幾何光學中光線傳輸矩陣 (2)

1、2.2 共軸球面腔的穩(wěn)定性條件光線傳輸矩陣(optical ray matrices or ABCD matrices)腔內(nèi)光線往返傳播的矩陣表示共軸球面腔的穩(wěn)定性條件常見的幾種穩(wěn)定腔、非穩(wěn)腔、臨界腔穩(wěn)區(qū)圖一 光線傳輸矩陣腔內(nèi)任一傍軸光線在某一給定的橫截面內(nèi)都可以由兩個坐標參數(shù)來表征：光線離軸線的距離r、光線與軸線的夾角。規(guī)定：光線出射方向在腔軸線的上方時， 為正；反之，為負。光線在自由空間行進距離L時所引起的坐標變換為球面鏡對傍軸光線的變換矩陣為（R為球面鏡的曲率半徑）球面鏡對傍軸光線的反射變換與焦距為f=R/2的薄透鏡對同一光線的透射變換是等效的。用一個列矩陣描述任一光線的坐標，用一個二階

2、方陣描述入射光線和出射光線的坐標變換。該矩陣稱為光學系統(tǒng)對光線的變換矩陣。Ray optics-by which we mean the geometrical laws for optical ray propagation, without including diffraction-is a topic that is not only important in its own right, but also very useful in understanding the full diffractive propagation of light waves in optical re

3、sonators and beams.Ray matrices or paraxial ray optics provide a general way of expressing the elementary lens laws of geometrical optics, or of spherical-wave optics, leaving out higher-order aberrations, in a form that many people find clearer and more convenient.Ray optics and geometrical optics

4、in fact contain exactly the same physical content, expressed in different fashion.Ray matrices or “ABCD matrices” are widely used to describe the propagation of geometrical optical rays through paraxial optical elements, such lenses, curved mirrors, and “ducts”. These ray matrices also turn out to b

5、e very useful for describing a large number of other optical beam and resonator problems, including even problems that involve the diffractive nature of light.Since a ray is, by definition, normal to the optical wavefront, an understanding of the ray behavior makes it possible to trace the evolution

6、 of optical waves when they are passing through various optical elements. We find that the passage of a ray (or its reflection) through these elements can be described by simple 2x2 matrices. Furthermore, these matrices will be found to describe the propagation of spherical waves and of Gaussian bea

7、ms such as those which are characteristics of the output of lasers.Ray propagation through cascaded elements:A single 4-element ray matrix equal to the ordinary matrix product of the individual ray matrices can thus describe the total or overall ray propagation through a complicated sequence of casc

8、aded optical elements. Note, however, that the matrices must be arranged in inverse order from the order in which the ray physically encounters the corresponding elements.二 腔內(nèi)光線往返傳播的矩陣表示由曲率半徑為R1和R2的兩個球面鏡M1和M2組成的共軸球面腔，腔長為L，開始時光線從M1面上出發(fā)，向M2方向行進當凹面鏡向著腔內(nèi)時，R取正值；當凸面鏡向著腔內(nèi)時，R取負值光線從M1面上出發(fā)到達M2面上時當光線在曲率半徑為R2的鏡

9、M2上反射時當光線再從鏡M2行進到鏡M1面上時然后又在M1上發(fā)生反射傍軸光線在腔內(nèi)完成一次往返，總的坐標變換為傍軸光線在腔內(nèi)完成一次往返總的變換矩陣為The sign of R is the same as that of the focal length of the equivalent. This makes R1 (or R2) positive when the center of curvature of mirror 1(or 2) is in the direction of mirror 2 (or 1), and negative otherwise.三 共軸球面腔的穩(wěn)定性

10、條件(mode stability criteria)傍軸光線能在腔內(nèi)往返任意多次而不橫向逸出腔外，要求n次往返變換矩陣Tn的各個元素An、Bn、Cn、Dn對任意n值均保持有限引入g參數(shù)，可寫成簡單共軸球面腔穩(wěn)定腔非穩(wěn)腔臨界腔或或The ability of an optical resonator to support low (diffraction) loss modes depend on the mirrors separation L and their radii of curvature R1 and R2.四 常見的幾種穩(wěn)定腔、非穩(wěn)腔、臨界腔雙凹穩(wěn)定腔、非穩(wěn)腔凹凸穩(wěn)定腔、非穩(wěn)

11、腔平凹穩(wěn)定腔、非穩(wěn)腔（如果L=R/2，稱為半共焦腔；如果L=R，稱為半共心腔）雙凸腔、平凸腔都是非穩(wěn)腔(a)、(b) 雙凹穩(wěn)定腔(c) 凹-凸穩(wěn)定腔(d) 平-凹穩(wěn)定腔 (e) 半共焦腔對稱共焦腔(confocal) R1=R2=L平行平面腔(plane-parallel) R1=R2=共心腔 R1+R2=L實共心腔 R1、R2均為正值，當R1=R2=L/2時，稱為對稱共心腔(symmetric concentric)虛共心腔 R1、R2異號臨界腔(a) 對稱共焦腔(b) 平行平面腔(c) 實共心腔(d) 對稱共心腔(e) 虛共心腔五 穩(wěn)區(qū)圖 (stability diagram of optical resonator)任意一個球面腔唯一地對應(yīng)于g1-g2平面上的一個點。由g1=0、g2=0和g1g2=1雙曲線的兩支圍成的區(qū)域?qū)儆谇坏姆€(wěn)定工作區(qū)域，其余的區(qū)域?qū)儆诜欠€(wěn)區(qū)。如果滿足g1=0、g2=0 或g1g2=1 ，則是臨界腔。任意一個具有確定（R1、R2、L）值的球面腔唯一地對應(yīng)于圖中一個點，但反過來，圖中每個點并不單值地代表某一具體尺寸的球面腔。對稱共