Intensity dependent nonlinear refractive index of fused silica

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Intensity dependent nonlinear refractive index of fused silica

James Amoani

Master’s Thesis April 2019

Department of Physics and Mathematics

University of Eastern Finland

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James Amoani Intensity dependent nonlinear refractive index of quartz, 48 pages University of Eastern Finland

Master’s Degree Programme in Photonics Supervisors Prof. Yuri Svirko

Dr. Viatcheslav Vanyukov

Abstract

Fused silica (quartz) is widely used in telecommunication, electronics, laser physics, sensors technology and many other areas. It is also used as a reference to characterize nonlinear optical properties of various materials as well as a substrate for thin films.

However, the experimental measured value of nonlinear refraction coefficient of fused silica presented in literature is widely distributed.

We employed z-scan technique with femtosecond laser excitation of 800nm and pulse duration of 160f s to measure the nonlinear refractive index of quartz. The conventional z-scan technique was extended to measure nonlinear absorption and nonlinear refraction coefficient simultaneously. The average value of nonlinear refractive index of quartz was measured to be 2.3×10⁻¹⁶cm²/W. It was found that n2 is proportional to the applied pulse energy which has a contribution to intensity.

Keywords: nonlinear refractive index; fused silica; nonlinear absorption; Kerr effects; z-scan.

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Preface

Glory be to Almighty God for the strength and wisdom to come to this far. I express my heartfelt gratitude to my supervisors, Prof. Yuri Svirko and Dr. Viatcheslav Vanyukov for their fatherly and brotherly guidance, advice and encouragement to come out with this work. Its been pleasure working with you. The next appreciation goes to all the lectures and staff members of institute of photonics for their support and assistance during my study in the University of Eastern Finland.

Special thanks to my lovely wife Mrs.Vida Rivers Amoani, my kids Jeffrey and Nelda. Also to my parents Mr and Mrs Amoani and entire Amoani farmily for their prayers and always standing for me.

I acknowledge Miss Marrian Baah, Benjamin Asamoah, Issa Ibrahim, Tweneboah Aderkson, Mrs Toku-Boateng, not forgetting my mother-in-law Mrs. Sarah Boafo, Dora Oppong(Ama Kumah) and all my friends for their diverse support. God bless you all.

Joensuu, the 30th of April 2019 James Amoani

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Chapter I

Introduction

1.1 Background

Optical properties of materials are very vital as far as industrial and scientific applications are concern. When the light intensity is low, the optical response of material is determined by refraction index and absorption coefficient. These material parameters describe reflection, absorption and transmission. The underlying microscopic mechanism of the light refraction and absorption is light-induced oscillations of the electrons belonging to the atoms and molecules of the medium [1]. These oscillations change the phase velocity of the light wave inside the material. This phenomenon is termed as linear refraction [2]. The refractive index is a measure of the linear refraction, i.e. it is a ratio of speed of light in the vacuum to the speed of light in the medium [3]. The refractive index depends on the light wavelength; however, it does not depend on light intensity. For instance, at the wavelength of 622nm air, water, silica glasses, and salt have linear refractive indices of 1.00, 1.33, 1.458 and 1.54 respectively [2].

However, when the intensity of the incident light is high, the refractive index and absorption coefficient show dependence on the light intensity. This is because the electric field magnitude in the light wave becomes non-negligible in comparison with intra-atomic electric field, which bound electrons to the molecular skeleton. As a result, the oscillations of the electrons become nonlinear function of the electric field amplitude and correspondingly, the refraction and absorption become intensity dependent [4].

The dependence of the refractive index on the light intensity manifest itself as

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Self-focusing. This phenomenon is conventionally described assuming that the refraction index of the medium is a linear function of the light intensity,n(I) = n+n2I, where n2 is the nonlinear refractive index, which is a quantitative measure of the optical nonlinearity. [4, 5] Since the intensity I is bigger in the center of the beam, the refractive index at the beam periphery is smaller than that at the center of the beam (at positiven2). Due to this, the material acts as a positive lens, i.e. it focuses the beam inside the material. [4, 6]

Self-focusing is used for laser mode locking applications [7]. Specifically, several longitudinal modes (resonant frequency) can overcome the losses and hence be generated in the laser cavity. However, in the active medium (sapphirine crystal), the refractive index for each mode will depend on its intensity, i.e. one may design the cavity in such a way that only the mode with higher intensity is amplified. This approach is often referred to as Kerr lens mode locking. [8, 9]

Fused silica (quartz, SiO2) is widely used in telecommunication, electronics, laser physics, sensor technology among others [10]. It is transparent in a wide range and is strong to mechanical actions and therefore its finds numerous applications in optics.

Quartz is a reliable material for fabrication of lenses and windows for lasers that operate at high intensities at wavelengths ranging from ultraviolet (UV) to infrared (IR) [5]. It is also used as a reference to characterize nonlinear responses of most dielectric crystals [11].

As for many optical materials, the light intensity coming on fused silica should be lower than a damage threshold. Highly focused laser beam can cause the optical breakdown of quartz . For large beam coming on fused silica the self–focusing must be minimized which may result in overloading of the tiny holes in filters. This reduces the efficiency and injection of beam to the target [12].

The need to measure the nonlinear refractive index also known as second-order nonlinear refractive index (n2) is very important in many nonlinear applications.

Measurement of the nonlinear refractive index started as far back at 1970 where n2 of some bulk glasses were measured. n2 for fused silica was measured to be 2.73×10⁻¹⁶cm²/W with an accuracy of 0.1 at 1.06µm [13]. A list of measurements that were made from 248nmto 1550nmshowed that the nonlinear refractive index of fused silica values decreased at longer wavelengths [13] as depicted in table 1 . In line with this, the (n2) values changed slowly with wavelengths from 800nmto 1600nm, specifically decreased by 5% [13]. Three main techniques were used to measure the

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nonlinear refractive index at that time. Among those are self–phase modulation (SPM), cross–phase modulation (XPM), and four–wave mixing (FWM) [13].

