575

# Primer on Photonics

Authored by: Paul R. Prucnal , Bhavin J. Shastri , Malvin Carl Teich

# Neuromorphic Photonics

Print publication date:  May  2017
Online publication date:  May  2017

Print ISBN: 9781498725224
eBook ISBN: 9781315370590

10.1201/9781315370590-3

#### Abstract

Most of us have an intuitive understanding of light as a geometric ray: something that travels in straight lines in a uniform medium. Interfaces between different materials (e.g., glass and air) can alter the direction of these rays. In this section, we will focus on light propagation in waveguides: special material structures that confine and direct the propagation of light with near perfect efficiency. Waveguides can be thought of as wires for light; they allow us to send optical signals over long distances in fiber optics and allow the construction of larger photonic circuits and systems on integrated chips. Waveguides are poorly explained by the intuitive ray description, so a more appropriate description of light as a wave is introduced. Figure 3.1 summarizes some of the main results associated with waveguides. Figure 3.1 (a) A slab waveguide (WG) in 2D with a dielectric cladding and core where <i>n<sub>core</sub> </i> > <i>n</i> <sub>clad</sub>. <i>x</i> is the propagation direction. Three transverse electric (TE) modes are plotted in red. TE means the electric field vector is in the <i>z</i> direction. At a given frequency, there are a discrete number of field profiles that satisfy boundary conditions, (b) The dispersion diagram for the slab WG, plotting frequency (<i>ω</i>) vs. propagation wavenumber (<i>k<sub>x</sub> </i>). Blue curves correspond to different modes. At a given <i>ω</i>, there are a finite number of supported modes, which have symmetric forward and backward propagating versions. Guided modes cannot exist within the so-called light cone (yellow region) because they would leak out into the cladding. The fundamental mode TE<sub>0</sub> is always present in dielectric WGs, but higher order modes TE<sub>1</sub>, TE<sub>2</sub>, etc. have minimum cutoff frequencies, which depend on the relationship between frequncy and WG dimensions. This means single-mode waveguides can be designed below the cutoff of the second mode.

#### 3.1  Waveguides

Most of us have an intuitive understanding of light as a geometric ray: something that travels in straight lines in a uniform medium. Interfaces between different materials (e.g., glass and air) can alter the direction of these rays. In this section, we will focus on light propagation in waveguides: special material structures that confine and direct the propagation of light with near perfect efficiency. Waveguides can be thought of as wires for light; they allow us to send optical signals over long distances in fiber optics and allow the construction of larger photonic circuits and systems on integrated chips. Waveguides are poorly explained by the intuitive ray description, so a more appropriate description of light as a wave is introduced. Figure 3.1 summarizes some of the main results associated with waveguides.

Figure 3.1   (a) A slab waveguide (WG) in 2D with a dielectric cladding and core where ncore > n clad. x is the propagation direction. Three transverse electric (TE) modes are plotted in red. TE means the electric field vector is in the z direction. At a given frequency, there are a discrete number of field profiles that satisfy boundary conditions, (b) The dispersion diagram for the slab WG, plotting frequency (ω) vs. propagation wavenumber (kx ). Blue curves correspond to different modes. At a given ω, there are a finite number of supported modes, which have symmetric forward and backward propagating versions. Guided modes cannot exist within the so-called light cone (yellow region) because they would leak out into the cladding. The fundamental mode TE0 is always present in dielectric WGs, but higher order modes TE1, TE2, etc. have minimum cutoff frequencies, which depend on the relationship between frequncy and WG dimensions. This means single-mode waveguides can be designed below the cutoff of the second mode.

