Based on FresnelâKirchhoff diffraction theory, a diffraction model of nonlinear optical media interacting ... on the FresnelâKirchhoff diffraction integral equation,.
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Z-scan theory based on a diffraction model Baoli Yao State Key Laboratory of Transient Optics and Technology, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710068, China
Liyong Ren Department of Physics, Northwest University, Xi’an 710069, China
Xun Hou State Key Laboratory of Transient Optics and Technology, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710068, China Received August 27, 2002; revised manuscript received December 20, 2002 Based on Fresnel–Kirchhoff diffraction theory, a diffraction model of nonlinear optical media interacting with a Gaussian beam has been set up that can interpret the Z-scan phenomenon in a new way. This theory not only is consistent with the conventional Z-scan theory for a small nonlinear phase shift but also can be used for larger nonlinear phase shifts. Numerical computations indicate that the shape of the Z-scan curve is greatly affected by the value of the nonlinear phase shift. The symmetric dispersionlike Z-scan curve is valid only for small nonlinear phase shifts ( 兩 ⌬ 0 兩 ⬍ ), but, with increasingly larger nonlinear phase shifts, the valley of the transmittance is severely suppressed and the peak is greatly enhanced. The power output through the aperture will oscillate with increasing nonlinear phase shift caused by the input laser power. The aperture transmittance will attenuate and saturate with increasing Kerr constant. © 2003 Optical Society of America OCIS codes: 190.5940, 050.1970, 190.3270, 050.5080.
1. INTRODUCTION Nonlinear refractive index n 2 and Kerr constant ␥ are key parameters of the third-order nonlinearity of nonlinear optical materials. Among numerous techniques for the measurement of nonlinear refraction, the Z-scan method proposed by Sheik-Bahae et al.1 in 1989 is a simple, sensitive, single-beam method that uses the principle of spatial beam distortion to measure both the real and the imaginary parts of complex nonlinear refractive indices and their signs. In this technique the sample is scanned along the optical axis (designated the Z direction) in the focal region of a single focused laser beam, and the intensity transmitted through an aperture in the far field is recorded. For a purely refractive nonlinearity, the amplitude of the transmitted intensity changes as a function of the sample’s position because the nonlinear medium acts as a positive lens (for n 2 ⬎ 0) or a negative lens (for n 2 ⬍ 0). Thus a prefocal transmittance maximum (peak) followed by a postfocal transmittance minimum (valley) is the Z-scan signature of negative refractive nonlinearity. Positive nonlinear refraction, by the same analogy, gives rise to an opposite valley–peak configuration. Recently the Z-scan theory and technique have been extended for measurement of nondegenerate twophoton absorption coefficients and nondegenerate nonlinear refractive indices2 studies of the thermal lensing effect,3 properties of Gaussian beam propagation, and optical limiting of nonlinear media.4–6 As we know, the conventional Z-scan theory is based on the Gaussian decomposition method7 in which several ap0740-3224/2003/061290-05$15.00
proximations, i.e., the slowly varying envelope approximation, a thin sample, and a small nonlinear phase change (⌬ 0 ), must be satisfied; then a simple relation can be derived: T 共 z, ⌬ 0 兲 ⬇ 1 ⫹
4⌬ 0 x 共 x ⫹ 9 兲共 x 2 ⫹ 1 兲 2
,
(1)
where x ⫽ z/z 0 . Relation (1) represents only the closedaperture Z-scan normalized transmittance for purely nonlinear refraction. Obviously, for large phase shifts ( 兩 ⌬ 0 兩 ⬎ ), a symmetric equation does not fit the experimental data well. Considering this deficiency, Kwak et al.8 recently developed a new theory by employing an aberration-free approximation of the Gaussian beam, which can be applicable to a large nonlinear phase shift. In this paper we study the Z scan from a new approach. Here a thin nonlinear medium is regarded as a phasemodulated screen caused by an incident Gaussian beam for purely nonlinear refraction, and the thin nonlinear medium conversely diffracts the Gaussian beam. Based on the Fresnel–Kirchhoff diffraction integral equation, the complex electric field distribution on the aperture plane can be calculated; then the power through the aperture and the aperture transmittance can be obtained. This idea had been tested by Samad and Vieira,9 who tried to find an analytical solution, which normally would not be possible. By using some approximations they obtained an analytical result, but it applies only to the center point (on axis) of the aperture plane. For arbitrary points on the aperture plane they could not get an ana© 2003 Optical Society of America
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lytical resolution. So all their calculations for the Z-scan were based on on-axis intensity. Here we aim at obtaining a numerical solution by computation according to the basic Fresnel–Kirchhoff integral equation. Fewer approximations are used. The intensities of any points on the aperture plane can be calculated. Comparing our results with Sheik-Bahae’s theory and Kwak’s and Samad’s results, we found that for small phase shifts (⬍) and for large phase shifts (⬎) our results agree with theirs. By making numerous calculations we also found some interesting new results.
