CHAPTER 1 Introduction
1.3 Organization of the thesis
The thesis consists of four chapters. Chapter 1 gives an introduction of the second-order nonlinearity in optical fibers and explains our motivation for doing this research. Chapter 2 describes the basic concept of SHG, and the mechanisms of thermal poling and UV erasure in
optical fibers. Chapter 3 presents the experiment procedures and results. In Chapter 4 we make a brief conclusion and discuss possible future research directions.
Chapter 2 Basic concept of second-harmonic generation in optical fibers
2.1 Principle of nonlinear optics
The first working laser was made by Theodore H. Maiman in 1960 [13]. He used a solid-state flash-lamp-pumped ruby crystal to produce red laser lights at 694nm. The invention of the laser enabled us to examine the behavior of lights in optical materials at higher intensities. In fact, the beginning of the field of nonlinear optics is often taken to be the discovery of SHG by Franken et al. in 1961 [14].
Nonlinear optics is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system by the presence of lights [15, 16].
They occur when the response of a material system to an applied optical field depends on a nonlinear manner upon the strength of the optical field, and the intense laser field can cause the polarization of the medium to develop new frequency components that are not present in the incident field. These new frequency components of the polarization act as sources of new frequency components of the electromagnetic field.
The nonlinear processes also have to satisfy the conditions of conservation of energy and conservation of momentum, in order to get the optimum conversion efficiency. Fig. 2.1 illustrates the relation between the energy-level and the momentum vectors.
(a) (b) Fig. 2.1 (a) Energy-level, conservation of energy (=ω1+=ω2 ==ω3),
Fig. 2.1 (b) Momentum vectors, conservation of momentum ( kG1+ kG2 = kG3
= = = ).
As an example of nonlinear optical interaction, let us consider the process of sum-frequency generation, which is illustrated schematically in Fig. 2.2. A weak light (ω1) and an intense pump light (ω2) are incident into a nonlinear medium simultaneously. After the sum frequency generation process, a light (ω3) which frequency is the sum of ω1 and ω2,
emits from the nonlinear medium.
2 1
3 ω ω
ω = + (2.1)
This process is known as the up-conversion and equation (2.1) is the condition of conservation of energy.
is called the wave vector (or momentum) mismatch, which describes how good the phase
matching (momentum conservation) is achieved. If the process satisfy the phase matching condition (∆k =0), it will be under the optimum working condition.
Fig. 2.2 Sum-frequency generation, where d is the second-order nonlinear coefficient and χ(2) is the second-order nonlinear susceptibility.
2.1.1 Nonlinear susceptibility
A linear dielectric medium is characterized by a linear relation between the polarization (P) and the electric field (E),
E
P=ε0χ(1) (2.3)
whereε0 is the permittivity of free space andχ(1)is the linear electric susceptibility of the
medium. The relation between the dielectric constant, refractive index and χ(1)is as below,
) 1 ( 0
2 1 χ
ε ε = +
=
n (2.4)
But in a nonlinear dielectric medium, the polarization can be a power series of the
electric field, which can be expressed as
The first term, which is linear, dominates at small electric field E. The second term represents a quadratic effect, whereχ(2)is the second-order nonlinear susceptibility. The third term represents third-order nonlinearity andχ(3)is the third-order nonlinear susceptibility. Actually the linear susceptibilityχ(1)is a second-rank tensor and the n-order nonlinear susceptibility
( )n
χ is a (n+1)th - rank tensor.
The second-order nonlinear optical interaction can occur in a noncentrosymmetric media only. That is, they do not exist in symmetric media. On the other hand, the third-order
nonlinear optical interaction can occur in both centrosymmetric and noncenrosymmetric media.
