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Capitolo 5 – Cavità ottiche 5. Cavità ottiche Una cavità risonante ottica (o risonatore ottico) si può considerare come l’estensione a frequenze ottiche dei circuiti oscillanti in bassa frequenza e delle cavità risonanti impiegate alle frequenze delle microonde. Anche per le cavità ottiche si può parlare di modi di oscillazione, cioè di configurazioni di campo stazionarie all’interno della cavità. Non rientrano in questa definizione le cosiddette cavità instabili, usate nei laser di grande potenza. Le cavità ottiche sono costituite generalmente da specchi o altre strutture riflettenti, che hanno il compito di far sì che il campo percorra più volte il materiale attivo, risultandone amplificato. Modi lungitudinali Longitudinal modes involve light waves which travel parallel to the laser cavity axis. When such a wave is reflected from a cavity mirror, the reflected wave combines with the incident wave to give rise to a standing or stationary wave as indicated in Figure 5.1. To simplify the diagram, the amplifying medium has been omitted and, provided that it can sustain several modes of oscillation at the same time, it is not necessary to consider it to understand the origin of multiple longitudinal modes. Amplifying media that can support the simultaneous oscillation of a number of longitudinal modes are referred to as being inhomogeneously broadened. Figure 5.1. Standing waves can be sustained within the laser cavity only if the length of the cavity is equal to an integral number of half wavelengths. This condition can be written as: (5.1) nL = s λ/2, con s intero Nella 5.1 n indica l’indice di rifrazione del mezzo che riempie la cavità. For most lasers, values of s larger than a million are typical. Because of this, different longitudinal modes corresponding to successive values of s have wavelengths that are different, but only slightly different. It is a simple matter to write down the above condition in terms of the frequency of a mode rather than its wavelength and, from the resulting expression, calculate the difference in frequency between two successive longitudinal modes. La separazione in frequenza tra due modi longitinali adiacenti si ottiene dalla (5.1) e vale (5.2) Δν = c/2nL con c velocità della luce nel vuoto. In termini di lunghezza d’onda avremo ΔλMode = λ2 (5.3) 2nL Nel caso di un laser a He-Ne di 50 cm di lunghezza avremo: λ2 (632.8 × 10 −9 m) 2 ΔλMode = = = 6.7 × 10 −13 m 2nL 2 × 1 × 0.3 m ovvero ΔλMode 6.7 10 –4 nm. Considering that the half-width of the gain profile of He and Ne gases is about 2 10-3 nm, three longitudinal modes can be excited. 5/ Anno accademico 2007/2008 1 Capitolo 5 – Cavità ottiche Many types of laser emit light in more than one longitudinal mode at the same time. Depending upon the cavity length and the range of frequencies over which the amplifying medium amplifies, the number of longitudinal modes in the output of a multimode laser can be in the region of 100. The output of a laser emitting several longitudinal modes, represented as a plot of intensity against frequency, would have the form shown in Figure 5.2: Figure 5.2 The relative heights of the different modes illustrated in this diagram are determined by the extent to which the amplifying medium amplifies these different frequencies of light. While the frequency spacing in hertz between the different modes might appear large, it must be borne in mind that it is very small compared to the frequency of light in the visible region of the spectrum. Compared to a conventional thermal light source, the output of a multimode laser would still be regarded monochromatic. To make the output even more monochromatic, there are techniques for selecting out a single longitudinal mode so that the output of the laser consists of that one mode only. The simplest mode selection technique involves positioning in the laser cavity a component known as an etalon which has the effect of surpressing all modes except the one required. Mode selection is a technique that enables a laser to emit highly monochromatic light, but longitudinal modes are relevant to laser technology in another and quite different respect. If many longitudinal modes with the appropriate phase relationship are combined, the result is a light pulse. The duration of the pulse becomes shorter as the number of longitudinal modes increases. Techniques have been developed for isolating within the laser cavity a set of longitudinal modes that have the required phase relationship with the result that the output of what would otherwise be a continuous laser consists of a train of regularly spaced and very short pulses of light. The time interval between these pulses is itself very short and is equal to the time that it would take for light to make one round trip in the laser cavity. These techniques are referred to here as 'mode locking'. There are many ways of limiting the laser oscillation to the width of a single longitudinal mode. The most common technique is to insert a Fabry-Perot etalon, a pair of reflective surfaces, between the laser mirrors to form a second resonant cavity. Of the multiple longitudinal modes of the laser cavity, only the one that coincides with the etalon mode can oscillate; all other modes will be canceled by destructive interference when they pass through the etalon. As shown in figure IV-63, the etalon is generally tilted at a suitable angle to the axis of the cavity in order to coincide the oscillation mode of the Fabry-Perot etalon with the laser cavity. La Figure 5.3 mostra la tecnica di selezione di unb singolo modo longitudinale per mezzo di un etalon Fabry-Perot. 