为什么折射率与波长相关的菲涅尔方程的图未按预期显示？

Question

我想重现复折射率与波长相关的薄膜的反射光谱（复折射率数据，在代码中名为N2，可以从这里获得）。

使用 菲涅耳方程 对于介质 1：空气、介质 2：薄膜和介质 3：空气： ;

反射系数

其中 \delta 定义为

最后是反射率方程

由于折射率与波长有关，因此折射角 也具有相同的依赖性，使用斯涅尔方程：

我写了以下代码：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

## Function definition
def Ang_Refrac(na,nb,angle_a):
ang_refrac = np.arcsin( (na/nb).real * np.sin(angle_a) )
return ang_refrac

# Data
Thin_film = pd.read_csv("Si_refractive index.txt", delimiter='\t')

## Variable definition

# thin film thickness, [d]: nm
d = 200

# Wave-length
lamb = Thin_film[Thin_film.columns[0]]

# Complex refractive index
N1 = np.ones(len(lamb))
N2 = Thin_film[Thin_film.columns[1]] + Thin_film[Thin_film.columns[1]]*1j
N3 = np.ones(len(lamb))

# Angle:
ang_1_s = 0  #sexagesimal

ang_1 = ang_1_s*np.pi/180  #radians
ang_2 = [] #radians
ang_3 = [] #radians
for i in range(len(N2)):
    ang_refrac_12 = Ang_Refrac(N1[i],N2[i],ang_1)
    ang_refrac_23 = Ang_Refrac(N2[i],N3[i],ang_refrac_12)
    ang_2.append(ang_refrac_12)
    ang_3.append(ang_refrac_23)

## Reflectance
R_s = np.zeros(len(lamb))
R_p = np.zeros(len(lamb))

for i in range(len(lamb)):
    # S-type polarization
    r12_s = (N1[i]*np.cos(ang_1) - N2[i]*np.cos(ang_2[i]))/(N1[i]*np.cos(ang_1) + N2[i]*np.cos(ang_2[i]))
    r23_s = (N2[i]*np.cos(ang_2[i]) - N3[i]*np.cos(ang_3[i]))/(N2[i]*np.cos(ang_2[i]) + N3[i]*np.cos(ang_3[i]))

    # P-type polarization
    r12_p = (N2[i]*np.cos(ang_1) - N1[i]*np.cos(ang_2[i])) / (N2[i]*np.cos(ang_1) + N1[i]*np.cos(ang_2[i]))
    r23_p = (N3[i]*np.cos(ang_2[i]) - N2[i]*np.cos(ang_3[i])) / (N3[i]*np.cos(ang_2[i]) + N2[i]*np.cos(ang_3[i]))

    # Phase shift
    delta = 2 * np.pi * (1/lamb[i]) * d * np.sqrt(abs(N2[i])**2 - (np.sin(ang_1)**2))

    # Reflection coefficient
    r123_s = (r12_s + r23_s*np.exp(2j*delta)) / (1 - r12_s*r23_s*np.exp(2j*delta))
    r123_p = (r12_p + r23_p*np.exp(2j*delta)) / (1 - r12_p*r23_p*np.exp(2j*delta))

    # Reflectance
    R_s[i] = abs(r123_s)**2
    R_p[i] = abs(r123_p)**2


# Reflectance normalization
R_s_normalized = (R_s - min(R_s)) / (max(R_s) - min(R_s))
R_p_normalized = (R_p - min(R_p)) / (max(R_p) - min(R_p))

# Plotting
plt.title("Refletância $\phi=" + str(ang_1_s) + "$°")
plt.xlim(300, 1000)  # Definindo os limites do eixo x
plt.plot(lamb, R_s_normalized, 'go', label="Polarização S", markersize=4)
plt.plot(lamb, R_p_normalized, 'b-', label="Polarização P")
plt.legend()
plt.show()