Fig. 3.1(a) shows a diagram of a slab waveguide, which consists of a core layer of high index material between two cladding layers with lower index. The structure is translationally invariant in the xz plane. We assume the electromagnetic field profile has no z dependence and that light propagates in the x direction. Since the slab waveguide is invariant in time and x, we expect the conservation of two physical quantities corresponding to energy and momentum. In the case of lightwaves, these quantities are frequency, w (or ν = w/(2π)), and wavenumbers, kx and ky, which describe the spatial derivative of the optical field. Just as in free-space, we will be able to separate the solutions into two polarizations depending on the field orientation. The electromagnetic problem of the slab waveguide stemming from Maxwell’s equations can be framed as a partial differential equation with boundary conditions at the material interfaces. Instead of reproducing the derivation here, we refer the reader to [1, 2] or numerous other texts and course notes.

If the solutions to Maxwell’s equations in free space are best described as rays, then the solutions in a waveguide are best described by modes. At a given optical frequency, there is a continuum of ray solutions whose fields do not approach zero anywhere; however, in the case of a waveguide, there are also guided mode solutions that do approach zero at large y values, meaning that they are localized to the waveguide core and can propagate indefinitely without leaking out. Modes are discrete solutions to the field equations in a waveguide which are identified by a polarization and mode number. The first three transverse-electric (TE) modes’ field profiles are shown in Fig. 3.1. Modes do not interact, so optical intensity in a particular mode will stay in that mode. Each mode has a propagation wavenumber, kx , which depends on optical frequency and waveguide geometry. This means each mode also has an effective phase velocity and effective index:

3.1()

In some ways, a propagating guided mode has very similar characteristics to a plane wave in a uniform material with index neff . The main results about modes, cutoff frequencies, and guiding found in the 2D example are still valid for real 3D waveguides, although solving the boundary conditions requires numerical methods.

Fig. 3.1(b) plots the dispersion relations of the 2D slab WG. Each blue curve corresponds to a mode, which has a propagation wavenumber that depends on frequency. The light cone describes a continuum of unguided ray states. From this figure, we see that the fundamental TEo mode is present at all frequencies, while higher order modes have minimum cutoff frequencies. This is an important property for signal processing with waveguides, since a single-mode waveguide can keep guided light in a known state. In fiber optics, the single-mode condition is important to prevent signal spreading caused by different modal velocities. In integrated circuits, single-mode waveguides are ideal for constructing interferometers and resonators because they keep all guided light in a single coherent phase state. These devices based on single-mode waveguides will be described in Section 3.1.3 and Section 3.3.

Clearly, a single straight waveguide is not useful for building larger systems. Single-mode waveguides can be extended with three key elements: bends, couplers, and interferometers. While simple, these three elements are intimately related to the characteristics of waveguiding and have a substantial impact on the performance possibilities of the passive photonic circuits that can be made.

#### 3.1.1  Bends

In a photonic Integrated circuit (PIC), waveguides guide light between different components of the PIC. Bent waveguides are important in PICs for their use as couplers, or even filters and delay lines. For effective PIC systems and the integration of multiple components on a semiconductor wafer, we need waveguides that have sharp bends to quickly change wave propagation direction in a short distance with low loss. Small bends with minimal loss can be obtained due to the index contrast between the core and the cladding materials. Bends in waveguides with more than one mode cause a change in the propagation direction, so that the propagation wavenumber of modes is not conserved, meaning that intermodal mixing can occur. This is one of the primary reasons why single-mode waveguides are desirable for fibers and waveguides.

If the waveguide bends too quickly or tightly, then total internal reflection (TIR) will not guarantee that the light stays inside the waveguide. This leads to an effective minimum bending radius allowed for a certain waveguide cross-section design. The minimum bending radius is a key determinant of the compactness of PIC layouts. Bending loss is also a limiting factor in the design of microring resonators [3], which are further described in Section 3.3.

#### 3.1.2  Interwaveguide Couplers

Once the ability to guide light on-chip is established, the next capability for photonic processing is the ability to have signals interact with one another. Interwaveguide couplers transfer optical fields coherently, which enables interferometric circuits (Section 3.1.3). There are several different ways to couple optical fields between modes of two or more integrated waveguides, which we will compare based on several criteria.

