Embed Size (px)
Citation preview
670 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Tsigaridas et al.
Z-scan analysis for near-Gaussian beams throughHermite–Gaussian decomposition
G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas
Department of Physics, University of Patras, Patras 26500, Greece
Received July 29, 2002; revised manuscript received November 12, 2002
Gaussian decomposition is used as a theoretical infrastructure with which Z-scan experiments are analyzed.This procedure is extended here to the interesting, from a practical point of view, case in which the laser beamused is not perfectly Gaussian. We follow a perturbative approach to obtain the far-field pattern of the beamafter the beam passes through a nonlinear sample. The procedure is based on the decomposition of the elec-tric field at the exit plane of the sample to a linear combination of Hermite–Gaussian functions. To a first-order approximation, each mode of the incident beam is decomposed to a linear combination of different-ordermodes that do not exceed the order of the original mode. Finally, the effects of the simultaneous presence offirst and higher-order refractive nonlinearities or first-order refractive nonlinearity and nonlinear absorptionare studied. © 2003 Optical Society of America
OCIS codes: 000.3860, 140.3300, 260.5950, 350.5500.
1. INTRODUCTIONThe Z-scan technique has been widely used over the pastdecade as a simple and sensitive method for measuringthe nonlinear optical parameters of several materials.1,2
In the original version of this technique1 the beam inci-dent upon the sample was considered to be a circularGaussian beam (TEM00). Unfortunately, the beams pro-duced by most laser systems are not perfect Gaussiansbut contain also a small amount of higher-order modes(TEMlm). Here we refer to such beams as near-Gaussianbeams. Although several efforts toward the extension ofthe Z-scan technique to more-general input beam profileshave been made to date,2–8 the interesting case of a near-Gaussian input beam had not to our knowledge ad-equately been studied, at least by an analytic approach.We attempt to do that here.
The Gaussian decomposition method, introduced byWeaire et al.9 and further developed by Sheik-Bahaeet al.,1 is employed. We extend this method adequatelyto near-Gaussian incident beams by decomposing theelectric field pattern at the exit plane of the sample to alinear combination of Hermite–Gaussian modes. Thesemodes constitute eigenfunctions of the propagation opera-tor in free space and therefore can straightforwardly givethe electric field pattern of the distorted beam at any de-sired distance D from the exit plane of the sample. A per-turbative approach is followed, considerably simplifyingthe calculations involved.
The method is also extended to the case of existence ofsecond-order refractive nonlinearities or nonlinear ab-sorption. Finally, we perform analytical simulations todetermine the effects of high-order modes in actualZ-scan experiments.
2. GENERAL THEORYIn general, the electric field of a near-Gaussian beam isgiven by
0740-3224/2003/040670-07$15.00 ©
E~x, y, z, t ! 5 c0E0~x, y, z, t ! 1 «Eh~x, y, z, t !,(1)
where E0(x, y, z, t) is the zero-order Gaussian mode(TEM00) and Eh(x, y, z, t) is a sum over the higher-orderHermite–Gaussian modes, given by
Eh~x, y, z, t ! 5 (l,m
clmElm~x, y, z, t !. (2)
Here « is a constant with small value such that «2 is neg-ligible compared to «, permitting a perturbative approach.The coefficients c0 and clm are normalized according to
uc0u2 1 «2 (l,m
uclmu2 5 1. (3)
The electric field of each mode is given by10,11
Elm~x, y, z, t !
5 A~z, t !(2 l2ml!m!)21/2HlF A2x
wx~z !GHmF A2y
wy~z !G
3 expF2x2
wx2~z !
2y2
wy2~z !
