Degenerate multi-wave mixing inside a 4f imaging system in presence of nonlinear absorption

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DOI 10.1007/s00340-010-4004-z

Degenerate multi-wave mixing inside a 4f imaging system in presence of nonlinear absorption

K. Fedus·G. Boudebs·H. Leblond

Received: 27 November 2009 / Revised version: 28 January 2010 / Published online: 16 April 2010

We report on multiwave mixing inside a 4f coherent imaging system using an I-scan configuration and taking into account nonlinear absorption inside materials. A simple math- ematical calculation is presented in order to explain the nonlinearly induced diffracted beam distribution in the image plane. The influence of nonlinear absorption is discussed and a study on the sensitivity of the technique and experimental methodology for absorbing materials is proposed.

We provide also a simple quadratic relation to characterize the cubic nonlinear refraction for absorbing media.

1 Introduction

We presented recently a wave-mixing process combined with a Z-scan technique [1] for measuring nonlinear refraction (n2) in nonabsorbing materials moving around the focal region of a 4f coherent imaging system (Fig. 1a) [2].

An object composed of three circular apertures is located at the entry while a field stop blocks the geometrical image of the object at the output of the 4f setup. The object was illuminated with an expanded laser beam giving a uniform (top-hat [3]) field distribution inside the apertures. A few in- tense diffracted orders were recorded (by a CCD) (Fig.1b) in the image plane outside the field stop, as a result of the self-diffraction of the fundamental beams on a phase grat-

K. Fedus (

⁾^·^{G. Boudebs}^·^{H. Leblond}

Laboratoire de Photoniques d’Angers, EA 4464, UFR Sciences, Université d’Angers, 2 Boulevard Lavoisier,

49045 Angers Cedex 01, France e-mail:[email protected] Fax: +33-(0)2-41735216

ing nonlinearly induced inside the sample. We defined the diffraction efficiency as the quantity to be measured:

η=(E_DNL−E_DL)

E_TOT , (1)

whereEDNL is the diffracted energy in nonlinear regime, E_DL is the diffracted energy in linear regime, and E_TOT is the total energy detected in the nonlinear regime without any field stop at the output. We found that the diffraction efficiency (η) is independent of the geometrical para- meters characterizing the object. Thus, it was possible to use this physical quantity for nonlinear characterization. We provided a simple quadratic relation

η=2.41×ϕ_NL0² ×10⁻², (2)

connecting the diffracted energy with a low nonlinear on- axis phase shiftϕ_NL0=2π n2LI₀/λ, where λis the wave- length, I₀ the on-axis peak intensity at the focus, n₂ the nonlinear refraction, and L the thickness of the sample.

This method can be considered as a generalized case of the degenerate forward four-wave mixing (DFFWM) technique [4], in which only one diffracted forward fourth wave is used in order to measure the third-order susceptibility.

Later we have shown [5] that the same results could be obtained without moving the sample, following the I-scan configuration [6]. Moreover, changing simultaneously the objects at the entry and the corresponding field stops at the output of the 4f system, it was possible to perform different multiwave mixing experiments. We compared numerically the sensitivity of the 4f system with a sample fixed at the focus using different objects (composed of one, two, three, or four circular apertures), in order to optimize this nonlinear characterization technique [5]. For each wave-mixing process, we provided quadratic relationships relatingηand

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Fig. 1 (a) Schematic of the 4f imaging system. The nonlinear medium (NL) is in the focal plane. The labels refer to lenses (L1–L2). The object composed of three circular apertures is placed at the entry plane O(x, y), and the L shaped field stop is located at the output plane U (x, y). (b) The experimental image of the diffracted beams in a for- ward four wave mixing process (from 2). The “L” pattern is the field stop. The coordinatesxandyare expressed in number of pixels and the natural logarithm is used to enhance lower intensity

ϕ_NL0. It was found that the sensitivity increases with de- creasing number of apertures, when the size of the field- stops blocks exactly the geometrical images of the apertures composing the object. However, when the radius of the opaque disks (r) in the field stop is at least twice larger than the radius of the circular apertures (R) of the object, the sensitivity is enhanced for higher number of apertures. The latter case is much easier to realize experimentally since the optical alignment is simplified with the large field stop. The field stop appearing on the experimental image in Fig.1b and having “L” shape is equivalent to three opaque disks with radiir=2R.

