光束的矢量性、非傍轴性和像差对光谱移动和光谱开关的影响
|
摘要
众所周知，光谱分析是自然科学中一种重要的研究手段。长期以来，人们进行光谱测量时，总是坚信一个理所当然的假设，即光在自由空间中传输时光谱保持不变。在实际操作过程中，光谱仪与待测对象往往相距一段距离，该假设的正确与否对光谱测量精度的影响是至关重要的。此外，发光体发出的光是在经历了自由空间、透镜和光阑等光学系统后才到达光谱仪的，研究光束通过这些光学系统时光谱的变化规律，具有明确的理论和实际意义。
光在自由空间中传输时光谱保持不变的假设，在许多情况下被证明是正确的。直到1986年，Wolf教授首次证明，仅当光源的光谱相干度满足所谓的定标律时，由这种光源辐射的光在自由空间中传输时才保持光谱不变。自那时起，广大学者对此进行了深入的理论研究和实验证实。1999年，蒲继雄教授等在研究一类典型的部分相干光(谢尔模型光束)被硬边光阑衍射的光谱变化时，发现了一种被称为“光谱开关”的新现象。2002年，Gbur等人将“奇点光学”的研究范围扩展到衍射引起多色完全空间相干光的光谱异变问题，并将完全空间相干光的“光谱开关”归结于“奇点光学”中衍射引起的光谱异变新现象，使“奇点光学”这一现代光学的前沿研究领域有了更为丰富的内涵。本论文在前人工作的基础上，以高功率激光的应用为背景，针对一些尚未完全清楚且有重要意义的问题进行了研究，主要工作内容有：
基于空间-频率域中交叉谱密度函数的传输理论，采用Zernike多项式描述像差光束畸变波前相位，以有球差多色高斯光束被硬边光阑衍射和多色高斯-谢尔模型(GSM)光束通过硬边光阑像差透镜为例，详细研究了像差存在时对光谱移动和光谱开关的影响。结果表明：像差存在时会对光谱移动和光谱开关产生影响，像差会引起实验结果和理想情况下的理论计算间出现差异。相对于无像差情况的光谱，像差会引起光谱蓝移或红移。特别是，像差会对光谱开关产生影响，产生光谱开关的临界位置随像差发生变化。在透镜的几何焦面上，光谱开关效应随球差或像散因子的增加而减弱，且当大于某值时，光谱开关消失；而被硬边光阑衍射的多色高斯光束轴上光谱开关，有时选取恰当的球差因子却可得到最好的效果(光谱开关极小值为零，跃迁量最大)。光束通过含有像差的非理想光学系统的传输变换是实际工作中经常遇见的问题。因此，研究像差对光谱移动和光谱开关的影响对提高光谱测量精度具有重要意义。所得结果可作为光谱异变的实验和理论结果定量上出现差异的一个可能的物理解释。
利用交叉谱密度矩阵和部分相干光的传输公式，首次推导出了多色矢量GSM光束通过硬边光阑透镜光学系统焦平面上光谱表达式。将光谱移动和光谱开关的研究从各向均匀偏振的标量场拓展到非均匀偏振的矢量场。详细讨论了影响光束偏振特性的空间自相关长度、空间互相关长度、偏振片的旋转角度对光谱移动和光谱开关的影响。在插入偏振片前，矢量GSM光束与标量GSM光束传输特性相同，光谱开关临界位置、极小值和跃迁量只依赖于光束的自相关长度，而与互相关长度无关；插入偏振片后，光束的自相关和互相关长度共同影响光谱开关的行为。偏振片的旋转角度对归一化光谱产生影响，可以控制光谱开关的产生与消失。利用上述相同的原理，推导出了PGSM光束和多色GSM光束通过硬边光阑透镜分离光学系统的光强表达式，据此研究了轴上光场的聚焦行为。首次将焦移和焦开关的研究从准单色的标量场拓展到多色矢量场。插入偏振片之前，多色矢量GSM光束通过硬边光阑透镜分离光学系统后，有焦移，但无焦开关；而插入偏振片之后，不仅有焦移，还会产生焦开关。改变偏振片的旋转角度可以控制焦开关的特性。多色矢量完全空间相干光和标量GSM光束可作为特例含于所得结论之中。
透镜是重要和常见的光学元件，且单个透镜的球差是无法完全消除的，所以研究光束通过透镜尤其是球差透镜的光谱变化对其推广应用是非常重要的。本论文将多色GSM光束通过球差透镜后的光谱与通过理想薄透镜和像散透镜后的光谱作了比较，发现通过球差透镜后的光谱不同于通过理想薄透镜和像散透镜后的光谱。通过理想薄透镜和像散透镜后的轴上光谱相对于源光谱只有蓝移，而通过球差透镜后的轴上光谱不仅蓝移，还有红移，在某些特定的位置还将产生光谱开关。产生这些现象的物理原因是通过理想薄透镜和像散透镜后的光谱调制函数是单调的，而通过球差透镜后的光谱调制函数出现了震荡。透镜的球差可以诱导光谱开关的产生。
近年来，随着二极管激光器、微腔激光器和光子晶体等技术的发展，对束宽与波长可相比拟和／或有大发散角的非傍轴光束的研究变得日益重要，引发了精确求解光频电磁波传输的大量理论研究。根据实际工作的需要，从一般的瑞利衍射积分公式出发，推导出了多色完全空间相干光被硬边光阑衍射和用部分空间相干光照明杨氏双缝的光谱表达式。首次讨论了光束的非傍轴性对光谱移动和光谱开关的影响。所得公式在傍轴近似条件下，与原有的傍轴理论结果一致。非傍轴光束的光谱与傍轴近似下的光谱不同，前者相对后者可能蓝移或红移，傍轴近似出现误差主要表现在光谱开关的临界位置发生了变化，而光谱开关的极小值和跃迁量则基本不变。完全空间相干光(高斯光束)光场的非傍轴性由光束束宽和截断参数共同决定。只有光束束宽和截断参数满足某种条件时，傍轴近似计算结果才与非傍轴结果一致。而傍轴近似计算部分空间相干光(GSM)光谱开关的精度主要受光束束宽的影响，源的相对空间相关度对其影响较小。因此，当研究束宽与波长可相比拟的强聚焦激光束和／或有大发散角的多色光的光谱移动和光谱开关时，考虑光束非傍轴性的影响是十分必要的。
以傍轴近似下杨氏双缝实验中多色完全空间相干光(高斯光束)和部分空间相干光(GSM)的光谱为例，对光谱开关是否为“奇点光学”效应这一重要问题做了深入的比较研究。首次明确提出多色场“奇点光学”效应的判据应当是光谱开关出现时的谱强度极小值为零，或等效地，光谱调制函数为零，而不是总光强极小值为零。当谱强度极小值为零时，总光强极小值不一定为零。其物理意义是当产生光谱开关时频率ω_(min)(光谱开关极小值的频率)对应的谱强度为零，位相不确定，这与单色场或准单色场的情况是一致的。光谱开关在总光强极小值附近产生，而ω_(min)位于源光谱中心频率附近。光束的空间相关参数、谱宽、光学系统参数和场点位置都会对谱强度是否为零产生影响。由于本论文研究的是光谱的奇异行为，直接相关的物理量是谱强度，不涉及在相关奇点光学中光谱相干度的奇异行为，因此也不用光谱相干度是否有零值点作为“奇点光学”效应的判据。所得结果可推广用于分析其他类型多色光入射不同光学系统时产生的光谱开关是否属于“奇点光学”效应，对多色场“奇点光学”效应的判断和研究可以澄清一些人们容易模糊的认识。
光在传输过程中与物质相互作用时(大气、水面等)，光的散射也是不可避免的。对粗糙面散射光波的研究在医学检测、地球物理和太空遥感等领域有着重要的民用和军用价值。尽管各种研究方法已日见成熟，散射现象也被人们所熟知，但散射诱导的光谱变化却始终没有在“奇点光学”概念的框架内被充分理解。本论文在稳态散射的框架内，对高斯光束透过表面粗糙的绝缘板散射诱导的远场光谱变化作了研究，着重强调了绝缘板粗糙表面各点高度相对平均高度值最大差值、相对折射率和源光谱谱宽对光谱变化的影响。结果表明：散射可能导致光谱相对源光谱红移和蓝移，某些特定条件下会产生光谱开关。改变折射率的值对光谱开关的跃迁量和最小值影响不大。而光谱开关跃迁量随谱宽的减小而减少，最小值不随谱宽发生变化。折射率越小或谱宽越大，产生光谱开关所需要的绝缘板表面越粗糙。在影响散射诱导光谱变化的诸参数中，表面粗糙度扮演主要角色。完全空间相干的高斯光束透过表面粗糙绝缘板时散射诱导的远场光谱开关属于奇点光学效应。
实际光学系统中，光阑是不可缺少的光学元件。光束通过各种光阑的传输变换也是激光光学的重要研究内容。当有硬边光阑时，若光阑位于入射面上，利用将硬边光阑窗口函数展开为有限个复高斯函数叠加的方法，常可推出近似解析传输公式。但若硬边光阑不在入射面上，而置于复杂光学系统内时，直接利用柯林斯公式遇到困难。利用矩阵分解的原理，可得到光束通过内含硬边光阑ABCD光学系统传输变换的解析方法。此外，光束传输时常常受到多硬边光阑的限制，这使得在微机上计算变得相当困难甚至不可能。本论文首次推导出了高斯光束通过多个硬边光阑ABCD光学系统的近似解析传输公式。所得到的解析公式不仅可以直观地分析光束的传输变换规律、影响光场分布的物理因素及光阑对光束质量的影响等，还可大大节约机时，便于在微机上操作。但由于有限个复高斯函数展开不能很好地拟合硬边光阑的窗口函数，因此存在计算误差，且在近场误差还比较大。针对这一问题，提出了一种改进算法，利用傅立叶级数和复高斯函数构建一个新的函数，用以更精确地拟合硬边光阑窗口函数。通过比较，证实了改进算法的优点是兼顾计算精度和机时，并有较大适用范围。文中所用方法可进一步推广用于研究其它类型光束，对于研究光束通过含硬边光阑光学系统的光谱变化问题也是适用的，具有较为普遍的应用意义。
上述研究内容涉及光场的相干性理论和在传输中因衍射、空间相关性等多种物理机制引起的光谱异常行为以及实验测量等问题，并在光学、计量学和天文学中精密光谱测量和谱选择的光学互连等方面有诱人的应用前景而引起国际和国内光学界的普遍重视。本学位论文拓展了光谱移动和光谱开关的研究范围，对于提高光谱测量精度、合理解释实验与理论结果之间出现的差异以及在非傍轴光场中讨论光谱异变问题等方面具有明确的现实意义，为各种物理机制诱导光谱变化的推广应用提供必要和充分的理论依据。
It is well known that the spectral analysis of radiation is one of the mostimportant analytic methods in science. Implicit in its use is the assumption that thespectrum of light does not change as the radiation propagation in free space. Onlyrecently has it been called into question. Apparently, this assumption is important todetect the spectrum of light source. Otherwise, the spectral detector is usually placedin a distance from the source. Therefore, before the light reaches the detector, it hasactually experienced some processes such as propagation in free space, focused bylenses, diffracted by aperture etc. Studying the spectral changes of beams passingthrough the optical systems not only has a theoretical meaning, but also has anapplied one.