2. THEORETICAL ANALYSIS Assume that a TEM00 Gaussian beam travels in the ⫹Z direction, coordinate zero is at the beam waist, and the incident plane of a nonlinear medium is at Z 1 . If nonlinear absorption is negligible and if the slowly varying envelope approximation is met, the complex electric field in the sample can be written as E 共 r, z 兲 ⫽ E 共 0, 0 兲
冋
⌬0 ⫽
(2)
where E(0, 0) denotes the on-axis amplitude of the electric field at the focus, ␣ is the linear absorption coefficient, 0 is the beam waist radius, (z) ⫽ 0 关 (1 ⫹ z 2 /Z 0 2 ) 兴 1/2 is the beam radius at z, R(z) ⫽ z(1 ⫹ Z 0 2 /z 2 ) is the radius of curvature of the wave front at z, Z 0 ⫽ 0 2 / is the diffraction length of the beam, k ⫽ 2 / is the wave number, is the laser wavelength, and (z) contains the entire radially uniform phase variation. In the following derivation we define 1 ⫽ (Z 1 ) and R 1 ⫽ R(Z 1 ). According to the optical Kerr effect, the refractive-index change induced by laser irradiance is ⌬n ⫽ ␥ I,
冋 册
2r 2 exp关 ⫺␣ 共 z ⫺ Z 1 兲兴 exp ⫺ . 2 共 z 兲 2共 z 兲 (4)
The transverse phase variation on the exit surface of the sample is the integration of Eq. (4) over the length of the sample (d): ⌬ 共 r, Z 1 兲 ⫽
2
冕
Z 1 ⫹d
␣ 12
.
(7)
˜ 共 r, Z 兲 ⫽ exp关 ⫺i⌬ 共 r, Z 兲兴 . U 1 1
(8)
The radial electric field distribution of the Gaussian beam on the exit surface of the sample can be written as
˜ 共 r, Z 兲 ⫽ E 共 0, 0 兲 V 1
0 1
exp共 ⫺␣ d/2兲
冉 冊 冉 冊
r2 ikr 2 ⫻ exp ⫺ exp ⫺ . 2R 1 12
(9)
So the complex electric field distribution in the far-field aperture plane can be obtained from the Fresnel– Kirchhoff diffraction integral equation10:
(3)
where the laser intensity is I ⫽ ⑀ 0 cn 0 兩 E 兩 2 /2, the on-axis intensity at the focus is I(0, 0) ⫽ 2P/ 0 2 , n 0 is the linear refractive index, P is the laser power, and c and ⑀ 0 are, respectively, the speed of light and its permittivity in vacuum. The laser-induced refractive-index distribution thus can be expressed as 2␥P
4 ␥ P 关 1 ⫺ exp共 ⫺␣ d 兲兴
Expression (6) indicates that the transverse phase variation on the exit surface of the sample is a Gaussianlike distribution. We can regard this phase distribution as a phase hole. According to diffraction theory, the incident laser beam will be diffracted by this phase hole just as in standard pinhole diffraction, but here the diffraction screen function is
r2 exp关 ⫺␣ 共 z ⫺ Z 1 兲 /2兴 exp ⫺ 共 z 兲 2共 z 兲
册
(6)
where the peak nonlinear phase shift is [i.e., ⌬ 0 (Z 1 )]
冋 册
0
ikr 2 ⫻ exp ⫺ ⫺ i共 z 兲 , 2R 共 z 兲
⌬n 共 r, z 兲 ⫽
冉 冊
2r 2 , ⌬ 共 r, Z 1 兲 ⬇ ⌬ 0 exp ⫺ 12
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⌬n 共 r, z 兲 dz.
