写在前面

? ? ? ? 光路是很重要的,用mma可以很好得模拟它.但是很遗憾,mma跑得不快,如果是因为sanction而用不上正版的话...

? ? ? ? 嗯,所以会python是很重要的

? ? ? ? 如果有能力自己去解一个方程的话,也可以...

? ? ? ? 但是题主选用了Scipy??

? ? ? ? 至于可视化吗,题主选用了matplotlib

? ? ? ? 有些地方为了算的快一点,也用了multiprocessing? 和 numpy

几何光学:在连续折射率介质中进行光线追踪

光线方程
- 模拟光在大气中的折射
- 模拟光在光纤中的传播
  - 纤芯的折射率模型 $n(r)=n_1[1-\frac{n_1-n_2}{n_1}(\frac{r}{a})^2]$
  - 纤芯中心折射率n1? 包层区域折射率n2
    - 对于单模光纤? n2/n1? 通常在99.4%~99.7%
    - 对于多模光纤? n2/n1? 通常在98%~99%
  - 光线围绕纤芯中心周期性地向前传播,表现出自聚焦的特性
  - 光线路径的周期大小与初始入射角有关,不同的光线将不严格聚集在相同的点上,一个入射的光脉冲将逐渐弥漫散开

几何光学:在折射率跃变介质中进行光线追踪

Snell 定律
球面凸透镜
- 球面方程
  - $f_1(x,y)=R^2-[y^2+(x-R+a)^2]=0$
- 透镜半径
  - $r=\sqrt{a(2R-a)}$
- 曲面的法线方向
  - $\Omega =\frac{\bigtriangledown f}{\begin{vmatrix} \bigtriangledown f \end{vmatrix}}$
- 平行光束正入射
  - 光束的粗细相对于凸透镜半径r来说? y方向上宽度为 -0.1r~0.1r
- 平行光斜入射
- 轴上光源
- 轴外光源
- 凸透镜的近轴焦点
- 厚透镜的形式焦距的倒数
  - $\frac{1}{f}=(n-1)[\frac{1}{r_1}-\frac{1}{r_2}+\delta \frac{n_2-1}{n_2r_1r_2}]$
- 物点经过凸透镜之后的像距
  - 待写
三棱镜
?白光的色散和色光的合成
- 单个三棱镜的色散
- 两个三棱镜的色散??
消色差透镜(凹凸组合透镜)

波动光学:光衍射的计算

根据夫琅禾费衍射理论,衍射场的振幅分布为:
- $U(\alpha,\beta)=\iint_{\sum}u(x,y)e^{2\pi i (xsin\alpha + ysin\beta)/\lambda}dxdy$
- $u(x,y)$ 衍射屏的透射函数
- 衍射强度的分布 $I(\alpha,\beta)=\begin{vmatrix} U(\alpha,\beta) \end{vmatrix}^2$
- $U=U_1+iU_2=\iint_{\sum}ucosfdxdy+i\iint_{\sum}sinfdxdy$
单个圆孔的衍射
- $u(x,y)=UnitStep[r^2-(x-x_0)^2-(y-y_0)^2]$
- 硬核的来了,这个程序等他跑完也就该下班了,几个小时也不一定跑的完,我们还是换Python取编写.当然,如果读者会C或者Fortran啥的那更好的,不过,写个几百行累死人了,不是吗?
- 热力图是表征强度的一个很好的工具,
- 下面给出单个圆孔的夫琅禾费衍射计算程序（以方向角形式展示的，所以他不是一个标砖的圆，这个演示完会有用X，Y坐标表示的程序）
- 当然,以Python语言的一般特性,这个n取到100的时候,,,,

import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
from scipy.integrate import dblquad
import math