Table 1.1: Measured values of n2 for fused silica [5]

number λ/nm n2/cm²W⁻¹ 1 248 5.6±0.8×10⁻¹⁶ 2 248 3.4±1.6×10⁻¹⁶ 3 266 7.8±0.17×10⁻¹⁶ 4 308 3.0±0.22×10⁻¹⁶ 5 351 1.7±0.7×10⁻¹⁶ 6 351 2.5±1.2×10⁻¹⁶ 7 355 2.41±0.48×10⁻¹⁶ 8 355 2.62±0.6×10⁻¹⁶ 9 402 3.42±0.37×10⁻¹⁶ 10 514 3.0±0.45×10⁻¹⁶ 11 532 1.72±0.34×10⁻¹⁶ 12 694 2.24±0.46×10⁻¹⁶ 13 804 3.3±1.7×10⁻¹⁶ 14 1053 2.48±0.23×10⁻¹⁶ 15 1053 2.77±0.14×10⁻¹⁶ 16 1064 2.83±0.14×10⁻¹⁶ 17 1064 1.9±0.95×10⁻¹⁶ 18 1064 2.73±0.27×10⁻¹⁶ 19 1064 1.92±0.4×10⁻¹⁶ 20 1064 3.3±1.0×10⁻¹⁶ 21 1064 2.14±0.43×10⁻¹⁶ 22 1319 2.66±0.13×10⁻¹⁶ 23 1550 2.79±0.14×10⁻¹⁶

Recently, other techniques such as beam distortion measurements, photoacoustic experiment, optical third-harmonic generation, ellipse experiment , nonlinear inter- ferometry, four-wave mixing rotation and z-scan [12] have been used to measuren2.

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quickness [14]. In contrast, most measurement done using this method in nanosecond and picosecond pulse lasers had a challenge of thermal effect. To remedy this challenge, we use the femtosecond laser pulse with 160fs pulse duration.

1.2 Aim of the thesis

The aim of this work is to measure the nonlinear refractive index of fused silica (Quartz) at 800nm using z-scan technique. We investigate the dependence of the nonlinear refractive index of quartz on incident light intensity. Our results will be compared with the literature values also measured with femtosecond laser pulses at the wavelength of 800nm provided in table 1.2.

This work has been organized in the following order. Chapter 1 is about the introduction which emphasize on the background, aim of the thesis and the review of n2 values at 800nm for fused silica. Chapter 2 entails the concept of nonlinear optics and nonlinear optical effects, while chapter 3 focuses on the theory of z–scan technique. In chapter 4 we introduce the experimental techniques and equipment employed in this research. In chapter chapter 5 we present the results of the measurements and provide the analysis of the data obtained from the experiments. The final chapter 6, is the conclusions pointing out the main outcomes of this research.

1.3 Review of nonlinear refractive index values of quartz

Several measurement techniques have been unfolded in a lot of publications, and each technique measured value differ and far from the theoretical value. Some of the techniques and the n₂ values obtained are as follows.

Z–scan has been used to measure n2 of fused silica in nanoseconds for infrared and visible wavelengths of 1064nm and 532nm respectively. The outcome of these measurements was reported as (4.9±0.6)× 10⁻¹⁶cm²/W at 1064nm and (3.4± 0.5)× 10⁻¹⁶cm²/W at 532nm for 20ns. For 7ns pulse duration the values of n₂ changed to (3.9±0.5)×10⁻¹⁰cm²/W at 1064nm and (2.6±0.4)×10⁻¹⁶cm²/W respectively [15]. Again, other values have been found to be 2.36×10⁻¹⁶cm²/W at 1.3µm and 3.0 ×10⁻¹⁶ at 514.5nm [16] A publication to assess and review of the measured values of fused silica indicate currently available values in ultraviolet (351nm), visible (527nm), near infrared (804nm), and infrared (1053nm) regions of light spectrum. The values used in most applications for n2 with femtosecond

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laser pulse are (3.6±0.64)×10⁻¹⁶cm²/W at 351nm,(3.0±0.25)×10⁻¹⁶cm²/W at 527nm, (2.48±0.23)×10⁻¹⁶cm²/W at 804nmand (2.14±0.17)×10⁻¹⁶cm²/W at 1053nm [5]. Some literature values for femtosecond pulses at wavelength of 800nm are shown in table 1.2. It could be seen from table 1.2 that the values are quite different from each other and therefore it is of interest and practical importance to evaluate the n2 value of quartz to define the trustworthy one. In order to achieve this task we use femtosecond laser pulse with central wavelength of 800nm with a pulse duration of 160f s.

Table 1.2: Some literature values of n₂ of fused silica with femtosecond laser pulses at wavelength of around ±800nm and 804

number λ/nm τ /f s n₂/10⁻16cm²W⁻¹ reference

1 800 160 2.85 [17]

2 804 130 2.40 [5]

3 800 150 2.7 [18]

4 800 100 7.75 [12]

5 804 100 2.48 [19]

6 800 100 3.0 [11]

7 800 130 2.3 [20]

8 800 18 4.95 [21]

9 800 37 5.62 [22]

10 800 100 3.0 [23]

.