G3 expF2
ikx2
2Rx~z !2
iky2
2Ry~z !G
3 exp@2ikz 1 i~l 1 m 1 1 !u~z !#. (4)
Here z is the propagation direction, k is the wave number,and Hl@A2x/wx(z)# and Hm@A2y/wy(z)# are Hermitepolynomials of orders l and m, respectively. wx,y(z) arethe principal semiaxes of the near-Gaussian beam foreach dimension (x, y), Rx,y(z) are the radii of curvature,and u(z) is the on-axis phase shift; they are given by
wx,y~z ! 5 w0x,yF1 1 S z 2 z0x,y
zRx,yD 2G1/2
, (5)
2003 Optical Society of America
Tsigaridas et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 671
Rx,y~z ! 5 z 2 z0x,y 1zRx,y
2
z 2 z0x,y, (6)
u~z ! 51
2tan21S z 2 z0x
zRxD
11
2tan21S z 2 z0y
zRyD , (7)
where z0x and z0y denote the locations of the beam waistsfor the principal dimensions x and y, w0x,y are the beamwidths at the waists, and zRx,y are the Rayleigh lengths,defined as
zRx,y 5kw0x,y
2
2. (8)
Finally, A(z, t) is a normalization quantity that is relatedto input power P(t) by
ce0n0
2uA~z, t !u2 5
2P~t !
pwx~z !wy~z !, (9)
where c is the speed of light in vacuum, n0 is the linearrefractive index of the medium in which the beam ispropagating, and e0 is the permittivity of vacuum.
If the near-Gaussian beam described above passesthrough a thin nonlinear material of length L that is char-acterized only by first-order refractive nonlinearity andlinear absorption, a nonlinear phase shift is induced, andthe electric field pattern at the exit plane of the samplebecomes1
Ee~x, y, zs , t !
5 E~x, y, zs , t !exp~2a0L/2!exp@iDw~x, y, zs , t !#,
(10)
where
Dw~x, y, zs , t ! 5 2kg~1 !I~x, y, zs , t !Leff~1 !. (11)
Here zs is the location of the sample plane, a0 is the linearabsorption coefficient, g (1) is the first-order nonlinear re-fractive index, defined by
n~x, y, zs , t ! 5 n lin 1 g~1 !I~x, y, zs , t !, (12)
where n(x, y, zs , t) is the total refractive index and n linis the linear refractive index of the nonlinear material,Leff
(1) is the first-order effective length, defined as
Leff~1 ! 5
1 2 exp~2a0L !
a0, (13)
and I(x, y, zs , t) is the intensity of the incident beam,given by
I~x, y, zs , t ! 5c«0n0
2uE~x, y, zs , t !u2
5c«0n0
2uc0E0~x, y, zs , t !
1 «Eh~x, y, zs , t !u2. (14)
When phase term exp@iDw(x, y, zs , t)# is expanded in aTaylor series, Eq. (10) becomes
Ee~x, y, zs , t ! 5 E~x, y, zs , t !exp~2a0L/2!
3 (q50
` iq
q!@Dw~x, y, zs , t !#q. (15)
Phase term Dw(x, y, zs , t) given by Eq. (11) can also bewritten in the form
Dw~x, y, zs , t ! 5 2kg~1 !c«0n0
2uc0E0~x, y, zs , t !
1 «Eh~x, y, zs , t !u2Leff~1 !
5 2kg~1 !c«0n0
2uA~zs , t !u2uc0
3 E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !u2Leff~1 !, (16)
where
E08~x, y, zs , t ! 5E0~x, y, zs , t !
A~zs , t !,
Eh8~x, y, zs , t ! 5Eh~x, y, zs , t !
A~zs , t !. (17)
If we define
Dw0~1 !~zs , t ! 5 2kg~1 !
c«0n0
2uA~zs , t !u2Leff
~1 !, (18)
Eq. (16) becomes
Dw~x, y, zs , t ! 5 Dw0~1 !~zs , t !uc0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !u2. (19)
Substituting Eqs. (17) and (19) into Eq. (15) yields for theelectric field pattern at the exit plane of the sample
Ee~x, y, zs , t ! 5 A~zs , t !@c0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !#exp~2a0L/2!
3 (q50
`@iDw0
~1 !~zs , t !#q
q!
3 uc0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !u2q. (20)
Bearing in mind that « is a very small quantity in thesense that terms of second and higher order in « can beneglected, we find for Eq. (20) that
672 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Tsigaridas et al.
Ee~x, y, zs , t !
5 A~zs , t !exp~2a0L/2!(q50
`@iDw0
~1 !~zs , t !#q
q!
3 $uc0u2qc0uE08~x, y, zs , t !u2qE08~x, y, zs , t !