Up to now the case of absorbing materials was not taken into account. Here we will discuss the influence of the linear and nonlinear absorption on the sensitivity of the wave mixing process inside the 4f imaging system, as well as the validity and the usefulness of the quadratic expressions re- latingηtoϕ_NL0, which were derived previously in the purely refractive case. The method of processing the experimental data obtained in presence of absorption will be described.

We will also provide a theoretical analysis explaining the distribution of the diffracted intensity in the image plane as it is shown experimentally in Fig.1b.

The theoretical model describing the image formation inside the 4f system with the absorbing medium placed in the

focal plane [7] is briefly recalled in Sect.2. Section3deals with the analytical calculations explaining the distribution of the diffracted intensity for the three apertures objects considered here. Section4is devoted to the numerical simulations and discussion about the characterization of nonlinear refraction in presence of absorption using the I-scan wave- mixing technique.

2 Theoretical frame

The image formation inside the 4f system [8] in the low- excitation regime can be described by scalar Fourier optics [7]. Let us consider a two-dimensional object defined by a transmittancet (x, y)and placed in the front focal plane of first lens L1(Fig.1a). The object is illuminated at normal in- cidence by a linearly polarized monochromatic plane wave defined byE=E0(t )exp[−j (ωt−kz)] +c.c., where ω is the angular frequency,kis the wave vector, andE0(t )is the amplitude of the electric field containing the temporal envelope of the laser pulse. Using the slowly varying envelope approximation to describe the propagation of the electric field in the nonlinear medium [9], and since we are con- cerned with the image intensity, the temporal terms will be omitted. The amplitude of the electric field just behind the plane where the object is placed isO(x, y)=E×t (x, y), and thus its spatial profile in the back focal plane(z=0)of lens L1is given by

S(u, v)= ˜F

O(x, y)

− _+∞

−∞ O(x, y)exp

−j2π(ux+vy) dxdy,

(3) whereF˜denotes the Fourier transform operation,u=x/λf andv=y/λf are the spatial frequencies, andf is the focal length of the lens L1. The electric field in the image plane at the output of the 4f system (back focal plane of second lens L2) is obtained by means of the inverse Fourier transform:

U (x, y)= ˜F⁻¹

SL(u, v)

= ˜F⁻¹

T (u, v)S(u, v)

, (4)

whereS_L(u, v) is the electric field at the exit face of the sample. It is assumed that L, the thickness of the nonlin- ear medium (NL) placed atz=0, is less than the diffraction length of the focussed beam. We assume also a cubic nonlinearity, and we consider medium exhibiting (i) linear absorption defined byα(m⁻¹), (ii) two-photon absorption defined byβ (m/W), and (iii) nonlinear refraction defined byn₂(m²/W). The nonlinear transmittance of the medium regarded as thin is defined [10] as

T (u, v)=S_L(u, v) S(u, v)

= e^αL

1+q(u, v)₋1/2

exp j ϕ^eff

NL(u, v)

, (5)

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whereq(u, v)=βL_effI (u, v)withL_eff=(1−e⁻^αL)/αrep- resenting the effective length, andI (u, v)denotes the intensity of the laser beam within the sample. The latter quantity is related to the electric field through the expression I (u, v)=2ε0n₀c|S(u, v)|², whereε₀is the vacuum permit- tivity, n0 is the linear refractive index, and c is the speed of light. Hereϕ^eff

NL(u, v)can be considered as the effective nonlinear phase shift,

ϕ^eff_NL(u, v)=2π

λ n₂L_effI_eff(u, v), (6) whereIeff(u, v)=I (u, v)ln(1+q(u, v))/q(u, v)is the effective intensity. In the particular case when the linear absorption α=0 and the nonlinear one q(u, v)=0 (no losses), ϕ_NL^eff reduces to ϕ_NL(u, v)=2π n2LI (u, v)/λ describing phase shift in nonabsorbing medium [2,5]. The idea of the effective length (Leff)and the effective intensity (Ieff) is commonly used to simplify the calculation when linear and nonlinear absorptions are present [7,11], respectively.

It allows the nonlinearity to be computed using the same expressions as for nonabsorbing materials.