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Chapter II

THEORETICAL BACKGROUND

When low–intensity light is transmitted within a material the change in optical properties is different from that of high–intensity light. We discussed the concept of nonlinear optics in this chapter, where we highlight the difference between the responses of linear and nonlinear materials. The chapter also considers second harmonic generation and some phenomenon in χ⁽²⁾ media. Derivation of Maxwell’s nonlinear equations will be looked at briefly and finally explain some χ⁽³⁾ effects.

2.1 Nonlinear Optics

Nonlinear optics refers to studies of light intensity and matter interaction when the intensity is high enough to introduce nonlinear phenomena such as nonlinear refraction and nonlinear absorption [6]. Nonlinear optics studies a phenomenon that results in the modification of optical properties of the material. This modification occurs when a high–intensity light source such as laser beam irradiates a medium.

When light with electric fieldnEincident on a medium, there is a displacement of the atoms in the medium and the electric charge molecules which induces secondary light electric field known asPolarization [6]. In the linear case, this dipole (polarization) has a linear relationship with the applied electric field.

P =ǫoχ⁽¹⁾E (2.1)

whereP is the polarization,ǫo is permittivity of free space,χ⁽¹⁾ is the linear susceptibility andE is the electric field. However, in the nonlinear medium, the relation is not linear. This is due to the anharmonic oscillation of bound electrons caused by

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the applied electric field.

P =ǫo[χ⁽¹⁾E+χ⁽²⁾E²+χ⁽³⁾E²E] (2.2) In terms of polarization it could be written as,

P(t) =ǫo[P⁽¹⁾+P⁽²⁾+P⁽³⁾...] (2.3) [4] The first term is the linear term and the second and the third terms are the nonlinear terms. Diagrammatic representation of the linear and nonlinear relationship of polarization verse electric field is shown in figure 2.1 [4].

P

E

Linear Nonlinear

Figure 2.1: Graphical representation showing the relation between polarization and the applied electric field in linear and nonlinear cases.

χ⁽¹⁾,χ⁽²⁾ andχ⁽³⁾are the first, second and third order susceptibilities respectively.

Since the electric field E is a vector, χ⁽¹⁾ is a second-rank tensor, χ⁽²⁾ is third-rank tensor and χ⁽³⁾ is fourth-rank tensor [4].

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2.2 Nonlinear susceptibility of a classical anharmonic oscil- lator

2.2.1 For Noncentrosymmetric system

The equation of motion for an electron at a positionx can be expressed as,

−eE(t)/m = δ²x

δt² + 2γδx

δt +ωox+ax² (2.4)

E(t) is the applied electric field and −2mγ^δy_δx is the damping of the electron. The restoring force is written as

FR=−mω_o²x−max² (2.5)

a, characterizes the strength of the nonlinearity and m is the mass of the electron.

The restoring force correspond to potential energy as shown in figure 2.2 is given as U(x) = 1

2mω_o²x²+ 1

3max³ (2.6)

Assuming that the applied electric field is given as

actual U(x)

potential parabola

parabola

Figure 2.2: Potential energy function of nocentrosymmetric system.

U(x) = 1

2mω_o²x²+ 1

3max³ (2.7)

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since there is no general solution, by representing E(t) as−̺E(t) where̺ is perturbation strength and assuming that the electric field applied is weak enough, we can solve by perturbation. The equation is now written as

−̺E(t)/m= δ²x

δt² + 2γδx

δt +ωox+ax² (2.8)

Using the power series of perturbation strength to solve for the solution of the equation, [4]

x=̺x⁽¹⁾+̺⁽²⁾x⁽²⁾+̺⁽³⁾x⁽³⁾+... (2.9) The solution for the̺ term , ̺⁽²⁾ and ̺⁽³⁾ are expressed respectively as

δ²x⁽¹⁾

δt² + 2γδx⁽¹⁾

δt +ω_o²x⁽¹⁾ = −eE(t)

m (2.10)

δ²x⁽²⁾

δt² + 2γδx⁽²⁾

δt +ω_o²x⁽²⁾+a(x⁽¹⁾)² = 0 (2.11) δx⁽³⁾

δt² + 2γδx⁽³⁾

δt +ω_o²x⁽³⁾+ 2ax⁽¹⁾x⁽²⁾+b(x⁽¹⁾)³ = 0 (2.12) In equation 2.10 , x⁽¹⁾ is in the same form as the Lorentz model, this implies that the field that drives displacement is given as

x⁽¹⁾(t) =x⁽¹⁾₁ e⁻^iω¹^t+x⁽¹⁾₂ e⁻^iω²^t (2.13) [24]

x⁽¹⁾(t) = −eEj

mDj

(2.14) where the complex denominator Dj is express as

Dj =ω_o²−ω_j²−2iωjγ (2.15) when squaringx⁽¹⁾ and substitute into equation 2.11 we obtain the amplitude of the lower correction term x⁽²⁾. x⁽²⁾ comprises of different responses which include ±2ω

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, (ω1+ω2), (ω1−ω2) and 0. Solving equation 2.11 with the square ofx⁽¹⁾ we obtain the various responses as

x⁽²⁾(2ω₂) = −a(e)E₂²

m²D(2ω2)D²(ω2) (2.16) For (ω₁−ω₂) frequency response,

x⁽²⁾(ω1−ω2) = −2ae²E1E₂^∗

m²D(ω₁−ω₂)D(ω₁)D(−ω₂) (2.17) where E1 ,E₂² and E₂^∗ are complex amplitudes at the various frequencies response.