1 «~q 1 1 !uc0u2quE08~x, y, zs , t !u2q
3 Eh8~x, y, zs , t ! 1 «quc0u2~q21 !c02u
3 E08~x, y, zs , t !u2~q21 !
3 @E08~x, y, zs , t !#2@Eh8~x, y, zs , t !#* %. (21)
When the electric field components E08(x, y, zs , t) andEh8(x, y, zs , t) defined through Eqs. (17) are written intheir explicit form [from Eq. (4)], the electric field patternat the exit plane of the sample takes the form
Ee~x, y, zs , t !
5 A~zs , t !exp~2a0L/2!(q50
`@iDw0
~1 !~zs , t !#q
q!
3 expF2~2q 1 1 !x2
wx2~zs!
2~2q 1 1 !y2
wy2~zs!
G3 expF2
ikx2
2Rx~zs!2
iky2
2Ry~zs!G
3 exp@2ikzs 1 iu~zs!#
3 H uc0u2qc0 1 «~q 1 1 !uc0u2q
3 (l,m
clm~2 l2ml!m! !21/2
3 HlF A2x
wx~zs!GHmF A2y
wy~zs!G
3 exp@i~l 1 m !u~zs!# 1 «quc0u2~q21 !c02
3 (l,m
clm~2 l2ml!m! !21/2HlF A2x
wx~zs!GHmF A2y
wy~zs!G
3 exp@2i~l 1 m !u~zs!#J . (22)
To express this field as a linear combination of Hermite–Gaussian functions we perform the transformation
wx8 5wx~zs!
A2q 1 1, wy8 5
wy~zs!
A2q 1 1. (23)
In terms of the new variables (wx8, wy8), the Hermitepolynomials Hl@A2x/wx(zs)# and Hm@A2y/wy(zs)# become
HlF A2x
wx~zs!G 5 (
r50
l
prF A2x
wx~zs!G r
5 (r50
l
pr~2q 1 1 !2r/2S A2x
wx8D r
, (24)
where pr are the coefficients of the Hermite polynomials.Analogous relations hold for Hm@A2y/wy(zs)#. Note thatthe polynomials in the new basis A2x/wx8 [right-handside of Eq. (24)] cease to be Hermitian. However, theycan quite easily be written as a linear combination of Her-mite polynomials up to order l (m) for the x ( y)polynomial.12
In this way, through the transformation given by Eqs.(23) in combination with the transformation of the Her-mite polynomials, the electric field pattern at the exitplane of the sample [Eq. (22)] has now been expressed asa linear combination of Hermite–Gaussian functions.Each term in the sum over q in the new expression of Eq.(22) can be associated with Hermite–Gaussian beamswhose principal semiaxes wx,y
(q) and radii of curvatureRx,y
(q) are the previously defined parameters wx,y8 andRx,y(zs), respectively. Thus we have
wx,y~q !~zs! 5 w0x,y
~q !H 1 1 F zs 2 z0x,y~q !
zRx,y~q ! G 2J 1/2
5 wx,y8,
(25)
Rx,y~q !~z ! 5 z 2 z0x,y
~q ! 1@zRx,y
~q !#2
z 2 z0x,y~q !
5 Rx,y~zs!.
(26)
Solving the system of Eqs. (25) and (26) for fundamentalparameters z0x,y
(q) and zRx,y(q) 5 k@w0x,y
(q)#2/2, namely,the location of the beam waist and the Rayleigh length foreach beam and each principal dimension, we have
z0x,y~q ! 5 zs 2
@Bx,y~q !#2Rx,y~zs!
@Bx,y~q !#2 1 Rx,y
2~zs!, (27)
zRx,y~q ! 5
Bx,y~q !Rx,y
2~zs!
@Bx,y~q !#2 1 Rx,y
2~zs!, (28)
where
Bx,y~q ! 5
k
2
wx,y2~zs!
2q 1 1. (29)
Afterward, the propagation of each beam to any desireddistance D can easily be calculated, and finally, their sumgives the electric field pattern at this position. Specialcare must be taken that the recomposed electric field beconsistent with the field at the exit plane of the sample(D 5 zs). More details are given in Section 3.
3. APPLICATIONSWe now apply the theory developed above to the practi-cally interesting case in which only the first three sym-metric modes exist in the incident beam, the electric fieldof which is given in the form
Tsigaridas et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 673
E~x, y, zs , t ! 5 c0E0~x, y, zs , t !
