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传输矩阵法[14-15, 19]与矢量法[20]是基于理想的单色平行光假设建立的薄膜光学特性计算方法。然而在理想的假设条件下,模型忽略了薄膜对光束频谱线宽的响应特性。因此,需要将理想条件修正为准单色平行光,建立新型光束与薄膜相互作用模型。
Chen G等人建立了准单色光入射条件下的单层膜光学特性计算方法,其关键在于利用部分相干理论计算薄膜界面处准单色光场的辐照度,并且引入归一化功率谱密度函数来定量地描述入射光束的线宽及其频谱分布。具体方法如下:首先,由波动光学原理推导获得透射光场以及反射光场与入射光场之间数量关系;其次,基于准单色光场的功率谱密度函数以及部分相干理论计算其辐照度;最后,利用光场之间的数量关系以及辐照度计算公式推导出薄膜的透射率、反射率以及吸收率公式。
在实际应用中,研究人员更加关注多层膜的光学特性,准单色光入射条件下多层膜光学特性计算方法可以由单层膜方法推广获得。如图1所示,假设基底表面上的多层膜共有k层,入射介质折射率为N1,基底折射率为NS,第j层薄膜的折射率为Nj,厚度为Lj。
对第k层薄膜进行分析,该膜层界面处光场的分布情形与单层薄膜相同,即下表面仅有出射光场,没有来自基底的入射光场;上表面处有入射光场以及反射光场。基于薄膜辐射特性的部分相干理论[18]获得Tk−1、Rk−1,其中Tk、Rk以及rk取决于基底与第k层膜的光学常数。由等效界面理论可知,第k层薄膜与基底可以由第k−1界面等效,此时对于第k−1层薄膜来说,k−1界面处同样仅有出射光场。因此,重复上述步骤,由基底表面向入射表面递归计算,记录中间变量Tj−1、Rj−1以及rj,最终获得多层膜整体的透射率T1以及反射率R1。
多层膜透射率公式如下:
$$ {T_j} = \frac{{{n_{j + 2}}}}{{{n_j}}}\frac{{{T_{j,j + 1}}{T_{j + 1}}{\beta _j}}}{{1 - 2{Re} (r_{j + 1,j}^*{{{γ }}_0}(\tau ;j)){{\left( {{R_{j + 1}}{\beta _j}{{\beta '}_j}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}} + {R_{j + 1,k}}{R_{j + 1}}{\beta _j}{{\beta '}_j}}} $$ (1) 式中:
$$ {\beta _j} = \exp \left( - \frac{{4\pi {k_j}}}{\lambda }{L_j}\right) $$ (2) 多层膜反射率公式如下:
$$ \begin{split} & R_j=R_{j, j+1}+\left(T_{j+1}+2 {Re}\left\{\frac{r_{j, j+1} t_{j+1, j}^*}{t_{j, j+1}}\left[\frac{\gamma_0(\tau ; j)}{\left(R_{j+1} \beta_j \beta_j^{\prime}\right)^{1 / 2}}-r_{j+1, j}\right]\right\}\right) \times \\ &\frac{T_{j, j+1} R_{j+1} \beta_j \beta_j^{\prime}}{1-2 {Re}\left(r_{j+1, j}^* \gamma_0(\tau ; j)\right)\left(R_{j+1} \beta_j \beta_j^{\prime}\right)^{1 / 2}+R_{j+1, j} R_{j+1} \beta_j \beta_j^{\prime}} \\ \end{split} $$ (3) 公式(1)和公式(3)中:
$$ \gamma_0(\tau , j)=r_{j+1, j}\left(R_{j+1} \beta_j \beta_j^{\prime}\right)^{1 / 2}+\frac{\displaystyle\int_0^{\infty}\left[t_{j, j+1}\left(-r_{j, j+1}+r_j\right)^* / t_{j+1, j}^*\right] S(v) {\rm{d}} v}{\left(\displaystyle\int_0^{\infty}\left|\left(1-r_{j, j+1} r_j\right) / t_{j+1, j}\right|^2 S(v) {\rm{d}} v \displaystyle\int_0^{\infty}\left|\left(-r_{j, j+1}+r_j\right) / t_{j+1, j}\right|^2 S(v) {\rm{d}} v\right)^{1 / 2}}\\ $$ (4) 式中:${t_{i j}}$为界面i处的菲涅耳振幅透射系数,方向为由i层入射至j层;${r_{i j}}$为界面i处的菲涅耳振幅反射系数,方向为由i层入射至j层;$ {r_i} $为i界面处,从第i至最后一层多层膜振幅反射系数;${R_{i j}}$为界面i处的菲涅耳振幅反射系数的平方,方向为由i层入射至j层;${T_{i j}}$为界面i处的菲涅耳振幅透射系数的平方,方向为由j层入射至i层;${R_i}$为界面i处至基底的多层膜反射率;${T_i}$为界面i处至基底的多层膜透射率;$N = n - ik$为薄膜的复折射率,n为折射率,k为消光系数。
公式(4)中,入射光场${E_i}(v)$的归一化功率谱密度函数$S(v)$为:
$$ S(v) = \frac{{{E_i}(v)E_i^*(v)}}{{\displaystyle\int_0^\infty {{E_i}(v)E_i^*(v){\rm{d}}v} }} $$ (5) 在之前的研究中,为了简化分析的复杂度,常利用一个理想化的矩形线型函数表征准单色光束的功率谱密度分布[7, 10]:
$$ S(\nu ) = \left\{ \begin{gathered} \frac{1}{{\Delta v}}\;\;{\text{ }}{\nu _0} - \frac{{\Delta \nu }}{2} < \nu < {\nu _0} - \frac{{\Delta \nu }}{2} \\ 0\;\;\;\;\;\;\;\;{\text{ other}} \\ \end{gathered} \right. $$ (6) 然而,矩形线型函数重点考虑了准单色光束线宽特征,却忽略了光谱能量的分布特征。激光是一种广泛应用的相干光源,其出射光束的归一化功率谱密度函数有两种基本线型分布:高斯线型和洛伦兹线型[21-22],这两种线型函数涵盖了光束的线宽以及光谱能量分布特征。因此,有必要对上述三种线型函数进行对比分析,进一步获得光束线宽与光谱线型对多层膜光学元件的光学特性的影响。
高斯线型函数如下:
$$ S(v) = \dfrac{{2\sqrt {\ln 2} }}{{\sqrt \pi \Delta v}}\exp \left[ { - {{\left( {2\sqrt {\ln 2} \dfrac{{v - {v_0}}}{{\Delta v}}} \right)}^2}} \right] $$ (7) 洛伦兹线型函数如下:
$$ S(v) = \frac{{2{{\left( {\pi \Delta v} \right)}^{ - 1}}}}{{1 + \left( {2 ({{v - {v_0}}})/{{\Delta v}}} \right)}} $$ (8) 式中:${\nu _0}$为谱线的中心频率;$\Delta \nu $为谱线的线宽。$S(v)$也被称为光谱线型。线宽是指谱线的宽度,共有两种表达式,分别对应于波长谱以及频率谱,它们满足如下关系:
$$ \Delta \lambda = \dfrac{c}{{\nu _0^2}}\Delta v $$ (9) 在下文中,光束线宽是指其波长谱表达式$\Delta \lambda $。
由上述分析可知,利用准单色光入射条件下多层膜光学特性计算方法计算多层膜元件的光学特性,需要获得以下参数:光束的频谱特征,包含中心频率、线宽与线型函数;多层膜的结构,包含膜层的厚度与光学常数。区别于传输矩阵法中要求膜层厚度是必须小于光束的相干长度,该方法对于膜层的厚度没有限制条件。
Study on the response characteristics of multilayer optical elements to beam linewidth
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摘要: 多层膜光学元件的设计是基于理想的单色光进行的,然而光源发出的光束均存在光谱线宽,在非单色光条件下工作,元件的光学特性会偏离理论值。为了分析光束线宽对多层膜光学元件的光学特性影响,首先,基于薄膜辐射特性的部分相干理论,提出了准单色光束入射条件下多层膜光学特性计算方法;其次,通过数值模拟实验研究了光束线宽对窄带滤光片光学特性的影响。研究结果表明:随着线宽增大,滤光片透射通带的矩形度逐渐降低,半高全宽先降低后增大,其在滤光片的理论半高全宽附近取得最小值;光谱线型主要改变透射通带的透射率,对于透射谱线的矩形度以及半高全宽影响较小;为了保证窄带滤光片的通带形状,入射光束的线宽应当小于滤光片透射谱线的理论半高全宽的一半,光谱线型应当趋近于矩形线型函数。Abstract:
Objective Multilayer films are critical optical elements for beam energy control in optical systems, and their quality directly influences optical system performance. Ideal monochromatic light serves as the foundation for the design of multilayer optical elements. But there is spectral linewidth in the actual beam. The optical characteristics of the element will differ from the theoretical value and may even result in total failure when operating in non-monochromatic light. In the established convolution model, the calculation technique utilizes monochromatic light conditions to determine the interference superposition of beams in the film, hence disregarding the quasi-monochromatic light interference effect. Based on this, the author proposes a calculation method for the optical properties of multilayer films under quasi-monochromatic light conditions and uses partial coherence theory to calculate the interference superposition of quasi-monochromatic beams. Methods A technique for estimating the optical characteristics of multilayer films in quasi-monochromatic lighting is proposed as a solution to this issue. To quantitatively quantify the linewidth and spectrum distribution of quasi-monochromatic light beams, the normalized power spectral density function is developed. Additionally, partial coherence theory is used to determine the irradiance of quasi-monochromatic light fields at the film interface. This study presents the design of a narrow-band filter with a passband ripple without collapse, a bandwidth of 4.29 nm, a center wavelength of 1 064 nm, and a Rectangle degree of 0.66 (Fig.2). Numerical simulation experiments are used to discuss how substrate thickness, spectral line-shape profile, and beam linewidth affect the optical characteristics of narrow-band filters. Results and Discussions As illustrated in Figure 4, for Gaussian, Lorentz, and rectangular line-shape, the Full Width at Half Maximum (FWHM) drops initially and subsequently increases as linewidth increases. The corresponding lowest values are 3.85 nm, 4.08 nm, and 3.74 nm. And these minimal values correspond to linewidth of 4.5 nm, 2.5 nm, and 5.5 nm, respectively. There is a similar shift trend for the three line-shape conditions for the transmission line's Rectangle Degree (RD). When the linewidth is less than 4 nm, the RD decreases rapidly with the increase of the linewidth. RD steadily diminishes