Four types of couplers are shown in Fig. 3.2. Perhaps the most basic design in theory is the directional or evanescent coupler, wherein two parallel waveguides are brought close enough for their evanescent field profiles to reach one another. This can be viewed as the effective index degeneracy lifting of symmetric and asymmetric normal modes in the two-mode system, which results in a beating transfer of power back and forth between the two waveguides as the two supermodes propagate [4]. Two-mode interference couplers, also sometimes called zero gap directional couplers (ZGDC), are fundamentally similar to directional couplers, but they have no gap in their coupling region. The coupling region supports two modes (symmetric and antisymmetric), which beat when propagating.

Figure 3.2   Four different approaches to interwaveguide coupling.

Multi-mode interference couplers (MMI), are fundamentally different from the above coupling schemes because they operate based on a reimaging effect in specially designed two dimensional cavities. Energy is not transferred simply between two interacting modes but is injected to a large area with reflective boundaries. In a 2 × 2 MMI coupler, light injected in one input waveguide will diffract and interfere with reflections, which results in the formation of several focal planes. The image of the input mode reappears after beat length L. Power is completely transferred to the opposite mode at length L/2, and the pseudoimage at L/4 is an equal distribution between the two sides. The output waveguides are placed so as to correspond to the refocused pseudoimage of the input modes.

Y-junction splitter/combiners come in two-mode or multi-mode forms. Essentially, due to their geometric symmetry, they act as precise 3 dB splitters, which are often convenient for the requirements of interferometry. However, transmission efficiency and compactness require careful design, as in Ref. [5].

For a given waveguide design (fixed effective index and mode profile), the coupling rate of directional couplers is determined by the gap width of the coupling section. The power splitting ratio after one pass through the directional coupler is then fixed by the length of the coupling section. Directional couplers are ideal when a small coupling ratio is required, making them ideal for microring resonators. The coupling ratio of directional couplers is easily controlled by simply changing their lengths. Because the beat length is a nontrivial function of waveguide geometry, mode distribution, and gap width, in practice additional simulation or calibration experiments must be performed to achieve precise coupling ratios.

MMI couplers and Y-junctions are more appropriate when precise coupling ratios are desired. MMI couplers with uneven field splitting distributions can be designed but can become much more complicated to design when non-rectangular coupling regions are considered. Because the power splitting is based on the self-imaging property, only discrete coupling ratios are available (85:15, 72:28, 50:50, 27:73, 15:85, and 0:100). Several sources have proposed modifications for achieving arbitrary coupling ratios by using two MMI sections [6], angled MMI sections [7], patterned MMI sections [8], or slotted MMI sections [9]. MMI coupler imaging length depends on the lateral dimension of the MMI region, which in turn depends on the separation of input waveguides. Once this dimension reaches a lower limit, there is very little flexibility to change the size of the coupling region. Several approaches for further shrinking MMI elements have been proposed, including access waveguide tapering [10, 11] and MMI section tapering [12].

For directional couplers, the beat length is determined by the propagation constant splitting Δβ between the symmetric and asymmetric supermodes. The symmetric eigenmode is concentrated near the center of the waveguide, while the asymmetric eigenmode has zero intensity in the center. That is why a very narrow gap of low refractive index material increases the coupling rate and in turn reduces the required length of the coupling region. On the other hand, directional couplers that are very compact will lose significant energy to mode conversion loss because the index profile changes rapidly, nonadiabatically. The extent of this inefficiency is directly linked to the beat length, creating a tradeoff between compactness, efficiency, and strong coupling. TMI couplers are very similar to directional couplers, except without the center gap to increase mode splitting. For this reason, they are not very compact in the straight geometry. Because of the relative freedom in fabrication bent TMI couplers with odd but compact shapes have recently been proposed [13].

Coupling always includes some conversion between different guided modes. If that conversion occurs non-adiabatically, energy will transfer into other modes, some of which may not be guided. In the case of directional couplers, that means the waveguides must transition between decoupled and coupled regions over a long distance, or loss will occur. Traditionally, directional couplers have been viewed as the simple low-loss, controllable coupling ratio device; however, the adiabatic requirement vastly complicates microring design situation by creating a fundamental tradeoff between quality and compactness. Rings with very small mode confinement and quality factor can only be very weakly evanescently coupled to bus waveguides. For high-Q filtering applications, this is not a large issue; however, this tradeoff poses a potential problem for high-bandwidth nonlinear signal processing. This issue was studied in detail by [14].