2.1.2 Tensor
In a noncenrosymmetric medium, we can describe the Cartesian components of the
second-order nonlinear polarization density and the second-order nonlinear susceptibility as
(2)
Pi and χijk, respectively, where i, j and k are the Cartesian coordinates x, y and z. We introduce the second-order susceptibility tensor
ijk
dijk = 12χ (2.6)
For sum frequency generation,
(2) (2)
, , 0 ( ( ); , ) ( ) ( )
i ijk m n m n j m k n
j k m n
P = Σ Σε χ − ω +ω ω ω E ω E ω (2.7)
and for second harmonic generation,
(2) (2)
According to the Kleinman symmetry condition, the dijk is with the intrinsic
permutation symmetry, where the last two indices j and k can be interchanged freely without altering the nonlinear susceptibility tensor, i.e. dijk =dikj. Therefore the following contracted
notation is used
For sum frequency generation,
⎥⎥
2
The material of our thermally poled optical fibers is fused silica, which the point group is
∞ mm and the interaction frequencies are far away from the absorption frequencies.
According to the Kleinman symmetry condition, the second-order susceptibility tensor is
31
therefore the SHG of thermal-poled optical fibers is
31
Stolen and Tom [2] proposed that an effective χ(2) originates from the interaction between the third-order susceptibility χ(3) and a built-in static electrical field E , that is l
(2) (3)
χ = χ for the centrosymmetric medium, therefore
33 3 31
d = d (2.15)
2.1.3 Coupled wave equation
Maxwell’s equations,
is the nonlinear polarization vector.
Because ∇×∇× = ∇ ∇ ⋅EG ( EG)− ∇2EG
According to the slowly varying envelope approximation (SVEA),
2
From equations (2.17) and (2.18), we get
0
2 2 0 of nonlinear coefficient. Equations (2.22), (2,23) and (2,24) are the coupled waves equations.
2.1.4 Second-harmonic generation
For SHG, if a wave propagating in the z direction with frequencyω ω ω= 1= 2 is normal incident into a lossless and nonlinear medium, its SH wave is with the frequency
3 1 2 2
where L is the length of the medium.
We can express the SH power as
2 2
3 2
Fig. 2.3 Second-harmonic generation.
2.1.5 Quasi-phase matching
From equation (2.28), we observe that the SH power P2ω is proportional to deff2, L 2 and sinc2(∆2kL), Fig. 2.4 shows the relation between sinc2(∆2kL) and ∆2kL. If the process satisfy the phase matching condition (∆k =0), we can get the maximum of SH power.
Fig. 2.4 sinc2(∆2kL) vs ∆2kL.
Typically, three-wave mixing is done in a birefringent crystalline material, in which the refractive index depends on the polarization and direction of the lights that passes through.
For satisfying the phase-matching condition, the polarization of the fields and the orientation of the crystal have to be chosen correctly. Such a phase-matching technique is called the angle tuning.
One undesirable effect of angle tuning is that the optical frequencies involved are not collinear with each other. This is due to the fact that the extraordinary wave propagating through a birefringent crystal possesses a Poynting vector which is not parallel with the propagation vector (critical phase-matching). This would lead to the beam walkoff which
Another phase-matching technique is called temperature tuning. The crystal is controlled at a certain temperature to achieve the phase matching condition. Beam walkoff is avoided by forcing all frequencies to propagate at the angle 90o with respect to the optical axis of the crystal(non-critical phase-matching).
Quasi-phase matching scheme allows the choice of the largest nonlinear coefficient with non-critical phase-matching. To obtain a quasi-phase matching structure, one may periodically modulate the nonlinear coefficient by altering the crystal symmetry in the nonlinear optical material. A class of materials called ferroelectric crystal (such as lithium niobate and lithium tantalate) possesses a spontaneous polarization inside the crystal. If one applies an external field larger than the intrinsic field in the crystal, it may flip the crystal symmetry and change the sign of the nonlinear coefficient. Fig. 2.5(a) shows the periodic structure, the relation between the SH intensity and the interaction length for ferroelectric crystals, and Fig. 2.5(b) also shows the periodic structure, the relation between the SH intensity and the interaction length for thermally poled silica fibers as a comparison.