In linear optics, polarization at frequency ω is written as,

P(ω) =ǫoχ⁽¹⁾(ω)E(ω) (2.18)

also,

P(ω) =−Nex (2.19)

hence linear and nonlinear susceptibility are evaluated respectively as [4]

χ⁽¹⁾_(ω) = −Ne²

mǫoD(ω) (2.20)

χ⁽²⁾_(2ω₁_,ω₁_,ω₁₎ = aNe³

m²ǫoD(2ω1)D²(ω1) (2.21) In the same manner, the susceptibility of centrosymmetric media yeilds

χ⁽³⁾_(ω₄_,ω₁_,ω₂_,ω₃₎ = aNe⁴

m³ǫoD(ω4)D(ω2)D(ω3) (2.22) [4]

Second–order nonlinear phenomena in a media with inversion symmetry (such as gas) are forbidden (see [6] for details). However, third-order nonlinearity occurs in both centrosymmetric and non-centrosymmetric media. The constitutive equation that relates total polarization and electric field can be expressed as power series as, PT =ǫo[χ⁽¹⁾E+χ⁽²⁾E⁽²⁾+χ⁽³⁾E⁽³⁾...] (2.23)

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By introducing a dimensionless quantity (E/Eat), whereEat is the interatomic electric field, the constitutive equation can be presented in the following form;

PT =ǫoχ⁽¹⁾[1 +E/Eat+ (E/Eat)²+ (E/Eat)³...]E (2.24) If the electronic mechanism dominates the nonlinear properties of the medium, the interatomic field can be approximated as ,Ea=e/4πǫoa², whereais the Bohr radius and e is electron charge

2.3 Second–harmonic generation

If an electric field,

E =Ee⁻^iwt+E^∗e^iwt (2.25)

is applied intoχ⁽²⁾media, the nonlinear polarization(PLN) is written mathematically as [4],

ω χ ⁽²⁾ ^ω

2 ω

Figure 2.3: Diagram that depicts the second harmonic generation

PN L = 2χ⁽²⁾E² (2.26)

PN L = 2χ⁽²⁾[Ee⁻^iwt+E^∗e^iwt]² (2.27)

PN L = 2χ⁽²⁾[EE^∗+E²e⁻^2iwt+E^∗e^2iwt] (2.28) The first term does not generate energy (frequency) because its second derivative goes to zero. This is known as optical rectification. The second term generates radiation at frequency 2w which is the second-harmonic generation.

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χ

⁽²⁾

ω

ω ω₃

=

^ω1+ω₂

a

1

2

χ

⁽²⁾

3

=

1^- 2

2 1

b

Figure 2.4: Schematic diagram for frequency generations. (a) is sum frequency generation and (b) is difference frequency generation

2.4 Sum and difference frequency generation

Considering two waves with electric fieldE1e⁻^iw¹^tandE2e⁻^iw²^tpropagating through χ⁽²⁾ media, due to frequency mixing some frequencies can add together to generate electromagnetic radiation with frequency w1+w2 and other energy is formed with frequency w1−w2 and can be illustrated as [25]

P_{N L}⁽²⁾ = 2χ⁽²⁾[(E₁²e⁻^2iw¹^t+c.c+E2e⁻^2w²^t+c.c+E1E2e⁻^i(w¹^+w²^)t+c.c+E1E^∗e⁻^i(w¹⁻^w²^)t] (2.29) The third and the fourth terms generate energy which vibrates at sum and difference frequencies respectively.

2.5 Nonlinear wave equation

From linear optics, displacement D is given as

D=ǫoE+P (2.30)

whereǫois the permittivity of free space,E is the electric field andP is polarization.

P =PL+PN L. (2.31)

PL and PLN are the linear and nonlinear polarization respectively [24]. Also,

Pl =ǫoχ⁽¹⁾E (2.32)

Therefore

D =ǫo(1 +χ⁽¹⁾)E+PN L (2.33)

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D=ǫE+PN L (2.34) Now from linear optics,

∇ ×E = −∂B

∂t (2.35)

∇ ×H = ∂D

∂t (2.36)

Taking curl of both sides for equation 2.35

∇ × ∇ ×E = −∂(∇ ×B)

∂t (2.37)

B =µoH, this implies that

∇ × ∇ ×E =µo

∂(∇ ×H)

∂t (2.38)

∇ × ∇ ×E = −µo∂²D

∂t² (2.39)

Through substitutions and simplification, we arrived at the nonlinear wave equation as

∇²E− µoǫ∂²E

∂t² = µo∂²PN L

∂t² (2.40)

[4]

2.6 Third–order nonlinear optics

This occurs in centrosymmetric systems. For centrosymmetric media such as fused silica (SiO2) and silicon (Si), as shown in figure 2.5, the nonlinear polarization of χ⁽³⁾ processes is proportional to cube of the input electric field. For lossless or dispersion less system, polarization is given as

P =ǫo[χ⁽¹⁾E+χ⁽²⁾E⁽²⁾+χ⁽³⁾E⁽³⁾] (2.41) The first term is the linear part, the second and the third terms are the nonlinear terms. But since ǫoχ⁽²⁾E⁽²⁾ goes to zero (inversion symmetry) [4]. This means we can write the nonlinear polarization for χ⁽³⁾ as,

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x,-y,-z

x,y,z

Figure 2.5: Schematic diagram to showing centrosymmetric system

2.7 Third–harmonic generation

When a wave with electric field E = ¹₂[Eoe⁻^iwt+c.c] transmit in a χ⁽³⁾ media as illustrated in figure 2.6, the nonlinear polarization can be written as [4, 25],

PN L = ǫoχ⁽³⁾

8 [E_o³e⁽⁻^3iwt)+ 3Eo|Eo|²e⁻^iwt+c.c+...] (2.43) As we can see from equation 2.43, there is the generation of electromagnetic radia-

ω χ ⁽³⁾ ^ω

3 ω

Figure 2.6: Schematic diagram for third–order harmonic generation

tion with amplitude E_o³ oscillating at frequency 3w [4]. This is the third–harmonic generation.