Mode conversion in MMI couplers is very different in that it is discrete. Input waveguides immediately enter a multi-mode section instead of continuously approaching one another. The input field distribution can be decomposed into the eigenmodes of the coupling region, though not all of these modes are guided. The guided eigenmodes form a subspace that is lossless. This non-adiabatic conversion from single to multi-mode can therefore be efficient only if the component of the input field orthogonal to the guided eigenbasis is minimized. Loss minimization in MMI couplers thus becomes a process of lateral wavefunction engineering, which can come into play during low level waveguide design. There are many techniques under investigation for improving the mode matching properties including tapered access waveguides [15], modified coupling region waveguide cross-sections [16], or processes for countering the mode offset in bent waveguides [17].

In practice, the fabrication of directional couplers is the most difficult, since two waveguides must be very near one another to experience a significant coupling rate. For example, to obtain a strong enough coupling rate for compact devices, photonic nanowires might need to be 100–150 nm apart. If they are etched to a depth of 240 nm, this gap feature has a high contrast ratio. Depending on the fabrication process, non-idealities in the coupler geometry can be introduced in the acute angle formed by the incoming and outgoing waveguide pairs. It can become partially filled in by unetched silicon or a high surface tension liquid. Fabrication of MMI devices is the least demanding of the coupler geometries due to low contrast ratio.

#### 3.1.3  Interferometers

Interferometers couple light from a single source into two paths before recombining them. Since coupling is sensitive to phase, optical path length differences between the arms are reflected in output powers. In converting phase information to amplitude information, interferometers are key tools in sensing small changes in refractive index. The Mach–Zehnder Interferometer (MZI) is the most common interferometer used in integrated circuits, particularly for electrical modulation of optical signals. MZI modulator devices will be covered in Section 7.3. Let us now analyze the basic theory in a simplified MZI pictured in Fig. 3.3.

Figure 3.3   A Mach–Zehnder interferometer with two 50:50 directional couplers that split power evenly. The two arms have optical path lengths of n 1 L 1 and n 2 L 2, respectively. The output power coupling depends on the optical path length difference of the arms.

The effect of a single directional coupler can be described by a matrix with particular symmetries:

3.2()
where r is called the amplitude reflection coefficient and t is the amplitude transmission coefficient. This terminology is inhereted from semi-reflective mirrors. In a waveguide coupler, t described the field amplitude that is coupled across the evanescent coupler into the other waveguide; i is the square root of –1, which indicates that a π/2 phase shift is effectively acquired by fields that cross the coupler. Due to power conservation in a lossless coupler, there is a relationship between t and r so that t 2 + r 2 = 1. For the purposes of this MZI example, we set The waveguides between couplers can also be described by a matrix. This matrix is diagonal because there is no interaction between the waveguides. Concatenating these three transmission matrices results in the expression of a MZI:
3.3()
where k is the propagation wavenumber of the optical mode. The phase factor exp(−ikn 1 L 1) has been removed from the second matrix. Simplifying this expression results in another matrix relating all four ports, but let us focus just on the amplitude and power transfer function from input port 1 to output 1.
3.4()
3.5()

The power transfer function depends on the cosine of the phase difference between the two arms. Since loss was not modeled, the output from port 2 would be the complement of that of port 1. It can be seen from this expression that interferometers can be useful for detecting minute effects. To reach a π phase shift required to go from 0% to 100% transmission, for example, requires an effective length change of wavelength π/2, which is much smaller than the arm length L.