1 «@c11E11~x, y, zs , t !
1 c22E22~x, y, zs , t !
1 c33E33~x, y, zs , t !#. (30)
In this case the electric field pattern at the exit plane ofthe sample [Eq. (22)] becomes
Ee~x, y, zs , t !
5 A~zs , t !exp~2a0L/2!(q50
`@iDw0
~1 !~zs , t !#q
q!
3 expF2~2q 1 1 !x2
wx2~zs!
2~2q 1 1 !y2
wy2~zs!
G3 expF2
ikx2
2Rx~zs!2
iky2
2Ry~zs!Gexp@2ikzs 1 iu~zs!#
3 H uc0u2qc0 1 «~q 1 1 !uc0u2q (l 51
3 c l l
2 l l !
3 H l F A2x
wx~zs!GH l F A2y
wy~zs!Gexp@2il u~zs!#
3 «quc0u2~q21 !c02 (
l 51
3 c l l
2 l l !H l F A2x
wx~zs!GH l F A2y
wy~zs!G
3 exp@22il u~zs!#J , (31)
where l 5 l 5 m. As described in Section 2, each termin the sum over q can be associated with Hermite–Gaussian functions with parameters
wx,y~q !~zs! 5
wx,y~zs!
A2q 1 1, (32)
Rx,y~q !~z ! 5 Rx,y~zs!. (33)
In terms of the new variables wx(q)(zs) the Hermite poly-
nomials Hl@A2x/wx(zs)# become
H1F A2x
wx~zs!G 5
1
~2q 1 1 !1/2 H1F A2x
wx~q !~zs!
G , (34)
H2F A2x
wx~zs!G 5
1
2q 1 1H H2F A2x
wx~q !~zs!
G 2 4qJ ,
(35)
H3F A2x
wx~zs!G 5
1
~2q 1 1 !3/2 H H3F A2x
wx~q !~zs!
G2 12qH1F A2x
wx~q !~zs!
G J . (36)
Analogous relations hold for the y polynomials. Underthese considerations, the electric field pattern at the exit
plane of the sample, given by Eq. (31), can be written interms of Hermite–Gaussian functions to take the form
Ee~x, y, zs , t !
5 A~zs , t !exp~2a0L/2!exp@2ikzs 1 iu~zs!#
3 (q50
`@iDw0
~1 !~zs , t !#q
q! Xuc0u2qc0E0~q !~x, y, zs!
1 «~q 1 1 !uc0u2qH c11
2~2q 1 1 !exp@2iu~zs!#
3 E11~q !~x, y, zs! 1
c22
8~2q 1 1 !2 exp@4iu~zs!#
3 @E22~q !~x, y, zs! 2 4qE20
~q !~x, y, zs! 2 4q
3 E02~q !~x, y, zs! 1 16q2E0
~q !~x, y, zs!#
1c33
48~2q 1 1 !3 exp@6iu~zs!#@E33~q !~x, y, zs!
2 12qE31~q !~x, y, zs! 2 12qE13
~q !~x, y, zs!
1 144q2E11~q !~x, y, zs!#J
1 «quc0u2~q21 !c02H c11
2~2q 1 1 !exp@22iu~zs!#
3 E11~q !~x, y, zs! 1
c22
8~2q 1 1 !2 exp@24iu~zs!#
3 @E22~q !~x, y, zs! 2 4qE20
~q !~x, y, zs!
2 4qE02~q !~x, y, zs! 1 16q2E0
~q !~x, y, zs!#
1c33
48~2q 1 1 !3 exp@26iu~zs!#@E33~q !~x, y, zs!
2 12qE31~q !~x, y, zs! 2 12qE13
~q !~x, y, zs!
1 144q2E11~q !~x, y, zs!#J C, (37)
where
Elm~q !~x, y, zs!
5 HlF A2x
wx~q !~zs!
GHmF A2y
wy~q !~zs!
GexpH 2F x
wx~q !~zs!
G2
2 F y
wy~q !~zs!
G2J expF2ikx2
2Rx~q !~zs!
2iky2
2Ry~q !~zs!