The assumption, which it is refered to as the assumption of spectral invariance onpropagation in free space, has proven by experiment to be justified in the vastmajority of circumstance. In 1986, the first theoretical prediction that the spectrum oflight may differ from the spectrum of the source, and that may change on propagationeven in free space, was made by professor Wolf. He showed that only when thespectral degree of coherence of a source satisfies the so-called scaling law, does thespectrum of the radiation from the source keep the spectral invariance during itspropagation. Conversely, when the source does not satisfy the scaling law, the spectrum of the radiation from the source will change. Since then, the extensivetheoretical and experimental studies have been made on this subject. In 1999, Pu andhis collaborators found a new phenomenon that is defined as a spectral switchtheoretically. In 2002, the study of singular optics was extended to the spectralanomalies of spatially fully coherent polychromatic beams, and the spectral switchwas regarded as a new effect in singular optics. These results broaden the developingfield of singular optics. Based on the work at home and abroad, some original studieshave been done in this dissertation. They are summarized as follows:
By using Zemike polynomials to discribe the wavefront distortion of beams, thespectral behaviors of spherically aberrated polychromatic Gaussian beams diffractedat an aperture and polychromatic GSM beams passing through an aperture lens withaberration are studied based on the propagation law of the cross-spectral densityfunction in the space-frequency. Emphasis is given on the influence of aberration onthe spectral shifts and spectral switches. The aberration-free results are obtained asspecial cases treated in our papers. Detailed numerical calculations and physicalanalysis are presented. It is shown that the spectrum of aberrated polychromaticbeams may be redshifted and blueshifted in comparison with the aberration-free case,and the spectral switch may also take place. At the geometrical focal plane ofpolychromatic GSM beams diffracted at an aperture lens with aberration, theaberration affects the behavior of spectral switches. The spectral minimum S_(min)increases as spherical aberration coefficient C_4 and astigmatic coefficient C_6 increase,the transition height△decreases with increasing C_4 and C_6, the spectral switchvanishes when C_4 and C_6 are greater than a certain value, respectively. As usual,S_(min)≠0, only for the spatially fully coherent and aberration-free case we have S_(min)=0.However, for the case of the on-axis spectrum of spherically aberratedpolychromatic Gaussian beams diffracted at an aperture, S_(min)=0 and△reachesmaximum corresponding with a apt C_4. The results would be useful for understandingsome discrepancies between the experimentally measured spectral behavior andnumerical calculation results without consideration of phase aberrations.
On the basis of the polarization matrix and propagation equation of the cross-spectral density matrix, the spectral changes including the spectral shifts and spectralswitches of vector GSM beams passing through an aperture lens are studied. Theattention has been focused on the influence of correlation and polarizer on thebehavior of the spectral switch at the focal plane. In comparison with the previouspublications, it has been found that the auto-correlation of vector GSM beams affectsthe spectral switch, but the cross-correlation does not if there is no polarizer. Thecritical position u_c shifts toward the z axis, the spectral minimum S_(min) decreases andtransition height△increases as the relative auto-correlationσ_a/σ_1 increases.Specifically, for the spatially fully coherent caseσ_a/σ_1=∞, S_(min) equals to zero and△reaches its maximum value. On the other hand, the polarizer affects the spectralswitch of vector GSM beams, whereas it does not affect the spectral behavior ofscalar GSM beams. In the presence of a polarizer both auto- and cross- correlations ofvector GSM beams affects u_c, S_(min) and△. There exists a certain value of the rotatingangleθof the polarizer, at the corresponding critical position u_c, S_(min) takes itsminimum value and△reaches its maximum value. The above conclusions would beuseful for understanding the spectral behavior of vector GSM beams anddistinguishing it from scalar GSM beams. Otherwise, a detailed study of the focalshift and focal switch of polychromatic vector GSM beams passing through asystemwith the aperture and lens separated is also performed. Before inserting a polarizer thefocal shift appears, there is no focal switch, and after inserting the polarizer the focalswitch can take place. By rotating the polarization angle, the behavior of the focalswitch can be cont rolled.
As we all know, the lens is a usual component in optical system, and sphericalaberration of a single lens is unavoidable, so it is very important to research theinfluencxe of spherical aberration of the lens on the spectral changes. To ourknowledge, there are only two papers dealing with the axial spectra of GSM beamspassing through an aberration-free lens and an astigmatic lens, in which the axialspectra are only blueshifted in comparison with the source spectrum. To provide a theoretical basis for the application of spectral switches, in this dissertation, spectralshift and spectral switching of the on-axis spectrum of vector GSM beams focused bya spherically aberrated lens is studied in detail. Due to oscillation of the spectralmodifier with frequency, the spectrum of vector GSM beams passing through aspherically aberrated lens may be blueshifted and redshifted, in comparison with thesource spectrum, and spectral switching may also take place at the critical positions.
With the development and wide application of diode lasers and practicaltechnology, there exist optical beams of very small spot size and/or large far-fielddivergence angle, for which the paraxial theory is invalid. Starting from thegeneralized Rayleigh-Sommerfeld diffraction integral, the expression for thespectrum of polychromatic Gaussian beams diffracted at an aperture and GSM beamsin Young's experiment are derived, where the influence of nonparaxiality of opticalfield on the spectral shifts and spectral switches is first stressed. The results obtainedin our papers are in consistence with the results under the paraxial approximation.There is a spectral shift in the nonparaxial case relative to the paraxial one and thecritical position, at which the spectral switch occurs, changes. Only when W_0/λ_0 andδsatisfy certain conditions, are the paraxial approximation results of Gaussian beamsdiffracted at an aperture consistent with the nonparaxial ones. The smaller w_0/λ_0 is,the larger the difference between the nonparaxial and paraxial results appears. Theeffect of width on spectral switch of polychromatic GSM beams in Young'sexperiment is the largest of all parameters. Therefore, in studying the spectralbehavior of fully and partially coherent light with width comparable to thewavelength, the effect of beam nonparaxiality has to be taken into consideration.
Within the framework of static light scattering, scattering-induced spectralchanges is studied in the forward-scattered component of polychromatic Gaussianbeams transmitted through a colourless transparent plate with a surface roughnesscomparable with the wavelength. Our main attention is focused on the effect of themaximum deviation of the surface profile from the mean surface line H, relativerefraction index n and bandwidthσ_0 on the spectral shift and spectral switch. The spectrum of the forward-scattered component may be blueshifted and redshifted, incomparison with the source spectrum, and spectral switching may also take place atthe critical positions. The effect of n on the spectral minimum S_(min) and transitionheight△is relatively small, but△decreases with decreasing bandwidthσ_0. H is thegoverning parameter in any case. Scattering-induced spectral switches belong to theeffect in singular optics in the forward-scattered component of polychromaticGaussian beams transmitted through a colourless transparent plate with a surfaceroughness.
The expression for the spectral density of polychromatic Gaussian and GSMbeams propagating through double slits in Young's experiment are derived, whichenables us to study the behavior of spectral switches in Young's experimentilluminated by fully and partially coherent, polychromatic light beams and to judge iftheir spectral switches belong to the effect in singular optics. It is shown that thecriterion for the effect in singular optics with polychromatic light field should beS_(min)=0(S_(min)—minimum of the spectral density at spectral switches) rather thanI_(min)=0(I_(min)—minimum of the total intensity). In Young's experiment illuminated bypolychromatic GSM beams the spectral switches in both near and far zones are notattributed to the effect in singular optics because S_(min)≠0. If the double slits in Young'sexperiment are illuminated by polychromatic Gaussian beams, the spectral switchesin the far zone belong to the effect in singular optics, whereas it is not the case in thenear zone.
By means of the expansion of the window function of the hard aperture into afinite sum of complex Gaussian functions, approximate closed-form propagationequations are derived, in which the successive integral calculations are replaced bythe finite elementary summation operation. For the case that the aperture number isgreater than 3, the straightforward integral of the Collins formula becomes difficulton a personal computer, whereas the method obtained in this dissertation is stillapplicable. Numerical results have found an agreement with those by straightforwardintegral of the Collins formula, but the computing time is greatly reduced. Discrepancies appear only in the vicinity of the aperture-lens caused by Wen'smethod, an improved algorithm is proposed. It is shown that the improved algorithmprovides a good trade-off between the computational accuracy and computer time,and is applicable to the far field and near field.
This field concerns with the coherent theory and spectral anomalies induced byvarious physical reasons. In recent several years, people have generated considerableinterests in studying on spectral changes due to a potentially applicative viewpoint inmany realm such as spectrum-selective interconnects. Applications of these studieshave been shown for the determination of the angular diameter of the stars, angularseparation of a double star etc. The results obtained in this dissertation broaden thestudies on spectral shifts and spectral switches, it is useful to improve the accuracy ofmeasuring spectrum, understand some discrepancies between the experimentallymeasured spectral behavior and theoretically numerical calculation results, and studythe spectral changes in nonparaxial field.