#### 3.1.4  Modulators

In an optical communication system, it is necessary to first convert the analog electric waveform into some property of an optical wave in such a way that it can be recovered by an optoelectronic detector, thus making it suitable to transmission through optical fibers and waveguides. This is called modulating the signal onto an optical carrier. A carrier wave is a pure, continuous optical wave with a unique wavelength. The Fourier spectrum of the modulated signal is centered around the carrier wavelength with a spectral width of at least as much as the modulating signal. The modulation changes some property of the wave—usually its amplitude, phase, or frequency—to encode the data being transmitted. There are three main strategies for modulation:

• Amplitude modulation changes the power/amplitude of the carrier wave (originally a sine wave) to be reflective of the value of the input signal.
• Frequency modulation occurs when the frequency of the carrier wave is changed in proportion to the value of the input signal.
• Phase modulation is a type of modulation where the phase of the carrier signal is modulated according to changes in input signal value.

Devices for inducing a voltage-dependent index change in a waveguide are discussed in Section 7.3. These devices can be used as phase modulators. A common use of Mach–Zehnder interferometers is as amplitude modulators. By loading the MZI arms in Eq. (3.4) with a phase modulator, the Mach–Zehnder modulator acts as an amplitude or power envelope modulator.

Figure 3.4   Modulation of an input signal. A carrier wave, in this case an simple sine wave, is modulated by a RF signal. The output shown is a modulated carrier wave.

#### 3.1.5  Multiplexing

As one can imagine, it would be highly inefficient to have one fiber dedicated for each optical communication channel. Multiplexing takes multiple inputs, combines them into one single signal for transmission, and then recovers the original signals at the receiving end. Multiplexing allows for signals to be transmitted much more densely, thereby greatly decreasing transmission costs. When a channel is multiplexed, its large bandwidth is broken down and shared between multiple virtual channels with lower bandwidths.

One of the most common multiplexing techniques used in optics is wavelength division multiplexing (WDM). Different wavelengths of light—or different colors—can easily be combined and then split once again. WDM multiplexes different signals onto different carrier wavelengths. Filters for certain wavelengths are inserted before receivers, so that they can direct the wavelength carriers to the correct receiver. This method takes advantage of the full transmission bandwidth window of optical fibers or waveguides. WDM also adds flexibility into communications systems. Data channels can be injected or extracted from the transmission path at any location by using add-drop multiplexers. These multiplexers make it possible to reconfigure the system to provide data connections for a large number of send and receive nodes.

The second major type of multiplexing used in optics and electronics is time division multiplexing (TDM), where separation is based not on wavelength but on difference in arrival time. In TDM, different signal streams are interleaved in the time domain of a single carrier wave to form a high speed compact signal. At the receiver end, an ultra-fast clock and data recovery extracts the data. This technique differs from WDM, requiring very fast serializers/deserializers to function effectively. TDM is heavily used in fiber optic telecommunication links and metropolitan networks.

Figure 3.5   Wavelength division multiplexing, where different signals are modulated onto carriers of different wavelengths.

Figure 3.6   Time division multiplexing, where multiplexing is based on difference in arrival time.

#### Optical Absorption

In its essence, optical absorption is a process during which electrons absorb energy in the form of light and transfer this energy to an electrical circuit. This conversion is fundamental to the operation of photodetectors.

Electrons absorb incident light in the form of photons, whose energy is given by E = . Given the energy band structure of the material, some electrons transition to the conduction bands from the valence bands, creating a free electron-hole pair.

When a voltage is applied across the ends of a semiconductor device, this sets up an electric field within it. Free e / h pairs then drift under the influence of the electric field, in what is observed as current. The current is proportional to the magnitude of the power of the incident light:

3.6()
where R is the constant of proportionality known as the responsivity, and has a dependence on the frequency v.

#### PN Junctions

The effects of optical absorption are particularly interesting in P-N junctions. P-N junctions consist of two oppositely doped semi-conductor materials inside a single semiconductor crystal. When the p- and n- type semiconductors are joined in a PN junction, the high concentration of electrons from the n-type semiconductor diffuses to fill holes in the p-type semiconductor, leaving positively charged ions in the n-region and negatively charged ions in the p-region. Together, this charged region consisting of a high density of positively charged ions at one end and negatively charged ions at the other is called the depletion region (Fig. 3.7). As a result of the charged ions, there is a potential difference between the ends of the space charge region, which creates an electric field in the region directed from the positively charged ions to the negatively charged ones.

