穩(wěn)定球面腔中模結(jié)構(gòu)

1、穩(wěn)定球面腔中模結(jié)構(gòu)He-Ne激光器基模Nd:YAG激光器多模一、激光的橫模(transverse modes)反射鏡的有限大小會(huì)引起衍射損耗，而且在決定開腔中激光振蕩能量的空間分布方面，衍射將起主要作用非選擇性損耗將使橫截面內(nèi)各點(diǎn)的場(chǎng)按同樣的比例衰減，對(duì)場(chǎng)的空間分布不會(huì)發(fā)生重要影響衍射主要發(fā)生在鏡的邊緣上，將對(duì)場(chǎng)的空間分布發(fā)生重要影響；而且，只要鏡的橫向尺寸是有限的，這種影響將永遠(yuǎn)存在。為突出特征、簡(jiǎn)化分析，提出理想的開腔模型：兩塊反射鏡片沉浸在均勻的、無(wú)限的、各向同性的介質(zhì)中。沒(méi)有側(cè)壁的不連續(xù)性，決定衍射效應(yīng)的孔徑由鏡的邊緣所構(gòu)成?？紤]在開腔中往返傳播的一列波用波在孔闌傳輸線中的行進(jìn)來(lái)模擬它

2、在平面開腔中的往復(fù)反射（暫不考慮干涉效應(yīng)）。這種孔闌傳輸線由一系列同軸的孔徑構(gòu)成，這些孔徑開在平行放置著的無(wú)限大完全吸收屏上，相鄰兩個(gè)孔徑的距離等于腔長(zhǎng)，孔徑大小等于鏡的大小。光從一個(gè)孔徑傳播到另外一個(gè)孔徑，就等效于光在開腔中從一個(gè)反射鏡面?zhèn)鞑サ搅硪粋€(gè)鏡面。在通過(guò)每一個(gè)孔闌時(shí)光將發(fā)生衍射，射到孔的范圍以外的光將被屏所吸收（對(duì)應(yīng)于損耗）。每經(jīng)過(guò)一個(gè)孔，波的振幅和相位分布就經(jīng)歷一次改變；在經(jīng)過(guò)若干個(gè)孔以后，其振幅和相位不可避免地逐次發(fā)生畸變，逐漸被改變成這樣的形狀，以致于它們受到衍射的影響越來(lái)越小，或者說(shuō)，它們逐漸趨于一定的穩(wěn)定分布狀態(tài)。當(dāng)通過(guò)的孔闌數(shù)足夠多時(shí)，鏡面上場(chǎng)的相對(duì)振幅和相位分布將不再發(fā)

3、生變化，或者說(shuō)不再受衍射的影響，在腔內(nèi)往返一次能夠“再現(xiàn)”出發(fā)時(shí)的場(chǎng)分布。把開腔鏡面上經(jīng)一次往返能再現(xiàn)的穩(wěn)態(tài)場(chǎng)分布稱為開腔的自再現(xiàn)?；驒M模。自再現(xiàn)模一次往返所經(jīng)受的能量損耗稱為模的往返損耗，所發(fā)生的相移稱為往返相移，該相移等于2的整數(shù)倍。并非任何形態(tài)的電磁場(chǎng)都能在開腔中長(zhǎng)期存在，只有那些不受衍射影響的場(chǎng)分布才能最終穩(wěn)定下來(lái)（特點(diǎn)1：非任意性）由不同的初始入射波所得到的最終穩(wěn)態(tài)場(chǎng)分布可能是各不相同的，這預(yù)示了開腔模式的多樣性。實(shí)際的物理過(guò)程是，開腔中的任何振蕩都是從某種偶然的自發(fā)輻射開始的，而自發(fā)輻射服從統(tǒng)計(jì)規(guī)律，因而可以提供各種不同的初始分布。（特點(diǎn)2：多樣性）理解激光的空間相干性：即使入射在