光在自由空间中传输时光谱保持不变的假设，在许多情况下被证明是正确的。直到1986年，Wolf教授首次证明，仅当光源的光谱相干度满足所谓的定标律时，由这种光源辐射的光在自由空间中传输时才保持光谱不变。自那时起，广大学者对此进行了深入的理论研究和实验证实。1999年，蒲继雄教授等在研究一类典型的部分相干光(谢尔模型光束)被硬边光阑衍射的光谱变化时，发现了一种被称为“光谱开关”的新现象。2002年，Gbur等人将“奇点光学”的研究范围扩展到衍射引起多色完全空间相干光的光谱异变问题，并将完全空间相干光的“光谱开关”归结于“奇点光学”中衍射引起的光谱异变新现象，使“奇点光学”这一现代光学的前沿研究领域有了更为丰富的内涵。本论文在前人工作的基础上，以高功率激光的应用为背景，针对一些尚未完全清楚且有重要意义的问题进行了研究，主要工作内容有：
基于空间-频率域中交叉谱密度函数的传输理论，采用Zernike多项式描述像差光束畸变波前相位，以有球差多色高斯光束被硬边光阑衍射和多色高斯-谢尔模型(GSM)光束通过硬边光阑像差透镜为例，详细研究了像差存在时对光谱移动和光谱开关的影响。结果表明：像差存在时会对光谱移动和光谱开关产生影响，像差会引起实验结果和理想情况下的理论计算间出现差异。相对于无像差情况的光谱，像差会引起光谱蓝移或红移。特别是，像差会对光谱开关产生影响，产生光谱开关的临界位置随像差发生变化。在透镜的几何焦面上，光谱开关效应随球差或像散因子的增加而减弱，且当大于某值时，光谱开关消失；而被硬边光阑衍射的多色高斯光束轴上光谱开关，有时选取恰当的球差因子却可得到最好的效果(光谱开关极小值为零，跃迁量最大)。光束通过含有像差的非理想光学系统的传输变换是实际工作中经常遇见的问题。因此，研究像差对光谱移动和光谱开关的影响对提高光谱测量精度具有重要意义。所得结果可作为光谱异变的实验和理论结果定量上出现差异的一个可能的物理解释。
利用交叉谱密度矩阵和部分相干光的传输公式，首次推导出了多色矢量GSM光束通过硬边光阑透镜光学系统焦平面上光谱表达式。将光谱移动和光谱开关的研究从各向均匀偏振的标量场拓展到非均匀偏振的矢量场。详细讨论了影响光束偏振特性的空间自相关长度、空间互相关长度、偏振片的旋转角度对光谱移动和光谱开关的影响。在插入偏振片前，矢量GSM光束与标量GSM光束传输特性相同，光谱开关临界位置、极小值和跃迁量只依赖于光束的自相关长度，而与互相关长度无关；插入偏振片后，光束的自相关和互相关长度共同影响光谱开关的行为。偏振片的旋转角度对归一化光谱产生影响，可以控制光谱开关的产生与消失。利用上述相同的原理，推导出了PGSM光束和多色GSM光束通过硬边光阑透镜分离光学系统的光强表达式，据此研究了轴上光场的聚焦行为。首次将焦移和焦开关的研究从准单色的标量场拓展到多色矢量场。插入偏振片之前，多色矢量GSM光束通过硬边光阑透镜分离光学系统后，有焦移，但无焦开关；而插入偏振片之后，不仅有焦移，还会产生焦开关。改变偏振片的旋转角度可以控制焦开关的特性。多色矢量完全空间相干光和标量GSM光束可作为特例含于所得结论之中。
透镜是重要和常见的光学元件，且单个透镜的球差是无法完全消除的，所以研究光束通过透镜尤其是球差透镜的光谱变化对其推广应用是非常重要的。本论文将多色GSM光束通过球差透镜后的光谱与通过理想薄透镜和像散透镜后的光谱作了比较，发现通过球差透镜后的光谱不同于通过理想薄透镜和像散透镜后的光谱。通过理想薄透镜和像散透镜后的轴上光谱相对于源光谱只有蓝移，而通过球差透镜后的轴上光谱不仅蓝移，还有红移，在某些特定的位置还将产生光谱开关。产生这些现象的物理原因是通过理想薄透镜和像散透镜后的光谱调制函数是单调的，而通过球差透镜后的光谱调制函数出现了震荡。透镜的球差可以诱导光谱开关的产生。
近年来，随着二极管激光器、微腔激光器和光子晶体等技术的发展，对束宽与波长可相比拟和／或有大发散角的非傍轴光束的研究变得日益重要，引发了精确求解光频电磁波传输的大量理论研究。根据实际工作的需要，从一般的瑞利衍射积分公式出发，推导出了多色完全空间相干光被硬边光阑衍射和用部分空间相干光照明杨氏双缝的光谱表达式。首次讨论了光束的非傍轴性对光谱移动和光谱开关的影响。所得公式在傍轴近似条件下，与原有的傍轴理论结果一致。非傍轴光束的光谱与傍轴近似下的光谱不同，前者相对后者可能蓝移或红移，傍轴近似出现误差主要表现在光谱开关的临界位置发生了变化，而光谱开关的极小值和跃迁量则基本不变。完全空间相干光(高斯光束)光场的非傍轴性由光束束宽和截断参数共同决定。只有光束束宽和截断参数满足某种条件时，傍轴近似计算结果才与非傍轴结果一致。而傍轴近似计算部分空间相干光(GSM)光谱开关的精度主要受光束束宽的影响，源的相对空间相关度对其影响较小。因此，当研究束宽与波长可相比拟的强聚焦激光束和／或有大发散角的多色光的光谱移动和光谱开关时，考虑光束非傍轴性的影响是十分必要的。
以傍轴近似下杨氏双缝实验中多色完全空间相干光(高斯光束)和部分空间相干光(GSM)的光谱为例，对光谱开关是否为“奇点光学”效应这一重要问题做了深入的比较研究。首次明确提出多色场“奇点光学”效应的判据应当是光谱开关出现时的谱强度极小值为零，或等效地，光谱调制函数为零，而不是总光强极小值为零。当谱强度极小值为零时，总光强极小值不一定为零。其物理意义是当产生光谱开关时频率ω_(min)(光谱开关极小值的频率)对应的谱强度为零，位相不确定，这与单色场或准单色场的情况是一致的。光谱开关在总光强极小值附近产生，而ω_(min)位于源光谱中心频率附近。光束的空间相关参数、谱宽、光学系统参数和场点位置都会对谱强度是否为零产生影响。由于本论文研究的是光谱的奇异行为，直接相关的物理量是谱强度，不涉及在相关奇点光学中光谱相干度的奇异行为，因此也不用光谱相干度是否有零值点作为“奇点光学”效应的判据。所得结果可推广用于分析其他类型多色光入射不同光学系统时产生的光谱开关是否属于“奇点光学”效应，对多色场“奇点光学”效应的判断和研究可以澄清一些人们容易模糊的认识。
光在传输过程中与物质相互作用时(大气、水面等)，光的散射也是不可避免的。对粗糙面散射光波的研究在医学检测、地球物理和太空遥感等领域有着重要的民用和军用价值。尽管各种研究方法已日见成熟，散射现象也被人们所熟知，但散射诱导的光谱变化却始终没有在“奇点光学”概念的框架内被充分理解。本论文在稳态散射的框架内，对高斯光束透过表面粗糙的绝缘板散射诱导的远场光谱变化作了研究，着重强调了绝缘板粗糙表面各点高度相对平均高度值最大差值、相对折射率和源光谱谱宽对光谱变化的影响。结果表明：散射可能导致光谱相对源光谱红移和蓝移，某些特定条件下会产生光谱开关。改变折射率的值对光谱开关的跃迁量和最小值影响不大。而光谱开关跃迁量随谱宽的减小而减少，最小值不随谱宽发生变化。折射率越小或谱宽越大，产生光谱开关所需要的绝缘板表面越粗糙。在影响散射诱导光谱变化的诸参数中，表面粗糙度扮演主要角色。完全空间相干的高斯光束透过表面粗糙绝缘板时散射诱导的远场光谱开关属于奇点光学效应。
实际光学系统中，光阑是不可缺少的光学元件。光束通过各种光阑的传输变换也是激光光学的重要研究内容。当有硬边光阑时，若光阑位于入射面上，利用将硬边光阑窗口函数展开为有限个复高斯函数叠加的方法，常可推出近似解析传输公式。但若硬边光阑不在入射面上，而置于复杂光学系统内时，直接利用柯林斯公式遇到困难。利用矩阵分解的原理，可得到光束通过内含硬边光阑ABCD光学系统传输变换的解析方法。此外，光束传输时常常受到多硬边光阑的限制，这使得在微机上计算变得相当困难甚至不可能。本论文首次推导出了高斯光束通过多个硬边光阑ABCD光学系统的近似解析传输公式。所得到的解析公式不仅可以直观地分析光束的传输变换规律、影响光场分布的物理因素及光阑对光束质量的影响等，还可大大节约机时，便于在微机上操作。但由于有限个复高斯函数展开不能很好地拟合硬边光阑的窗口函数，因此存在计算误差，且在近场误差还比较大。针对这一问题，提出了一种改进算法，利用傅立叶级数和复高斯函数构建一个新的函数，用以更精确地拟合硬边光阑窗口函数。通过比较，证实了改进算法的优点是兼顾计算精度和机时，并有较大适用范围。文中所用方法可进一步推广用于研究其它类型光束，对于研究光束通过含硬边光阑光学系统的光谱变化问题也是适用的，具有较为普遍的应用意义。
上述研究内容涉及光场的相干性理论和在传输中因衍射、空间相关性等多种物理机制引起的光谱异常行为以及实验测量等问题，并在光学、计量学和天文学中精密光谱测量和谱选择的光学互连等方面有诱人的应用前景而引起国际和国内光学界的普遍重视。本学位论文拓展了光谱移动和光谱开关的研究范围，对于提高光谱测量精度、合理解释实验与理论结果之间出现的差异以及在非傍轴光场中讨论光谱异变问题等方面具有明确的现实意义，为各种物理机制诱导光谱变化的推广应用提供必要和充分的理论依据。
It is well known that the spectral analysis of radiation is one of the mostimportant analytic methods in science. Implicit in its use is the assumption that thespectrum of light does not change as the radiation propagation in free space. Onlyrecently has it been called into question. Apparently, this assumption is important todetect the spectrum of light source. Otherwise, the spectral detector is usually placedin a distance from the source. Therefore, before the light reaches the detector, it hasactually experienced some processes such as propagation in free space, focused bylenses, diffracted by aperture etc. Studying the spectral changes of beams passingthrough the optical systems not only has a theoretical meaning, but also has anapplied one.
The assumption, which it is refered to as the assumption of spectral invariance onpropagation in free space, has proven by experiment to be justified in the vastmajority of circumstance. In 1986, the first theoretical prediction that the spectrum oflight may differ from the spectrum of the source, and that may change on propagationeven in free space, was made by professor Wolf. He showed that only when thespectral degree of coherence of a source satisfies the so-called scaling law, does thespectrum of the radiation from the source keep the spectral invariance during itspropagation. Conversely, when the source does not satisfy the scaling law, the spectrum of the radiation from the source will change. Since then, the extensivetheoretical and experimental studies have been made on this subject. In 1999, Pu andhis collaborators found a new phenomenon that is defined as a spectral switchtheoretically. In 2002, the study of singular optics was extended to the spectralanomalies of spatially fully coherent polychromatic beams, and the spectral switchwas regarded as a new effect in singular optics. These results broaden the developingfield of singular optics. Based on the work at home and abroad, some original studieshave been done in this dissertation. They are summarized as follows:
By using Zemike polynomials to discribe the wavefront distortion of beams, thespectral behaviors of spherically aberrated polychromatic Gaussian beams diffractedat an aperture and polychromatic GSM beams passing through an aperture lens withaberration are studied based on the propagation law of the cross-spectral densityfunction in the space-frequency. Emphasis is given on the influence of aberration onthe spectral shifts and spectral switches. The aberration-free results are obtained asspecial cases treated in our papers. Detailed numerical calculations and physicalanalysis are presented. It is shown that the spectrum of aberrated polychromaticbeams may be redshifted and blueshifted in comparison with the aberration-free case,and the spectral switch may also take place. At the geometrical focal plane ofpolychromatic GSM beams diffracted at an aperture lens with aberration, theaberration affects the behavior of spectral switches. The spectral minimum S_(min)increases as spherical aberration coefficient C_4 and astigmatic coefficient C_6 increase,the transition height△decreases with increasing C_4 and C_6, the spectral switchvanishes when C_4 and C_6 are greater than a certain value, respectively. As usual,S_(min)≠0, only for the spatially fully coherent and aberration-free case we have S_(min)=0.However, for the case of the on-axis spectrum of spherically aberratedpolychromatic Gaussian beams diffracted at an aperture, S_(min)=0 and△reachesmaximum corresponding with a apt C_4. The results would be useful for understandingsome discrepancies between the experimentally measured spectral behavior andnumerical calculation results without consideration of phase aberrations.
