非均匀波的反射与透射研究
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摘要
在每层的厚度足够小而层数足够多的情况下,折射率连续改变的介质实质上相当于由许多层折射率均匀的薄膜堆积而成,可以采用解析转移矩阵方法代替数值计算来精确确定非均匀介质的反射系数。通过定义总波数,其包括了在折射率连续变化的介质中传播的主波和子波的累计贡献,最终在没有引入任何近似的情况下得到了非常简洁的常规形式的公式,且具有清晰的物理意义。
所得公式表明,除了环境与该介质的上下表面之间的折射率之外,反射系数只依赖于主波和子波所累计的总的位相贡献。
与传统的WKBJ比较,由于WKBJ忽略了子波反射,所以是近似的结果而本文得到的结果是精确结果。
根据所得的公式,本文研究了光波在单层介质、双层介质、三层介质、折射率分别呈指数分布、高斯分布、正弦分布的介质的反射和透射情况并对结果进行了讨论。实际上本文的结果可以计算任意折射率分布的介质。
本文对光的反射与透射的研究结果具有广泛的应用前景,如各种具有很高的灵敏度和精确度的传感器或者测量装置。其中最引人注目的应当是可以根据其来指导薄膜工艺中的厚度控制或材料选择,只要厚度的改变或者折射率的改变满足所推导出条件,则反射率与透射率保持不变。这些结果对诸如红外薄膜、太阳能薄膜等热点具有重要意义。
在分析物质波在非均匀势场中的反射及透射几率时,由于薛定谔方程与光波导方程的近似性,含时薛定谔方程类似于菲涅尔方程而与时间无关的薛定谔方程则类似于亥姆霍兹方程,势场分布类似于折射率分布,能量本征值对应于光波的传播常数,高折射率区域相当于势阱而低折射率区域相当于势垒,因此将分析光波导中所应用的解析转移矩阵方法应用于分析薛定谔方程中。最终在没有求解薛定谔方程,而且没有引入任何近似的情况下,通过定义总波数,得到了非常简洁的表达式且物理意义清晰。
在所得公式中,整个的反射与透射几率除了环境、势垒的开始点和结束点的参数外,只依赖于主波和子波总的累积相移。同时公式不受德布罗意波长和能量范围大小的局限。
本文得到的结果可以广泛地应用于许多基本的量子现象,例如,量子隧道效应、量子反射、量子粒子和隧道共振等。利用该结果分别研究了抛物线型单势垒情况、方型双势垒情况、抛物线型双势垒情况以及置于外场中的双势垒情况。
从光波反射系数公式和物质波的反射系数公式来看,两者一个统一的形式,这样就将光波、物质波在非均匀介质中的反射及透射问题统一起来,理所当然具有十分重要的意义。
Since a continuous varying index profile is nothing more than a stack of thin layers which have homogeneous refractive indeices when the thickness of layers is very very small while the number of layers is very very large, in the present investigation, instead of the numerical calculations with transfer matrix approach, the analytical transfer matrix method is employed to determine the reflection coefficient from an inhomogeneous stratified media. With the help of the definition of general wavenumber, which contains both the main waves and the subwaves propagated in the structure with continuously varying index profile, an explicit and exact expression with clear physical insight is obtained without introducing any approximation.
The result shows that, besides the ambience and the refraction indices of both front-surface and back-surface of the structure, the unique dependence of the reflection coefficient is the total phase shift accumulated by the main wave and the subwaves.
The tranditional WKBJ approach is an approximation in which the subwaves refrected from the interior of the structure are neglected while the result presented here is exact.
The reflection from one-layer planar structure, two-layer planaer structure and three-layer structure is studied using the expression. Even more, the refractive index profile is an exponential function, or Gauss function, or periodi sine functions is calculated and the results are discussed. In fact the expression obtained in the paper can be applied to arbitrary refractive index profile.