3 Analytical results

Let us consider now the object composed by three circular apertures as seen in Fig. 1a. The transmittance t (x, y) of this object can be described by three circular functions as in [2,5], where only the numerical calculations were done to explain the intensity distribution in the image plane due to the wave mixing phenomenon inside the purely refractive medium. In the analytical calculations presented here, we will simplify the problem by approximating the three apertures to three single points placed at the center of each circle.

Hence, the transmittance of the object can be expressed by t (x, y)=δ(x−d, y)+δ(x, y−d)+δ(x−d, y−d), (7) whereδ(x, y)is the Dirac distribution, andd is the shift of these distributions from the origin of the coordinate system (see Fig.2) alongx- andy-directions. The amplitudeS(u, v) of the electric field in the back focal plane(z=0)of lens L1

is given by (3). Calculating the Fourier transform, we get S(u, v)=E

e⁻^{j2π ud}+e⁻^{j2π vd}+e⁻^j^2π(u⁺^v)d

. (8)

The electric field at the exit face of the sample isS_L(u, v)= T (u, v)S(u, v), where the transmittance is given by (5). For convenience, we rewriteq(u, v)defined in this equation as

q(u, v)=2Q S(u, v) ², (9)

where Q =βLeffK/2 with K = 2ε0n0c. We also set ξ =2π n2L_effK/λ. Assuming both a small nonlinear absorption (q(u, v)1) and a small effective phase shift

Fig. 2 The amplitude U (x, y) of the output electric field. Each star corresponds to a quantitye^−αL/2(j ξ−Q)E³, the circles rep- resent the output geometrical image of the apertures with amplitude Ee^−αL/2(1+5(j ξ−Q)E²)

(|ϕ^eff_NL(u, v)| 1), we can expand the transmittanceT (u, v) to the first order in intensity, so that

S_L(u, v)≈e⁻^αL/2

1+(j ξ−Q)|S(u, v)|²

S(u, v). (10) By developing the latter we get

S_L(u, v)≈Ee^−αL/2

1+5(j ξ−Q)E²

×

e⁻^{j2π ud}+e⁻^{j2π vd}+e⁻^j2π(u⁺^v)d +e⁻^αL/2(j ξ−Q)E³

2+2e⁻^{j4π ud} +2e⁻^{j4π vd}+e^j2π(u⁻^v)d

+e^j2π(v⁻^u)d+e^j2π(v⁻^2u)d+e^j2π(u⁻^2v)d +e⁻^j2π(u⁺^2v)d+e⁻^j2π(v⁺^2u)d

. (11)

Then the output electric field is obtained through the inverse Fourier transformU (x, y)= ˜F⁻¹[S_L(u, v)], which, using (11), reduces to

U (x, y)=Ee⁻^αL/2

1+5(j ξ−Q)E² t (x, y) +e⁻^αL/2(j ξ−Q)E³

2δ(x, y)+2δ(x−2d, y) +2δ(x, y−2d)+δ(x+d, y−d)

+δ(x−d, y+d)

+δ(x−2d, y+d)+δ(x+d, y−2d) +δ(x−2d, y−d)+δ(x−d, y−2d)

, (12) wheret (x, y)(corresponding to the geometrical image) is defined by (7). The expression in (12) is schematically rep- resented in Fig.2. It can be seen that the spatial distribution of the intensity (squared electric field) is similar to the experimental one (Fig.1b). Moreover, three diffracted beams (with their centers located at (0,0), (0,2d), and (2d,0)) have their intensities four times higher than that of the other

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diffracted beams laying outside the geometrical image of the input apertures. This result is also in very good agreement with the experimental observation and the numerical sim- ulation performed in [2,5]. Furthermore, it can be seen in (12) that both the nonlinear refraction (ξ) and the nonlinear absorption (Q) contribute to the output signal, due to diffraction on the induced phase and amplitude gratings. The theoretical analysis presented here can be easily extended for any objects with a random number of the apertures.

4 The nonlinear characterization in presence of absorption

Since the absorptive (Q) and refractive (ξ) contributions to the output signal are coupled (see (12)), the contribution due to nonlinear refractive index (n2)cannot be sep- arated from that of nonlinear absorption coefficient (β).

