共軸球面腔的穩(wěn)定性條件實(shí)用教案

1、一 光線(gungxin)傳輸矩陣 腔內(nèi)任一傍軸光線在某一給定(i dn)的橫截面內(nèi)都可以由兩個(gè)坐標(biāo)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(shí)， 為正；反之，為負(fù)。 光線在自由空間行進(jìn)距離L時(shí)所引起的坐標(biāo)變換為101LTL第1頁/共24頁第一頁，共25頁。11011201fRTR球面鏡對(duì)傍軸光線(gungxin)的變換矩陣為（R為球面鏡的曲率半徑）球面鏡對(duì)傍軸光線的反射變換(binhun)與焦距為f=R/2的薄透鏡對(duì)同一光線的透射變換(binhun)是等效的。用一個(gè)列矩陣描述(mio sh)任一光線的坐標(biāo)，用一個(gè)二階方陣描述(mio sh)入射光線和出

2、射光線的坐標(biāo)變換。2121rrABCD該矩陣稱為光學(xué)系統(tǒng)對(duì)光線的變換矩陣。第2頁/共24頁第二頁，共25頁。 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 w

3、aves in optical resonators 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.第3頁/共24頁第三頁，共25

4、頁。 Ray optics and geometrical optics 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”.

5、 These ray matrices also turn out to be very useful for describing a large number of other optical beam and resonator problems, including even problems that involve the diffractive nature of light.第4頁/共24頁第四頁，共25頁。 Since a ray is, by definition, normal to the optical wavefront, an understanding of t

6、he ray behavior makes it possible to trace the evolution 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 describ

7、e the propagation of spherical waves and of Gaussian beams such as those which are characteristics of the output of lasers.第5頁/共24頁第五頁，共25頁。 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

8、the total or overall ray propagation through a complicated sequence of cascaded 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.第6頁/共24頁第六頁，共25頁。二 腔內(nèi)光線往返傳播的矩陣(j zhn)表示 由曲率半徑為R1和R2的兩個(gè)

9、球面鏡M1和M2組成的共軸球面腔，腔長為L，開始時(shí)光線從M1面上出發(fā)，向M2方向(fngxing)行進(jìn)當(dāng)凹面鏡向著(xing zhe)腔內(nèi)時(shí)，R取正值；當(dāng)凸面鏡向著(xing zhe)腔內(nèi)時(shí)，R取負(fù)值光線從M1面上出發(fā)到達(dá)M2面上時(shí)211211101LrrrLT第7頁/共24頁第七頁，共25頁。當(dāng)光線在曲率半徑(bnjng)為R2的鏡M2上反射時(shí)232232221021RrrrTR當(dāng)光線(gungxin)再從鏡M2行進(jìn)到鏡M1面上時(shí)334334101LrrrLT然后又在M1上發(fā)生(fshng)反射154454411021RrrrTR第8頁/共24頁第八頁，共25頁。傍軸光線在腔內(nèi)完成一次往返，

10、總的坐標(biāo)(zubio)變換為LRLRTTTTLRLRDCBAT212110112011011201511151111210101122110101rrrrLLABTCDRR傍軸光線在腔內(nèi)完成一次往返總的變換(binhun)矩陣為第9頁/共24頁第九頁，共25頁。221RLA)1 (22RLLB1212122RLRRC21121212RLRLRLDThe 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 cur

11、vature of mirror 1(or 2) is in the direction of mirror 2 (or 1), and negative otherwise.第10頁/共24頁第十頁，共25頁。三 共軸球面(qimin)腔的穩(wěn)定性條件(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(gè)元素An、Bn、Cn、Dn對(duì)任意n值均保持(boch)有限1)(211DA1)1)(1(021RLRL101,1212211ggRLgRLg引入g參數(shù)(cnsh)，可寫成簡單共軸球面腔第11頁/共24頁第十一頁，共25頁