#### Reverse Biasing

A PN junction is said to be in reverse bias when its p-side is connected to the negative terminal of a voltage source and the n-side is connected to the positive terminal. Under the influence of the external voltage, conduction band electrons on the n-side are pullled towards the positive terminal, with holes on the p-side simultaneously moving towards the negative terminal, widening the depletion region, as observed in Fig. 3.7. This widening increases the magnitude of the internal potential difference of the depletion region, strengthening the electric field across it.

Figure 3.7   Reverse biased PN junction converting photon absorption to electrical current. Republished with permission of John Wiley and Sons, Inc., from Fiber-Optic Communication Systems (2002) 3rd ed. by G. P. Agrawal Ref. [18]; permission conveyed through Copyright Clearance Center, Inc.

#### 3.2.1  Photodiodes

When light is incident on one side of such a reverse biased PN junction, e/h pairs are created through optical absorption. The free carriers produced in the depletion region are accelerated by the strong built-in E field, with the electrons and holes being swept to opposite sides of the region. This flow of charge carriers produces current. Since this current is proportional to the power of the incident light, measuring it can give us an idea of the incident optical power. This is the basic principle of p-n photodiodes.

#### Diode Metrics

For a photodetector, bandwidth is a measure of how fast it responds to variations in incident optical power. This rate of response can be inferred from a metric called rise time, which is defined as the time it takes for the current output to increase from 10% to 90% of its final value when the optical power of the incident light is instantaneously changed.

Mathematically, the rise time of a photodetector is given by:

3.7()
where, τRC is the RC time constant of the equivalent circuit, and τ tr is the transit time. The transit time accounts for the time it takes for the carriers generated through optical absorption to be collected and register as current. Here, it is noteworthy that reducing W, the width of the depletion region will reduce τtr , as the carriers have to traverse a shorter distance to be collected. From the metrics above, it is intuitive that the bandwidth would be inversely related to the rise time, and therefore, to the sum of the transit time and the RC time constant. This intuition is materialized in the following definition of a photodetector’s bandwidth, ∆ f:
3.8()

The bandwidths of p-n photodiodes are often limited by the transit time, τtr , since τRC can be engineered to a minimum value by minimizing parasitic capacitance and resistance. For a p-n photodiode where W is the width of the depletion region, and vd is the drift velocity, the transit time is given by:

3.9()

Thus, in order to maximize bandwidth, W and vd would need to be optimized. W depends on the doping concentrations of the two semiconductors and can be controlled to an extent through them. vd , on the other hand, can be increased by increasing the applied voltage, which accelerates the free carriers. However, vd can only be increased up to a point, called the saturation velocity, which is the maximum vd that a free carrier can attain.

#### Diffusion Current

So far, the photodetector principles discussed only account for carriers produced in the depletion region that drift under the influence of the electric field. However, optical absorption also occurs outside of the depletion region which results in a diffusive component to the photocurrent. Electrons generated on the p side and holes generated on the n side have to diffuse to the boundary of the depletion region before they can drift under the influence of the built-in field. Diffusion is an inherently slow process and is the bottleneck in the response time of a photodetector. This also distorts the temporal response of a photodetector, as shown in Fig. 3.8.

Figure 3.8   Diffusion current contribution to distortions of detected signal. Republished with permission of John Wiley and Sons, Inc., from Fiber-Optic Communication Systems (2002) 3rd ed. by G. P. Agrawal Ref. [18]; permission conveyed through Copyright Clearance Center, Inc.

In order to minimize the effects of diffusion, we can reduce the widths of the p and n regions, and increase the width of the space-charge region, W. When this happens, naturally, most light is incident in the depletion region, and the optical absorption taking place outside of it is minimized.