Hence we have to proceed as usually; β (together withα) has to be determined separately through transmission mea- surements by acquiring the output intensity versus the input one. Knowingα andβ, the nonlinear refractive index can be found using the wave-mixing technique. Generally the on-axis peak intensity at the focus (I0) is a critical parameter to estimate quantitatively the nonlinearities when the light is focused into the sample. This intensity is used via ϕ_NL0^eff =2π n2LeffIeff(q0), the on-axis maximal induced phase shift, whereq0=βLeffI0. It is advantageous to deter- mine n2by using simple quadratic expression relating the diffraction efficiency (η) with this maximal phase shift as in (2), which was obtained numerically and validated experimentally in [2,5] for nonabsorbing media. However, for the nonlinearly absorbing materials, we have in addition to take into account the coupling between the nonlinear absorption and the nonlinear refraction, both contributing to the diffracted signal. For this purpose, let us define a coupling factor asγ=Q/ξ=q₀/2ϕNL0=β/2kn2. In the transparency region this coefficient is generally less than one (γ≤1) in a wide variety of different materials [1]. The latter condition means that the imaginary part of the third-order susceptibility is not larger than the real part. Note thatγis related to the figure of merit (FOM) defined in [12], which is a geometry- independent factor widely adopted in the literature to clas- sify different nonlinear materials for optical switching or optical limiting applications: γ =FOM/8π. In order to find a relation betweenη, ϕ_NL0^eff , andγ, we have performed numerical simulations using (3) to (6). In the simulations we consider the transmittance of a more physical object, composed of three circular apertures instead of three Dirac distributions of (7):t (x, y)=C_R(x−d, y)+C_R(x, y−d)+ C_R(x−d, y−d). HereC_R(x, y)is defined as equal to one if the radius(x²+y²)^1/2is less thanR and zero elsewhere

(Ris the radius of the circular apertures). The diffracted en- ergyE_D is calculated after numerically integrating the image through a field stopfs(x, y), following the double in- tegralED=

|U (x, y)|²fs(x, y)dxdy, wherefs(x, y)= 1− [C2R(x−d, y)+C2R(x, y−d)+C2R(x−d, y−d)] represents three opaque disks with radius 2R blocking the geometrical image of the apertures and the diffraction that occurs in the vicinity. Under these conditions and in the low- excitation regime, we can simulate the effect of absorption considering circular apertures (as it has been done in [13] for Gaussian beams). Whenγ≤1, we found that the following relation describes the variations ofη within 10% accuracy for|ϕ_NL0| ≤1:

η=2.41×Ω²×10⁻², (13)

whereΩ=(1+γ²)^1/2ϕ_NL0^eff is defined as the generalized effective phase shift. In Fig.3a we can see the compari- son between this relation (solid lines) and the numerically calculatedη (through (5)) versus the effective phase shift (ϕ_NL0^eff ) forγ =0 (filled dots),γ=0.5 (empty circles), and

Fig. 3 Calculated diffraction efficiency (η) for γ =0 (filled dots), γ=0.5 (empty circles), andγ=1 (stars). The solid lines are described by (13). (a) Versusϕ^eff_NL0, the effective nonlinear phase shift; (b) versus Ω, the generalized effective nonlinear phase shift

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Table 1 Quadratic relationships relatingη, the diffraction efficiency toΩ, the generalized input effective phase shift in the wave-mixing techniques using the 4f system with different objects at the entry (composed of one, two, three, and four circular apertures)

Object r=R r=2R

One aperture η=9.8×Ω²×10⁻² –

Two apertures η=7.5×Ω²×10⁻² η=2.1×Ω²×10⁻² Three apertures η=5.4×Ω²×10⁻² η=2.4×Ω²×10⁻² Four apertures η=5.3×Ω²×10⁻² η=2.4×Ω²×10⁻²