4、第一個(gè)孔面上的光是空間非相干的，但由于衍射效應(yīng)，第二個(gè)孔面上任一點(diǎn)的波應(yīng)該看作是第一個(gè)孔面上所有各點(diǎn)發(fā)出的子波的疊加，這樣，第二個(gè)孔面上各點(diǎn)波的相位就發(fā)生了一定的關(guān)聯(lián)。在經(jīng)過(guò)了足夠多次衍射之后，光束橫截面上各點(diǎn)的相位關(guān)聯(lián)越來(lái)越緊密，因而空間相干性隨之越來(lái)越增強(qiáng)。在開腔中，從非相干的自發(fā)輻射發(fā)展成空間相干性極好的激光，正是由于衍射的作用。在無(wú)源開腔中，自再現(xiàn)模的形成過(guò)程和場(chǎng)的空間相干性的增強(qiáng)過(guò)程，都不可避免地伴隨著初始入射波能量的衰減。在激活腔中，只要某一自再現(xiàn)模能滿足閾值條件，則該模在腔內(nèi)就可以形成自激振蕩。這時(shí)，自再現(xiàn)模的形成過(guò)程將伴隨著光的受激放大，其結(jié)果是：光譜不斷變窄，空間相干性不斷增

5、強(qiáng)，同時(shí)，光強(qiáng)也不斷增大，最終形成高強(qiáng)度的激光輸出。取激光器的軸向作為z軸，以諧振腔的中心點(diǎn)為原點(diǎn)，并在與主軸垂直的平面上取x、y軸，用TEMmn符號(hào)來(lái)表示各種橫向模式。 m、n分別代表在橫截面內(nèi)的x、y軸方向出現(xiàn)的節(jié)線數(shù)，或，光強(qiáng)為零的那些零點(diǎn)的序數(shù)?？v模和橫模各自從一個(gè)側(cè)面反映了諧振腔內(nèi)能自再現(xiàn)的光場(chǎng)分布。腔內(nèi)光波往返傳播時(shí)，干涉和衍射效應(yīng)同時(shí)存在。一個(gè)完整的模式不但有確定的橫向分布，而且沿縱向形成駐波。用三個(gè)正整數(shù)m、n、q來(lái)標(biāo)志， TEMmnq 。一種模式的振蕩頻率不僅與縱模序數(shù)q有關(guān)，而且與橫模序數(shù)m、n也有關(guān)。研究表明：一方面，人們從理論上論證了開腔模的存在，并且用數(shù)值和解析的方法

6、求出了各種開腔模式；另外，又從實(shí)驗(yàn)上觀測(cè)到了激光的各種穩(wěn)定的強(qiáng)度花樣，而且理論分析與實(shí)驗(yàn)觀測(cè)的結(jié)果符合得很好。Rather than thinking of repeated round trips within a resonator, it can be helpful to think of the pulse as propagating instead through repeated sections of an iterated periodic optical system. In setting up such an iterated periodic lensguide,

7、curved mirrors in the original resonator are replaced by thin lenses of equal focusing power, and all other elements encountered in the lensguide are made the same as those encountered in the original resonator.The diffraction and aperturing effects that the pulse sees in a series of repeated round

8、trips around the original laser cavity will then be the same as in propagating through an equivalent number of segments in the periodic lensguide. This lensguide approach obviously adds no new physics to the problem, but it does convert the resonator problem into an equivalent waveguide problem, and

9、 it can sometimes be helpful in visualizing the behavior in an optical resonator.Suppose a pulse makes one complete round trip around an optical cavity, or travels through one complete period of the equivalent lensguide. After one complete round trip, the transverse field pattern as it arrives back

10、at its starting plane will in general be different from its starting pattern before the round trip, because of diffraction, reflection and aperturing effects; and after a second round trip the pattern may again be still different. (Note that we have dropped the z dependence in writing these patterns

11、, because we are only considering the transverse variation as observed at one arbitrarily chosen reference plane somewhere within the resonator, or at a set of such planes spaced one period apart in the equivalent lensguide.),()1(yxE),()0(yxE),()2(yxEWe can then ask if, to put the question in physic

12、al terms, there exist any transverse patterns, all them , such that if a pulse starts off with one of these transverse patterns, it will return one round trip later with exactly the same pattern? More precisely, we require that the pulse of radiation must return with exactly the same transverse form

13、, but possibly with a reduced amplitude because of diffraction and other losses during the round trip.If we can find any such self-reproducing transverse patterns, it certainly seems reasonable to call them transverse modes of the resonator. That is, a pulse which is launched with an initial transve

14、rse pro one of the transverse modes can then propagate repeatedly around the resonator, or propagate indefinitely down the lensguide, always getting weaker in amplitude, but always maintaining the same transverse pro the same reference plane in the resonator or the lensgiude.Do such lossy but self-r

15、eproducing transverse eigenmodes then really exist for open-sided and finite-diameter optical cavities, especially the very long slender cavities often used in practical lasers? The answer is that they do indeed exist, and that moreover the lowest-order transverse modes in properly designed (and ali