12、。 共軸球面腔的往返矩陣以及n次往返矩陣均與光線的初始坐標(biāo)無關(guān)，可以描述任意傍軸光線在腔內(nèi)往返傳播的行為。 隨著(su zhe)光線在腔內(nèi)的初始出發(fā)位置及往返一次的行進(jìn)次序的不同，矩陣T各元素的具體表達(dá)式也將各不相同。 可以證明，(A+D)/2對(duì)于一定幾何結(jié)構(gòu)的球面腔是一個(gè)不變量，與光線的初始坐標(biāo)、出發(fā)位置及往返一次的順序都無關(guān)。第12頁/共24頁第十二頁，共25頁。 穩(wěn)定(wndng)腔 非穩(wěn)腔 臨界腔1021gg121gg或021gg121gg或021ggThe ability of an optical resonator to support low (diffraction) los

13、s modes depend on the mirrors separation L and their radii of curvature R1 and R2.第13頁/共24頁第十三頁，共25頁。四 常見的幾種穩(wěn)定(wndng)腔、非穩(wěn)腔、臨界腔 雙凹穩(wěn)定(wndng)腔、非穩(wěn)腔 凹凸穩(wěn)定(wndng)腔、非穩(wěn)腔 平凹穩(wěn)定(wndng)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)腔第14頁/共24頁第十四頁，共25頁。(a)、(b) 雙凹穩(wěn)定(wndng)腔(c) 凹-凸穩(wěn)定(wndng)腔第15頁/共24頁第十五頁，共25頁。(d) 平-

14、凹穩(wěn)定(wndng)腔 (e) 半共焦腔（ L=R/2）第16頁/共24頁第十六頁，共25頁。a對(duì)稱共焦腔(confocal) R1=R2=La平行平面腔(plane-parallel) R1=R2=a共心腔 R1+R2=La實(shí)共心腔 R1、R2均為正值(zhn zh)，當(dāng)R1=R2=L/2時(shí)，稱為對(duì)稱共心腔(symmetric concentric)a虛共心腔 R1、R2異號(hào)臨界(ln ji)腔第17頁/共24頁第十七頁，共25頁。(a) 對(duì)稱(duchn)共焦腔(b) 平行(pngxng)平面腔第18頁/共24頁第十八頁，共25頁。(c) 實(shí)共心腔(d) 對(duì)稱(duchn)共心腔(e) 虛

15、共心腔第19頁/共24頁第十九頁，共25頁。五 穩(wěn)區(qū)圖 (stability diagram of optical resonator)-2-1012-2-1012g1g2任意一個(gè)球面(qimin)腔唯一地對(duì)應(yīng)于g1-g2平面上的一個(gè)點(diǎn)。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。第20頁/共24頁第二十頁，共25頁。 任意一個(gè)具有確定（R1、R2、L）值的球面腔唯一地對(duì)應(yīng)于圖中一個(gè)點(diǎn)，但反過來，圖中每個(gè)點(diǎn)并不單值地代表某一具體尺寸的球面腔。 對(duì)稱共焦腔（本屬于臨界(ln ji)腔

16、g1=0，g2=0），其中任意傍軸光線均可在腔內(nèi)往返多次而不橫向逸出，而且經(jīng)兩次往返即自行閉合。在這種意義上，共焦腔屬于穩(wěn)定腔之列。共軸球面(qimin)腔的穩(wěn)定性條件改寫為：0102121gggg第21頁/共24頁第二十一頁，共25頁。 From this diagram, for example, it can be seen that the symmetric concentric (R1=R2=L/2), confocal (R1=R2=L), and the plane-parallel (R1=R2=) resonator are all on the verge of instability and thus may become extremely lossy by small deviations of the parameters in the direction of instability.第22頁/共24頁第二十二頁，共25頁。1)(211DA小結(jié)(xioji)：可以證明，(A+D)/2對(duì)于一定幾何結(jié)構(gòu)的球面(qimin)腔是