Chapter 1 Introduction
1.1.3 Other types of OCT
To achieve real-time imaging, imaging speed becomes an important parameter in OCT. Recently, two types of Fourier domain detection have been developed to increase the imaging speed dramatically. These techniques are known as Fourier/spectral domain detection (SD-OCT) [8~11] and swept source/Fourier domain detection (SS-OCT) [12~14].
Unlike TD-OCT, SD-OCT performs high-resolution detection by measuring the interference spectrum with a low-coherence light source and a spectrometer. SS-OCT utilizes an interferometer with a narrow-bandwidth, frequency-swept light source and detectors to measure the interference signal as a function of time. The main reason that Fourier domain detection has a powerful sensitivity advantage over the time domain detection is that it essentially measures all the backscattered light simultaneously [14~16]. On the other hand, only the transverse scanning has to be performed. The depth information is provided by an inverse Fourier transform of the spectrum of the interference between the
θ w0
2w0 f
d
2w0
zR
b
Fig. 1.4 The relationship between minimum spot size and depth of field.
backscattered light from fixed reference and sample. Such a sensitivity enhancement in Fourier-domain OCT corresponds to a boom in image speed.
For more information and details about OCT, see the reference [17].
1.2 Broadband emitters for OCT
In order to meet the requirement of broad spectrum, superluminescent diodes (SLDs) and femtosecond laser are two kinds of major low-coherence light sources in OCT.
1.2.1 Superluminescent Diodes (SLDs)
Briefly, SLD is the light output from laser structure below threshold.
Operating at amplified spontaneous emission (ASE) and suppressing stimulated emission, SLD achieves wider spectrum than laser does by tilting the ridge structure. Compared to its competitive light sources, SLDs are now considered the most attractive light emitters due to their small size, easy use, and much lower price.
In the past, SLD could not become an ideal light source for OCT application because it was unable to meet all the demands simultaneously, including high optical power, broad spectrum, and negligible parasitic spectrum modulation. Early medical OCT application employed commercially available GaAs-based SLD, which operated at near 800nm with bandwidths of ~30 nm to achieve axial resolutions of ~10 μm in tissue. Besides, there have been advances in multiplexed SLD.
Bandwidths broader than 150 nm but with limited power, which corresponds to axial resolutions as fine as 3-5 μm, can be achieved [18].
However, for scattering tissues, light sources with longer wavelengths are
essential to get enough penetration depth. At the same time, increasing of spectral wavelength results in increasing of coherence length which is proportional to square of wavelength. Therefore, SLD with centered wavelength at 1300 nm must be at least 2.5 times broader than 800nm counterpart to achieve the same resolution. Although much broader spectrum in multiple quantum well (MQW) and quantum dot (QD) SLDs at 1300~1550 nm have been demonstrated, spectral width of the most powerful (10~20 mW from SM fiber) commercial SLDs at 1300 nm is currently around 60~70 nm [19~20].
1.2.2 Femtosecond Lasers
Femtosecond lasers are the other powerful light sources for applications of ultra-high resolution OCT because they can generate extremely broad bandwidth spectrum at near infrared (IR) region. The Kerr lens modelocked (KLM) Ti:Al2O3 is the most commonly used technique in femtosecond optics and ultra-short pulse application.
Through the development of double-chirped mirror technology (DCM) [21], high-performance Ti:Al2O3 laser has been implemented recently.
Using state-of-the-art Ti:Al2O3 lasers, axial resolution of ~1 μm can be demonstrated [22]. Fig. 1.5 shows a comparison of the spectrum, autocorrelation function and its demodulated envelope (coherence length) of a typical SLD and a Ti:Al2O3 laser. A pioneering study of UHR-OCT in the retina reported by Drexler et al. shows axial resolution of ~3 μm by using a femtosecond laser light source [23]. In combination with nonlinearity, air-silica microstructure fibers or tapered fibers, Ti:Al2O3 laser can generate broadband continuum and achieve axial resolution of
~1 μm in the different spectral region ranging from visible to near IR [24~26]. Despite of their outstanding performance in bandwidth,
femtosecond lasers are criticized due to its relatively high cost and complex. Recently, new Ti:Al2O3 lasers operate with much lower pump power and therefore greatly reducing cost has been developed [27].
1.3 Overview of Semiconductor Quantum Dot Lasers
Self-assembled quantum confined nanostructures, in the form of quantum dots, have been paid lots of attention due to its near delta-function of density of states (DOS) [28]. Ultralow threshold current density Jth [29], better temperature stability T0 [30], high modulation bandwidth [31], low chirp [32], and small linewidth enhancement factor have been demonstrated [33]. Limited by the number of quantum dot (QD), the gain in QD saturates rapidly at a certain level, gst, with increasing current density [34], which is the main difference of QD laser behavior from quantum well (QW) system. After the gain saturation at ground state (GS), carriers start accumulating at excited states (ES) and then achieve its threshold condition. For a specific laser device, GS
Fig.1.5 Optical bandwidth, interference signal and point spread function using a typical SLD and a femtosecond Ti:Al2O3 laser. For ultra-high resolution OCT, a broadband femtosecond laser with spectrum bandwidth ~260 nm can achieve a free space axial resolution of 1.5 μm.
and/or ES lasing will be determined by its cavity length, density of QD, driving current, and other structural factors.