16、gned) laser cavities can have remarkably low diffraction losses, as well as remarkably good propagation properties.二、衍射理論的分析方法（diffraction theory approach）求解開腔模式，歸結(jié)為求解菲涅耳基爾霍夫衍射積分；公式表明：如果知道了光波場(chǎng)在其所達(dá)到的任意空間曲面上的振幅和相位分布，就可以求出該光波場(chǎng)在空間其它任意位置處的振幅和相位分布。sdeyxuikyxuikS)cos1 (),(4),(意義：觀察點(diǎn)P處的場(chǎng)可以看作是S面上各子波源所發(fā)出的非均勻球

17、面子波的疊加Kirchhoff-Fresnel integral已知某一鏡面上的場(chǎng)分布 ，在衍射作用下經(jīng)腔內(nèi)一次渡越而在另一個(gè)鏡面上生成的場(chǎng)),(1yxusdeyxuikyxuikS)cos1 (),(4),(112就將一個(gè)鏡面上的場(chǎng)通過(guò)菲涅耳基爾霍夫積分與另一個(gè)鏡面上的場(chǎng)聯(lián)系起來(lái)。經(jīng)過(guò)j次渡越后所生成的場(chǎng)uj+1與產(chǎn)生它的場(chǎng)uj之間亦應(yīng)滿足類似的迭代關(guān)系sdeyxuikyxuikSjj)cos1 (),(4),(11、考慮對(duì)稱開腔中的自再現(xiàn)模。按照模式再現(xiàn)概念，當(dāng)式中的j足夠大時(shí)，除了一個(gè)表示振幅衰減和相位移動(dòng)的常數(shù)因子以外， uj+1應(yīng)能將uj再現(xiàn)出來(lái)，即12111jjjjuuuu當(dāng)j足夠

18、大時(shí)此即模式再現(xiàn)概念的數(shù)學(xué)表達(dá)),(1)cos1 (),(4),(1yxusdeyxuikyxujikSjjsdeyxuikyxuikSjj)cos1 (),(4),(11sdeyxuikyxuikSjj)cos1 (),(4),(以v(x,y)表示開腔中這一不受衍射影響的穩(wěn)態(tài)場(chǎng)分布函數(shù)（即uj 、uj+1、），有)cos1 (),(4),(),(yxyxeikyxyxKyxyxik滿足方程的任意一個(gè)分布函數(shù)v(x,y)就描述腔的一個(gè)自再現(xiàn)?；驒M模。一般地， v(x,y)應(yīng)為復(fù)函數(shù)，它的模v(x,y) 描述鏡面上場(chǎng)的振幅分布，而其輻角 arg v(x,y) 描述鏡面上場(chǎng)的相位分布。積分方程的核

19、(Kernel)sdyxvyxyxKyxvS),(),(),(),(),(yxyxikeLiyxyxKLyxyxaRaL2),()cos1 (,sdyxvyxyxKyxvS),(),(),(Such an integral equation can be shown to posses a discrete set of solutions (eigenmodes), which we denote by vmn(x,y). The function can be shown to approach in the limit of large Fresnel numbers the Hermi

20、te-Gaussian solutions. The solution yields also the associated complex eigenvalue mn. mn corresponds physically to the factor by which the amplitude changes in one round trip. 復(fù)常數(shù)的意義(complex eigenvalue)ie將復(fù)常數(shù)表示為 ，代入的定義，得到ijjjeueuu)(11e-量度每經(jīng)單程渡越時(shí)自再現(xiàn)模的振幅衰減(the loss in mode amplitude per round trip)，愈大

21、,衰減愈甚， 0時(shí)，自再現(xiàn)模在腔內(nèi)能無(wú)損耗地傳播。表示每經(jīng)一次渡越模的相位滯后(the phase shift per round trip)， 愈大，相位滯后愈多。222212111euuujjjd愈大，模的單程損耗愈大自再現(xiàn)模在腔內(nèi)經(jīng)單程渡越的總相移定義為jjuuargarg1在對(duì)稱開腔的情況下，1arg自再現(xiàn)模在腔內(nèi)經(jīng)單程渡越所經(jīng)受的相對(duì)功率損耗稱為模的單程損耗，通常以d表示。在對(duì)稱開腔情況下,在腔內(nèi)存在激活物質(zhì)的情況下，為了使自再現(xiàn)模在往返傳播過(guò)程中能形成穩(wěn)定振蕩，還必須滿足多光束干涉條件：在腔內(nèi)一次往返的總相移等于2的整數(shù)倍(the phase delay per round tri