The study on the refrection and transmission of lights can be adopted in many application, such as high sensitive sensor or measurement instructions. Especially it can be used in the control of thickness of materials selection in the thin film process, the reflection coefficient will keep unchange only if the change in the thickness or refractive index satisfied the derived condition presented here. The conclusions are very useful for the hot application in the infra-red thin film or solar thin film today.
In the alaysis of reflection and transmission probability of the particle wave from inhomogeneous potential field, since the Shrodinger equation is similar with the light waveguide equation, such as the time-dependent Schrodinger equation is similar with Fresnel equation while the time-independent Schrodinger equation is similar with Helmhoz equation, it can be regarded that the potential filed equates to refractive index profile, energy eigenvalues equates to light propagation constant, high refractive index section to potential well while low refractive index section to potential barrier. So the analytical transfer matrix method can be also applied to analyzing the Schrodinger equation. Without solving the Schrodinger equation, and with the help of defining the general wavenumber, an exact and general expression for the transmission and reflection probabilities are presented in a very explicit way.
Different from the WKB method and its refined versions, subwaves, which inherently exist in a inhomogeneous systemand is always neglected in the semiclassical approaches, is taken into account, results in the total phase shift of a quantum particle across an arbitrary potential barrier. Moreever, it is not subject to the requirement of the de Broglie wavelength and the range of the particle energy.
As a consequence, the expression obtained here may extensively be applied to many basic quantum phenomena, such as, quantum tunneling, quantum reflection, the time related to a tunneling particle and the resonant tunneling. The parabolic barrier, double barrier with a rectangular well, double barrier with a parabolic well structure and the potential barrier placed in an external field are discussed here.
The reflection and the transmission of light and quantum particle are unified in the expressions, obviously it is very important in physics.
所得公式表明,除了环境与该介质的上下表面之间的折射率之外,反射系数只依赖于主波和子波所累计的总的位相贡献。
与传统的WKBJ比较,由于WKBJ忽略了子波反射,所以是近似的结果而本文得到的结果是精确结果。
根据所得的公式,本文研究了光波在单层介质、双层介质、三层介质、折射率分别呈指数分布、高斯分布、正弦分布的介质的反射和透射情况并对结果进行了讨论。实际上本文的结果可以计算任意折射率分布的介质。
本文对光的反射与透射的研究结果具有广泛的应用前景,如各种具有很高的灵敏度和精确度的传感器或者测量装置。其中最引人注目的应当是可以根据其来指导薄膜工艺中的厚度控制或材料选择,只要厚度的改变或者折射率的改变满足所推导出条件,则反射率与透射率保持不变。这些结果对诸如红外薄膜、太阳能薄膜等热点具有重要意义。
在分析物质波在非均匀势场中的反射及透射几率时,由于薛定谔方程与光波导方程的近似性,含时薛定谔方程类似于菲涅尔方程而与时间无关的薛定谔方程则类似于亥姆霍兹方程,势场分布类似于折射率分布,能量本征值对应于光波的传播常数,高折射率区域相当于势阱而低折射率区域相当于势垒,因此将分析光波导中所应用的解析转移矩阵方法应用于分析薛定谔方程中。最终在没有求解薛定谔方程,而且没有引入任何近似的情况下,通过定义总波数,得到了非常简洁的表达式且物理意义清晰。
在所得公式中,整个的反射与透射几率除了环境、势垒的开始点和结束点的参数外,只依赖于主波和子波总的累积相移。同时公式不受德布罗意波长和能量范围大小的局限。
本文得到的结果可以广泛地应用于许多基本的量子现象,例如,量子隧道效应、量子反射、量子粒子和隧道共振等。利用该结果分别研究了抛物线型单势垒情况、方型双势垒情况、抛物线型双势垒情况以及置于外场中的双势垒情况。
从光波反射系数公式和物质波的反射系数公式来看,两者一个统一的形式,这样就将光波、物质波在非均匀介质中的反射及透射问题统一起来,理所当然具有十分重要的意义。
Since a continuous varying index profile is nothing more than a stack of thin layers which have homogeneous refractive indeices when the thickness of layers is very very small while the number of layers is very very large, in the present investigation, instead of the numerical calculations with transfer matrix approach, the analytical transfer matrix method is employed to determine the reflection coefficient from an inhomogeneous stratified media. With the help of the definition of general wavenumber, which contains both the main waves and the subwaves propagated in the structure with continuously varying index profile, an explicit and exact expression with clear physical insight is obtained without introducing any approximation.