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2.8 Nonlinear optical Effects

Third-order optical nonlinearity manifests itself as dependence of the refractive index and absorption coefficient depending on the intensity of the light beam. [6]. When a coherent laser beam propagates through a nonlinear medium, the refractive index is modified (or changed). The changed in the refractive index is proportional to the intensity of the light [6].

∆n =no+n2I (2.44)

where ∆n is the change in refractive index, no is linear refractive index, n2 is the nonlinear refractive index and I is the intensity of light. This can be explained mathematically as follows. Considering the first and second terms of equation 2.3, total polarization (PT) can be expressed as [4],

PT =PL+PN L (2.45)

PT =ǫo[χ⁽¹⁾E+3

4|Eo|²E] (2.46)

PT =ǫo[χ⁽¹⁾+3

4|E_o²|]E (2.47)

We can write the polarization as

P =ǫoχ⁽³⁾_{ef f}E (2.48)

where χef f = (χ⁽¹⁾+ ³₄χ⁽³⁾|Eo|²) |Eo|² is directly proportional to the intensity (I) and refractive index in terms of susceptibility is written as, [4]

n²= 1 +χ⁽³⁾_{ef f} = 1 +χ⁽¹⁾ + ³₄χ⁽³⁾E²

n²_o = 1 +χ⁽¹⁾ (2.49)

where no is the linear refractive index. This implies n² =n²_o[1 + 3

4n²χ⁽³⁾|Eo|²] (2.50)

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Since |Eo|² is proportional to I and I = 1

2ǫon²_oc|Eo|² (2.51)

we arrived at

n=no+ 3χ⁽³⁾

4ǫon²_ocI (2.52)

[4] From equation 2.52 it could be seen that in the presence ofχ⁽³⁾ and intensity I, the refractive index is modified.

n2 = 3χ⁽³⁾

4ǫon²_oc (2.53)

[4] Table 2.1 gives n2 value ranges for some materials [6].

Table 2.1: n2 values ranges for some materials number Material Range/cm²W⁻¹ 1 glass 10⁻¹⁶−10⁻¹⁴ 2 doped−glass 10⁻¹⁴−10⁻⁷ 3 organic−material 10⁻¹⁸−10⁻⁸ 4 semiconductor 10⁻¹⁰−10⁻²

2.8.1 Thermal mechanism of the third-order nonlinearity

One important factor that affectn2 is the thermal effects. When light is transmitted in a nonlinear media, part of the light is absorbed by the material. The temperature of the illuminating part rises causing a change in refractive index due to expansion from the energy dissipation. In gases, the rise in temperature leads to drop a in the refractive index at constant pressure [4]. In liquids and solids, an increase in temperature may either cause a rise or drop in temperature depending on the molecular structure of the medium. By assuming ∆n dependence on temperature, the refractive index can be expressed by [4],

∆n=no+ [∂n

∂T]Ti (2.54)

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I(r)

r

(a.) (b.)

Figure 2.7: Schematic sketches to illustrate self–focusing. (a) Gaussian beam intensity profile and (b) Gaussian beam through Kerr material

where [_∂T^∂n] is the n dependent on temperature and Ti is the change in temperature of the material caused by the laser [4]. Another important optical effects as far as this research concerns are self–focusing and self–defocusing. Gaussian beam has a high intensity at the center than the peripheral as depicted in figure 2.7. If a Gaussian beam propagates through optical Kerr material, due to Kerr effect the refractive index becomes unequal in the radial direction within the medium. As a result, the medium behaves as a lens and the beam profile continually to divergence and convergence depending on n2. The medium converges the beam when n2 > 0 (self focusing) and diverges the beam ifn2 <0 (defocusing). [6, 24]

2.8.2 Nonlinear absorption

This is also known as two-photon absorption, which occurs when an electron with two or more photons having the same energies are excited to higher state [26].

Whenχ⁽³⁾ is complex which usually occurs at a frequency near the absorption band, it contains the fingerprint of nonlinear absorption (β) andn2. This can be expressed as,

χ⁽³⁾ =Reχ⁽³⁾+Imχ⁽³⁾ (2.55)

where Reχ⁽³⁾ is the real and Imχ⁽³⁾ is the imaginary parts. The imaginary part carries information on nonlinear absorption. TheImχ⁽³⁾ relates to change in amplitude of the beam when the intensity is altered. Wave propagation equation for field amplitude in z-position is given as,

∂A = −3ω

imχ⁽³⁾A³ (2.56)

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using

I = 2n|A|² Zo

(2.57)

∂A

∂z = −3ωZo

4cn² imχ⁽³⁾I² (2.58)

we can write change in absorption as,

∆α=βI² (2.59)

where

β = −3ωZo

4cn² imχ⁽³⁾ (2.60)

is the nonlinear absorption coefficient. This is for materials when both linear and nonlinear absorption are present. ∆α may reduce or rise depending on the sign of β which rely on the type of medium. If ∆α reduces with increasing intensity, it is saturable absorption and if in an opposite case it is the reverse saturable absorption.

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Chapter III

Z-Scan Technique

There are many methods that could be used to measure the nonlinear refractive index of isotropic materials, but the most commonly used technique is the z-scan.

This chapter emphasizes the principle behind this technique and the theoretical equations used in determining then2 from the experimental data.

3.1 The principle of z-scan measurement method

This method is not only used to measure nonlinear refractive index of materials but also to measure the protein concentration in blood [14]. It is also used to study semiconductors, biological materials and liquid crystals [14]. The z-scan technique can measure both the nonlinear refractive index and nonlinear absorption at the same time. It is based on a single beam that gives both the sign and magnitude of the nonlinear refractive index of materials simultaneously [27]. This method can also be used to determine whether the nonlinear absorption is saturable or reverse saturable [6]. A schematic diagram of conventional z-scan technique is shown in figure 3.1 below.

