---
# System prepended metadata

title: NCKU 波導光學(Guided wave optics) 作業3

---

# NCKU 波導光學(Guided wave optics) 作業3
contributed by <[`tintinjian12999`](https://github.com/tintinjian12999)>
## 1. Consider the MMI device fabricated on the silicon-on-insulator platform as shown in the figure, the refractive index of the SiO2 substrate is 1.46, the refractive index of the Si core is 3.6, and the refractive index of the air cover is 1. The width of the multimode section Wmmi is 18um, the waveguide height H is 3um, the slab height h is 2um, and the oxide thickness d is 1um (assume it’s infinitely thick in the effective index calculation).

### a )For TE mode at a wavelength of 1.55m, we can use the effective index method to obtain a 2D representation of the MMI as in Fig. 2.21 of the textbook. Calculate the effective indices, neff and no.
利用
![](https://hackmd.io/_uploads/ryNQWLoWp.png)
得到 neff = 3.5916，n0 = 3.5819。
## b) How many modes are supported in the multimode section?
用 HW2 的 mode solver，可以算得 cos(phi) 為
```
[0.99738447 0.99760194 0.99781728 0.99802525 0.99822431 0.99841373
 0.99859311 0.99876218 0.99892075 0.99906871 0.99920595 0.99933239
 0.99944798 0.99955267 0.99964642 0.99972921 0.999801   0.99986177
 0.99991152 0.99995022 0.99997787 0.99999447]
```
共 22 個 mode
## c
![](https://hackmd.io/_uploads/r1h3vBsWp.png)
如圖
![](https://hackmd.io/_uploads/S1RDKLib6.png)
利用 python 的 poly fit 得到 B = 7.7， A = 1.45591605e+07
反推得到的 neff = 3.5916，相差不大。
## d) Calculate the beat length Lpi of the MMI.
利用![](https://hackmd.io/_uploads/BkaFwUibT.png)
得到
Lpi = 13 mm
## 2.請參考範例程式，畫出課本p.168頁的兩張圖。
F = 1.0
![](https://hackmd.io/_uploads/Byx_mpq-p.png)

F = 0.2
![](https://hackmd.io/_uploads/BJJLm6cW6.png)
如圖
## 3 Consider a coupler having three identical waveguides shown in the figure. The three waveguides are spaced evenly on a circle of radius R. Each waveguide, when isolated from the others, has a propagation constant β.
### (a) Write down the coupled-mode equations for the coupler with three identical waveguides. Let the coupling constant and coupler length be κ and L, respectively.
![](https://hackmd.io/_uploads/ByYfU6qb6.png)
如上，C 為 第三根波導的場。
### (b)At the input (z = 0), a power Pin is fed to one of the waveguides and none to the other two waveguides. Based on the coupled-mode equations obtained in (a), find expressions for power transferred to the other two waveguides at z = L.

### ( c ) Is it possible that all power fed to the input waveguide is transferred to the other two waveguides? If it is possible, what is the condition of a complete power transfer?
無法

### (d) Modify the MATLAB scripts to solve the coupled mode equations for this system.
![](https://hackmd.io/_uploads/S1Kr0xjWT.png)
可以看到在三個波導都相同的情況下能量不會完全由一根波導轉移到其他兩根上。
