , P2 ( z ) and P3 ( z ) at different z-positions are almost identically for both of the BPM and
the TMM methods.
Note that the TMM method only considers the guided modes and ignores the radiation (leakage) modes. The
results given by the TMM method are only accurate after it travels a distance along the z-direction where only the
power of the guided modes is preserved.
Since the TMM method calculates the power distribution profile P ( x, z )  E ( x, z )
2
, which is an
interference of the optical modes in the waveguides, this suggests that the mechanism of the power coupling
between the waveguides are due to the interference of the modes in the waveguides. To our best knowledge, such
interference-based power coupling mechanism has not been reported before.