5/ Anno accademico 2007/2008 2 Capitolo 5 – Cavità ottiche Figure 5.3 Modi trasversali Transverse modes are stable modes of oscillation involving light waves that are not propagating parallel to the laser cavity axis. Such modes arise from the use of cavity mirrors with a spherical rather than plane contour and from the effects of diffraction. They are condiderably more difficult to analyse mathematically than are longitudinal modes and resonators are usually designed with the intention of ensuring that high order transverse modes do not occur. Except in the case of the simplest mode, their effect on the distribution of intensity over a cross section of the output beam is to give rise to two or more regions of high intensity separated by regions of lower intensity. Examples of such intensity distributions are illustrated in Figure 5.4. Figure 5.4 In the simplest case, the resonator modes are Hermite-Gaussian modes. At a beam waist, the electric field distribution can be written as a product of two Hermite polynomials with orders n and 5/ Anno accademico 2007/2008 3 Capitolo 5 – Cavità ottiche m (non-negative integers, corresponding to x and y direction) and two Gaussian functions. One also calls these modes TEMnm modes; the article on higher order modes describes the exact mathematical form. The intensity distribution of such a mode (see the Figure 5.4) has n nodes in the horizontal direction and m nodes in the vertical direction. For a resonator mode, there is in addition an axial mode number q, because the resonance condition is satisfied only for certain discrete frequencies; one may use a notation like TEMnmq to include the axial mode number in cases where it is important. Modes with n = m = 0 are called axial modes (or fundamental modes, Gaussian modes), while all other are called higher-order modes or higher-order transverse modes. La Figura 5.5 mostra la distribuzione della fase per i modi trasversali di ordine piùbasso. Figure 5.5 The simplest, or the lowest-order transverse mode is TEM00, which has a smooth crosssection profile with a Gaussian peak in the middle, as shown in Figure 5.6. Figure 5.6 La Figura 5.7 mostra, infine, lo spettro dei modi longitudinali (o modi assiali) con associati i vari modi trasversali (o modi di ordine superiore). Figure 5.7 Il numero di mosi assiali che entrano in oscillazione dipende dalla larghezza della riga di guadagno del mezzo attivo. La Figura 5.8 illustra la situazione dei modi che cadono entro la curva di guadagno del laser. 5/ Anno accademico 2007/2008 4 Capitolo 5 – Cavità ottiche Figure 5.8 Il modo TEM00 presenta le minori perdite per diffrazione ed è, quindi, favorito nell’oscillazione laser. Gli altri modi di ordine superiore hanno ai bordi un’intensità più elevata del modo TEM00 e subiscono perdite per diffrazione maggiori. La Figura 5.9 illustra come si giunge ad una distribuzione d’intensità tipo modo TEM00 a seguito delle riflessioni multiple tra i due specchi della cavità laser: in questo caso, si simula l’effetto facendo attraversare il fascio laser attraverso una sequenza di diaframmi di apertura pari alla sezione degli specchi. I lobi di diffrazione vengono via via attenuati, fino ad ottenere una distribuzione d’intensità con il massimo al centro dell’apertura. Figura 5.9 Il fascio gaussiano Un fascio di luce è detto gaussiano quando il suo profilo di intensità su un piano perpendicolare alla direzione di propagazione, segue una distribuzione gaussiana, come nel caso del modo fondamentale TEM00. L'importanza nello studio dei fasci gaussiani sta nel fatto che ben descrivono, in termini ondulatori, la radiazione emessa da una sorgente laser; infatti non sempre, e specialmente nell'interazione con elementi ottici, si può considerare valida l'approssimazione di considerare un fascio laser come un'onda piana. L'intensità trasportata dal fascio gaussiano è data da I(x,y,z) = I0 [w0/w(z)]2 exp[-2r2 /w2(z)] Il parametro w(z) rappresenta la distanza dall'asse z alla quale l'ampiezza del campo si riduce di un fattore 1/e e l'intensità si riduce di 1/e2; in inglese prende il nome di spot size (letteralmente "dimensione del punto") del fascio. In pratica rappresenta il diametro dell'area intorno all'asse di propagazione del fascio in cui è concentrata la maggior parte della potenza. Il parametro w0 rappresenta il minimo, per z = 0, delle dimensioni trasverse del fascio gaussiano, cioè per z = 0 si ha il massimo di intensità. Il nome inglese è waist (letteralmente "cintura"). 