On the basis of the polarization matrix and propagation equation of the cross-spectral density matrix, the spectral changes including the spectral shifts and spectralswitches of vector GSM beams passing through an aperture lens are studied. Theattention has been focused on the influence of correlation and polarizer on thebehavior of the spectral switch at the focal plane. In comparison with the previouspublications, it has been found that the auto-correlation of vector GSM beams affectsthe spectral switch, but the cross-correlation does not if there is no polarizer. Thecritical position u_c shifts toward the z axis, the spectral minimum S_(min) decreases andtransition height△increases as the relative auto-correlationσ_a/σ_1 increases.Specifically, for the spatially fully coherent caseσ_a/σ_1=∞, S_(min) equals to zero and△reaches its maximum value. On the other hand, the polarizer affects the spectralswitch of vector GSM beams, whereas it does not affect the spectral behavior ofscalar GSM beams. In the presence of a polarizer both auto- and cross- correlations ofvector GSM beams affects u_c, S_(min) and△. There exists a certain value of the rotatingangleθof the polarizer, at the corresponding critical position u_c, S_(min) takes itsminimum value and△reaches its maximum value. The above conclusions would beuseful for understanding the spectral behavior of vector GSM beams anddistinguishing it from scalar GSM beams. Otherwise, a detailed study of the focalshift and focal switch of polychromatic vector GSM beams passing through asystemwith the aperture and lens separated is also performed. Before inserting a polarizer thefocal shift appears, there is no focal switch, and after inserting the polarizer the focalswitch can take place. By rotating the polarization angle, the behavior of the focalswitch can be cont rolled.
As we all know, the lens is a usual component in optical system, and sphericalaberration of a single lens is unavoidable, so it is very important to research theinfluencxe of spherical aberration of the lens on the spectral changes. To ourknowledge, there are only two papers dealing with the axial spectra of GSM beamspassing through an aberration-free lens and an astigmatic lens, in which the axialspectra are only blueshifted in comparison with the source spectrum. To provide a theoretical basis for the application of spectral switches, in this dissertation, spectralshift and spectral switching of the on-axis spectrum of vector GSM beams focused bya spherically aberrated lens is studied in detail. Due to oscillation of the spectralmodifier with frequency, the spectrum of vector GSM beams passing through aspherically aberrated lens may be blueshifted and redshifted, in comparison with thesource spectrum, and spectral switching may also take place at the critical positions.
With the development and wide application of diode lasers and practicaltechnology, there exist optical beams of very small spot size and/or large far-fielddivergence angle, for which the paraxial theory is invalid. Starting from thegeneralized Rayleigh-Sommerfeld diffraction integral, the expression for thespectrum of polychromatic Gaussian beams diffracted at an aperture and GSM beamsin Young's experiment are derived, where the influence of nonparaxiality of opticalfield on the spectral shifts and spectral switches is first stressed. The results obtainedin our papers are in consistence with the results under the paraxial approximation.There is a spectral shift in the nonparaxial case relative to the paraxial one and thecritical position, at which the spectral switch occurs, changes. Only when W_0/λ_0 andδsatisfy certain conditions, are the paraxial approximation results of Gaussian beamsdiffracted at an aperture consistent with the nonparaxial ones. The smaller w_0/λ_0 is,the larger the difference between the nonparaxial and paraxial results appears. Theeffect of width on spectral switch of polychromatic GSM beams in Young'sexperiment is the largest of all parameters. Therefore, in studying the spectralbehavior of fully and partially coherent light with width comparable to thewavelength, the effect of beam nonparaxiality has to be taken into consideration.
Within the framework of static light scattering, scattering-induced spectralchanges is studied in the forward-scattered component of polychromatic Gaussianbeams transmitted through a colourless transparent plate with a surface roughnesscomparable with the wavelength. Our main attention is focused on the effect of themaximum deviation of the surface profile from the mean surface line H, relativerefraction index n and bandwidthσ_0 on the spectral shift and spectral switch. The spectrum of the forward-scattered component may be blueshifted and redshifted, incomparison with the source spectrum, and spectral switching may also take place atthe critical positions. The effect of n on the spectral minimum S_(min) and transitionheight△is relatively small, but△decreases with decreasing bandwidthσ_0. H is thegoverning parameter in any case. Scattering-induced spectral switches belong to theeffect in singular optics in the forward-scattered component of polychromaticGaussian beams transmitted through a colourless transparent plate with a surfaceroughness.
The expression for the spectral density of polychromatic Gaussian and GSMbeams propagating through double slits in Young's experiment are derived, whichenables us to study the behavior of spectral switches in Young's experimentilluminated by fully and partially coherent, polychromatic light beams and to judge iftheir spectral switches belong to the effect in singular optics. It is shown that thecriterion for the effect in singular optics with polychromatic light field should beS_(min)=0(S_(min)—minimum of the spectral density at spectral switches) rather thanI_(min)=0(I_(min)—minimum of the total intensity). In Young's experiment illuminated bypolychromatic GSM beams the spectral switches in both near and far zones are notattributed to the effect in singular optics because S_(min)≠0. If the double slits in Young'sexperiment are illuminated by polychromatic Gaussian beams, the spectral switchesin the far zone belong to the effect in singular optics, whereas it is not the case in thenear zone.
By means of the expansion of the window function of the hard aperture into afinite sum of complex Gaussian functions, approximate closed-form propagationequations are derived, in which the successive integral calculations are replaced bythe finite elementary summation operation. For the case that the aperture number isgreater than 3, the straightforward integral of the Collins formula becomes difficulton a personal computer, whereas the method obtained in this dissertation is stillapplicable. Numerical results have found an agreement with those by straightforwardintegral of the Collins formula, but the computing time is greatly reduced. Discrepancies appear only in the vicinity of the aperture-lens caused by Wen'smethod, an improved algorithm is proposed. It is shown that the improved algorithmprovides a good trade-off between the computational accuracy and computer time,and is applicable to the far field and near field.
This field concerns with the coherent theory and spectral anomalies induced byvarious physical reasons. In recent several years, people have generated considerableinterests in studying on spectral changes due to a potentially applicative viewpoint inmany realm such as spectrum-selective interconnects. Applications of these studieshave been shown for the determination of the angular diameter of the stars, angularseparation of a double star etc. The results obtained in this dissertation broaden thestudies on spectral shifts and spectral switches, it is useful to improve the accuracy ofmeasuring spectrum, understand some discrepancies between the experimentallymeasured spectral behavior and theoretically numerical calculation results, and studythe spectral changes in nonparaxial field.
引文
1.E.Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett. 1986, 56: 1370-1372.
2.G.M.Morris and D. Faklis, Effects of source correlation on the spectrum of light, Opt.Commun.1987,62:5-11.
3.D. Faklis and G.M.Morris, Spectral shifts produced by source correlations, Opt. Lett. 1988,13:4-6.
4.M.F.Bocko, D.H.Douglass, and R.S.Knox, Observation of frequency shifts of spectral lines due to source correlations. Phys. Rev. Lett. 1987, 58:2649-2651.
5.D.L.Jacobson, S.A.Werner and H. Rauch, Spectral modulation and squeezing at high-order neutron interferences. Phys. Rev. A. 1994,49:3196-3202.
6.E.Wolf, D F V James, Correlation-induced spectral changes. Rep. Prog. Phys. 1996,59:771-818.
7. E.Wolf, Non-cosmological redshift of spectral lines.Nature,1987,326:363-365.
8.Z.Dacic and E.Wolf, Changes in the spectrum of partially coherent light propagation in free space. J. Opt.Soc.Am.A 1988, 5: 1118-1126.
9.H.C.Kandpal, J.S.Vaishya,K.Saxena,etc, Intensity distribution across a source from spectral measurements. J.Mod.Opt. 1995,42: 455-464.
10.C.Palma, G. Cincotti and G. Guattari, Spectral shift of a Gaussian Schell-model beam beyond a thin lens. IEEE J. Quant. Electron. 1998, 34:378-383.
11.Pu J, Zhang H, Nemoto S. Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt Commun 1999; 162: 57-63.
12.G. Gbur, T D Visser, E Wolf, Anomalous behavior of spectra near phase singularities of focused waves. Phys. Rev. Lett. 2002,88:013901-1-4
13. J. Foley, E. Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J. Opt.Soc.Am.A 2002, 19: 2510-2516.
14. H.C. Kandpal. Experimental observation of the phenomenon of spectral switch. J. Opt. A: Pure and Appl. Opt. 2001, 3: 296-299.
15.H.C. Kandpal, S. Anand, J.S. Vaishya. Experimental observation of phenomenon of spectral switch for a class of partially coherent light. IEEE J. Quant. Electron. 2002, 38: 336-339.
16. G .Popescu, A .Dogariu, Spectral anomalies at wave-front dislocations.Phys. Rev. Lett. 2002,88:183902-1
17.B.K.Yadav, S.A.M.Rizvi, H.C. Kandpal. Experimental observation of the phenomenon of spectral switch. J. Opt. A: Pure and Appl. Opt. 2006, 8: 72-76.
18.B.K.Yadav, S.A.M.Rizvi, S.Raman, R.Mehrotra, H.C.Kandpal. Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture. Opt. Commun. 2007, 269:253-260.
19.K.Vyacheslav, Polyanskii, V.Oleg, Angelsky and V.Peter. Scattering-induced spectral changes as a singular optical effect. Optica applicata. 2002,XXXII: 843- 848.
20 A. A. Tovar, Production and propagation of cylindrically polarized Laguerre - Gaussian laser beams. J Opt Soc Am A. 1998,15: 2705-2711.
21 C.J.R.Sheppard, High-aperture beams. J Opt Soc Am A. 2001,18:1579-1587.
22. A.Ciattoni, B.Crosignani and P.Di Porto, Vectorial analytical description of propagation of a highly nonparaxial beam. Opt Commun. 2002,202:17-20.
23. R.Borghi and M.Santarsiero,Nonparaxial propagation of spirally polarized optical beams. J Opt Soc Am A. 2004,21: 2029-2037.
24. G. Zhao, Y. Peng, B. Lü, Spectral changes of vector Gaussian Schell- model beams passing through an aperture lens. Opt.and Laser Tech. 2006. 38:101-106.
25. B. Lu, and K.Duan, Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture. Opt. Lett. 2003,28:2440-2442.
26.K.Duan and B. Lü, Partially coherent nonparaxial beams Opt. Lett. 2004, 29:800-802.
27.E. Wolf, T. Shirai, H. Chen, W. Wang. Coherence filters and their uses: 1.Basic theory and examples. J. Mod. Opt. 1997, 44: 1345-1353.
28.T. Shirai, E. Wolf, H. Chen, W. Wang. Coherence filters and their uses: 2.One-dimensional realizations. J. Mod. Opt. 1998,45: 799-816.
29. S.A Ponomarenko, E.Wolf, Spectral anomalies in a Fraunhofer diffraction pattern. Opt.Lea. 2002,27:1211-1213.
30. J.Pu, C.Cai, S.Nemoto, Spectral anomalies in Young's double-slit interference experiment. Optics Express.2004,21:5131-5139.
31. H.C.Kandpal, W.Aiay, J.S.Vaishya, et.al., Application of Spatial-coherent spectroscopy for determining the angular diameters of stars feasibility experiment. Indian J.Pure Appl.Phys. 1998,36:665-674.
32. D.F.V.James, H.C.Kandpal and E.Wolf, A new method for determining the angular separation of double stars. Astrophys.J. 1995,445:406-410.
33. T.Yatagai, S.Kawai and H.Huang, Optical computing and interconnects. Proc. IEEE. 1996,84:825-852.
34. M.V.Berry, Exploring the colors of dark light. New J.Phys. 2002, 4:74.1-74.14.
35. M.V.Berry, Colored phase singularites. New J.Phys. 2002, 4:66.1-66.14.
36. G. Gbur, T D Visser, E Wolf, Singular behavior of the spectrum in the neighborhood of focus. J Opt Soc Am A. 2002, 19:1694-1700.
