共軸球面腔的穩(wěn)定性條件ppt課件

1、2.2 共軸球面腔的穩(wěn)定性條件共軸球面腔的穩(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)圖一一 光線傳輸矩陣光線傳輸矩陣 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(gè)坐標(biāo)參數(shù)來表征：光線離軸線的可以由兩個(gè)坐標(biāo)參數(shù)來表征：光線離軸線的間隔間隔r、光線與軸線的夾角、光線與軸線的夾角。規(guī)定：光線。規(guī)定：光線出射方向在腔軸線的上方時(shí)，出射方向在腔軸線的上方時(shí)， 為正；反為正；反之，為負(fù)。之，為負(fù)。 光線在自在空間行

2、進(jìn)間隔光線在自在空間行進(jìn)間隔L時(shí)所引起的坐標(biāo)時(shí)所引起的坐標(biāo)變換為變換為101LTL11011201fRTR球面鏡對(duì)傍軸光線的變換矩陣為R為球面鏡的曲率半徑球面鏡對(duì)傍軸光線的反射變換與焦距為f=R/2的薄透鏡對(duì)同一光線的透射變換是等效的。用一個(gè)列矩陣描畫任一光線的坐標(biāo)，用一個(gè)二階方陣描畫入射光線和出射光線的坐標(biāo)變換。2121rrABCD該矩陣稱為光學(xué)系統(tǒng)對(duì)光線的變換矩陣。 Ray optics-by which we mean the geometrical laws for optical ray propagation, without including diffraction-is a

3、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 resonators and beams. Ray matrices or paraxial ray optics provide a general way of expressing the elementary lens laws of geometrical optics, or of sphe

4、rical-wave optics, leaving out higher-order aberrations, in a form that many people find clearer and more convenient. 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 pr

5、opagation of geometrical optical rays through paraxial optical elements, such lenses, curved mirrors, and “ducts. 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

6、 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 of optical waves when they are passing through various optical elements. We find that the passage of a ray (or its reflection) through these elemen

7、ts can be described by simple 2x2 matrices. Furthermore, these matrices will be found to describe the propagation of spherical waves and of Gaussian beams such as those which are characteristics of the output of lasers. Ray propagation through cascaded elements: A single 4-element ray matrix equal t

8、o the ordinary matrix product of the individual ray matrices can thus describe 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 th

9、e corresponding elements.二二 腔內(nèi)光線往返傳播的矩陣表示腔內(nèi)光線往返傳播的矩陣表示 由曲率半徑為由曲率半徑為R1和和R2的兩個(gè)球面鏡的兩個(gè)球面鏡M1和和M2組成的共軸球面腔，腔長為組成的共軸球面腔，腔長為L，開場光陰線從，開場光陰線從M1面上出發(fā)，向面上出發(fā)，向M2方向行進(jìn)方向行進(jìn)當(dāng)凹面鏡向著腔內(nèi)時(shí)，當(dāng)凹面鏡向著腔內(nèi)時(shí)，R取正值；當(dāng)凸面鏡向取正值；當(dāng)凸面鏡向著腔內(nèi)時(shí)，著腔內(nèi)時(shí)，R取負(fù)值取負(fù)值光線從M1面上出發(fā)到達(dá)M2面上時(shí)211211101LrrrLT當(dāng)光線在曲率半徑為R2的鏡M2上反射時(shí)232232221021RrrrTR當(dāng)光線再從鏡M2行進(jìn)到鏡M1面上時(shí)33433

10、4101LrrrLT然后又在M1上發(fā)生反射154454411021RrrrTR傍軸光線在腔內(nèi)完成一次往返，總的坐標(biāo)變換為LRLRTTTTLRLRDCBAT212110112011011201511151111210101122110101rrrrLLABTCDRR傍軸光線在腔內(nèi)完成一次往返總的變換矩陣為221RLA)1 (22RLLB1212122RLRRC21121212RLRLRLDThe sign of R is the same as that of the focal length of the equivalent. This makes R1 (or R2) positive w

11、hen the center of curvature of mirror 1(or 2) is in the direction of mirror 2 (or 1), and negative otherwise.三三 共軸球面腔的穩(wěn)定性條件共軸球面腔的穩(wěn)定性條件(mode stability criteria) 傍軸光線能在腔內(nèi)往返恣意多次而不橫向逸傍軸光線能在腔內(nèi)往返恣意多次而不橫向逸出腔外，要求出腔外，要求n次往返變換矩陣次往返變換矩陣Tn的各個(gè)元的各個(gè)元素素An、Bn、Cn、Dn對(duì)恣意對(duì)恣意n值均堅(jiān)持有限值均堅(jiān)持有限1)(211DA1)1)(1(021RLRL101,1212211

12、ggRLgRLg引入引入g參數(shù)，可寫成參數(shù)，可寫成簡單共軸球面腔 共軸球面腔的往返矩陣以及共軸球面腔的往返矩陣以及n次往返矩陣均次往返矩陣均與光線的初始坐標(biāo)無關(guān)，可以描畫恣意傍與光線的初始坐標(biāo)無關(guān)，可以描畫恣意傍軸光線在腔內(nèi)往返傳播的行為。軸光線在腔內(nèi)往返傳播的行為。 隨著光線在腔內(nèi)的初始出發(fā)位置及往返一隨著光線在腔內(nèi)的初始出發(fā)位置及往返一次的行進(jìn)次序的不同，矩陣次的行進(jìn)次序的不同，矩陣T各元素的詳細(xì)各元素的詳細(xì)表達(dá)式也將各不一樣。表達(dá)式也將各不一樣。 可以證明，可以證明，(A+D)/2對(duì)于一定幾何構(gòu)造的球?qū)τ谝欢◣缀螛?gòu)造的球面腔是一個(gè)不變量，與光線的初始坐標(biāo)、面腔是一個(gè)不變量，與光線的初始坐

13、標(biāo)、出發(fā)位置及往返一次的順序都無關(guān)。出發(fā)位置及往返一次的順序都無關(guān)。 穩(wěn)定腔 非穩(wěn)腔 臨界腔1021gg121gg或021gg121gg或021ggThe ability of an optical resonator to support low (diffraction) loss modes depend on the mirrors separation L and their radii of curvature R1 and R2.四四 常見的幾種穩(wěn)定腔、非穩(wěn)腔、臨界腔常見的幾種穩(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)腔 平凹穩(wěn)定腔、非穩(wěn)腔假設(shè)L=R/2，稱為半

14、共焦腔；假設(shè)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 L=R/2a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，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)臨界腔臨界腔(a) 對(duì)稱共焦腔(b) 平行平面腔(c) 實(shí)共心腔(d) 對(duì)稱共心腔(e) 虛共心腔五五 穩(wěn)區(qū)圖穩(wěn)區(qū)圖 (stability diagram of optical

15、 resonator)-2-1012-2-1012g1g2恣意一個(gè)球面腔獨(dú)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)定任務(wù)區(qū)域，其他的區(qū)域?qū)儆诜欠€(wěn)區(qū)。假設(shè)滿足g1=0、g2=0 或g1g2=1 ，那么是臨界腔。 恣意一個(gè)具有確定R1、R2、L值的球面腔獨(dú)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詳細(xì)尺寸的球面腔。 對(duì)稱共焦腔本屬于臨界腔g1=0，g2=0，其中恣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。共軸球面腔的穩(wěn)定性條件共軸球面腔的穩(wěn)定性條件改寫為：改寫為：0102121gggg 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