#### PIN Detectors

One way to increase the effective width of the depletion region is to insert a piece of intrinsic semiconductor material between the p and n semiconductors, to form a structure called a p-i-n photodiode. Under reverse bias, most voltage drops across the intrinsic region due to the resistance offered by its intrinsic nature. This leads to a strong electric field being set up in the i- region, effectively extending the depletion region, as shown in Fig. 3.9.

Figure 3.9   PIN detector for minimizing diffusion current distortions. Republished with permission of John Wiley and Sons, Inc., from Fiber-Optic Communication Systems (2002) 3rd ed. by G. P. Agrawal Ref. [18]; permission conveyed through Copyright Clearance Center, Inc.

Extending the depletion in this manner leads to most of the optical power being absorbed in the i- region of the semiconductor, as a result of which current from carrier drift dominates that from carrier diffusion. Their ability to reduce diffusion current makes PIN photodetectors preferable to those consisting of simply PN junctions.

#### 3.2.2  Detection Noise

Noise occurs when a detector generates currents that are not caused by the intended optical signal and plays an important role in receiver system design. The two kinds of noise that are fundamental to receivers are shot noise and thermal noise. The presence of noise naturally affects the signal-to-noise ratio (SNR) of the photodetector, as is mathematically derived below.

The origins of shot noise lie in the fact that an electric current consists of a stream of a discrete number of electrons generated probabilistically at random times. Thus, photodiode current generated when constant optical power is incident is represented as a fluctuating current, is over the average current, Ip , so

3.10()
which introduces an element of randomness in the current that is produced at a photodiode in response to a constant optical signal. Mathematically, shot noise follows Poisson statistics, and its auto-correlation function is related to its spectral density by the Wiener-Khinchin theorem:
3.11()
Here, S(f) is the spectral density of shot noise, which is modeled as white noise (i.e., constant spectrum) and is therefore given by S(f) = qIp between –∆ f and +∆ f, where q is the electron charge. Then, the noise variance, or noise power due to shot noise is given by:
3.12()
where ∆ f is the effective noise bandwidth of the receiver, and the dark current, Id , is the leakage current that flows when no light is incident.

At any finite temperature, electrons are constantly in random motion. This motion in a resistor is observed as a fluctuating current in a resistor. In a load resistor such as one at the front end of a receiver, this fluctuation is observed as noise, which is called thermal, Johnston, or Nyquist noise. Mathematically, this is treated as a stationary Gaussian random process which is white up to frequencies as high as 1 THz. Its two-sided spectral density is given by:

3.13()
where kB is the Boltzmann constant, and RL is the load resistor. As above, the noise variance due to thermal noise can be calculated by integrating its spectral density over all frequencies:
3.14()

Thermal noise ocuring in other electrical components in a receiver, such as amplifiers, invariably add noise to the system. This is accounted for using the following formula:

3.15()
where Fn represents the factor by which thermal noise is changed due to presence of resistors in other electrical components.

Together, these two types of noise contribute to the total noise as:

3.16()
where variances from the two have been linearly added since the two have approximately Gaussian distributions.

Signal to noise ratio (SNR) can be defined as the ratio of the signal power to noise power. In photodiodes, this quantity is naturally affected by the presence of the two types of fundamental noise discussed above, shot noise and thermal noise.

3.17()

Using Ip = RPin , the SNR of a detected signal is given by:

3.18()

Detector SNR is a key point of analysis for energy use in photonic links since optical signals are always ultimately detected. Linear optical devices, such as filters, and transmission modulators, such as MZI modulators, have no theoretical minimum optical power needed to perform their functions. Their impact on noise and signal power is therefore quantitatively understood in regard to the detected SNR. Supposing the introduction of an optical device or effect causes power attenuation. The performance of this device can be stated as a link power penalty, or the additional optical power required to recover the SNR prior to the device’s introduction. From the expression for detector SNR, we can see that there are two primary regimes of operation. At high average photocurrent, RPin , the system is typically in a shot noise-limited regime, in which SNR increases linearly with average photocurrent. For low received photocurrent, thermal noise limits SNR, which increases quadratically with average photocurrent.