5/ Anno accademico 2007/2008 5 Capitolo 5 – Cavità ottiche Per r2 << z2 il fronte d'onda di un fascio gaussiano può essere approssimato con un fronte d'onda sferico di raggio R(z). In z = 0, R diverge, cioè il fronte d'onda è piano, mentre in valore assoltuo è minimo per z = ± z0; a grande distanza dal punto di waist si può considerare , cioè un'onda sferica. Tipi di cavità ottiche Alcune cavità risonanti sono mostrate in Figura 5.10. La forma più semplice di cavità risonante è quella costituita da due specchi piani paralleli (Fig. 5.10.1) di cui uno semitrasparente; vi possono essere però anche cavità costituite da specchi sferici confocali (Fig. 5.10.2) o da uno specchio piano e uno sferico (Fig. 5.10.3) disposti in maniera tale che il fuoco dello specchio sferico cada sullo specchio piano. Tutte le cavità costruite in questa maniera sono dette cavità stabili, in quanto vi sono alcuni raggi, che seguono particolari percorsi, che si richiudono, in termini di ottica geometrica, su se stessi e che escono dal sistema risonante per trasparenza. Figura 5.10: alcuni tipi di cavità ottiche stabili Vi sono invece altri tipi di cavità, dette instabili, per le quali non si possono individuare percorsi seguendo i quali i raggi si richiudano su se stessi; con questi tipi di cavità, la potenza utile è quella che sfugge a un'estremità (non trasparenza, ma perdita ai bordi dello specchio o attraverso fori praticati nello specchio stesso). Le cavità risonanti instabili vengono utilizzate per laser ad altissima potenza, in quanto, se la potenza sale molto, specchi costituiti da materiali trasparenti potrebbero degradarsi in seguito al passaggio del fascio laser. Gli specchi per laser di alta potenza sono generalmente costituiti da blocchi metallici dorati dotati di alta resistenza alla temperatura. Figura 5.11: alcuni tipi di cavità ottiche instabili Cavità stabili Illustriamo, ora, brevemente i tipi di cavità ottiche più diffuse (Figura 5.12). Figura 5.12 Plane Parallel Optical Cavity At both ends there are two plan mirrors, parallel to each other, and perpendicular to the laser optical axis. 5/ Anno accademico 2007/2008 6 Capitolo 5 – Cavità ottiche Advantages •Optimal use of all the volume of the active medium. Thus, used in pulsed lasers which need the maximum energy. •No focusing of the laser radiation inside the optical cavity. In high power lasers such focusing can cause electric breakdown, or damage to the optical elements. Disadvantages •High diffraction losses. •Very high sensitivity to misalignment. Thus, very difficult to operate. Concentric Circular Cavity At both ends there are two spherical mirrors with the same radiuses. The distance between the center of the mirrors is equal to twice the radius of curvature of each of them (R1 = R2 = L/2). This arrangement cause focusing of the beam at the center of the cavity. The properties of this cavity are the opposite of those of the plan parallel cavity. Advantages •Very low sensitivity to misalignment. Thus, very easy to align. •Low diffraction losses. Disadvantages •Limited use of the volume of the active medium. Used in optical pumping of continuous Dye lasers. In these lasers the liquid dye is flowing in the region of the beam focusing (the flow direction is perpendicular to the optical axis of the laser). Thus very high power density is used to pump the dye. •Maximum focusing of the laser radiation inside the optical cavity. Such focusing can cause electric breakdown, or damage to the optical elements. Confocal Cavity This cavity is a compromise between plan parallel and circular optical cavities. At both ends there are two spherical mirrors with the same radiuses. The distance between the center of the mirrors is equal to the radius of curvature of each of them (R1 = R2 = L). This arrangement cause much less focusing of the beam at the center of the cavity. Advantages •Little sensitivity to misalignment. Thus, easy to align. •Low diffraction losses. •No high focusing inside the cavity. •Medium use of the volume of the active medium. The main difference between the confocal cavity and the spherical cavity is that in the confocal cavity the focal point of each mirror is at the center of the cavity, while in spherical cavity the center of curvature of the mirrors is in the center of the cavity. Cavity with Radius of Curvature of the mirrors Longer than Cavity length. 5/ Anno accademico 2007/2008 7 Capitolo 5 – Cavità ottiche This cavity is a better compromise than confocal cavity between plan parallel and circular optical cavities. At both ends there are two spherical mirrors with big radiuses of curvature (does not need to be the same). The distance between the center of the mirrors is much less then the radius of curvature of each of them (R1 , R2 >> L). This arrangement cause much less focusing of the beam at the center of the cavity. Advantages •Medium sensitivity to misalignment. •Medium diffraction losses. •No high focusing of the beam inside the cavity. •Good use of the volume of the active medium Hemispherical Cavity The cavity is created by one plan mirror, and one spherical mirror with radius of curvature equal to the length of the cavity. This cavity is similar in properties to circular optical cavity, with the advantage of the low price of the plan mirror. Most Helium-Neon lasers use this cavity which have low diffraction losses, and is relatively easy to align. Advantages •Low sensitivity to misalignment. •Low diffraction losses Half Curve with longer than cavity radius of curvature. The cavity is created by one plan mirror, and one spherical mirror with radius of curvature much larger than the length of the cavity. This cavity is similar in properties to Confocal cavity, with the advantage of the low price of the plan mirror. Cavità instabili Advantages • High volume of the modes inside the active medium (The entire volume). • All the power inside the cavity is emitted out of the laser, not just a small fraction of it. The laser radiation is emitted out of the laser around the edges of the small mirror. This cavity is used in high power lasers, which can not use standard output coupler. Disadvantages The beam shape has a hole in the middle. 5/ Anno accademico 2007/2008 8