Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators

Optics & Laser Technology 31 (1999) 239±245

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E€ect of chromatic aberration on Gaussian beams: non-dispersive laser resonators Luis Martõ -LoÂpez*, Omel Mendoza-Yero Cuban Neuroscience Centre. Laboratory of Optoelectronics, A.P. 6880, C.P. 10600 Havana, Cuba Received 22 October 1998; received in revised form 13 April 1999; accepted 19 April 1999

Abstract The propagation of a non-monochromatic (polychromatic) TEM00 Gaussian beam in vacuum, its passage through a thin plate and its transformation by a thin lens are studied in the case of a non-dispersive laser resonator. The basic assumptions of the model are as follows: optical ®elds are stationary and plane-polarized, the paraxial wave equation is valid, an equivalent nondispersive hemiconfocal resonator represents the lasing medium and its stable resonator, the laser emits in a single mode. It is also assumed that the plate and the lens have large transverse dimensions. Mathematical expressions, for beam radius, divergence, radius of curvature and beam parameter product, are obtained. A beam quality factor for polychromatic Gaussian beams is de®ned and its value calculated in each case of interest. It is proposed to simulate a dispersive laser resonator by a non-dispersive resonator complemented with a plate and/or a thin lens. # 1999 Elsevier Science Ltd. All rights reserved. PACS: 42.60.Jf (Beam characteristics: pro®le intensity and power; spatial pattern formation) Keywords: Gaussian beams; Chromatic aberration

1. Introduction Laser beams are usually regarded to be monochromatic because their spectral bandwidths may be extremely small. In such an approximation, the optical properties of the medium remain almost constant within the spectral range of the beam. The propagation of monochromatic Gaussian beams in the paraxial approach is well known [1±4]. When the paraxial approach is not held, some extensions of the theory should be used [5,6]. The e€ect of wave front aberrations on monochromatic Gaussian beams has also been studied [2,7±11]. If the monochromatic approximation is not valid, the framework of the theory of partial coherence can be applied for studying laser beam propagation. Unfortunately, the quasi-monochromatic approximation l  Dl (where l is the * Corresponding author. Fax: +53-7-336-321. E-mail address: [email protected] (L. Martõ -LoÂpez)

mean wavelength in vacuum and Dl is the spectral bandwidth) is often used for the study of laser beams (see, e. g. Refs. [12,13]). As in the monochromatic approximation, it is equivalent to considering that the optical properties of the medium do not change within the spectral range of the beam. As a consequence, the e€ect of some optical phenomena on laser beam parameters is neglected. For example, some commercially available molecular lasers (with no tuning device in their resonators) may have a bandwidth of Dl 0 300 nm and more (it is equal or greater than the bandwidth of the visible spectrum!). Although the condition l  Dl may hold for these lasers, the quasi-monochromatic approximation cannot be applied if one wants to take into account the e€ect of chromatic aberration on beam parameters. Obviously, in the later case the laser beam has to be considered non-monochromatic (polychromatic). In this work, we study the propagation of a polychromatic Gaussian beam in vacuum, its passage through a thin plate and its transformation by a thin

0030-3992/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 9 ) 0 0 0 4 7 - X

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…1 …1 ÿ1 ÿ1

Fig. 1. Propagation of a monochromatic TEM00 Gaussian beam in vacuum. n is the index of refraction of the active medium; it does not depend on coordinates x, y and z.

lens. We use a simpli®ed model of the laser resonator, the non-dispersive laser resonator. Extensions of the known de®nitions of beam parameter and beam quality factors are introduced and applied in order to characterize the e€ect of dispersion on the laser beam. Attention is focused on polychromatic TEM00 Gaussian beams.

ju…x, y, z, l†j2 dx dy ˆ 1:

…2†

Function u(x, y, z, l ) is a solution of the paraxial wave equation. Examples of this kind of functions are normalized Hermite±Gaussian and Laguerre±Gaussian TEM p modes. It should be noted that function r…l†u…x, y, z, l† is also a solution of the paraxial wave equation. The irradiance of the beam, I(x, y, z ), the spectral power density, r(l ) and its radiant ¯ux, P, satisfy the following formulae. …1 Il …x, y, z, l† dl …3† I…x, y, z† ˆ 0

r…l† ˆ

Pˆ

…1 …1 ÿ1 ÿ1

…1 …1 ÿ1

ÿ1

Il …x, y, z, l† dx dy

I…x, y, z† dx dy ˆ

…1 0

r…l† dl :

…4†

…5†

2. Physical model

It will be also supposed that the plate and the lens have large transverse dimensions (so, we may neglect di€raction e€ects) and the lens is free from wave front aberrations. Attenuation is neglected too.