#lambda 用k表示
r = 3.0
k = 0.6
q = 2*math.pi / k
#alpha 用a表示 beta用b表示
a0 = math.pi
b0 = math.pi
n = 10

def fun(D,k):
    a = D[0]
    b = D[1]
    u = "1"
    x1 = 3
    x2 = -3
    y1 = lambda x: (9-x**2)**0.5
    y2 = lambda x: -(9-x**2)**0.5
    U1 = dblquad(lambda x,y:eval(u)*math.cos(2*math.pi*(x*math.sin(a)+y*math.sin(b))/k)
                 ,x1,x2,y1,y2)
    U2 = dblquad(lambda x,y:eval(u)*math.sin(2*math.pi*(x*math.sin(a)+y*math.sin(b))/k)
                 ,x1,x2,y1,y2)
    U = U1[0]**2 + U2[0]**2
    return U

data0 = np.array([[[i,j] for j in np.linspace(-a0,a0,n)] for i in np.linspace(-b0,b0,n)])
data = [[fun(j,k) for j in i] for i in data0]
#注意!
data =  np.array(data)#设置二维矩阵
f,ax = plt.subplots(figsize=(11,12))
#数据

f = open("diffraction.txt","w")
f.write(str(data))
f.close()
 
#注意!
sns.heatmap(data, ax=ax,vmin=0,vmax=6,cmap='YlOrRd',linewidths=2,cbar=True)
#热力图绘制代码
 
 
plt.title('heatmap') 
plt.ylabel('y_label')  
plt.xlabel('x_label')
plt.savefig("diffraction.jpg")
plt.show()

这时候必须multiprocessing辅助一下了

import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
from scipy.integrate import dblquad
import math
import datetime
import multiprocessing as mp
import time

a0 = math.pi/10
b0 = math.pi/10
n = 100
k = 0.6


def final_fun(name, param):
    result = []
    global a0
    global b0
    global n
    global k
    A = np.linspace(-a0,a0,n)
    B = np.linspace(-b0,b0,n)
    for params in param:
        a = A[params[0]]
        b = B[params[1]]
        u = "1"
        x1 = 3
        x2 = -3
        y1 = lambda x: (9-x**2)**0.5
        y2 = lambda x: -(9-x**2)**0.5
        U1 = dblquad(lambda x,y:eval(u)*math.cos(2*math.pi*(x*math.sin(a)+y*math.sin(b))/k)
                     ,x1,x2,y1,y2)
        U2 = dblquad(lambda x,y:eval(u)*math.sin(2*math.pi*(x*math.sin(a)+y*math.sin(b))/k)
                     ,x1,x2,y1,y2)
        U = U1[0]**2 + U2[0]**2
        result.append(U)
    return {name:result}