37. M. Born, E. Wolf, Principles of Optics. Eingland: Cambrige University Press, 7thed., 1999.
38. G P. Agrawal, E. Wolf, Propagation-induced polarization changes in partially coherent optical beams. J Opt Soc Am A 2000; 17: 2019-2023.
39. F.Gori, M.Santarsiero, G.Piquero,et al: partially polarized Gaussian Schell-model beams. J. Opt.A: Pure Appl. Opt. 2001,3: 1-9.
40. G.Zhao, X.Xiao, B.Lu, Effect of astigmatism on spectral switches of partially coherent beams. Chinese Physics. 2004,13: 2064-2070.
41. G.Zhao, B.Lu. Influence of aperture lens with spherical aberration on the spectral behavior of polychromatic Gaussian Schell-model beams. Optik. 2004, 115: 181-185:
42．赵光普，肖希，吕百达．光场的非傍轴性对光谱移动和光谱开关的影响．物理学报．2007，56(8)．
43. Z.Wang, Z.Zhang, Z.Xu, et al. Spectral and temporal properties of ultrashort light pulse in the far zone. Opt. Commun. 1996,123: 5-10.
44.B.Lü, L.Pan, Propagation properties of ultrashort pulsed Bessel beams in dispersive media. J. Opt. Soc. Am. A. 2003, 20: 582-587.
45.M.S.Soskin, M.V.Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier, Amsterdam) 2001,42:219-276.
46.F. Zernike, Physica 5 (1938) 785, see also, Proc. Phys. Soc. (London) 61 (1948) 158.
47.H.Roychowdhury, E.Wolf , Young's interference experiment with light of any state of coherence and of polarization. Opt Commun. 2005, 252: 268-274.
48.J.Pu, Spectral shifts of partially coherent light produced by passing through an annular aperture. J.Opt.(Paris) 1993,24:141-144.
49.J.D.Farina, L.M.Narducci and E.Collet, Generation of highly directional beams from a globally incoherent source. Opt. Commun. 1980, 32:203-208.
50.M.Bertolotti, F.Scudieri and S. Verginelli, Spatial coherent of light scattered by media with large correlation length of refractive index fluntuation. 1976, 15:1842 -1844.
1. A.T.Friberg and J.Turunen, Imaging of Gaussian Schell-model sources. J Opt Soc AreA, 1988, 5: 713-719.
2. L.Mandel and E.Wolf, Optical Coherence and Quantum Optics, Cambridge University Press. Cambridge, 1995.
3. F. Gori, M.Santarsiero, S.Vicalvi et al, Beam coherence-polarization matrix J. Opt.A: Pure Appl. Opt 1998, 7:941-51
4. B.Lu and K.Duan, Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture, Opt.Lett. 2003, 28: 2440-2442.
5. M.J. Bastianns. The Wigner distribution function applied to optical signals and systems. Opt. Commun. 1978, 25: 26-30.
6. M.J. Bastianns. Wigner distribution function and its application to first-order optics. J. Opt. Soc. Am. 1979, 69: 1710-1716.
7. M.J. Bastianns. Application of the Wigner distribution function to partially coherent light. J. Opt. Soc. Am. A 1986, 3: 1227-1238.
8. H.H.Hopkins, Tneory of Aberrration, Oxford University, London. 1995.
9. W.T.Welford, Aberration of the symmetric Optical system, Academatic, New York, 1974.
10. M. Born, E. Wolf, Principles of Optics. Eingland: Cambrige University Press, 7thed., 1999.
11 T. Yura, S.G Hanson. Optical beam wave propagation through complex optical systems. J Opt Soc Am A 1987, 4: 1931-1948.
12 A. Gerrard, J.M. Burch. Introduction to Matrix Methods in Opticss. London, New York: John Wiley & Sons, 1975.
13 A Belanger. Beam propagation and ABCD ray matrices. Opt. Lett. 1991, 16: 196-198
14 A.A. Tovar. Generalized beam matrices: Application to diffraction analysis. 1996, J. Opt. Soc. Am. A 14: 2239-2246.
15 吕百达，激光光学，高等教育出版社，2003．
16 J.W. Goodman. Introduction to Fourier Optics. London. New York: John Willy & Sons,1974
17 S.A. Colins Jr. Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 1970, 60: 1168-1177.
18 R.K.Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, Calif., 1966.
19 J W.Goodman, Introduction to Fourier Optics , McGraw-Hill Book Company, Inc. New York, 1968.
20 K .Duan and B. Lu, Propagation of Hermite- and Laguerre-Gaussian beams beyond the paraxial approximation. J Opt Soc Am A. 2005. 22. 1976-1980.
21 W.H.Carter and E. Wolf, Coherence and radiometry with quasihomogeneous planar sources, J Opt Soc Am A. 1977,67: 785-796.
22 E.Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett. 1986, 56: 1370-1372.
1. R. Bariheau, M. Rioux. Influence of speckle on laser range finders. Appl. Opt. 1991, 30: 2873-2878. 2.P. Ostlund, A.T. Friberg. Radiometry and far-zone polarization for partially coherent electromagnetic planar sources. Opt. Commun. 2001,197: 1-8.
3 E.Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett. 1986, 56: 1370-1372.
4 E .Wolf and D F V Jamest,Correlation-induced spectral changes. Rep.Prog.Phys. 1996,59:771-818.
5 J. T. Foley, The effect of an aperture on the spectrum of partially coherent light. Opt. Commun. 1990,75 :347
6 Pu J, Zhang H, Nemoto S. Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt Commun 1999; 162: 57-63.
7 J. T. Foley, E.Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J Opt Soc Am A2002; 19: 2510-2516.
8 G P. Agrawal, E. Wolf, Propagation-induced polarization changes in partially coherent optical beams. J Opt Soc Am A 2000; 17: 2019-2023.
9 F.Gori, M.Santarsiero, G.Piquero,et al: partially polarized Gaussian Schell-model beams. J. Opt. A: Pure Appl. Opt. 2001,3: 1-9
10 T.Friberg ,J.Turunen , Imaging of Gaussian Schell-model sources. J Opt Soc Am A 1988; 5: 713-719
11 A. E. Siegman, Analysis of laser beam quality degradation caused by quartic phase aberration. Appl.Opt. 1993, 32: 5893-5901.
12 L .Mandel and E.Wolf, Optical Coherence and Quantum Optics, Cambridge University Press. Cambridge, 1995.
13 Y.Li, E.Wolf. Focal shifts in diffracted converging spherical waves. Opt. Commun. 1981,39:211-215
14 W.H.Carter, M.F. Aburdence. Focal shift in Laguerre-Gaussian beams.J. Opt. Soc. Am 1987,4:1949-1952.
15 Y.Li and H.Platzer. An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems. Opt. Acta, 1983,30(10):1621-1643.
16 M.Martinez-Corral, V. Climent. Focal switch: a new effect in low-Fresnel-number systems. App1.Opt. 1996,35:24-27.
17 Y.Li. Focal shift and focal switch in dual-focus systems J.opt.Soc.Am.A. 1997,14:1297-1304.
18 B.Lu and B. Zhang. Focusing of a Gaussian Schell-model beam through a circular lens. J.Mod.Opt. 1995,42:289-298.
19 R.Borrghi,M.Santarsiero,S.Vicalvi. Focal shift of focused fiat-topped beams. Opt. Commun. 1998,154:243-248.
20 彭润伍，吕百达。厄米-高斯光束的焦开关。物理学报2003，52：2795-2802
21 F. Gori, Matrix treatment for partially polarized, partially coherent beams. Opt.Lett. 1998,23:241-243
22 F .Gori, M .Santarsiero, S .Vicalvi et al. Beam coherence-polarization matrix. Pure Appl. Opt. 1998,7:941-951
1 J. Pu, H.Zhang, S.Nemoto, Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt. Commun. 1999,162:57-63.
2 J .Pu, Nemoto.S: Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture. IEEE J.Quant.Electron. 2000,36:1407-1411.
3 B.Lü, L.Pan, Spectral switching of Gaussian-Schell model beams passing through an aperture lens. IEEE J. Quant. Electron. 2002, 38: 340-344.
4. H C Kandpal, Experimental observation of the phenomenon of spectral switch. J. Opt. A: Pure and Appl. Opt. 2001 3: 296-299.
5 H C Kandpal, S.Anand, J.S Vaishya, Experimental observation of phenomenon of spectral switch for a class of partially coherent light. IEEE. J. Quant. Electron. 2002, 38:336-339.
6 G. Gbur, T D Visser, E Wolf, Anomalous behavior of spectra near phase singularities of focused waves. Phys. Rev. Lett. 2002,88:013901-1-4
7 S.A Ponomarenko, Wolf E: Spectral anomalies in a Fraunhofer diffraction pattern. Opt.Lett. 2002,27:1211-1213.
8 J. Foley, E. Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J. Opt.Soc.Am.A 2002,192510-2516.
9 S.A Ponomarenko, E.Wolf, Spectral anomalies in a Fraunhofer diffraction pattern. OptXett. 2002,27:1211-1213.
10 J.Pu, C.Cai, S.Nemoto, Spectral anomalies in Young's double-slit interference experiment. Optics Express.2004,21:5131-5139.
11 B.K.Yadav, S.A.M.Rizvi, S.Raman, R.Mehrotra, H.C.Kandpal. Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture. Opt. Commun. 2007, 269:253-260.
12 L.Mandel and E.Wolf. Optical Coherence and Quantum Optics(Cambridge U.Press,Cambridge,UK) 1995.
13 A. E. Siegman, Analysis of laser beam quality degradation caused by quartic phase aberration. Appl.Opt. 1993, 32: 5893-5901.
14 J.Turunen, T .Friberg, Matrix representation of Gaussian-Shell model beams in optical system. Opt. and Laser Tech. 1986,18:259-267.
15 L .Pan, B .Lü, Spectral changes and spectral switches of partially coherent beams focused by an aperture lens. J. Opt.Soc.Am.A 2004 ,21:140-148.
16 L .Pan, B .Lü, Spectral shift of Gaussian Schell-model beams in passage through an astigmatic lens. Optik. 2003,114:551-555.
17 C .Palma, G .Cincotti and Guattari. Spectral Shift of a Gaussian Schell-Model Beam Beyond a Thin Lens. IEEE J.Quant.Electron. 1998, 34:378-383.
18 J. Turunen, E. Tervonen, A.T. Friberg, Acousto-optic contral and modulation of optical coherence by electronically synthesized holographic gratings J. Appl. Phys. 1990,67:49-59.
19 D. Mendlovic, G. Shabtay, A.W. Lohmann, Synthesis of spatial coherence Opt. Lett. 1999,24:361.
20 X. Hu, J. Pu, Spectral anomalies of polychromatic, spatially coherent light diffracted by an annular aperture in the far field, Chin. Opt. Lett. 2005,2:418-421.
1 T.Takenaka, M.Yokota and O.Fukumitsu, Propagation of light beams beyond the paraxial approximation. J Opt Soc Am A. 1985, 2: 826-829.
2 G.P.Agrawal and D.N.Pattanayak, Gaussian beam propagation beyond the paraxial approximation. J Opt Soc Am. 1979, 69: 575-578.
3 C.G.Chen, P.T.Konkola, J.Ferrera, F.K.Heilmann and M.L.Schattenburg, Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations. J Opt Soc Am A. 2002, 19: 404-412.