The present model for studying the propagation of polychromatic Gaussian beams is based on the following assumptions:

3. De®nitions of beam radius and divergence

1. Optical ®elds are stationary and plane polarized. Evanescent waves are neglected. The paraxial wave equation is valid for each component of the polychromatic beam. 2. An equivalent hemiconfocal resonator represents the lasing medium and its stable resonator. The Rayleigh range of this equivalent resonator is zR0. The Rayleigh range zR0 does not depend on l and the waist of each monochromatic component of the beam is located in the plane mirror of the resonator. In other words, we deal with a non-dispersive laser resonator. For simplicity, it will be considered that the beam leaves the resonator through the plane mirror. The beam propagates along the z-axis. For further details see Fig. 1. 3. The spectral irradiance of output laser beam can be described by a function Il(x, y, z, l ) given by the expression Il …x, y, z, l† ˆ r…l†ju…x, y, z, l†j2 ,

…1†

where, x, y and z are coordinate axes, l is the wavelength (in vacuum) of a monochromatic component of the beam, r(l )r 0 is the spectral power density and u(x, y, z, l ) describes a normalized transverse mode, such that

The standard ISO/DIS 11146 [14] de®nes the beam radius, s, by the expression …1 …1 21 r2 I…x, y, z† dx dy, …6† s P ÿ1 ÿ1 p where r ˆ x 2 ‡ y2 . It was supposed that the centre of irradiance has coordinates x=y = 0. We de®ne the beam radius of a monochromatic component of the beam (r(l )>0), sl, by the expression …1 …1 r2 ju…x, y, z, l†j2 dx dy s2l ˆ ÿ1 ÿ1

ˆ

1 r…l†

…1 …1 ÿ1 ÿ1

r2 Il …x, y, z, l† dx dy:

…7†

By using formulae (3), (4) and de®nition (7) we can transform the expression (6) into expression … 1 1 2 r…l†s2l dl ˆ s2l : …8† s ˆ P 0 The squared far-®eld divergence of the beam, y 2, is de®ned by the expression

L. MartõÂ-LoÂpez, O. Mendoza-Yero / Optics & Laser Technology 31 (1999) 239±245

!

y2 ˆ

lim

zÿz0 41

s2 , …z ÿ z0 †2

…9†

where z0 is the position of the beam waist. A further transformation of expression (9) yields … 1 1 r…l†y2l dl ˆ y2l , …10† y2 ˆ P 0 where yl is the divergence of a monochromatic component of the beam, given by the expression ! 2 s l : …11† y2l ˆ lim zÿz0 41 …z ÿ z †2 0

4.1. Beam radius According to de®nition (7) and assumption (b), the squared radius, sl2, of a monochromatic TEM00 Gaussian beam in a plane z is given by the expression [1±3] ! w2l …z, l† …z ÿ z0 †2 2 2 ˆ sl0 1 ‡ , …12† sl ˆ 2 z2R0 where, wl(z, l ) is the beam radius de®ned according to the e ÿ2 criterion [2], z0 is the coordinate of the waist in the z-axis and sl0 is the waist radius, de®ned according to expression (7). The waist radius according to the e ÿ2 criterion, wl0 and variables sl0, l, zR0 are linked by the expression s2l0 ˆ

2

ˆ

lzR0 : 2p

…13†

Substitution of expression (12) in expression (8) yields ! …z ÿ z20 † 2 2 …14† s ˆ s0 1 ‡ z2R0 where s20 ˆ s2l0 1 l ˆ P

w2 l zR0 ˆ l0 ˆ 2 2p

…1 0

r…l†l dl:

4.2. Far-®eld divergence From expressions (10)±(12) it follows that,

l : 2pzR0

…17†

4.3. Radius of curvature The radius of curvature, R, of a monochromatic Gaussian beam in vacuum is given by the expression [3] ! z2R0 : R ˆ …z ÿ z0 † 1 ‡ …z ÿ z0 †2

…18†

In this case the radius of curvature does not depend on wavelength. 4.4. Beam parameter product and beam quality factor

4. Polychromatic TEM00 Gaussian beam in vacuum

w2l0

y2 ˆ

241

…15†

…16†

Let us consider a beam parameter product of a monochromatic TEM00 beam, MBP, given by the expression, MBP ˆ sl0 yl :