γ =1 (stars). The sameηis plotted versus the generalized effective phase shift Ω, proving that whatever the absorption is (γ≤1), the diffraction efficiency is always described by (13). Moreover, we have checked that this general expression is valid in the presence of linear absorption. Note that in the limiting case where there is no absorption (α=0 and q0=0), Ω reduces to ϕNL0, and then (13) is identi- cal to (2). In [5], which deals with the more simple case of lossless materials, we investigated different multi-waves mixing processes considering various objects composed of one, two, three, and four circular apertures. For each object, we provided a quadratic relationship between ηandϕNL0. We have checked numerically that the extension of these relations is also valid, provided that ϕNL0 is replaced byΩ. These relations are summarized in Table1, where column 2 contains the expressions which are valid when the radiusr of the opaque disks composing the field stop (placed at the output of the system) is equal to the radiusRof the circular apertures composing the object. In column 3 we can find the relations which hold whenris twice larger thanR(advantageous for optical easy alignment). These simple quadratic relations have an interesting implication for the evaluation of the sensitivity of the wave-mixing techniques. Generally, the sensitivity is defined by the slope of the curve giving the output signal versus the input one. Thus, as in [1], for pure refractive materials the sensitivity is given byS=dη/dϕNL0. In our general case, when the absorption is present and be- cause the contribution to the signal is due to both absorptive and refractive grating effects, we should useΩ as the input, therefore generalizing the sensitivity to S_a=dη/dΩ. Fol- lowing this definition and according to the relations given in Table1, one can easily notice that the nonlinear absorption does not affect the sensitivity of the wave-mixing techniques when compared to the results obtained in [5]. Physically, this could be understood by the fact that the sensitivity should be independent from the material. The absorption in the low- excitation regime should not change the performance of the measurement system.

In summary, for a given material, the nonlinear characterization should begin by measuringαandβusing the square modulus of (5) and by performing simple transmission mea- surements without any object at the entry and any field stop at the output. After that, the wave mixing experiment pro- vides the diffraction efficiency using the 4f imaging system (see the detail of the experimental technique in [2,5]). Fi- nally, the nonlinear coefficientn2can be determined by sim- ply fitting the quadratic relation corresponding to the object used (see Table1) to the measured data ofη. The only un- known parameter in this relation isn2, since the on-axis in- tensityI₀can be calibrated using well known reference nonabsorbing materials as in [14].

5 Conclusions

We have provided a theoretical analysis explaining the distribution of the diffracted intensity in the image plane obtained in multiwave mixing experiments using a 4f coherent imaging system. The influence of both the linear and third- order nonlinear absorptions on nonlinear characterization has been studied. We have generalized the simple quadratic relations relating the diffraction efficiency to the effective nonlinear phase shift. Consequently, we have shown that the sensitivity of the measurement is not affected by the presence of nonlinear absorption inside the material in the low- excitation regime.

References

1. M. Sheik-Bahae, A.A. Said, T.H. Wei, D. Hagan, E.W. Stryland, IEEE J. Quantum Electron. 26, 760 (1990)

2. G. Boudebs, K. Fedus, C. Cassagne, H. Leblond, Appl. Phys. Lett.

93, 021118 (2008)

3. W. Zhao, P. Palffy-Muhoray, Appl. Phys. Lett. 63, 1613 (1993) 4. S.-Y. Tseng, W. Cao, Y.-H. Peng, J.M. Hales, S.-H. Chi, J.W.

Perry, S.R. Marder, C.H. Lee, W.N. Herman, J. Goldhar, Opt. Ex- press. 14, 8737 (2006)

5. K. Fedus, G. Boudebs, J. Opt. Soc. Am. B 26, 244 (2009) 6. B. Taheri, H. Liu, B. Jassemnejad, D. Appling, R.C. Powell,

J.J. Song, Appl. Phys. Lett. 68, 1317 (1996)

7. G. Boudebs, M. Chis, A. Monteil, Opt. Commun. 150, 287 (1998) 8. J.W. Goodman, Introduction to Fourier Optics, 2nd edn.

(McGraw-Hill, New York, 1996)

9. Y.R. Shen, Nonlinear Optics (Wiley, New York, 1984). Chap. 3 10. J.A. Hermann, J. Opt. Soc. Am. B 1, 729 (1984)

11. Y. Liu, H.K. Tsang, Appl. Phys. Lett. 90, 211105 (2007) 12. V. Mizrahi, K.W. Delong, G.I. Stegeman, M.A. Saifi, M.J. An-

drejco, Opt. Lett. 14, 1140 (1989)

13. S. Cherukulappurath, J.L. Godet, G. Boudebs, J. Nonlinear Opt.

Phys. Mater. 14, 49 (2005)

14. G. Boudebs, K. Fedus, J. Appl. Phys. 105, 103106 (2009)