4 H.Laabs. Propagation of Hermite-Gaussian-beams beyond the paraxial approximation. Opt Commun. 1998, , 147: 1-4.
5 A.A.Tovar, Production and propagation of cylindrically polarized Laguerre- Gaussian laser beams. J Opt Soc Am A. 1998, 15: 2705-2711.
6 C.J.R.Sheppard. High-aperture beams. J Opt Soc Am A. 2001, 18: 1579-1587.
7 A.Ciattoni, B.Crosignani and P.Di Porto. Vectorial analytical description of propagation of a highly nonparaxial beam. Opt Commun. 2002, 202: 17-20.
8 H.Laabs. Propagation of Hermite-Gaussian-beams beyond the paraxial approximation. Opt Commun. 1998, 147: 1-4.
9 H.Laabs and A.T.Friberg. Nonparaxial Eigenmodes of Stable Resonators. IEEE J Quan Elect. 1999, 35: 198-207.
10 C.J.R.Sheppard and S.Saghafi, Beam modes beyond the paraxial approximation: A scalar treatment. Phys Rev.A. 1998, 57: 2971-2979.
11 C.J.R.Sheppard, Aberrations in high aperture optical systems. Optik. 1997, 105: 29-33.
12 R.Borghi, M .Santarsiero and M.A.Alonso, Highly focused spirally polarized beams. J. Opt Soc Am. A. 2005, 22: 1420-1431.
13 P.Torok, C.J.R.Sheppard and P.Varga. Study of evanescent waves for transmission near-field optical microscopy. 1996, 43: 1167-1183.
14 D. W.Parther Design and application of subwavelength diffractive lenses for integration with infrared photodetectors. Opt Eng. 1999, 38: 870-878.
15 K.Hirayama, E.N.Glytsis and T.K.Gaylord. Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings. J Opt Soc Am A. 1997, 14: 907-917.
16 G.Gbur, T.D.Visser, E.Wolf, Anomalous Behavior of Spectrum near Phase Singularities of Focused Waves.Phys Rev Lett, 2002, 88(1): 013901-1-4
17 Guangpu Zhao, Yuanjie Peng, Baida Lü, Spectral changes of vector Gaussian Schell-model beams passing through an aperture lens. Opt.and Laser Tech.2006, 38: 101-106
18 B.Lü and K.Duan. Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture. Opt. Lett. 2003,28:2440-2442
19 K. Duan and B. Lü, Vectorial Nonparaxial Propagation Equation of Elliptical Gaussian Beams in the Presence of a Rectangular Aperture. J. Opt. Soc. Am A. 2004.21. 1613-1620
20 L.Mandel and E.Wolf. Optical. Coherence and Quantum Optics(Cambridge U.Press,Cambridge,UK) 1995.
21 G P. Zhao, X. Xiao and B.D.Lü, Effect of astigmatism on spectral switches of partially coherent beams, Chinese Physics, 2004,13:2064-2067.
22 E. Wolf, Invariance of the spectrum of light on propagation Phys. Rev. Lett. 1986, 56: 1370-1372
23 H.C.Kandapl, J.S.Vaishya, M.Chander,et al., Field correlations from spectral measurements in Young's interference experiment, Phys.Lett.A 1992, 167: 114
24 L ,Z,Pan and B.D.LU, The spectral switch of partially coherent light in Young's experiment. IEEE J. Quant. Electron. 2001,37:1377-1381.
25 KDuan, BLü, Partially coherent nonparaxial beams, Opt. Lett. 2004, 29: 800-802.
26 L, Pan and B. Lü, Spectral changes and spectral switches of partially coherent beams focused by an aperture lens J. Opt.Soc.Am.A 2004,21:140-145
27 B.K.Yadav,S.A.M.Rizvi and H.C.Kandpal, Experimental observation of spectral changes of partially coherent light in Young's experiment. Journal of Optics A.Pure and Applied Optics.2006,8:72-76.
1 J.T.Foley, E.Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J. Opt. Soc. Am. A.2002,19:2510-2516.
2. J. Pu, H.Zhang, S.Nemoto, Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt. Commun. 1999, 162:57-63.
3. J.Pu, S.Nemoto, Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture. IEEE J.Quant.Electron. 2000,36:1407-1411.
4. J.Pu, S.Nemoto,Spectral changes and 1 ~N spectral switches in the diffraction of partially coherent light by an aperture. J. Opt. Soc. Am. A. 2002,19:339-344.
5 M.Bom, E.Wolf, Principles of Optics (Cambridge U. Press, Cambridge,UK) 1999
6 M.S.Soskin, M.V.Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier, Amsterdam) 2001,42:219-276.
7 L.Pan and B.Lu, The spectral switch of partially coherent light in young's experiment, IEEE J. Quant. Electron.2001, 37:1377-1381.
8. L.Pan and B.Lfi, The aperture effect on the spectral shift of partially coherent light in Young's experiment. J. Opt. A: Pure and Appl. Opt. 2002,4:349-352.
9. J.Pu, C.Cai, S.Nemoto, Spectral anomalies in Young's double-slit interference experiment. Optics Express.2004,21:5131-5139.
10 赵光普，吕百达．有球差多色高斯光束衍射引起的光谱开关，物理学报，2004，53：2974-2979
11. G.Popescu and A.Dogariu, Spectral anomalies at wave-front dislocations, Phys. Rev. Lett. 2002,88:183902/1-4.
12 S.Anand, M.N. Kamalasanan, Experimental study of the spectral anomalies in Young's interference experiment, Opt. Commun.2006,266:469-472.
13. B. K. Yadav, S. A. M. Rizvi and H. C. Kandpal, Experimental observation of Spectral changes of partially coherent light in Young's experiment, J. Opt. A: Pure and Appl. Opt. 2006,8:72-76.
14. Yadav B K, Rizvi S A M,Raman S et al.,Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture, Opt.Commun. 2007,269:253-260.
15 E. Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett., 1986, 56:1370-1372.
16 H.C.Kandapl, J.S.Vaishya and M.Chander,et al., Field correlations from spectral measurements in Young's interference experiment, Phys.Lett.A.1992,167:120-122.
17.L .Mandel, E.Wolf ,Optical Coherence and Quantum Optics (Cambridge U .Press,Cambridge,UK) 1995.
18 G.Gbur, T.D.Visser, E.Wolf, 'Hidden' singularities in partially coherent wavefields J. Opt. A: Pure and Appl. Opt 2004,6:239-242
1 R.F.Harrigton.Field Computation by Moment Method.New York: Macmillan Company,1968.
2 C.Eftimiu, Scattering by a rough dielectric interface: A modified Wiener-Hermite expansion approach. J Opt Soc Am A 1990, 7: 875-884.
3. D.S.Weile,B.Shanker and E.Michielssen. An accurate scheme for the numerical solution of 'the time domain electric field integral equation. IEEE International Antennas and Propagation Symposium.2001,7:516-519.
4 H.C van de Hulst.,Light Scattering by Small Particles, Wiley, New York 1957
5 V.K. Polyanskii, Spectral characteristic of colorless glass with rough surface, J.Appl.Spectr. 1970,13:1039-1042.
6 L.V.Strinadko, V.K. Polyanskii, M.T. Strinadko, Structure of field scattered by a phase transparent. Opt.Spectrosk. 1980, 49:521-524.
7 K.Vyacheslav, Polyanskii,V.Oleg,Angelsky and V.Peter.Scattering-induced spectral changes as a singular optical effect. Optica applicata. 2002,ⅩⅩⅫ: 843-848.
8 吕百达 激光光学．高等教育出版社，2003．
9 E.Wolf and D F V Jamest,Correlation-induced spectral changes. Rep.Prog.Phys. 1996, 59:.771-818.
1 S A Collins: Lens-system diffraction integral written terms of matrix optics. J. Opt. Soc. Am. A. 1970, 60: 1168-1177.
2 J.J.Wen, M.A Breazeale, A diffraction beam field expressed as the superposition of Gaussian beams, J. Acoust.Soc.Am., 1988,83:1752-1756.
3 王喜庆，梁国栋，吕百达．高斯光束通过有硬边光阑ABCD光学系统的近似解析传输公式，强激光与粒子束，2001，4：418-422．
4. D .Ding, X:.Liu, Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture. J. Opt. Soc. Am. A. 1999,16:1286-1293
5. K .Tanaka, K .Yoshida, M .Taguchi, Analytical and experimental investigations of the diffraction field of a gaussian beam through a sequence of apertures: Applicability of the beam mode expansion method. Appl. Opt. 1988, 27: 1310-1312
2.G.M.Morris and D. Faklis, Effects of source correlation on the spectrum of light, Opt.Commun.1987,62:5-11.
3.D. Faklis and G.M.Morris, Spectral shifts produced by source correlations, Opt. Lett. 1988,13:4-6.
4.M.F.Bocko, D.H.Douglass, and R.S.Knox, Observation of frequency shifts of spectral lines due to source correlations. Phys. Rev. Lett. 1987, 58:2649-2651.
5.D.L.Jacobson, S.A.Werner and H. Rauch, Spectral modulation and squeezing at high-order neutron interferences. Phys. Rev. A. 1994,49:3196-3202.
6.E.Wolf, D F V James, Correlation-induced spectral changes. Rep. Prog. Phys. 1996,59:771-818.
7. E.Wolf, Non-cosmological redshift of spectral lines.Nature,1987,326:363-365.
8.Z.Dacic and E.Wolf, Changes in the spectrum of partially coherent light propagation in free space. J. Opt.Soc.Am.A 1988, 5: 1118-1126.
9.H.C.Kandpal, J.S.Vaishya,K.Saxena,etc, Intensity distribution across a source from spectral measurements. J.Mod.Opt. 1995,42: 455-464.
10.C.Palma, G. Cincotti and G. Guattari, Spectral shift of a Gaussian Schell-model beam beyond a thin lens. IEEE J. Quant. Electron. 1998, 34:378-383.
11.Pu J, Zhang H, Nemoto S. Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt Commun 1999; 162: 57-63.
12.G. Gbur, T D Visser, E Wolf, Anomalous behavior of spectra near phase singularities of focused waves. Phys. Rev. Lett. 2002,88:013901-1-4
13. J. Foley, E. Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J. Opt.Soc.Am.A 2002, 19: 2510-2516.
14. H.C. Kandpal. Experimental observation of the phenomenon of spectral switch. J. Opt. A: Pure and Appl. Opt. 2001, 3: 296-299.
15.H.C. Kandpal, S. Anand, J.S. Vaishya. Experimental observation of phenomenon of spectral switch for a class of partially coherent light. IEEE J. Quant. Electron. 2002, 38: 336-339.
16. G .Popescu, A .Dogariu, Spectral anomalies at wave-front dislocations.Phys. Rev. Lett. 2002,88:183902-1
17.B.K.Yadav, S.A.M.Rizvi, H.C. Kandpal. Experimental observation of the phenomenon of spectral switch. J. Opt. A: Pure and Appl. Opt. 2006, 8: 72-76.
18.B.K.Yadav, S.A.M.Rizvi, S.Raman, R.Mehrotra, H.C.Kandpal. Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture. Opt. Commun. 2007, 269:253-260.
19.K.Vyacheslav, Polyanskii, V.Oleg, Angelsky and V.Peter. Scattering-induced spectral changes as a singular optical effect. Optica applicata. 2002,XXXII: 843- 848.