…19†

Substituting expressions (11)±(13) in expression (19), we get MBP ˆ

l : 2p

…20†

For a polychromatic beam, the beam parameter product, BP, can be de®ned by the formula BP ˆ s0 y :

…21†

It is easy to show that BP ˆ

l : 2p

…22†

Let us consider a beam quality factor, M2P, de®ned by the expression M2P ˆ

BP : l =2p

…23†

The value l =2p is the beam parameter product of a monochromatic TEM00 Gaussian beam of wavelength l0 ˆ l in vacuum. It can be demonstrated that the de®nition (23) contains the usual de®nition [15] of M 2, as a particular case, for which r(l )=Pd(lÿl0), where letter d denotes a Dirac delta function. For a monochromatic TEM00 Gaussian beam, the beam quality factor takes the value, M2P=1. From the expressions (22) and (23) it follows that the beam quality factor of a polychromatic TEM00 Gaussian beam emitted by a non-dispersive laser resonator is M2P=1.

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Qi ˆ

1 P

…1 0

  1 i r…l†s2l0 1 ÿ dl i ˆ 1, 2: n…l†

…27†

It is easy to show that the position z=zC of the plane that cuts the narrowest part of the polychromatic beam and the beam radius in that plane, sC, are given by the expressions z C ˆ z0 ‡ Fig. 2. Passage of a monochromatic TEM00 Gaussian beam trough a plate.

5. Polychromatic TEM00 Gaussian beam transmitted through a plate We will consider a non-absorbing plane-parallel plate placed behind the plane mirror of the resonator. The laser beam, as shown in Fig. 2 normally illuminates the plate. In the paraxial approach, the longitudinal displacement of the waist of a monochromatic Gaussian beam, due to its passage through a plate is given by the expression [3],   1 tˆ)z0l ˆ z0 Dz0l ˆ z0l ÿ z0 ˆ 1 ÿ n…l†   1 t, ‡ 1ÿ n…l†

5.1. Beam radius The squared radius of a monochromatic TEM00 beam, transmitted through the plate is, ! …z ÿ z0 †2 2 2 ÿ2 sl ˆ sl0 1 ‡ z2R0 

   1 …z ÿ z0 †t 1 2 t2 2 1ÿ ‡ sl0 1 ÿ : n…l† n…l† z2R0 z2R0

…25†

To obtain expression (25) we substituted expression (24) in expression (12). The formula (25) is valid for the zone z>zp. The substitution of expression (25) in expression (8) yields, ! …z ÿ z0 †2 …z ÿ z0 †t t2 2 2 Q ‡ Q2 , …26† s ˆ s0 1 ‡ ÿ 2 1 z2R0 z2R0 z2R0 where

2

ˆ s …zC † ˆ

…28†

s20

! t2 Q21 ‡ 2 Q2 ÿ 2 : zR0 s0

…29†

The beam radius sC is the virtual waist radius of the transmitted polychromatic beam. 5.2. Far-®eld divergence It is easy to show that, after the passage through the plate, the far-®eld divergence of a monochromatic component of the beam, yl and the far-®eld divergence of the polychromatic beam itself, y, are given by the expressions y2l ˆ

l 2pzR0

…30†

y2 ˆ

l 2pzR0

…31†

…24†

where Dz0l is the displacement of the waist of a monochromatic component of the Gaussian beam, z0l is the position on the z-axis of the virtual waist of the transformed beam, n(l ) is the index of refraction of the plate and t is its width. See Fig. 2.

s2l0

s2C

tQ1 s20

5.3. Radius of curvature The radius of curvature Rl of a transmitted monochromatic component of the Gaussian beam can be calculated by using expression (18). It yields ! z2R0 : …32† Rl ˆ …z ÿ z0l † 1 ‡ …z ÿ z0l †2 If z>>zP+(1ÿ(1/n(l )))t, then Rl 1 R. It means, that in far-®eld the plate does not a€ect the radius of curvature of the transmitted beam. 5.4. Beam parameter product and beam quality factor From expressions (13), (19) and (30) it follows that the monochromatic beam parameter product is an invariant. That is MBP ˆ MBPP ,

…33†

where MBP and MBPP are the monochromatic beam products of the non-perturbed (before the plate was placed in the beam path) and the transmitted beams, respectively.