Broad-spectrum, high-power and highly efficient semiconductor light sources are strongly desirable for sensing and low-coherence imaging applications such as OCT. With higher power than SLD and more compactness than femtosecond laser, broadband semiconductor laser is more competitive if real supercontinuum broad bandwidth can be performed. Nevertheless, because of active material growth technology and fundamental physics, a conventional QW laser generally produces a narrow spectrum with a spectral width of the subnanometer order for a single-frequency laser to a few nanometers for multi-longitudinal mode lasers. Recently, simultaneous two-state lasing from the GS and ES has been observed from QD lasers with well-separated wavelength emissions [35]. Except for incomplete gain clamping at threshold stated as before, this behavior is also attributed to the retarded carrier relaxation process in QDs, which is also known as the phonon-bottleneck [36]. On the other hand, to achieve real broad laser spectrum of tens of nanometer scale from the QD laser, high dispersion in QD size and small energy spacing between quantized energy states that lead to the broadened optical gain characteristics play important roles as well. In fact, optical line broadening of the emission spectrum of a transition depends on both homogeneous and inhomogeneous broadening mechanisms. Besides the contribution of large dot size fluctuation towards inhomogeneous broadening phenomenon, mechanisms such as acoustic and optical phonon-carrier interaction, lifetime broadening, and carrier-carrier interaction greatly affect the homogeneous linewidth in semiconductors [37]. Lately, NL Nanosemiconductor demonstrated the extended lasing spectral width with uniform intensity distribution (only a 3dB modulation) of about 75 nm [38~39]. A broad lasing spectrum of 40 nm was also
demonstrated by H. S. Djie et al [40].
In this study, specially designed chirped-multilayered QD (CMQD) lasers with three kinds of different wavelength InAs-InGaAs QD stacks were grown. Besides, we have set up a fiber-based, delay-tunable Mach-Zehnder interferometer to measure the interferograms or autocorrelation function which can be used to quantitatively determine the coherence length of a light source. Except for achieving low threshold current density and high saturated modal gain, we demonstrate broad spectrum of 29 nm in this structure and expect that it could be improved to satisfy all the demands for a suitable OCT light source someday.
Chapter 2
Theoretical Fundamentals
2.1 Coherence Theory
In optics, the original sense of the word coherence was attributed to the ability of radiation to produce interference phenomena. Nowadays, the term of coherence is defined more specifically by the correlation properties between quantities of an optical field. Common interference is the simplest phenomenon showing correlations between light waves. It is convenient to divide coherence effects into two classifications, temporal and spatial. The former relates directly to the finite bandwidth of the light source, the latter to its finite extent in space. Experimentally, Michelson introduced a technique for measuring the temporal coherence by Michelson interferometer. Spatial coherence is illustrated by the double-slit experiment of Young. When it comes to coherence length, what we mainly consider here is temporal coherence.
2.1.1 Temporal Coherence
First of all, we consider the path of rays in a Michelson interferometer which is shown in Fig. 2.1. The incident light is divided into two beams by a beam splitter. One beam is reflected back onto itself by a fixed mirror, the other one is also reflected back by a mirror which can be shifted along the beam. Both reflected beams are divided again into two by the beam splitter, whereby one beam from each mirror propagates to a photodetector (PD). The idea of this arrangement is to superimpose a light wave with a time-shifted copy of itself.
Let E be the light wave that reaches the screen via the fixed mirror 1 and E be the one that reaches the PD via the movable mirror. Then we 2 have at a point on the PD, when the incoming wave has been split evenly at the beam splitter, mirror displacement d according to
c 2d
τ = . (2.2) On the PD, the interference of both waves given by the superposition of the wave amplitudes: This superposition is not directly visible. A photodetector with a finite integration time detects the superposed light beams and provides the integrated/averaged intensity:
Fig.2.1 The Michelson interferometer.
} the first wave and I of the second wave and an additional term, the 2 interference term. The meaningful information is contained in the expression 〈E1*E2〉. With Eq. (2.1) this gives rise to the definition:
Γ is known as the complex self coherence function. It is the autocorrelation function of the complex light wave E1(t). We then get
)}
The magnitude γ(τ) is called the complex degree of self coherence.