22、p be some integer of 2)，即復(fù)常數(shù)的模量度自再現(xiàn)模的單程損耗，它的輻角量度自再現(xiàn)模的單程相移，從而也決定模的諧振頻率。21arg22q2、在非對(duì)稱開腔中，按場(chǎng)在腔內(nèi)往返一次寫出模式再現(xiàn)條件及相應(yīng)的積分方程。其中的復(fù)常數(shù)的模量度自再現(xiàn)模在腔內(nèi)往返一次的功率損耗，它的輻角量度自再現(xiàn)模的往返相移，并從而決定模的諧振頻率。求解思路將尋求開腔振蕩模的問(wèn)題歸結(jié)為求解菲涅耳基爾霍夫衍射積分方程這樣一個(gè)數(shù)學(xué)問(wèn)題（積分本征值問(wèn)題）根據(jù)各類開腔的具體幾何結(jié)構(gòu)，寫出方程的具體形式，根據(jù)問(wèn)題的對(duì)稱性引入適當(dāng)?shù)淖鴺?biāo)系考慮到波長(zhǎng)、鏡的線度以及腔長(zhǎng)的相互數(shù)量級(jí)關(guān)系，將方程簡(jiǎn)化（將積分核展開，舍去無(wú)關(guān)緊

23、要的高階小量）對(duì)常見(jiàn)的幾何結(jié)構(gòu)，實(shí)現(xiàn)變量分離，將關(guān)于二元函數(shù)的積分方程化成兩個(gè)單元函數(shù)的積分方程求出積分方程的本征值(m、n)與本征函數(shù)(vm(x) 、 vn(y) )，得到開腔自再現(xiàn)模的全部特征（包括場(chǎng)分布及傳輸特性）nmmnnmmnyxyx)()(),(一般地， vmn(x,y)應(yīng)為復(fù)函數(shù)，它的模vmn(x,y)描述鏡面上場(chǎng)的振幅分布，而其輻角arg vmn(x,y) 描述鏡面上場(chǎng)的相位分布。復(fù)常數(shù)mn的模量度自再現(xiàn)模的單程損耗，它的輻角量度自再現(xiàn)模的單程相移，從而也決定模的諧振頻率。例：平行平面腔模的迭代解法平行平面腔的優(yōu)點(diǎn)：光束方向性極好（發(fā)散角?。?、模體積大、比較容易獲得單橫模缺點(diǎn)：

24、調(diào)整精度要求極高，與穩(wěn)定腔比較，損耗也較大，對(duì)小增益器件不大適用平行平面腔振蕩模所滿足的自再現(xiàn)積分方程至今尚得不到精確的解析解利用迭代公式 直接進(jìn)行數(shù)值計(jì)算。首先，假定在某鏡面上存在一個(gè)初始場(chǎng)分布u1，將它代入上式，計(jì)算u2、u3、u4等。如此反復(fù)運(yùn)算經(jīng)過(guò)足夠多次后，判斷能否滿足下述關(guān)系式如果直接數(shù)值計(jì)算得出了這種穩(wěn)定的場(chǎng)分布，則可認(rèn)為找到了腔的一個(gè)自再現(xiàn)模或橫模。1jjuKu ds12111jjjjuuuuThe round-trip wave propagation in a real laser cavity can be studied by carrying out analytic

25、al or computer calculations of the manner in which the transverse field pattern of the optical beam changes on repeated round trips within a given resonator. Optical resonator mode calculations of this type were first pioneered in the early 1960s by A.G. Fox and T. Li at the Bell Telephone Laborator

26、ies, and are often referred to as “Fox and Li” calculations. Such calculations are usually carried out with the laser gain omitted for simplicity. It then turns out that for any given laser cavity, employing either finite-diameter planar or (more usually) finite-diameter curved end mirrors, one will

27、 always find a certain discrete set of transverse eigenmodes, or distinct amplitude and phase patterns for the circulating beam in the cavity, which will reproduce themselves in form, though slightly reduced in overall amplitude, after one round trip. These self-reproducing transverse field patterns

28、 represent the characteristic set of lowest-order and higher-order transverse eigenmodes or transverse spatial modes characteristic of that particular laser resonator. These self-reproducing transverse eigenmodes, with amplitude and phase patterns that depend on the specific curvature and shape of the laser mirrors, are closely analogous to the lowest-order and higher-order propagation modes in a leaky optical lensguide. Indeed, we can view the repeated round trips in either a standing-wave or a rin