G . (38)
At a distance D the Hermite–Gaussian functionsElm
(q)(x, y, zs) become
674 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Tsigaridas et al.
Elm~q !~x, y, D !
5 A ~q !~D !HlF A2x
wx~q !~D !
GHmF A2y
wy~q !~D !
G3 expH 2F x
wx~q !~D !
G2
2 F y
wy~q !~D !
G2J3 expF2
ikx2
2Rx~q !~D !
2iky2
2Ry~q !~D !
G3 exp$2ik~D 2 zs! 1 i~l 1 m 1 1 !
3 @u~q !~D ! 1 d ~q !#%. (39)
Phase parameter d (q) has been added for consistency tothe electric field pattern at the exit plane of the sample,whereas normalization constant A (q)(D) has been addedto ensure the conservation of energy ** uElm
(q)u2dxdy in-cluded in each Hermite–Gaussian mode during its propa-gation. It is given by
A ~q !~D ! 5 Fwx~q !~zs!wy
~q !~zs!
wx~q !~D !wy
~q !~D !G 1/2
. (40)
If we define
w0x,y~q ! 5 F2zRx,y
~q !
k G1/2
, (41)
the remaining parameters wx,y(q)(D), Rx,y
(q)(D), andu (q)(D) are written as
wx,y~q !~D ! 5 w0x,y
~q !H 1 1 FD 2 z0x,y~q !
zRx,y~q ! G 2J 1/2
, (42)
Rx,y~q !~D ! 5 @D 2 z0x,y
~q !#H 1 1 F zRx,y~q !
D 2 z0x,y~q !G 2J ,
(43)
u~q !~D ! 51
2tan21FD 2 z0x
~q !
zRx~q ! G
11
2tan21FD 2 z0y
~q !
zRy~q ! G , (44)
d ~q ! 5 21
2tan21F zs 2 z0x
~q !
zRx~q ! G
21
2tan21F zs 2 z0y
~q !
zRy~q ! G . (45)
Equations (37)–(45) fully define the electric field patternof the beam exiting the sample at any desired distance D.
Analytical simulations of close-aperture Z-scan plotswith different input beam profiles of the general form ofEq. (30) are shown in Figs. 1 and 2. It is obvious that, inall cases, the involvement of the odd-order modes (TEM11and TEM33) does not significantly alter the Z-scan plots,whereas the effects of the TEM22 mode are more drastic.The reason is that the odd-order modes do not have brightspots in the central region of the beam where the irradi-ance is high. Therefore, because the induced phase shiftis analogous to the irradiance that is incident upon the
Fig. 1. Theoretical close-aperture Z-scan plots (S 5 0.4) for acircular near-Gaussian incident beam. The location of the beamwaist is at z0 5 0. The curves correspond to the modes involvedin the near-Gaussian beam, as shown. For all cases « 5 0.1 andDw0
(1) (z0 , t) 5 1.
Fig. 2. Theoretical close-aperture Z-scan plots (S 5 0.4) for el-liptical near-Gaussian incident beams of increasing waist sepa-ration: (a) z0y 2 z0x 5 2zR and (b) z0y 2 z0x 5 3zR . The beamwidths at the waists for the two principal dimensions have beenset equal to each other (w0x 5 w0y 5 w0), leading to equal val-ues for the Rayleigh lengths (zRx 5 zRy 5 zR). The beam waistsare symmetrically located near z0 5 0. The curves correspondto the modes involved in the near-Gaussian beam, as shown.For all cases « 5 0.1 and Dw0
(1) (z0 , t) 5 1.
Tsigaridas et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 675
sample, the phase disturbance and consequently the al-terations to the far-field pattern caused by these modesare limited. However, the TEM22 mode has a centralbright spot, and therefore its influence on the inducedphase shift, and consequently on the far-field pattern ofthe beam, is more significant.
4. EXTENSIONSA. Higher-Order Refractive NonlinearitiesIn the more general case when both of the two first ordersof refractive nonlinearity are present, i.e., when
n~x, y, zs , t ! 5 n lin 1 g~1 !I~x, y, zs , t !
1 g~2 !I2~x, y, zs , t !, (46)
the nonlinear phase shift induced in the incident beamtakes the form1
Dw~x, y, zs , t ! 5 2kg~1 !I~x, y, zs , t !Leff~1 !