From the Fourier expansion, the periodically modulated second-order nonlinearity d z ( )
can be written as
= Λ is the periodically modulated grating wave vector , Λ is the modulated
period and G is the Fourier coefficient. m
From equation (2.29), we have
When km= ∆ , we can get the quasi-phase matching. The modulated period is k
2 2 c
m ml
k Λ = π =
∆ , where the coherent length is
4( 2 )
For periodically poled lithium niobate, assuming the duty cycle of the modulated deff( )z
is 50% ( l 50%
D= =
Λ , where l is the length of second-order nonlinearity) and the second-order nonlinearity is a constant, then the Fourier coefficient can be calculated to be
2 sin( )
Gm m D
m π
= π (2.31)
where m is the order of quasi-phase matching. In this way, the effective nonlinearity of quasi-phase matching structure is
QPM eff m
d =d G (2.32)
For 1st order quasi-phase matching, because it is a (1,-1) type quasi-phase matching, and the effective second-order nonlinearity is
2
QPM eff
d d
=π (2.33)
On the other hand, the quasi-phase matching structure of periodically poled optical fiber is (1,0) type. The effective second-order nonlinearity is thus
2
Fig. 2.5 (a) First-order quasi-phase matching of ferroelectric crystal
Fig. 2.5 (b) First-order quasi-phase matching of thermally poled silica fiber
2.2 Quasi-phase-matched second-harmonic generation in periodically poled fibers
The modulation for the second-order nonlinearity in thermally poled is the type (1, 0), which is different from ferroelectric crystal. Quasi-phase matching occurs when the period Λ for the modulation of the nonlinear coefficient d is a multiple integer of 2l , where c l is the c coherent length. The unpoled sections permit free evolution of the interacting fields without energy exchange and their relative phase shift due to propagation can compensate the phase
The QPM SHG in thermally poled optical fibers is different from the thermally poled glass. The quasi-phase matching waveguide condition establishes the dependence of the period Λ as a function of the fundamental wavelength (λω), core radius (a) and numerical aperture (NA) [17, 18],
For the quasi-phase matching waveguide geometry with a low fundamental light depletion, the SH power P2ω is given by
where P2ω is the fundamental power, deff is the nonlinear coefficient which includes the overlap factor between the poled region and the interacting modes as well as the 1
mπ
reduction factor associated with the mth-order quasi-phase matching. L is the length of the periodical poled region, 1 2
OVL OVL
A = I is an equivalent area that depends on the overlap factor
IOVL between the interacting fields, ,2 ,
and ρ is an enhancement factor that takes account of the multimode nature of our fundamental source.
Because of the large quasi-phase matching bandwidth and low group-velocity mismatch (GVM) between pulses at different frequencies in optical fibers, it allows us to use long
devices without compromising the frequency stability and is suitable for pulsed frequency conversion.
2.3 Mechanism of thermal poling
Silica optical fibers are amorphous material with macroscopic inversion symmetry and inherently have no second-order nonlinearity. In 1986, the first report of SHG in silica fibers was reported by Osterberg and Margulis [1], who discovered that prolonged exposure of fibers to infrared light causes the self-organized growth of green SH lights. Since then, wide-ranging studies are engaged on the mechanism and properties of this unexpected photoinduced phenomenon.
Since 1991, second-order nonlinearities of order 1 pm/V have been achieved in glasses using a variety of different techniques including thermal poling [4,9,10], corona poling [5]
and electron implantation [6]. Here, we just discuss about the mechanism of second-order optical nonlinearity in thermally poled optical fibers.
The mechanism behind the formation of second order nonlinearity in thermally poled optical fibers is not yet fully understood. Until now, there are three types of explanation for the mechanism. The first explanation is most widely used.
(1) During the thermal poling procedure, ions such as Na+, H+,H O3 + or holes migrate toward the negative electrode with a relatively high mobility at a high
temperature, which leaves ≡ −Si O− in a near-surface layer contacted with the positive electrode. The ions reaching the negative electrode must be neutralized by electrons injected from the negative electrode. A high electric field arises in this region, between the negatively charged region and the anode, and this field creates second-order nonlinearity. The second-order nonlinear susceptibility χ(2) is related to the third-order nonlinear susceptibilityχ(3) [2,4] and the relation between
them is
(2) The orientation of hyperpolarizable entities (bonds or defects) in the optical fibers is realigned under the applied electric field and the formation of second-order optical nonlinearity χ(2) is [19],
(2) N L( )
χ ∝ β ρ (2.39)
where N is the concentration of hyperpolarizable entities, β is the second-order hyperpolarizability, and L( )ρ is an orientation factor (0≤ ≤L 1) under the total electric-field E within the optical fiber. L can be written as a sum of Langevin functions. m E.