The detector D1 measures the incident light intensity. This is to avoid fluctuations within the beam energy [6]. The detector D2 measures light intensity at the far field after the focal point. Closed aperture (CA) is the signature of the nonlinear refractive index due to its sensitivity ton2. However, the open aperture (OA) carries the information of the nonlinear absorption (β). β measurement requires the whole light beam to get into the detector [28].

The main idea behind this method is to analyze the changes in the far field

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Figure 3.1: Schematic diagram of z–scan setup. BS is a beam splitter, D1 is the reference detector and D2 is the signal detector

sample [28]. Nonlinear refraction is a phenomenon caused by the change in the refractive index with a change in intensity through the medium [29]. The sample moves within the beam in order to change the focal point within the sample. At low intensity in the material (thus, sample far from the focal point of the beam), the intensity is insufficient to cause recognizable nonlinear refraction and therefore the transmittance does not depend on the intensity [29]. In contrary, if the intensity is high the nonlinear refraction causes a shift in the focal point of the beam towards the lens. Beam divergence at the far field increases due to the shifting of the beam waist and the optical power measured by the detector is decreased. This is illustrated in figure 3.2. In other words, if the sample is displaced towards +z from left to right, the transmittance curve measured for n2 > 0 samples, light intensity through the focal point gradually decreases and forms minimum-maximum curve (self–focusing or positive n2). On the other hand, for n2 < 0 samples, the light intensity gradually increased and form peak-valley curve (self-defocusing or negative n2) [6]. The two cases have been shown in figure 3.3. Presence multiphoton nonlinear absorption may lead to suppression of the maximum point and increases the minimum which results in peak-valley unequal (unequal amplitude). The opposite occurs for

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Lens sample

Aperture

-Z

+z

Detector

Figure 3.2: Schematic diagram showing the propagation of intense beam through a sample exhibiting self-focusing phenomenon before focal point of the lens. The dash lines indicate the shifted focal point

nonlinear saturable absorption [28].

NormalizedTransmittance

Position z(cm) n>02

Position z(cm) n<0₂

Figure 3.3: Representations of positive (self-focusing) and negative (self- defocusing) nonlinear refractive index respectively

.

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3.2 Calculations

Gaussian beam which can be mathematically expressed as E(z, r, t) =Eo(t) wo

w(z)exp[−r²

w(z)− −ikr²

2R(z)] exp−iφ(z, t) (3.1) [27] wherew(z) and R(z) are beam waist radius and radius of curvature, respectively, which are given by the following equations;

w(z) =w_o²(1 + z²

z²_o) (3.2)

R(z) =z(1 + z_o²

z²) (3.3)

Herewo is the beam waist, zo is the diffraction length of the beam.

zo = kw_o²

2 (3.4)

k is the wave vector,

k= 2π

λ (3.5)

λ is the wavelength, Eo(t) and φ are the amplitude and phase of the light wave respectively. For a very small sample length, the diffraction that causes a change in beam diameter can be ignored and the sample is considered to be a thin material and therefore the phase shift is calculated as [30]

∆φ(z, r, t) = ∆φo(t) 1 + ^z_z²²

o

exp[−2r²

w²(z)] (3.6)

∆φ is defined as

∆φ(t) =kn2Io(t)Lef f (3.7)

where Io(t) is the irradiance at the focus when z = 0 and Lef f is the effective propagation length which is expressed as

Lef f = (1− exp(−αL)

α ) (3.8)

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α is the linear absorption coefficient and L is the length of the sample. The transmittance power through an aperture can be computed by integrating the electric field through the radius as shown below

Pr(∆φo(t)) = cεonoπ Z r

0

|Eo(r, t)|²rdr (3.9) cis the speed of light,εois the permittivity of free space andno is the linear refractive index The variation of the power measured with respect to the position z (transmittance) is proportional to phase shift passing through the sample. The transmittance through the sample can be calculated from the expression

T(z) = R^∞

−∞PT∆φo(t)dt SR^∞

−∞Pi(t)dt (3.10)

[14] Pi is the incident power which is expressed as P(i) = πw²_oIo

2 (3.11)

S is the aperture transmittance computed as S= 1−exp(−2r²_a

wa

) (3.12)

where wa is the beam waist at the aperture and ra is the aperture radius.

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The difference of the maximum (peak) and minimum (valley) can be determined as

Position z(cm)

Tpv 2

n>02

n<0

Figure 3.4: Difference between peak and valley transmittance ∆T_pv

∆Tpv =Tp−Tv (3.13)

∆Tpvcan be expressed in terms of phase shift by the expression

∆Tpv = 0.406(1−S)^0.27|∆φo| (3.14) One can evaluate n2 as

n2 = ∆Tpv×λ×ω²_o×τ

0.406(1−S)^0.27×2×Ein×Lef f

(3.15) In the presence of nonlinear absorption, the normalized transmittance is given by well-established relation

T(x) = 1− 4x

(x²+ 9)(x²+ 1)∆φ− 2(x²+ 3)

(x²+ 9)(x²+ 1)∆Θ (3.16)

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where x = z/zo. ∆φ and ∆Θ describe the phase shift as a result of nonlinear refraction and absorption determined as

∆φ =kn2IoLef f (3.17)

where

k= 2π

λ (3.18)

and

∆Θ = 1

2βIoLef f (3.19)

This equation is applicable for small phase change or distortion and CA is narrow enough.