2 kg~2 !I2~x, y, zs , t !Leff~2 !, (47)
where
Leff~2 ! 5
1 2 exp~22a0L !
2a0. (48)
In terms of electric field components E80(x, y, zs , t) andE8h(x, y, zs , t) the induced phase shift is written in theform
Dw~x, y, zs , t ! 5 Dw0~1 !~zs , t !uc0E08~x, y, zs , t !
1 «Eh8~x, y, zs, t !u2
1 Dw0~2 ! 3 ~zs , t !u
3 c0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !u4, (49)
where
Dw0~2 !~zs , t ! 5 2kg~2 !F c«0n0
2u A~zs , t !u 2G2
Leff~2 !.
(50)
The electric field pattern at the exit plane of the samplenow becomes
Ee~x, y, zs , t !
5 A~zs , t !@c0E08~x, y, zs , t ! 1 «Eh8~x, y, zs , t !#
3 exp~2a0L/2!(q50
` 1
q!@iDw0
~1 !~zs, t !
3 uc0E08~x, y, zs , t ! 1 «Eh8~x, y, zs , t !u2
1 iDw0~2!~zs, t!uc0E08~x, y, zs , t ! 1 «Eh8~x, y, zs , t !u4#q.
(51)
If we further assume that Dw0(2)(zs , t) ! Dw0
(1)(zs , t),Eq. (51) takes the form
Ee~x, y, zs , t !
5 A~zs , t !@c0E08~x, y, zs , t ! 1 «Eh8~x, y, zs , t !#
3 exp~2a0L/2!(q50
`@iDw0
~1 !~zs, t !#q
q!
3 uc0E08~x, y, zs , t ! 1 «Eh8~x, y, zs , t !u2q
1@iDw0
~1 !~zs, t !#q21@iDw0~2 !~zs, t !#
~q 2 1 !!
3 uc0E08~x, y, zs , t ! 1 «Eh8~x, y, zs , t !u2~q11 !.
(52)
Comparing Eq. (52) with Eq. (20), which considers onlyfirst-order refractive nonlinearity, we can conclude thatthe analysis that was followed in that case is applicable tohigher-order nonlinearities as well. The only modifica-tion is that, as far as the second term in the sum of Eq.(52) is concerned, q should simply be replaced by q 1 1.
B. Nonlinear AbsorptionWhen first-order nonlinear absorption is present togetherwith a single first-order refractive nonlinearity, the inten-sity profile of the beam and the induced phase shift at theexit plane of the sample become1
lc~x, y, zs , t ! 5I~x, y, zs , t !exp~2a0L !
1 1 Q~x, y, zs , t !, (53)
Dw~x, y, zs , t ! 5kg~1 !
b~1 !ln@1 1 Q~x, y, zs , t !#,
(54)
respectively; where b (1) is the first-order nonlinear ab-sorption coefficient and
Q~x, y, zs , t ! 5 b~1 !I~x, y, zs , t !Leff~1 !. (55)
Under these conditions the electric field pattern at theexit plane of the sample takes the form
Ee~x, y, zs , t ! 5 A~zs , t !@c0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !#exp~2a0L/2!
3 @1 1 Q~x, y, ,zs , t !# ikg~1 !/b~1 !21/2.
(56)
Expanding the term @1 1 Q(x, y, zs , t)# ikg(1)/b(1)21/2 in abinomial series, provided that uQ(x, y, zs , t)u , 1, wehave
@1 1 Q~x, y, zs , t !# ikg~1 !/b~1 !21/2
5 (q50
`
F ~q !@Q~x, y, zs , t !#q
q!, (57)
where F (0) 5 1 and
F ~q ! 5 )r51
q
@ikg~1 !/b~1 ! 2 1/2 2 r 1 1#, q > 1.
(58)
676 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Tsigaridas et al.