L. MartõÂ-LoÂpez, O. Mendoza-Yero / Optics & Laser Technology 31 (1999) 239±245

2zE z2E ‡ z2RE bˆ 1‡ ‡ f f 2

243

!ÿ1 :

…40†

6.1. Beam radius The radius of a monochromatic component of the transformed beam, slS, can be calculated by using expressions (12), (37), (38) and (40). It yields Fig. 3. Transformation of a monochromatic TEM00 Gaussian beam by a thin lens.

s2lS ˆ

l z2 ‡ z2RE ‡ z2l ÿ 2zl zE 2pzRE E ! z2l zE ÿ zl …z2E ‡ z2RE † z2l …z2E ‡ z2RE † ‡2 ‡ , f f 2

The beam parameter product of the transmitted beam, BPP, can be found by using expressions (29) and (31). It yields !!1=2 t2 Q21 BP, …34† BPP ˆ sC y ˆ 1 ‡ 2 2 Q2 ÿ 2 zR0 s0 s0 where it was taken into account that, the factor l =2p is the beam parameter product of the non-perturbed beam. Expression (34) means that, in general, the beam parameter product is not an invariant. The substitution of expression (34) in expression (23) yields, !!1=2 t2 Q21 2 : …35† MP ˆ 1 ‡ 2 2 Q 2 ÿ 2 zR0 s0 sl0

6. Polychromatic TEM00 Gaussian beam transformed by a thin lens The waist radius, wl0E, its position, zE, its far-®eld divergence, ylE and the Rayleigh range, zRE, of a monochromatic component of the input polychromatic TEM00 Gaussian beam and their counterparts, wl0S, zlS, ylS, zRlS (see Fig. 3) of the output beam are linked by formulae [3,4] ! z2E ‡ z2RE …36† zlS ˆ b zE ‡ f wl0S ˆ b1=2 wl0E , sl0S ˆ b1=2 sl0E

…37†

zRlS ˆ bzRE

…38†

ylS ˆ bÿ1=2 ylE ,

…39†

where f is the focal length of the lens and the coef®cient b is given by the expression

…41†

where zl is the distance from the lens. See Fig. 3. The beam radius of the transformed polychromatic beam, sS, can be calculated by substituting expression (41) in de®nition (8). It yields s2S ˆ

s2l0E 2 z2l 2 ‡ z ‡ ÿ2 z RE bP z2RE E 

…z2 ‡ z2RE † z l zE ‡ E F1

! ,

…42†

where sl0E is the waist radius of the input beam (it is related to wl0E by expression (13)) and … 1 1 s2l0E ˆ r…l†s2l0E dl : …43† P 0 Coecients F1 F2 and bP are given by formulae … 1 1 1 r…l†l ˆ dl i ˆ 1, 2 …44† Fi l P 0 f i  ÿ1 2zE …z2E ‡ z2RE † ‡ : bP ˆ 1 ‡ F1 F2

…45†

After simple calculations it can be shown that, the coordinate zl=zlC of the plane, which cuts the narrowest part of the transformed beam and the beam radius in that plane, sSC, are given by the expressions   …z2 ‡ z2RE † zlC ˆ bP zE ‡ E …46† F1 s2SC ˆ s2S …zlC † ˆ  ÿbP

s2l0E 2 z ‡ z2RE z2RE E

…z2 ‡ z2RE † zE ‡ E F1

2 !

:

…47†

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L. MartõÂ-LoÂpez, O. Mendoza-Yero / Optics & Laser Technology 31 (1999) 239±245

Radius sSC may be considered as the waist radius of the transformed polychromatic beam. 6.2. Far-®eld divergence The substitution of formula (42) in expression (9) yields y2S ˆ

l : 2pbP zRE

…48†

6.3. Beam parameter product and beam quality factor Waist radii and far-®eld divergences of a monochromatic component of the Gaussian beam satisfy expression MBPE ˆ MBPS ,

…49†

where MBPE and MBPS are the monochromatic beam parameter products of the incident beam and the transformed beam, respectively. The expression (49) means that in an optical system free from wave front aberrations, the monochromatic beam parameter product is an invariant. The beam parameter product of the transformed polychromatic beam, BPS, is 2 2 zÿ1 BPS ˆ yS sSC ˆ bÿ1=2 P RE zE ‡ zRE

 ÿbP

…z2 ‡ z2RE † zE ‡ E F1

2 !1=2 BPE

M2P ˆ 1 ‡

It easy to demonstrate that obtained formulae for polychromatic Gaussian beam contains the formulae for monochromatic beams and for non-dispersive media (dn(l )/dl=0 for both the plate and the lens) as particular cases.