ρ = kT G G
, where m is the dipole moment, E is the electrostatic field, T is the absolute temperature, and k is Boltzmann constant.
(3) The last explanation combines explanation (1) and explanation (2) [20], for which the effective χ(2) is created via both the interaction of the intense electric field through χ(3) and the dipole orientation. When the electrostatic field E < 1 V/nm, the dipole orientation plays the main role in the formation ofχ(2), whereas the interaction through χ(3) is the dominant factor for larger fields. When E > 3V/nm, the poling temperature must be over a threshold value, and there exists an optimal temperature. Therefore, the second-order nonlinearity from the combination of the orientation of the dipole and χ(3) is the first-order and third order Langevin functions, respectively. When ρ is small, the L1( )ρ and L3( )ρ can be represented as,
dipole.
Fig. 2.6 illustrates the formation process of the depletion region near the anode surface of the thermally poled fused silica [21]. Under thermal poling process, the Na+ ions are with energies in excess of the potential barrier under the high temperature about 2800C and thus can move into the SiO2 network. While a high dc voltage is applied, alkali metal ions (such as Na+ ions) will drift to the cathode where most of them are neutralized by the incoming electrons, as shown in Fig. 2.6(a). If all Na+ ions within about 5 µm of the anode surface in the fused silica is depleted, an intense electric field E of about 107 V/cm is established at this very thin region, and H3O+ is generated by the chemical reaction of ≡Si OH− with H2O, as described by equation (2.44),
2 3
Si OH H O Si O− H O+
≡ − + ↔ − + (2.44)
Due to the thermal fluctuations, some of the ions will also drift to the cathode and are neutralized, as shown in Fig. 2.6(b). With both of the H3O+ and Na+ ions drifting to the cathode, then Si O− − ions are left. Over there a negative space charge is generated near the anode surface, and a negative depletion layer is also formed. After that, H3O+ continues drifting to the cathode and the negative depletion layer slowly extends into the interior of the fused silica. Thereafter, because of the thermal fluctuations, the water molecules H2O in the air diffuse into the fused silica and react with the fused silica network to form immobile hydroxyl (OH), ≡Si OH− ,
0 2 2
Si Si H O Si OH
≡ − − ≡ + ↔ ≡ − (2.45)
After that, ≡Si OH− react with H2O to generate H3O+ under an intense electric field near the anode surface, as by equation (2.44). Then this H3O+ builds a positively charged layer. On the other hand, the thermal fluctuations and the intense electric field cause the fused silica to ionize, and the ionized electrons drift to anode and are neutralized, as shown in Fig. 2.6(c).
After the thermal poling process is accomplished, the maximum electric field exist in the region between the positively charged region and the negative depletion layer in the depletion region, and thus this region is most greatly ionized, as shown in Fig. 2.6(d). Fig. 2.7 (a) and Fig. 2.7 (b) show the planar schematic diagram of fused silica network before poling and after poling, respectively.
Fig.2.6 Formation process of the depletion region near the anode surface of the thermally poled fused silica
Fig. 2.7 (a) Planar schematic diagram of fused silica network before poling
Fig. 2.7 (b) Planar schematic diagram of fused silica network after poling
2.4 Mechanism of ultraviolet erasure
The thermally poled optical fiber can be erased by heating [2], electron beam [8], ultraviolet [9,10], or near infrared [11], which could be used to achieve the QPM SHG in optical fibers. Here, we just discuss about the UV erasure. The second-order nonlinearity in thermally poled optical fibers can be erased by UV simply because of the vanishment of the built-in electric field. When the UV lights are exposed to the thermally poled optical fiber,
Si O−
≡ − is destroyed by the one-photon absorption process. Holes from the conduction
band are considered to be trapped at ≡ −Si O−sites and thus the space charges of Si O−
≡ − are neutralized. In this way the nonlinearity is erased after the UV exposure.