In terms of electric field components E08(x, y, zs , t) andEh8(x, y, zs , t), Eq. (57) can also be written in the form
@1 1 Q~x, y, zs , t !# ikg~1 !/b~1 !21/2
5 (q50
`
F ~q !@Q0~zs , t !#q
q!uc0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !u2q, (59)
where
Q0~zs , t ! 5 b~1 !c«0n0
2uA~zs , t !u2Leff
~1 !. (60)
Thus the electric field pattern at the exit plane of thesample finally becomes
Ee~x, y, zs , t ! 5 A~zs , t !@c0E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !#exp~2a0L/2!
3 (q50
`
F ~q !@Q0~zs , t !#q
q!uc0
3 E08~x, y, zs , t !
1 «Eh8~x, y, zs , t !u2q. (61)
Comparing Eq. (61) with Eq. (20), which considers onlyfirst-order refractive nonlinearity, we can conclude thatthe analysis followed in that case is also applicable here.The only modification is that @iDw0(zs , t)#q/q! shouldsimply be replaced by F (q)@Q0(zs , t)#q/q!.
5. CONCLUSIONSIn conclusion, the Z-scan technique used for measuringoptical nonlinearities was extended to the interestingcase of a near-Gaussian incident beam. Our approachwas based on the Gaussian decomposition method, appro-priately extended to Hermite–Gaussian input beams. Itwas shown that, to a first-order approximation, the effectof nonlinear refraction on a near-Gaussian beam is the re-distribution of energy to new modes up to the same orderas those contained in the incident beam.
Analytical simulations were achieved for a near-Gaussian incident beam that contained, apart from thedominant TEM00 mode, the three first-order modes,TEM11 , TEM22 , and TEM33 , as well. It has been shownthat only the even-order mode (TEM22) affects the Z-scancurves significantly. This result has been attributed to
the fact that the odd-order modes do not have bright spotsin the central region of the beam where the intensity ofthe dominant TEM00 is high. Therefore they do not sig-nificantly disturb the nonlinear phase shift induced onthe Gaussian beam, and consequently the far-field pat-tern employed in the Z-scan simulation remains mostlyunaffected.
The simultaneous presence of first- and second-orderrefractive nonlinearity or first-order refractive nonlinear-ity and nonlinear absorption has also been investigated,and it was shown that the analysis is similar to that fol-lowed for a single first-order refractive nonlinearity.
V. Giannetas’s e-mail address is [email protected].
REFERENCES1. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E.
W. Van Stryland, ‘‘Sensitive measurement of optical nonlin-earities using a single beam,’’ IEEE J. Quantum Electron.26, 760–769 (1990).
2. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J.McKay, and R. G. McDuff, ‘‘Single beam Z-scan: mea-surement techniques and analysis,’’ J. Nonlinear Opt. Phys.Mater. 6, 251–293 (1997).
3. W. Zhao and P. Palffy-Muhoray, ‘‘Z-scan using top-hatbeams,’’ Appl. Phys. Lett. 63, 1613–1615 (1993).
4. R. E. Bridges, G. L. Fischer, and R. W. Boyd, ‘‘Z-scan mea-surement technique for non-Gaussian beams and arbitrarysample thickness,’’ Opt. Lett. 20, 1821–1823 (1995).
5. S. M. Mian, B. Taheri, and J. P. Wicksted, ‘‘Effects of beamellipticity in Z-scan measurements,’’ J. Opt. Soc. Am. B 13,856–863 (1996).
6. B. K. Rhee, J. S. Byun, and E. W. Van Stryland, ‘‘Z scan us-ing circularly symmetric beams,’’ J. Opt. Soc. Am. B 13,2720–2723 (1996).
7. P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis,‘‘Two-dimensional Z scan for arbitrary beam shape andsample thickness,’’ J. Appl. Phys. 85, 7043–7050 (1999).
8. Y.-L. Huang and C.-K. Sun, ‘‘Z-scan measurement with anastigmatic Gaussian beam,’’ J. Opt. Soc. Am. B 17, 43–47(2000).
9. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith,‘‘Effect of low-power nonlinear refraction on laser-beampropagation in InSb,’’ Opt. Lett. 4, 331–333 (1979).
10. A. E. Siegman, Lasers (University Science Books, Mill Val-ley, Calif., 1986).
11. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York,1989).
12. M. Abramowitz and I. A. Stegun, eds., Handbook of Math-ematical Functions with Formulas, Graphs, and Math-ematical Tables (Dover, New York, 1972).