结果图：

预期图表：从这里获得

为什么结果不一样？

Answer 1

你可以试试这个。主要变化是：

• N2 由 n + i.k 给出，而不是像你那样由 n + i.n 给出

• delta的表达式中只有N2 ** 2，而不是abs(N2)**2，这会丢失一个重要的复杂部分

• 我认为反射系数的分母中应该有“+”，而不是“-”（尽管我可能是错的）。

我在https://ir.lib.nycu.edu.tw/bitstream/11536/27329/1/000186822000019.pdf

中发现了一些背景（和一些错别字）

代码：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

## Function definition
def Ang_Refrac(na,nb,angle_a):
    ang_refrac = np.arcsin( (na/nb) * np.sin(angle_a) )
    return ang_refrac

# Data
Thin_film = pd.read_csv("Si_refractive index.txt", delimiter='\t')

d = 200                                         # thin film thickness, [d]: nm
lamb = Thin_film[Thin_film.columns[0]]          # Wave-length

# Complex refractive index       n + ik
num = len(lamb)
N1 = np.ones(num)
N2 = Thin_film[Thin_film.columns[1]] + Thin_film[Thin_film.columns[2]]*1j      # <=========== NOTE ======
N3 = np.ones(num)

# Angle:
ang_1_s = 0
ang_1 = ang_1_s * np.pi / 180
ang_2 = [] #radians
ang_3 = [] #radians
for i in range(num):
    ang_refrac_12 = Ang_Refrac(N1[i],N2[i],ang_1)
    ang_refrac_23 = Ang_Refrac(N2[i],N3[i],ang_refrac_12)
    ang_2.append(ang_refrac_12)
    ang_3.append(ang_refrac_23)

## Reflectance
R_s = np.zeros(num)
R_p = np.zeros(num)

for i in range(num):
    # S-type polarization
    r12_s = (N1[i]*np.cos(ang_1) - N2[i]*np.cos(ang_2[i]))/(N1[i]*np.cos(ang_1) + N2[i]*np.cos(ang_2[i]))
    r23_s = (N2[i]*np.cos(ang_2[i]) - N3[i]*np.cos(ang_3[i]))/(N2[i]*np.cos(ang_2[i]) + N3[i]*np.cos(ang_3[i]))

    # P-type polarization
    r12_p = (N2[i]*np.cos(ang_1) - N1[i]*np.cos(ang_2[i])) / (N2[i]*np.cos(ang_1) + N1[i]*np.cos(ang_2[i]))
    r23_p = (N3[i]*np.cos(ang_2[i]) - N2[i]*np.cos(ang_3[i])) / (N3[i]*np.cos(ang_2[i]) + N2[i]*np.cos(ang_3[i]))

    # Phase shift
    delta = 2 * np.pi * ( d / lamb[i] ) * np.sqrt( N2[i]**2 - np.sin(ang_1)**2)       # <====== NOTE ====

    # Reflection coefficient
    r123_s = (r12_s + r23_s*np.exp(2j*delta)) / (1 + r12_s*r23_s*np.exp(2j*delta))    # <====== NOTE ====
    r123_p = (r12_p + r23_p*np.exp(2j*delta)) / (1 + r12_p*r23_p*np.exp(2j*delta))    # <====== NOTE ====

    # Reflectance
    R_s[i] = abs( r123_s ) ** 2
    R_p[i] = abs( r123_p ) ** 2


# Reflectance normalization
R_s_normalized = ( R_s - min( R_s ) ) / ( max( R_s ) - min( R_s ) )
R_p_normalized = ( R_p - min( R_p ) ) / ( max( R_p ) - min( R_p ) )

# Plotting
plt.title( r"Refletancia $\phi=" + str(ang_1_s) + "$ deg")
plt.xlim(300, 1000)  # Definindo os limites do eixo x
plt.plot(lamb, R_s_normalized, 'go', label="Polarizacao S", markersize=4)
plt.plot(lamb, R_p_normalized, 'b-', label="Polarizacao P")
plt.legend()
plt.show()

输出（注意：标准化与您的“预期”存在一些系统差异）