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Chapter IV

Experimental Techniques and Equipment

4.1 Z-scan measurement

Pulse laser

=800nm

BS BS

lens

Sample

D1 D2

D3

Aperture

Computer

-Z +z

Figure 4.1: Modified z-scan setup for measuring CA and OA transmittance at the same time. BS a is beam splitter, D1 is a reference detector to monitor fluctuations in energy of the beam, D2 is a signal detector to measure OA and D3 measures CA respectively.

The schematic set up for the measurement is shown in fig 4.1. The modification we made to the traditional z-scan setup is to introduce two detectors after the sample. This is to enable measurement of reflected beam (for nonlinear absorption) and transmitted beam through the aperture (for nonlinear refractive index) at the same time. The recorded data on D2 and D3 are divided by D1. The idea is to minimize or do away with misleading results cause by fluctuations in the laser input

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energy. The quotient (D2/D1 and D3/D1) is recorded by a computer for easy data analysis and calculations. Photo of the experimental setup is shown in figure 4.2.

Figure 4.2: Photo of experimental set up.

In this experiment, we used 800nm femtosecond pulse which is generated by Quantronix Integra–C showed in figure 4.3. Integra–C has a build in components that aid its functions. These include integrated pump laser, amplification stage, and compressor that generate pulse energy of about 3.5mJ [31]. This device supplies stable long–term performance high energy pulse and a clean beam profile due to its Pockel cell compartment that is capable of eradicating experimental errors. The stabilization system ensures steady energy over a wider temperature range for laser

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In the measurements we used 160 laser pulse. The laser beam was focused on the sample with the aid of focusing lens that gave a beam waist of about 17µm. The sample was positioned in the beam direction and translated along Z-axis from +Z to -Z.

Optical power transmitted through the sample was recorded with an optical power meter manually. We obtained the data for aperture linear transmittance (S) after taken and averaging of the five measurements of input and output powers through the aperture. Incident power needs to be controlled to avoid optical damaging of the characterized material. To achieve this, neutral density filters were used to ensure that the power through the sample is below the threshold. Equation 3.15 was used to calculate n2 from the data we obtained in the experiment.

Figure 4.3: Photo of Integra–C

4.2 Measurement of linear absorbance and transmittance

Perkin Elmer Lambda 9 UV/VIS/NIR spectrophotometer gives reliable data for linear absorbance and transmittance analysis. It’s wavelengths range spans from 185nm to 3200nm. The spectrophotometer employs deuterium lamp and halogen lamps for transmission measurements in the UV and VIS/IR range. Two built in detectors enable the device to detect wavelengths below and above 860nm respectively [32]. Photo of Perkin Elmer Lambda 9 UV/VIS/NIR spectrophotometer is shown in figure 4.4.

To ensure accuracy in measurement of the linear absorption coefficient(α) needed in our calculations, neutral filters were used to reduce irradiance. The device was

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Figure 4.4: Photo of Perkin Elmer Lambda 9 UV/VIS/NIR spectrophotometer for measuring linear absorbance and transmittance.

first calibrated when empty to remove/minimize the noise level. Quartz sample was placed in the measuring chamber and the absorbance and transmittance data were saved on a computer for our calculations.

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Chapter V

Results and Discussion

The open aperture detector (D2) measures the far field whole beam power, which is sensitive to only nonlinear absorption. The result of transmittance measurements is shown in figure 5.1a. The pronounced peak at the focus of the beam is an indication that there exists a purely nonlinear absorption in the sample.

Figure 5.1b is the transmittance curve recorded after the aperture by dedector D3. It’s unsymmetrical nature is an indication of both nonlinear absorption and nonlinear refraction present in the sample. To obtain pure nonlinear refraction, we divide data D3 by D2 (D3/D2). The quotient gives symmetric shape the prove of pure nonlinear refraction data as shown in figure 5.1c.

The normalized transmittance curve as illustrated in figure 5.3 is the transmission at wavelength of 800nm of quartz as a function of position (z). One can be see in figure 5.3 the normalizes transmittance is maximum in pre-focal region and minimum in post-focal region. This is an indication that n2 of quartz is negative, contrary to the literature sign. The reason for this is that in our experiment, the translation was from +Z to -Z. This explains the fact that the sign of n2 from the transmission curve depends on the direction of translation of the sample within the beam. As reported by [12] the sign of n2 for quartz is positive and it was manifested in our calculations and is also obvious from figure 5.3 looking from +Z to−Z.

As discussed in chapter 2 of this thesis the effect of nonlinearity may originate from the optical Kerr effect, thermal effect and so on. The nonlinear refraction index of the quartz investigated is as a result of thermal effect, because of the presence of absorption in the input beam by the quartz sample which led to energy deposition.

This was manifested in a rise in temperature of the illuminated part of the sample

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a

-15 -10 -5 0 5 10 15

Z /mm 0.935

0.94 0.945 0.95 0.955 0.96 0.965 0.97 0.975 0.98

Transmittance

Transmittance vs. Z fitted data data1

b

-15 -10 -5 0 5 10 15

Z /mm 0.7

0.75 0.8 0.85 0.9 0.95 1 1.05

Transmittance

Transmittance vs. Z fitted data data

c

-15 -10 -5 0 5 10 15

Z /mm 0.75

0.8 0.85 0.9 0.95 1 1.05 1.1

Transmission

Transmission vs. z fitted data data1

Figure 5.1: Experimental results recorded by D2, D3, and D3/D2 respectively. (a.) is the OA result, (b.) CA result and (c.) the quotient of CA and OA.

by non-radioactive decay from the excited state. The temperature change is clear evidence of the temperature coefficient of refractive index change ^∂n_∂T as presented in equation 2.54.