20 A. A. Tovar, Production and propagation of cylindrically polarized Laguerre - Gaussian laser beams. J Opt Soc Am A. 1998,15: 2705-2711.
21 C.J.R.Sheppard, High-aperture beams. J Opt Soc Am A. 2001,18:1579-1587.
22. A.Ciattoni, B.Crosignani and P.Di Porto, Vectorial analytical description of propagation of a highly nonparaxial beam. Opt Commun. 2002,202:17-20.
23. R.Borghi and M.Santarsiero,Nonparaxial propagation of spirally polarized optical beams. J Opt Soc Am A. 2004,21: 2029-2037.
24. G. Zhao, Y. Peng, B. Lü, Spectral changes of vector Gaussian Schell- model beams passing through an aperture lens. Opt.and Laser Tech. 2006. 38:101-106.
25. B. Lu, and K.Duan, Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture. Opt. Lett. 2003,28:2440-2442.
26.K.Duan and B. Lü, Partially coherent nonparaxial beams Opt. Lett. 2004, 29:800-802.
27.E. Wolf, T. Shirai, H. Chen, W. Wang. Coherence filters and their uses: 1.Basic theory and examples. J. Mod. Opt. 1997, 44: 1345-1353.
28.T. Shirai, E. Wolf, H. Chen, W. Wang. Coherence filters and their uses: 2.One-dimensional realizations. J. Mod. Opt. 1998,45: 799-816.
29. S.A Ponomarenko, E.Wolf, Spectral anomalies in a Fraunhofer diffraction pattern. Opt.Lea. 2002,27:1211-1213.
30. J.Pu, C.Cai, S.Nemoto, Spectral anomalies in Young's double-slit interference experiment. Optics Express.2004,21:5131-5139.
31. H.C.Kandpal, W.Aiay, J.S.Vaishya, et.al., Application of Spatial-coherent spectroscopy for determining the angular diameters of stars feasibility experiment. Indian J.Pure Appl.Phys. 1998,36:665-674.
32. D.F.V.James, H.C.Kandpal and E.Wolf, A new method for determining the angular separation of double stars. Astrophys.J. 1995,445:406-410.
33. T.Yatagai, S.Kawai and H.Huang, Optical computing and interconnects. Proc. IEEE. 1996,84:825-852.
34. M.V.Berry, Exploring the colors of dark light. New J.Phys. 2002, 4:74.1-74.14.
35. M.V.Berry, Colored phase singularites. New J.Phys. 2002, 4:66.1-66.14.
36. G. Gbur, T D Visser, E Wolf, Singular behavior of the spectrum in the neighborhood of focus. J Opt Soc Am A. 2002, 19:1694-1700.
37. M. Born, E. Wolf, Principles of Optics. Eingland: Cambrige University Press, 7thed., 1999.
38. G P. Agrawal, E. Wolf, Propagation-induced polarization changes in partially coherent optical beams. J Opt Soc Am A 2000; 17: 2019-2023.
39. F.Gori, M.Santarsiero, G.Piquero,et al: partially polarized Gaussian Schell-model beams. J. Opt.A: Pure Appl. Opt. 2001,3: 1-9.
40. G.Zhao, X.Xiao, B.Lu, Effect of astigmatism on spectral switches of partially coherent beams. Chinese Physics. 2004,13: 2064-2070.
41. G.Zhao, B.Lu. Influence of aperture lens with spherical aberration on the spectral behavior of polychromatic Gaussian Schell-model beams. Optik. 2004, 115: 181-185:
42．赵光普，肖希，吕百达．光场的非傍轴性对光谱移动和光谱开关的影响．物理学报．2007，56(8)．
43. Z.Wang, Z.Zhang, Z.Xu, et al. Spectral and temporal properties of ultrashort light pulse in the far zone. Opt. Commun. 1996,123: 5-10.
44.B.Lü, L.Pan, Propagation properties of ultrashort pulsed Bessel beams in dispersive media. J. Opt. Soc. Am. A. 2003, 20: 582-587.
45.M.S.Soskin, M.V.Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier, Amsterdam) 2001,42:219-276.
46.F. Zernike, Physica 5 (1938) 785, see also, Proc. Phys. Soc. (London) 61 (1948) 158.
47.H.Roychowdhury, E.Wolf , Young's interference experiment with light of any state of coherence and of polarization. Opt Commun. 2005, 252: 268-274.
48.J.Pu, Spectral shifts of partially coherent light produced by passing through an annular aperture. J.Opt.(Paris) 1993,24:141-144.
49.J.D.Farina, L.M.Narducci and E.Collet, Generation of highly directional beams from a globally incoherent source. Opt. Commun. 1980, 32:203-208.
50.M.Bertolotti, F.Scudieri and S. Verginelli, Spatial coherent of light scattered by media with large correlation length of refractive index fluntuation. 1976, 15:1842 -1844.
1. A.T.Friberg and J.Turunen, Imaging of Gaussian Schell-model sources. J Opt Soc AreA, 1988, 5: 713-719.
2. L.Mandel and E.Wolf, Optical Coherence and Quantum Optics, Cambridge University Press. Cambridge, 1995.
3. F. Gori, M.Santarsiero, S.Vicalvi et al, Beam coherence-polarization matrix J. Opt.A: Pure Appl. Opt 1998, 7:941-51
4. B.Lu and K.Duan, Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture, Opt.Lett. 2003, 28: 2440-2442.
5. M.J. Bastianns. The Wigner distribution function applied to optical signals and systems. Opt. Commun. 1978, 25: 26-30.
6. M.J. Bastianns. Wigner distribution function and its application to first-order optics. J. Opt. Soc. Am. 1979, 69: 1710-1716.
7. M.J. Bastianns. Application of the Wigner distribution function to partially coherent light. J. Opt. Soc. Am. A 1986, 3: 1227-1238.
8. H.H.Hopkins, Tneory of Aberrration, Oxford University, London. 1995.
9. W.T.Welford, Aberration of the symmetric Optical system, Academatic, New York, 1974.
10. M. Born, E. Wolf, Principles of Optics. Eingland: Cambrige University Press, 7thed., 1999.
11 T. Yura, S.G Hanson. Optical beam wave propagation through complex optical systems. J Opt Soc Am A 1987, 4: 1931-1948.
12 A. Gerrard, J.M. Burch. Introduction to Matrix Methods in Opticss. London, New York: John Wiley & Sons, 1975.
13 A Belanger. Beam propagation and ABCD ray matrices. Opt. Lett. 1991, 16: 196-198
14 A.A. Tovar. Generalized beam matrices: Application to diffraction analysis. 1996, J. Opt. Soc. Am. A 14: 2239-2246.
15 吕百达，激光光学，高等教育出版社，2003．
16 J.W. Goodman. Introduction to Fourier Optics. London. New York: John Willy & Sons,1974
17 S.A. Colins Jr. Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 1970, 60: 1168-1177.
18 R.K.Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, Calif., 1966.
19 J W.Goodman, Introduction to Fourier Optics , McGraw-Hill Book Company, Inc. New York, 1968.
20 K .Duan and B. Lu, Propagation of Hermite- and Laguerre-Gaussian beams beyond the paraxial approximation. J Opt Soc Am A. 2005. 22. 1976-1980.
21 W.H.Carter and E. Wolf, Coherence and radiometry with quasihomogeneous planar sources, J Opt Soc Am A. 1977,67: 785-796.
22 E.Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett. 1986, 56: 1370-1372.
1. R. Bariheau, M. Rioux. Influence of speckle on laser range finders. Appl. Opt. 1991, 30: 2873-2878. 2.P. Ostlund, A.T. Friberg. Radiometry and far-zone polarization for partially coherent electromagnetic planar sources. Opt. Commun. 2001,197: 1-8.
3 E.Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett. 1986, 56: 1370-1372.
4 E .Wolf and D F V Jamest,Correlation-induced spectral changes. Rep.Prog.Phys. 1996,59:771-818.
5 J. T. Foley, The effect of an aperture on the spectrum of partially coherent light. Opt. Commun. 1990,75 :347
6 Pu J, Zhang H, Nemoto S. Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt Commun 1999; 162: 57-63.
7 J. T. Foley, E.Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J Opt Soc Am A2002; 19: 2510-2516.
8 G P. Agrawal, E. Wolf, Propagation-induced polarization changes in partially coherent optical beams. J Opt Soc Am A 2000; 17: 2019-2023.
9 F.Gori, M.Santarsiero, G.Piquero,et al: partially polarized Gaussian Schell-model beams. J. Opt. A: Pure Appl. Opt. 2001,3: 1-9
10 T.Friberg ,J.Turunen , Imaging of Gaussian Schell-model sources. J Opt Soc Am A 1988; 5: 713-719
11 A. E. Siegman, Analysis of laser beam quality degradation caused by quartic phase aberration. Appl.Opt. 1993, 32: 5893-5901.
12 L .Mandel and E.Wolf, Optical Coherence and Quantum Optics, Cambridge University Press. Cambridge, 1995.
13 Y.Li, E.Wolf. Focal shifts in diffracted converging spherical waves. Opt. Commun. 1981,39:211-215
14 W.H.Carter, M.F. Aburdence. Focal shift in Laguerre-Gaussian beams.J. Opt. Soc. Am 1987,4:1949-1952.
15 Y.Li and H.Platzer. An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems. Opt. Acta, 1983,30(10):1621-1643.
16 M.Martinez-Corral, V. Climent. Focal switch: a new effect in low-Fresnel-number systems. App1.Opt. 1996,35:24-27.
17 Y.Li. Focal shift and focal switch in dual-focus systems J.opt.Soc.Am.A. 1997,14:1297-1304.
18 B.Lu and B. Zhang. Focusing of a Gaussian Schell-model beam through a circular lens. J.Mod.Opt. 1995,42:289-298.
19 R.Borrghi,M.Santarsiero,S.Vicalvi. Focal shift of focused fiat-topped beams. Opt. Commun. 1998,154:243-248.
20 彭润伍，吕百达。厄米-高斯光束的焦开关。物理学报2003，52：2795-2802
21 F. Gori, Matrix treatment for partially polarized, partially coherent beams. Opt.Lett. 1998,23:241-243
22 F .Gori, M .Santarsiero, S .Vicalvi et al. Beam coherence-polarization matrix. Pure Appl. Opt. 1998,7:941-951
1 J. Pu, H.Zhang, S.Nemoto, Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt. Commun. 1999,162:57-63.
2 J .Pu, Nemoto.S: Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture. IEEE J.Quant.Electron. 2000,36:1407-1411.
3 B.Lü, L.Pan, Spectral switching of Gaussian-Schell model beams passing through an aperture lens. IEEE J. Quant. Electron. 2002, 38: 340-344.
4. H C Kandpal, Experimental observation of the phenomenon of spectral switch. J. Opt. A: Pure and Appl. Opt. 2001 3: 296-299.
5 H C Kandpal, S.Anand, J.S Vaishya, Experimental observation of phenomenon of spectral switch for a class of partially coherent light. IEEE. J. Quant. Electron. 2002, 38:336-339.
6 G. Gbur, T D Visser, E Wolf, Anomalous behavior of spectra near phase singularities of focused waves. Phys. Rev. Lett. 2002,88:013901-1-4
7 S.A Ponomarenko, Wolf E: Spectral anomalies in a Fraunhofer diffraction pattern. Opt.Lett. 2002,27:1211-1213.
8 J. Foley, E. Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J. Opt.Soc.Am.A 2002,192510-2516.
9 S.A Ponomarenko, E.Wolf, Spectral anomalies in a Fraunhofer diffraction pattern. OptXett. 2002,27:1211-1213.
10 J.Pu, C.Cai, S.Nemoto, Spectral anomalies in Young's double-slit interference experiment. Optics Express.2004,21:5131-5139.