(5)
Z1
Equation (5) cannot be analytically integrated, but for a thin sample (i.e., d ⬍ Z 0 ),7 we have (z) ⬇ (Z 1 ); therefore Eq. (5) can be simplified as
˜ 共兲 ⫽ E
⫺i
冕冕
˜ 共 r 兲U ˜ 共r兲 F共 0 , 兲V
exp共 ikD 兲 D
rdrd . (10)
Here F( 0 , ) ⫽ (cos 0 ⫹ cos )/2 is the inclination factor, 0 is the angle formed by the incident light and the subwave source (for vertical incident light, 0 ⫽ 0), and is the angle formed by the point to be calculated on the aperture plane and the subwave source. D ⫽ 关 Z 2 2 ⫹ (r cos ⫺ )2 ⫹ (r sin )2兴1/2 is the distance from the diffraction point to the receipt point. r and are, respectively, the radial coordinates on the diffraction screen and the aperture plane. Z 2 is the distance between the diffraction screen and the aperture plane. According to the characteristics of a Gaussian function, when r ⫽ 冑5 (Z 1 ), exp关⫺r2/2(Z1)兴 and ⌬ (r) tend to zero. Therefore the integral limits can be selected as r ⫽ 0 → 冑5 (Z 1 ) and ⫽0→2. Substituting Eqs. (6)– (9) into Eq. (10), we obtain the intensity distribution on the aperture plane:
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I共 , Z1 , Z2兲 ⫽
P 2 2 12 ⫻
exp共 ⫺␣ d 兲
冏冕 冕 冉 冊 冉 冊 再冋 冉 冊 册冎 冏 2
0
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冑5 共 Z 1 兲
1
D
0
⫹
Z2
D2
r2 kr 2 2 ⫻ exp ⫺ exp i D⫺ 2R 1 12 2r 2
⫺ ⌬ 0 exp ⫺ 12
2
rdrd .
(11)
From Eq. (11), the laser power through the aperture (with radius a) can easily be calculated: P A 共 Z 1 , Z 2 , a, ⌬ 0 兲 ⫽ 2
冕
a
I 共 , Z 1 , Z 2 兲 d . (12)
0
Scanning Z 1 , we can compute the Z-scan curve from Eqs. (11) and (12).
Fig. 2. Theoretical plots of closed-aperture Z-scan transmittance for purely nonlinear refraction for several nonlinear phase shifts.
Table 1. Influence of the Aperture Size Z-Scan Curve
3. COMPUTATION AND DISCUSSION To confirm that the theoretical equations above and the computing programs are valid, we used three known facts in our computations. The first is that, when the input laser power is fixed, the total power calculated in the aperture plane should be a constant, wherever the sample is located, since normally the position of the sample will influence the total power in the aperture plane. That is, the open-aperture transmittance is 1 in the case of free nonlinear absorption. The second fact is that the openaperture output laser power should be proportional to the input laser power. The third is that, when the nonlinear phase shift is small, our computation should revert to the conventional Z-scan result. All these three facts have been well confirmed by our numerical computations. Figure 1 shows one of the numerical computations, in
a
on the
a (mm)
S
⌬T p – v
10 5 2 1
0.21 0.057 0.0094 0.0024
0.369兩 ⌬ 0 兩 0.389兩 ⌬ 0 兩 0.391兩 ⌬ 0 兩 0.393兩 ⌬ 0 兩
⌬ 0 ⫽ ⫺ , Z 2 ⫽ 1 m.
⌬Z p – v ⫽ 1.696Z 0 in all cases.
Table 2. Influence of the Distance between the Sample and the Aperture (Z 2 ) on the Z-Scan Curvea
a
Fig. 1. Comparison of the new Z-scan theory with the conventional theory for a small nonlinear phase shift, ⌬ 0 ⫽ ⫺ /2.
(a)
Z 2 (m)
⌬T p – v
2 1 0.5 0.2
0.392兩 ⌬ 0 兩 0.391兩 ⌬ 0 兩 0.392兩 ⌬ 0 兩 0.387兩 ⌬ 0 兩
⌬ 0 ⫽ ⫺ , a ⫽ 2 mm.