Figure 5.2 is the absorption spectrum of the quartz. At 800 nm the absorbance is about 0.305, which is a contributing factor to the increase in temperature in the

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400 450 500 550 600 650 700 750 800 Waveleng

0 0 0 0 0 0 0 0 0 0

1

A

Linear absorbaⁿ

Figure 5.2: Spectrum for linear absorption of fused silica from 400nm to 850nm.

-15 -10 -5 0 5 10 15

ZZ !!"

# $ %

#$ %&

# $ '

#$ '&

1 1.05 1.1 1.15

1$(

Transmi))*+,.

N / 23 456789: 2 4; <3 6<< 6/;= >rve

Tv ?@BCBD

fitted data data1

Figure 5.3: Normalized transmission curve

illuminating part of the sample. The induced change of refraction index in the quartz sample enables it to act as a lens in the z-scan that resulted in the distortion in the

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phase of the propagating beam.

Using the normalized transmission curve and equation 3.15 we calculated average n2 to be 2.284×10⁻¹⁶cm²/W which in disagreement with references [11, 12, 17–19, 21–23] and in agreement with the value of n2 in [20].

-15 -10 -5 0 5 10 15

Z / mm 0.8

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

EFGHalized transmission

Figure 5.4: Closed aperture z-scan transmission curves of quartz at three different powers of 0.66mW , 0.40mW and 0.175mW corresponding to intensities of 4.38×10¹⁵W/m² , 2.65×10¹⁵W/m² and 1.160×10¹⁵W/cm² respectively.

Table 5.1: Some calculated parameters

Experiment intensityIo(W/m²) ∆φo n2(cm²/W) ∆Tpv

1 4.38×10¹⁵ 0.9409 2.718×10⁻¹⁶ 0.3425 2 2.65×10¹⁵ 0.4840 2.311×10⁻¹⁶ 0.0609 3 1.16×10¹⁵ 0.1671 1.823×10⁻¹⁶ 0.1765

Optical power is proportional to the intensity. It can be seen from figure 5.4 that the intensity increases with phase shift and grows proportionally withn2 according

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the beam self–action. Looking at figure 5.4, one can observe that at low intensities, the phase shift decreases (see table 5.1.). The calculated values matched with the experimental data in figure 5.4. The obtained dependence of the n2 of quartz on pulse energy is illustrated in figure 5.5, which shows that n2 grows with intensity leading to the nonlinear dependence of the phase shift on the pulse energy. It is

1 1.5 2 2.5 I IJK L LJK

intensity w / m² ¹⁰¹⁵

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

nonlinear refractive index cm2 / W 10^-16

1 1.5 2 2.5 3 3.5 4 4.5

intensity W / m² 10¹⁵

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

pMOPQPMRST

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

UVWXYX V[\ ]

1.8

^_ `

2 2.1 2.2

a_ c a_ d

2.5

a_e a_ f

2.8

hjkl2mo

10^{qr s}

Figure 5.5: Graph of intensity dependent onn₂, intensity dependent on phase shift and n₂ dependent on phase shift respectively.

conspicuous from the graphs that n2 grows with intensity of light as well as the phase shift. From this, one can think that the phase shift in nonlinear case also depend on the intensity of the light beam.

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Chapter VI

Conclusions

The z-scan method has been used to measure the nonlinear refractive index of fused silica with femtosecond laser pulses of the central wavelength of 800nm at 160f s pulse duration. The conventional z-scan setup was modified to make closed and opened aperture measurements at the same time to ensure accuracy. In order to avoid optical damaging, we measured the linear transmittance and absorbance with spectrophometer to determine the intensity range to perform the z–scan experiment.

It was confirmed that the deduction of sign of then2 from the transmission curve depends on the direction of the translation of the sample in the Z-axis. The nonlinear refractive index of quartz was calculated to be 2.284×10⁻¹⁶cm²W⁻¹, which is in disagreement with some literature values while in agreement with value in [20]. It was experimentally proved that nonlinearity is mostly experienced at high intensities as discussed by [4]. The nonlinearity originates from thermal effect due to presence of absorption. This results in the dependence that of nonlinear refractive index on the light intensity that influences on the phase shift in the quartz.

In nonlinear applications, quartz will be more efficient at higher intensities than at low intensities. One must also be cautious of threshold intensity to avoid optical damaging.

6.1 Recommendations

We recommend the use of computer control optical power meter. This is to ensure that all measurements are done at the same time for accuracy. Again, in other to avoid optical damage we recommend the measurement of linear absorption and

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Appendix A

calculation of n

₂

For Experiment 1 power = 0.6579mW

∆Zo = 1.7Zo (A.1)

∆Zo = 2−0 (A.2)

∆Zo = 2 (A.3)

Zo= 2

1.7 (A.4)

Zo = 1.1765×10⁻³m (A.5)

λ= 795×100⁻⁹m (A.6)

ω =

rλZo

(A.7)

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ωo = 1.7309e⁻⁰⁵m (A.8) P = 0.6597×10⁻³W

Lef f = 1×10⁻³m

∆TP−v = 0.3425; and S = 0.3371

n2 = ∆TP −v ×λ×ω²_o×τ

0.406(1−S)^0.27×2×P ×Leff (A.9) n2 = 2.718e⁻¹⁶m²/W

For Experiment 2 power = 0.4mW

∆Zo = 1.7Zo (A.10)

∆Zo = 2−0 (A.11)

∆Zo = 2 (A.12)

Zo= 2

1.7 (A.13)

Zo = 1.1765×10⁻³m (A.14)

λ= 795×10⁻⁹m (A.15)

ωo=

rλZo

π (A.16)

Intensity dependent nonlinear refractive index of fused silica