11 B.K.Yadav, S.A.M.Rizvi, S.Raman, R.Mehrotra, H.C.Kandpal. Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture. Opt. Commun. 2007, 269:253-260.
12 L.Mandel and E.Wolf. Optical Coherence and Quantum Optics(Cambridge U.Press,Cambridge,UK) 1995.
13 A. E. Siegman, Analysis of laser beam quality degradation caused by quartic phase aberration. Appl.Opt. 1993, 32: 5893-5901.
14 J.Turunen, T .Friberg, Matrix representation of Gaussian-Shell model beams in optical system. Opt. and Laser Tech. 1986,18:259-267.
15 L .Pan, B .Lü, Spectral changes and spectral switches of partially coherent beams focused by an aperture lens. J. Opt.Soc.Am.A 2004 ,21:140-148.
16 L .Pan, B .Lü, Spectral shift of Gaussian Schell-model beams in passage through an astigmatic lens. Optik. 2003,114:551-555.
17 C .Palma, G .Cincotti and Guattari. Spectral Shift of a Gaussian Schell-Model Beam Beyond a Thin Lens. IEEE J.Quant.Electron. 1998, 34:378-383.
18 J. Turunen, E. Tervonen, A.T. Friberg, Acousto-optic contral and modulation of optical coherence by electronically synthesized holographic gratings J. Appl. Phys. 1990,67:49-59.
19 D. Mendlovic, G. Shabtay, A.W. Lohmann, Synthesis of spatial coherence Opt. Lett. 1999,24:361.
20 X. Hu, J. Pu, Spectral anomalies of polychromatic, spatially coherent light diffracted by an annular aperture in the far field, Chin. Opt. Lett. 2005,2:418-421.
1 T.Takenaka, M.Yokota and O.Fukumitsu, Propagation of light beams beyond the paraxial approximation. J Opt Soc Am A. 1985, 2: 826-829.
2 G.P.Agrawal and D.N.Pattanayak, Gaussian beam propagation beyond the paraxial approximation. J Opt Soc Am. 1979, 69: 575-578.
3 C.G.Chen, P.T.Konkola, J.Ferrera, F.K.Heilmann and M.L.Schattenburg, Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations. J Opt Soc Am A. 2002, 19: 404-412.
4 H.Laabs. Propagation of Hermite-Gaussian-beams beyond the paraxial approximation. Opt Commun. 1998, , 147: 1-4.
5 A.A.Tovar, Production and propagation of cylindrically polarized Laguerre- Gaussian laser beams. J Opt Soc Am A. 1998, 15: 2705-2711.
6 C.J.R.Sheppard. High-aperture beams. J Opt Soc Am A. 2001, 18: 1579-1587.
7 A.Ciattoni, B.Crosignani and P.Di Porto. Vectorial analytical description of propagation of a highly nonparaxial beam. Opt Commun. 2002, 202: 17-20.
8 H.Laabs. Propagation of Hermite-Gaussian-beams beyond the paraxial approximation. Opt Commun. 1998, 147: 1-4.
9 H.Laabs and A.T.Friberg. Nonparaxial Eigenmodes of Stable Resonators. IEEE J Quan Elect. 1999, 35: 198-207.
10 C.J.R.Sheppard and S.Saghafi, Beam modes beyond the paraxial approximation: A scalar treatment. Phys Rev.A. 1998, 57: 2971-2979.
11 C.J.R.Sheppard, Aberrations in high aperture optical systems. Optik. 1997, 105: 29-33.
12 R.Borghi, M .Santarsiero and M.A.Alonso, Highly focused spirally polarized beams. J. Opt Soc Am. A. 2005, 22: 1420-1431.
13 P.Torok, C.J.R.Sheppard and P.Varga. Study of evanescent waves for transmission near-field optical microscopy. 1996, 43: 1167-1183.
14 D. W.Parther Design and application of subwavelength diffractive lenses for integration with infrared photodetectors. Opt Eng. 1999, 38: 870-878.
15 K.Hirayama, E.N.Glytsis and T.K.Gaylord. Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings. J Opt Soc Am A. 1997, 14: 907-917.
16 G.Gbur, T.D.Visser, E.Wolf, Anomalous Behavior of Spectrum near Phase Singularities of Focused Waves.Phys Rev Lett, 2002, 88(1): 013901-1-4
17 Guangpu Zhao, Yuanjie Peng, Baida Lü, Spectral changes of vector Gaussian Schell-model beams passing through an aperture lens. Opt.and Laser Tech.2006, 38: 101-106
18 B.Lü and K.Duan. Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture. Opt. Lett. 2003,28:2440-2442
19 K. Duan and B. Lü, Vectorial Nonparaxial Propagation Equation of Elliptical Gaussian Beams in the Presence of a Rectangular Aperture. J. Opt. Soc. Am A. 2004.21. 1613-1620
20 L.Mandel and E.Wolf. Optical. Coherence and Quantum Optics(Cambridge U.Press,Cambridge,UK) 1995.
21 G P. Zhao, X. Xiao and B.D.Lü, Effect of astigmatism on spectral switches of partially coherent beams, Chinese Physics, 2004,13:2064-2067.
22 E. Wolf, Invariance of the spectrum of light on propagation Phys. Rev. Lett. 1986, 56: 1370-1372
23 H.C.Kandapl, J.S.Vaishya, M.Chander,et al., Field correlations from spectral measurements in Young's interference experiment, Phys.Lett.A 1992, 167: 114
24 L ,Z,Pan and B.D.LU, The spectral switch of partially coherent light in Young's experiment. IEEE J. Quant. Electron. 2001,37:1377-1381.
25 KDuan, BLü, Partially coherent nonparaxial beams, Opt. Lett. 2004, 29: 800-802.
26 L, Pan and B. Lü, Spectral changes and spectral switches of partially coherent beams focused by an aperture lens J. Opt.Soc.Am.A 2004,21:140-145
27 B.K.Yadav,S.A.M.Rizvi and H.C.Kandpal, Experimental observation of spectral changes of partially coherent light in Young's experiment. Journal of Optics A.Pure and Applied Optics.2006,8:72-76.
1 J.T.Foley, E.Wolf, Phenomenon of spectral switches as a new effect in singular optics with polychromatic light. J. Opt. Soc. Am. A.2002,19:2510-2516.
2. J. Pu, H.Zhang, S.Nemoto, Spectral shifts and spectral switches of partially coherent light passing through an aperture. Opt. Commun. 1999, 162:57-63.
3. J.Pu, S.Nemoto, Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture. IEEE J.Quant.Electron. 2000,36:1407-1411.
4. J.Pu, S.Nemoto,Spectral changes and 1 ~N spectral switches in the diffraction of partially coherent light by an aperture. J. Opt. Soc. Am. A. 2002,19:339-344.
5 M.Bom, E.Wolf, Principles of Optics (Cambridge U. Press, Cambridge,UK) 1999
6 M.S.Soskin, M.V.Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier, Amsterdam) 2001,42:219-276.
7 L.Pan and B.Lu, The spectral switch of partially coherent light in young's experiment, IEEE J. Quant. Electron.2001, 37:1377-1381.
8. L.Pan and B.Lfi, The aperture effect on the spectral shift of partially coherent light in Young's experiment. J. Opt. A: Pure and Appl. Opt. 2002,4:349-352.
9. J.Pu, C.Cai, S.Nemoto, Spectral anomalies in Young's double-slit interference experiment. Optics Express.2004,21:5131-5139.
10 赵光普，吕百达．有球差多色高斯光束衍射引起的光谱开关，物理学报，2004，53：2974-2979
11. G.Popescu and A.Dogariu, Spectral anomalies at wave-front dislocations, Phys. Rev. Lett. 2002,88:183902/1-4.
12 S.Anand, M.N. Kamalasanan, Experimental study of the spectral anomalies in Young's interference experiment, Opt. Commun.2006,266:469-472.
13. B. K. Yadav, S. A. M. Rizvi and H. C. Kandpal, Experimental observation of Spectral changes of partially coherent light in Young's experiment, J. Opt. A: Pure and Appl. Opt. 2006,8:72-76.
14. Yadav B K, Rizvi S A M,Raman S et al.,Information encoding by spectral anomalies of spatially coherent light diffracted by an annular aperture, Opt.Commun. 2007,269:253-260.
15 E. Wolf, Invariance of the spectrum of light on propagation, Phys. Rev. Lett., 1986, 56:1370-1372.
16 H.C.Kandapl, J.S.Vaishya and M.Chander,et al., Field correlations from spectral measurements in Young's interference experiment, Phys.Lett.A.1992,167:120-122.
17.L .Mandel, E.Wolf ,Optical Coherence and Quantum Optics (Cambridge U .Press,Cambridge,UK) 1995.
18 G.Gbur, T.D.Visser, E.Wolf, 'Hidden' singularities in partially coherent wavefields J. Opt. A: Pure and Appl. Opt 2004,6:239-242
1 R.F.Harrigton.Field Computation by Moment Method.New York: Macmillan Company,1968.
2 C.Eftimiu, Scattering by a rough dielectric interface: A modified Wiener-Hermite expansion approach. J Opt Soc Am A 1990, 7: 875-884.
3. D.S.Weile,B.Shanker and E.Michielssen. An accurate scheme for the numerical solution of 'the time domain electric field integral equation. IEEE International Antennas and Propagation Symposium.2001,7:516-519.
4 H.C van de Hulst.,Light Scattering by Small Particles, Wiley, New York 1957
5 V.K. Polyanskii, Spectral characteristic of colorless glass with rough surface, J.Appl.Spectr. 1970,13:1039-1042.
6 L.V.Strinadko, V.K. Polyanskii, M.T. Strinadko, Structure of field scattered by a phase transparent. Opt.Spectrosk. 1980, 49:521-524.
7 K.Vyacheslav, Polyanskii,V.Oleg,Angelsky and V.Peter.Scattering-induced spectral changes as a singular optical effect. Optica applicata. 2002,ⅩⅩⅫ: 843-848.
8 吕百达 激光光学．高等教育出版社，2003．
9 E.Wolf and D F V Jamest,Correlation-induced spectral changes. Rep.Prog.Phys. 1996, 59:.771-818.
1 S A Collins: Lens-system diffraction integral written terms of matrix optics. J. Opt. Soc. Am. A. 1970, 60: 1168-1177.
2 J.J.Wen, M.A Breazeale, A diffraction beam field expressed as the superposition of Gaussian beams, J. Acoust.Soc.Am., 1988,83:1752-1756.
3 王喜庆，梁国栋，吕百达．高斯光束通过有硬边光阑ABCD光学系统的近似解析传输公式，强激光与粒子束，2001，4：418-422．
4. D .Ding, X:.Liu, Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture. J. Opt. Soc. Am. A. 1999,16:1286-1293
5. K .Tanaka, K .Yoshida, M .Taguchi, Analytical and experimental investigations of the diffraction field of a gaussian beam through a sequence of apertures: Applicability of the beam mode expansion method. Appl. Opt. 1988, 27: 1310-1312