⌬Z p – v ⫽ 1.696Z 0 in all cases.
which the conventional Z-scan curve and our computed curve are compared at a nonlinear phase shift ⌬ 0 ⫽ ⫺ /2. Based on this theory, we studied the effect of the nonlinear phase shift on the shape of the Z-scan curve. Our computed results are plotted in Fig. 2 (just for ␥⬍0; they are similar for ␥⬎0). The parameters used were ⫽532 nm, 0 ⫽ 5.8 m, and Z 0 ⫽ 0.2 mm. It can be seen that, when 兩 ⌬ 0 兩 ⬍ , with increasing nonlinear phase shifts the dispersionlike curves (symmetric with respect to focus) broaden, and the peak-to-valley separations increase. However, as the nonlinear phase shifts increase, the shapes of the Z-scan curves differ entirely from those obtained with the small nonlinear phase shift and are asymmetric. The valley of the transmittance is severely
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suppressed (nearly to zero) and the peak is greatly enhanced. This result is coincident with those of Kwak8 and Samad.9 We also studied the influences of aperture size (a) and the distance between the sample and the aperture (Z 2 ) on the Z-scan’s characteristics. The calculated results are listed in Tables 1 and 2. Table 1 also lists the aperture linear transmittances (S) that correspond to the aperture sizes. The peak-to-valley transmittance difference ⌬T p–v and the distance between peak and valley ⌬Z p–v , respectively, are evaluated with ⌬ 0 and Z 0 for comparison with Sheik-Bahae’s theory.7 It is clear that the larger the aperture size, the smaller the peak-to-valley transmittance difference; but there is no effect on ⌬Z p–v in the range
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a ⫽ 1 – 10 mm. Distance Z 2 has nearly no influence on the Z-scan curve as long as the far-field condition is satisfied. If the sample is fixed at focus, when we change the input laser power such that ⌬ 0 is increased, we find an interesting phenomenon: The output power through the aperture will first go to a peak, then fall to a valley, and then go up again, making an oscillation as shown in Fig. 3. If the input laser power is also fixed, by changing Kerr constant ␥ we can also change ⌬ 0 . Figure 4 shows the dependence of the aperture transmittance on the Kerr constant or on ⌬ 0 . This curve indicates that large Kerr constant materials have much better optical limiting properties than those of smaller Kerr constant materials.
4. CONCLUSIONS
Fig. 3. Theoretical plot of the output power through the aperture versus the input laser power or the corresponding ⌬ 0 . The sample is fixed at focus; a ⫽ 2 mm and Z 2 ⫽ 1 m.
Based on Fresnel–Kirchhoff diffraction theory, we set up a diffraction model of nonlinear optical media interacting with a Gaussian beam for purely nonlinear refraction. This theory can explain the Z-scan phenomenon in a new way. Numerical computations indicate that the shapes of Z-scan curves are greatly affected by the values of nonlinear phase shifts. The peak-to-valley transmittance difference (⌬T p–v) and the distance between a peak and a valley (⌬Z p–v) become larger with increasing nonlinear phase shift, whereas the aperture size influences only the former, not the latter, when the aperture linear transmittance is S ⬍ 0.2. The distance between the sample and the aperture has almost no effect on the Z-scan curve as long as the far-field condition is satisfied. The symmetric dispersionlike Z-scan curve is valid only for small nonlinear phase shifts ( 兩 ⌬ 0 ⬍ ). The valley of the transmittance is severely suppressed and the peak is greatly enhanced with increasing nonlinear phase shift. The output power through the aperture will oscillate with the increasing nonlinear phase shift caused by the input laser power. The aperture transmittance will attenuate and saturate with increasing Kerr constant. This theory is also applicable to analysis of Gaussian beam propagation after the beam has passed through a nonlinear optical medium with negligible nonlinear absorption.
ACKNOWLEDGMENTS This research is supported by the National Natural Science Foundation of China under grant 60007009 and by the President Foundation of Chinese Academy of Sciences under grant 40007059.
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2.
Fig. 4. Theoretical plot of the aperture transmittance versus Kerr constant or ⌬ 0 . The sample is fixed at focus; a ⫽ 2 mm and Z 2 ⫽ 1 m.
3.
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