The result shows that, besides the ambience and the refraction indices of both front-surface and back-surface of the structure, the unique dependence of the reflection coefficient is the total phase shift accumulated by the main wave and the subwaves.
The tranditional WKBJ approach is an approximation in which the subwaves refrected from the interior of the structure are neglected while the result presented here is exact.
The reflection from one-layer planar structure, two-layer planaer structure and three-layer structure is studied using the expression. Even more, the refractive index profile is an exponential function, or Gauss function, or periodi sine functions is calculated and the results are discussed. In fact the expression obtained in the paper can be applied to arbitrary refractive index profile.
The study on the refrection and transmission of lights can be adopted in many application, such as high sensitive sensor or measurement instructions. Especially it can be used in the control of thickness of materials selection in the thin film process, the reflection coefficient will keep unchange only if the change in the thickness or refractive index satisfied the derived condition presented here. The conclusions are very useful for the hot application in the infra-red thin film or solar thin film today.
In the alaysis of reflection and transmission probability of the particle wave from inhomogeneous potential field, since the Shrodinger equation is similar with the light waveguide equation, such as the time-dependent Schrodinger equation is similar with Fresnel equation while the time-independent Schrodinger equation is similar with Helmhoz equation, it can be regarded that the potential filed equates to refractive index profile, energy eigenvalues equates to light propagation constant, high refractive index section to potential well while low refractive index section to potential barrier. So the analytical transfer matrix method can be also applied to analyzing the Schrodinger equation. Without solving the Schrodinger equation, and with the help of defining the general wavenumber, an exact and general expression for the transmission and reflection probabilities are presented in a very explicit way.
Different from the WKB method and its refined versions, subwaves, which inherently exist in a inhomogeneous systemand is always neglected in the semiclassical approaches, is taken into account, results in the total phase shift of a quantum particle across an arbitrary potential barrier. Moreever, it is not subject to the requirement of the de Broglie wavelength and the range of the particle energy.
As a consequence, the expression obtained here may extensively be applied to many basic quantum phenomena, such as, quantum tunneling, quantum reflection, the time related to a tunneling particle and the resonant tunneling. The parabolic barrier, double barrier with a rectangular well, double barrier with a parabolic well structure and the potential barrier placed in an external field are discussed here.
The reflection and the transmission of light and quantum particle are unified in the expressions, obviously it is very important in physics.
引文
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[1]. Z. Cao et al., J. Opt.Soc. Am. A, 16, 2209 (1999)
[2]. Z. Cao, L. Qiu and Q. Shen et al., Chin. Phys. Lett., 16, 413 (1999)
[3]. Q. Liu, Z. Cao, and Q. Shen et al., Chin. Phys. Lett., 17, 349 (2000)
[4]. Q. Liu, Z. Cao, and Q. Shen et al, Optical and Quantum Electron., 33, 675 (2001)
[5]. Z. Cao et al., Phys. Rev. A, 63, 054103 (2001)
[6]. Z. Cao et al., J. Opt.Soc. Am. A, 16, 2209 (1999)
[7]. Z. Cao, L. Qiu and Q. Shen et al., Chin. Phys. Lett., 16, 413 (1999)
[8]. Q. Liu, Z. Cao, and Q. Shen et al., Chin. Phys. Lett., 17, 349 (2000)
[9]. Q. Liu, Z. Cao, and Q. Shen et al, Optical and Quantum Electron., 33, 675 (2001)
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