Because Γ(0)=I1 is always real and is the largest value that occurs when we take the modulus of the autocorrelation function Γ(τ), we have
1 easy to determine the contrast K between interference fringes. This quantity has been used by Michelson who called it visibility and defined
it via the maximum and minimum intensity Imax and Imin as
Obviously, the maximum and the minimum intensity of the interference fringes do not occur at the same time shift of the light waves; that is, K is a function of τ . Let τ1 and τ2, τ2 > , be the time shifts belonging τ1
τ − corresponds to half a mean wavelength, which is small compared to the duration of the wave train to be investigated. Only in this case does the definition make sense. According to Eq. (2.6), the maximum intensity is attained at maximum Re{Γ(τ)}, occurring at τ1, and the minimum intensity at minimum Re{Γ(τ)}, occurring at τ2. It follows, for τ taken from this interval, that
) ( )}
(
Re{Γτ1 = Γτ and Re{Γ(τ2)}=−Γ(τ). (2.12) This leads to the intensities
) and to the contrast function
) The visibility then is identical to the modulus of the complex degree of
coherence. This is valid for two waves of equal intensity, otherwise some prefactors will arise. Fig.2.2 shows the self coherence function Γ(τ) in the complex plane (up) and contrast function K(τ) (down) for different light sources. Light in the large range in between the two limiting cases, completely coherent and completely incoherent, is called partially coherent. The following cases are distinguished (τ ≠0, γ(0) =1) :
All real light sources are partially coherent. Most part of natural and artificial light sources have a monotonously decreasing contrast function;
for instance, the light from a mercury lamp. To characterize the decay of the contrast function, the coherence time τc is introduced. It is defined as the time delay when the contrast function has dropped to 21/ or possibly 1 of its maximum value. In other words, it specifies the /e duration of time that a source maintains its phase. Equivalently to the coherence time, the coherence length
τc
Completely coherent Completely incoherent Partially incoherent
Fig.2.2 Graph of the self coherence function Γ(τ), and the contrast function )K(τ for light from completely coherent, completely incoherent, and partially coherent light, respectively.
is used for characterizing the interference properties of light. Typical values of the coherence length are some micrometers for incandescent light and some kilometers for single-mode laser light.
In a word, to determine the coherence length of some light source experimentally, all we have to do is to measure the interferogram or autocorrelation with enough path difference by a delay-tunable interferometer. After extracting the AC component and then detecting its envelope, we can define the FWHM of this profile as the coherence time.
Coherence length will be obtained after multiplying coherence time by the light speed, c .
2.1.2 Wiener-Khinchin Theorem
When it comes to coherence of a light source, its spectral bandwidth plays an important role. A connection between the autocorrelation of a function and its Fourier transform is established by the Wiener-Khinchin theorem. It states that the autocorrelation and the power spectrum of a function are Fourier transforms of one another. In the time domain, if
) (t
f is a radiated electric field, f(t)2 is proportional to the radiant flux or power, and the total emitted energy is proportional to f t dt
2
0 ( )
∫
∞ .In the frequency domain, F(ω) is the Fourier transform of f(t), which is called the field spectrum. F(ω)2 =F*(ω)F(ω) must be a measure of the radiated energy per unit frequency interval, which is sometimes called the power spectrum. In other words, F(ω)2 is just what we measure from the optical spectrum analyzer (OSA).
According to the definition, the autocorrelation of f(t) is denoted by
dt t
f t
∫
+∞∞ f +=
Γ(τ) - *( ) ( τ) (2.17)
0
Let )F(ω be the Fourier transform of f(t), denoted by FT{ tf( )}. The autocorrelation can be rewrite to
∫ ∫
Changing the order of integration, we obtain
ω
In the last integral, notice that
}
IFT means inverse Fourier transform. Then, Eq. (2.19) becomes ω and both sides are functions of the parameter τ. If we take the transform of both sides, Eq. (2.20) then becomes
)2 This is a form of the Wiener-Khinchin theorem. It allows for determination of the spectrum by way of the autocorrelation of the generation function. By the way, Wiener-Khinchin theorem is the main idea in Fourier-Transform Spectroscopy (FTS). The variable part of the interference function, on Fourier transformation, yields the spectrum. It is now used widely in the infrared region because of the improved signal-to-noise (S/N) ratio, high throughput, and high resolution possible with it.
2.2 Semiconductor Quantum Dots
In this chapter, I will discuss about the density of state (DOS) in semiconductor, the formation of quantum dots and the carrier transition between excited states and ground state in quantum dots.
2.2.1 Ideal Quantum Systems
In semiconductor, an interaction between electrons of neighboring atoms is sufficiently strong. This interaction splits an atomic energy level to as many sublevels as the number of atoms contained in the system.