Chapter 3 Experimental procedures and results
3.1 Introduction
Silica optical fibers are amorphous material with macroscopic inversion symmetry and inherently have no second-order nonlinearity. But in 1986, Osterberg and Margulis reported that the Ge-doped fiber irradiated by intense 1064nm laser can exhibit SHG with the SH conversion efficiency as high as 5% [1]. Later, Myers et al. found that large and stable second-order nonlinear susceptibility χ( 2) (~1 pm/V) could be created in fused silica by means of thermal poling [4]. Since then, there have been intensive researches on the poling of glass materials and glass fibers. In this chapter we report our experimental results on using D-shape optical fibers for achieving the second order nonlinearity through thermal poling.
3.2 D-shape optical fiber
In our experiment, the fibers we used were called the D-shape optical fibers. The shape of these fibers looks like a D character as shown in Fig. 3.1. The core of the D- shape fibers is in an elliptical shape and its index is higher than the surrounding cladding. The D-shape optical fibers are polarization maintaining fibers, which employ the property of geometrical birefringence to achieve the polarization preserving characteristics. The fibers are constructed of high-grade silica materials and various high purity dopants, such as germanium (Ge). This also offers significant advantages, including low loss and high polarization maintaining. Some
of the important parameters of the D-shape fibers are listed in the Table 3.1.
Fig. 3.1 D-shape fiber
Nominal Operating Wavelength (nm)
1550 Single Mode Operating Band (nm) 1360-1680 Cut-off Wavelength (nm) 1160±70
Attenuation, dB/km 2-5
Polarization Holding (h), dB-m ≧40 Normalized Birefringence 1.5×10-4 Fiber Diameter (microns) ± 3 125 Center of Core to Flat (microns) 16 Coating Diameter (microns) ± 15 245
Table 3.1 The parameters of the D-shape fibers
3.3 Experimental procedures
thermal poling, we used the BOE solution (HF: NH4F = 1: 6) to etch the optical fibers so that the plane surface is very close to the core region.
3.3.1 Etching of D-shape fibers
The D-shape fiber used in our experiment has an outer diameter of 125µm and the distance between the plane surface and the core region is 16 µm. Before proceeding the
thermal poling, we have to etch the fiber with the BOE solution so that the flat surface to fiber core distance is reduced to 1µm~5µm, as shown in Fig. 3.2. In this way, after poling, the
nonlinear layer under the anodic surface is inside the core region. The etching rate is about 1.75µm/10 min.
Fig. 3.2 Ecthed D-shape fiber
3.3.2 Thermal poling
The etched fiber was sandwiched between two electrodes with the anodic electrode on the flat surface of etched fiber. The electrodes were made by n-type {1, 0, 0} silicon wafer.
One of them was placed on the fused silica plate with a dimension 20mmX 20mmX 1mm (length X wide X depth). Fig. 3.3 shows the diagram of the thermal poling system. The temperature sensor is a K-type thermo-coupler and the maximum heating temperature is 400 0C , where the temperature is controlled by a temperature controller (Omega, CNi1633-c24). The heater is isolated by ceramic and its periphery is covered with thermal insulated cotton. The heating current is about ~6A.
According to the reported conditions of thermal-poled fused silica as shown in Fig 3.4 [18], our thermal poling is carried out at 4.0 kV and 2800C for 30 minutes. The temperature is varied from the room temperature to 2800Cby the heater for about 1 hour, and thereafter the electrodes are applied with a voltage of 4kV for 30 minutes. The heater is then turned off and cooled down to room the temperature in 120 minutes with the high voltage still applied. If the applied voltage is above 4.2kV, air breakdown is observed.
Fig. 3.3 The diagram of the thermal poling system.
Fig 3.4 SH signal vs poling time at 2800C, in Ref. [18]
3.3.3 Second-harmonic generation measurement
In the experimental setup, a 1064nmNd:YAG laser is used as the light source, which is operated in a pulsed mode and has a 2kHz repetition rate with ~20nspulse width. An isolator