when the linewidth is larger than 4 nm. Under different line-shape conditions, the relationship between the Tmax of the transmission spectrum and the linewidth has a significant difference. The beam linewidth and spectra line-shape profile have an important influence on the shape of the spectral line. The increase of the linewidth will lead to the decrease of the transmittance and the variation of the RD. The spectral line-shape profile establishes the precise link between the transmittance, FWHM, and RD with the linewidth. With the right choice of line width value, the FWHM can obtain the smallest value. Figure 7(a) illustrates how, for a given set of four beam linewidths, Tmax progressively drops as substrate thickness increases. This decline is limited to 0.8% and is dependent on both the substrate's thickness and extinction coefficient. The smaller the extinction coefficient of the substrate, the smaller the decrease. Tmax dramatically drops as linewidth increases when substrate thickness remains constant. The examination of Figure 7(b) demonstrates that, in the case of a constant line width, the FWHM essentially stays constant as the substrate's thickness increases. The FWHM first rises and then falls as the linewidth increases, while the thickness stays constant. It is evident that the transmission spectrum's FWHM, passband shape, and Tmax of the narrow-band filter are all significantly influenced by the beam linewidth, while the substrate's thickness primarily determines the transmission spectrum's passband shape. Conclusions This research offers a calculating technique for the optical properties of multilayer films under the condition of quasi-monochromatic light incidence based on partial coherence theory, which can be used to analyze the response characteristics of multilayer optical elements to beam linewidth. Numerical experiments are used to investigate how substrate thickness and beam linewidth affect narrow-band filter performance. Numerical results demonstrate that the narrowband filter's response characteristics are significantly influenced by the beam linewidth and the power spectral density function's line-shape. The incident beam must satisfy the following requirements in order to guarantee the narrow-band filter's passband form: the beam linewidth must be less than half of the theoretical FWHM of the filter, and the spectral line-shape profile must have a tendency toward a rectangular distribution. This study is informative for the design and application of multilayer optical elements in coherent optical systems. -
图 4 高斯线型、洛伦兹线型与矩形线型条件下,(a)中心波长、(b)透射通带峰值透射率、(c)半高宽度以及(d)矩形度随光束线宽的变化曲线
Figure 4. The curves of (a) central wavelength, (b) peak transmittance of transmission band, (c) full width at half maximum and (d) rectangle degree versus beam linewidth under Gaussian, Lorentzian and rectangular line-shape conditions
图 6 高斯线型条件下,线宽为(a) 0.5 nm;(b) 2.5 nm;(c) 4.5 nm;(d) 9.0 nm时,单层基底1064 nm处透射率随基底厚度的变化曲线
Figure 6. Variation curves of transmittance at 1064 nm with substrate thickness for monolayer substrates with linewidths of (a) 0.5 nm; (b) 2.5 nm; (c) 4.5 nm, and (d) 9.0 nm under the Gaussian line-shape condition
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