These sublevels are grouped in bands. The width of a band is controlled by the strength of interaction of electronic shells, that is, interatomic spacing. In semiconducting crystals, where the interatomic spacing is set by the lattice constant in the range of few angstroms, the width of the order of several electron volts is typical for the valence (the last occupied) and conduction (the first empty) bands. The energy gap, which exists between these two allowed bands, contains no electronic states and is known as the forbidden band. Its width ranges from several electron volts (insulators) to zero (metals).
The number of allowed states within a given band is roughly equal to the number of atoms inside the crystal. It is as many as few 10 22 electrons per cubic centimeter. However, these states are not distributed uniformly within the band. To characterize this distribution, a DOS is introduced. DOS is a function of energy ρ(E) and it is a measure of allowed electronic states within unit energy interval per unit volume of crystal:
) / )(
/ 1 ( )
(E = V dN dE
ρ . (2.22)
DOS has dimensionality of energy-1 volume-1 and shows how many states are available within unit energy interval around some energy E in a crystal of unit volume. Eq. (2.23)~(2.30) are the corresponding dispersion relation and DOS for four kinds of ideal quantum systems:
Bulk: where E is the energy of the band edge of the corresponding band; 0
) (E −En
Θ is step-function which returns zero if E <En or unity otherwise; δ(E − En, lm, ) is the delta-function; and nQwire (cm-1) and nQD
(cm-2) are the surface density of quantum wires and QDs, respectively.
Fig. 2.3 illustrates the schematic view and DOS of various ideal quantum nanostructures.
2.2.2 Structural Characteristics of Quantum Dots
To realize quantum dot structure, various techniques based on self-organization have been intensively investigated recently instead of conventional lithography-based or regrowth technology. The self-organized quantum dots have high crystal quality, and they are small enough to exhibit quantum confined effects. Among several techniques, the Stranski-Krastanow (S-K) growth in highly strained heteroepitaxial systems is most studied by using molecular beam epitaxy (MBE) or metalorganic chemical vapor deposition (MOCVD). Cluster formation occurred during epitaxial growth of a semiconductor material on top of another one that has a lattice constant several percent smaller, for instance, InAs on GaAs substrate. For the first few atomic layers, the atoms arrange themselves in a planar layer called the wetting layer. As the
3D : Bulk 2D : QW 1D : QWire 0D : QD
Fig. 2.3 Schematic view and DOS for various quantum nanostructures.
epitaxial overgrowth proceeds, the atoms tend to bunch up and form clusters (quantum dots). It is energetic favorable, because the lattice can elastically relax the compressive strain and thus reduce strain energy within the islands.
The size of the islands can be adjusted within a certain range by changing the amount of deposited dot material. The change in size corresponds to an energy shift of the emitted light, which is due to the altered carrier confinement and strain in the dots. On the other hand, to obtain the information about structural and morphological characteristics of QDs, atomic force microscopy (AFM), scanning tunneling microscopy (STM), transmission electron microscopy (TEM), and photoluminescence (PL) spectroscopy are the most well-known and widely adopted methods.
In fact, except for its shape and size, the energetic property of QDs also depends on the thickness and the highness of potential barrier of the capping layer.
2.2.3 Carrier transition in Quantum Dots
To achieve population inversion, fast intraband relaxation time is required. In quantum well systems, the typical intraband relaxation lifetime is about 0.1~1 picoseconds (ps). However, due to the lack of phonons needed to satisfy the energy conservation rule, the carrier relaxation into the discrete ground state is significantly slowed down in QDs (the so-called phonon bottleneck). Retarded carrier relaxation, with a lifetime ranging from several tens to hundreds of picoseconds, has been observed in many experiments using time-resolved PL. Even though Auger-like process and carrier-carrier scattering have been suggested for possible fast relaxation channels with a lifetime of 10 ps or less, the bottleneck effect still affects QD laser performance to some extent, such
as the degradation of threshold current and the external quantum efficiency.
2.3 Quantum Dot Lasers
Laser is an acronym meaning “light amplification by stimulated emission of radiation”. A semiconductor, owing to a possibility of direct injection of electrons to the active region, gives an opportunity to make compact and efficient laser device. Basically, laser consists of the light-amplifying region where inverse population takes place and the positive optical feedback.
2.3.1 Basic Principles of Diode Lasers
In semiconductor, photon emission due to electron return from the conduction band to the valence band can be considered as radiative recombination of electron-hole pair. However, electron can also lose its energy non-radiatively. For any light-emitting device, too non-radiative recombination results in additional consumption of excited electrons and has to be minimized. Actually, there are two possible mechanisms when photon emission occurs. One is spontaneous emission with arbitrary
In semiconductor, photon emission due to electron return from the conduction band to the valence band can be considered as radiative recombination of electron-hole pair. However, electron can also lose its energy non-radiatively. For any light-emitting device, too non-radiative recombination results in additional consumption of excited electrons and has to be minimized. Actually, there are two possible mechanisms when photon emission occurs. One is spontaneous emission with arbitrary