…50† 8. Conclusion

where BPE is the beam parameter product of the incident polychromatic beam, given by the expression (22). From the expressions (23) and (50) it follows that the beam quality factor of the polychromatic TEM00 beam is z4E ‡ 2z2E ‡ z2RE z2RE

Gaussian beam transmitted through a dispersive plate is displaced in a magnitude DzC ˆ zC ÿ z0 ˆ tQ1 =s20 . Its squared beam waist increases in a magnitude Ds2C ˆ s2C ÿ s20 ˆ t2 =z2R0 …Q2 ÿ …Q21 =s20 ††. The beam quality factor is equal or greater than unity, where equality holds for a monochromatic beam or a non-dispersive plate. 3. The waist of a polychromatic TEM00 Gaussian beam, transformed by a thin lens, is displaced in a  ˆ b …zE ‡ …z2 ‡ magnitude DzlC ˆ zlC ÿ zlS …l† P E 2 2   and its zRE †=F1 † ÿ b…l †…zE ‡ …zE ‡ z2RE †=f…l†† squared radius increases in a magnitude Ds2SC ˆ s2SC ÿ s2l0S …l † ˆ s2l0 =z2RE …z2E ‡ z2RE ÿ bP …zE ‡  2 …l †, if it is compared with …z2E z2RE †=F1 †2 † ÿ b…l†s l0E the position and the radius of the waist of a monochromatic TEM00 Gaussian beam, transformed by the lens. This monochromatic Gaussian beam has wavelength l0 ˆ l , Rayleigh range zRE and waist position zE. The magnitudes DzlC and Ds2SC can be considered as errors, if the monochromatic approximation is used. 4. The coecient bP of a lens is a generalization of the `monochromatic' coecient b. The beam quality factor of the transformed beam is equal or greater than unity, where equality holds for a monochromatic beam or a non-dispersive lens.

!

1 1 ÿ F2 F21

 !1=2 :

…51†

7. Analysis of results The main results can be summarized as follows. 1. In vacuum, a polychromatic TEM00 Gaussian beam, emitted by a non-dispersive laser resonator, behaves as a monochromatic one with a wavelength  The beam quality factor of the beam equals l0 ˆ l. unity. 2. The virtual waist of a polychromatic TEM00

The fact that the laser beam is polychromatic means that the chromatic aberration of the optical system has in¯uences on the transformed beam. This is why the beam parameter product is not an invariant, unless the beam is a monochromatic one or the optical system is a non-dispersive one (that is, the optical system is free from chromatic aberration). It also explains why the beam quality factor may become greater than unity after its passage through the plate or the lens. We studied the propagation of polychromatic Gaussian beams, with emphasis on TEM00 mode. Obtained results for radius, waist position and far-®eld divergence can be extended to a pure mode of higher order, since its beam radius in any plane is proportional to the beam radius of the corresponding TEM00 mode [3]. Although the physical model of the laser resonator is quite simple, it gives a useful insight into the e€ects of chromatic aberration on fundamental parameters of polychromatic Gaussian beams. Furthermore, it allows representing a dispersive laser resonator as an equival-

L. MartõÂ-LoÂpez, O. Mendoza-Yero / Optics & Laser Technology 31 (1999) 239±245

ent non-dispersive laser resonator complemented with a plate and/or a lens. These ®ctitious optical elements can mathematically simulate resonator dispersion e€ects. Such representation would be applied to extend obtained results to some laser resonators other than non-dispersive ones and may be of practical interest for the design of optical systems with lasers. It should also be noted that a similar idea has been applied to the experimental characterization of the thermal lensing e€ect in lasers (see e.g. Refs. [16,17] and references cited therein). Acknowledgements Author (LML) thanks J. NuÂnÄez (from the National Centre for Scienti®c Research, Havana, Cuba) for help and Dr. J.C. Soriano, J.A. Ramos y R. Estrela for hospitality at the AsociacioÂn Industrial de OÂptica (Valencia, Spain). References [1] Kogelnik H. Laser beams and resonators. Proc IEEE 1966;54:1312±29. [2] Siegman AE. Lasers. Mill Valley, CA: University Science Book, 1986. [3] Klimkov Yu M. Osnovy rasschiota opticheskij i electronnij priborov s lazerami (Fundamentals of calculus of optical and electronic devices with lasers). Moscow: Sovietskoye Radio, 1978 (in Russian).

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