if __name__ == '__main__': 
    start_time = time.time()
    num_cores = int(mp.cpu_count())
    pool = mp.Pool(num_cores)
    param_dict = {"task1":[[0,i] for i in range(n)],
"task2":[[1,i] for i in range(n)],
"task3":[[2,i] for i in range(n)],
"task4":[[3,i] for i in range(n)],
"task5":[[4,i] for i in range(n)],
"task6":[[5,i] for i in range(n)],
"task7":[[6,i] for i in range(n)],
"task8":[[7,i] for i in range(n)],
"task9":[[8,i] for i in range(n)],
"task10":[[9,i] for i in range(n)],
"task11":[[10,i] for i in range(n)],
"task12":[[11,i] for i in range(n)],
"task13":[[12,i] for i in range(n)],
"task14":[[13,i] for i in range(n)],
"task15":[[14,i] for i in range(n)],
"task16":[[15,i] for i in range(n)],
"task17":[[16,i] for i in range(n)],
"task18":[[17,i] for i in range(n)],
"task19":[[18,i] for i in range(n)],
"task20":[[19,i] for i in range(n)],
"task21":[[20,i] for i in range(n)],
"task22":[[21,i] for i in range(n)],
"task23":[[22,i] for i in range(n)],
"task24":[[23,i] for i in range(n)],
"task25":[[24,i] for i in range(n)],
"task26":[[25,i] for i in range(n)],
"task27":[[26,i] for i in range(n)],
"task28":[[27,i] for i in range(n)],
"task29":[[28,i] for i in range(n)],
"task30":[[29,i] for i in range(n)],
"task31":[[30,i] for i in range(n)],
"task32":[[31,i] for i in range(n)],
"task33":[[32,i] for i in range(n)],
"task34":[[33,i] for i in range(n)],
"task35":[[34,i] for i in range(n)],
"task36":[[35,i] for i in range(n)],
"task37":[[36,i] for i in range(n)],
"task38":[[37,i] for i in range(n)],
"task39":[[38,i] for i in range(n)],
"task40":[[39,i] for i in range(n)],
"task41":[[40,i] for i in range(n)],
"task42":[[41,i] for i in range(n)],
"task43":[[42,i] for i in range(n)],
"task44":[[43,i] for i in range(n)],
"task45":[[44,i] for i in range(n)],
"task46":[[45,i] for i in range(n)],
"task47":[[46,i] for i in range(n)],
"task48":[[47,i] for i in range(n)],
"task49":[[48,i] for i in range(n)],
"task50":[[49,i] for i in range(n)],
"task51":[[50,i] for i in range(n)],
"task52":[[51,i] for i in range(n)],
"task53":[[52,i] for i in range(n)],
"task54":[[53,i] for i in range(n)],
"task55":[[54,i] for i in range(n)],
"task56":[[55,i] for i in range(n)],
"task57":[[56,i] for i in range(n)],
"task58":[[57,i] for i in range(n)],
"task59":[[58,i] for i in range(n)],
"task60":[[59,i] for i in range(n)],
"task61":[[60,i] for i in range(n)],
"task62":[[61,i] for i in range(n)],
"task63":[[62,i] for i in range(n)],
"task64":[[63,i] for i in range(n)],
"task65":[[64,i] for i in range(n)],
"task66":[[65,i] for i in range(n)],
"task67":[[66,i] for i in range(n)],
"task68":[[67,i] for i in range(n)],
"task69":[[68,i] for i in range(n)],
"task70":[[69,i] for i in range(n)],
"task71":[[70,i] for i in range(n)],
"task72":[[71,i] for i in range(n)],
"task73":[[72,i] for i in range(n)],
"task74":[[73,i] for i in range(n)],
"task75":[[74,i] for i in range(n)],
"task76":[[75,i] for i in range(n)],
"task77":[[76,i] for i in range(n)],
"task78":[[77,i] for i in range(n)],
"task79":[[78,i] for i in range(n)],
"task80":[[79,i] for i in range(n)],
"task81":[[80,i] for i in range(n)],
"task82":[[81,i] for i in range(n)],
"task83":[[82,i] for i in range(n)],
"task84":[[83,i] for i in range(n)],
"task85":[[84,i] for i in range(n)],
"task86":[[85,i] for i in range(n)],
"task87":[[86,i] for i in range(n)],
"task88":[[87,i] for i in range(n)],
"task89":[[88,i] for i in range(n)],
"task90":[[89,i] for i in range(n)],
"task91":[[90,i] for i in range(n)],
"task92":[[91,i] for i in range(n)],
"task93":[[92,i] for i in range(n)],
"task94":[[93,i] for i in range(n)],
"task95":[[94,i] for i in range(n)],
"task96":[[95,i] for i in range(n)],
"task97":[[96,i] for i in range(n)],
"task98":[[97,i] for i in range(n)],
"task99":[[98,i] for i in range(n)],
"task100":[[99,i] for i in range(n)]}
    
    results = [pool.apply_async(final_fun, args=(name, param)) for name, param in param_dict.items()]
    results = [p.get() for p in results]
    end_time = time.time()
    use_time = end_time - start_time
    print("多进程计算 共消耗: " + "{:.2f}".format(use_time) + " 秒")
    data = [i["task"+str(results.index(i)+1)] for i in results]
    data = np.array(data)
    f = open("diffraction.txt","w")
    f.write(str(data))
    f.close()
    fig,ax = plt.subplots(figsize=(11,12))
    sns.heatmap(data, ax=ax,vmin=0,vmax=6,cmap='YlOrRd',\
        linewidths=2,cbar=True)


    plt.title('diffraction') 
    plt.ylabel('y_label')  
    plt.xlabel('x_label')   
    plt.savefig("diffraction.jpg")
    plt.show()

用x，y坐标表示，这个更符合人的观察