#### 3.3  Optical Resonators

An optical cavity is a volume that confines optical modes in all three dimensions. Microcavities can have transmission properties that vary appreciably over a narrow range of wavelengths, making them an important fundamental component for photonic signal processing. The most basic optical resonator is called a Fabry-Perot (FP) etalon. It consists of two partially reflecting mirrors separated by some effective optical length: the product of the physical length and cavity refractive index. A Gires-Tournois cavity (Fig. 3.10(a)) is a special case of the Fabry-Perot where one of the mirrors is 100% reflecting. By replacing the partially reflecting mirror with an evanescent coupler and the straight cavity by a ring, the Gires-Tournois cavity becomes a microring resonator (MRR) (Fig. 3.10(b)). An MRR is much easier to integrate on chip than the literal two mirror cavity, yet its behavior is completely analogous.

Figure 3.10   Two types of optical cavity that exhibit the all-pass characteristic. The steady state intensity of the outgoing field equals that of the incoming field, but effective phase shift and circulating intensity buildup is wavelength dependent

#### 3.3.1  Microring Resonator Analysis

The circulating mode in an MRR is at the root of some of the unique properties exhibited by microrings. Heebner, et al. [2] provide an excellent resource on microring resonators. In the MRR, there are three optical fields: input, output, and circulating. The input-output relationships of an optical cavity, like many linear optical components, are specified by a scattering matrix. The scattering matrix is complex-valued and describes the linear relationship between incoming and outgoing waves. The complex angle of the scattering matrix of a cavity as well as the amplitude of the circulating wave depend strongly on the wavelength of incoming light. If the optical path length is an integer multiple of the wavelength, successive passes will interfere constructively, leading to a substantial resonant power buildup inside the cavity. On the other hand, half integer multiples of the wavelength interfere destructively and do not build up significantly within the cavity.

Taking the coupler Eq. (3.2) and adopting the port labeling shown in Fig. 3.10(b), this coupler can be described by

3.19()

is the power ratio that remains in the same waveguide, and τ = t 2 is the evanescent power coupling ratio. Supposing power is conserved in the coupler, t 2 + r 2 = 1. The relation between field 4 and 3 is
3.20()
where
3.21()
and L is the distance around the ring, a is the ratio of power remaining after one round trip of propagation losses, n is the effective index of refraction, and λ = 2πc/ω is the vacuum wavelength.

#### Power Buildup and Effective Phase Shift

Equations (3.19) and (3.20) can be solved by summing the contribution of each circulating field. Because many circulating field components can interfere constructively, the circulating optical intensity can exceed the input intensity by more than an order of magnitude. The ratio is called the intensity buildup ℬ:

3.22()
3.23()

The complex transmission between the input (E 1) and output (E 2) fields can also be expressed.

3.24()

There are several features of this equation that grant key intuition into the behavior of coupled MRRs. The magnitude-squared of the complex transmission will be one as a approaches one, which should make sense because there is only one way for power to leave the system. The power transmission is minimum on resonance (ϕ = 0), where the buildup factor is maximum. The absorption depth provides an easily measurable quantity for experimentally determining a and r. On resonance, the net effective phase transfer Φ is very sensitive to small detunings in ϕ. These features are stated mathematically by

3.25()
and
3.26()

Equations (3.23) and (3.26) are plotted in Figs. 3.11(b) and 3.11(a), respectively, for an example MRR with r = 0.9 and a = 0.99. Notice that the circulating field is about 19 times more intense than the input power at perfect resonance. Also at this resonant point, the slope of the phase transfer function is much greater than 1, meaning it is very sensitive to changes in the single-pass phase accumulation.

Figure 3.11   Linear microring transfer functions as a function of the detuning ϕ, the phase acquired by a single-pass. ϕ is linearly related to applied optical wavelength, so the spectral response of both values is a scaled version of the same plots. r = 0.9, a = 0.99

It is worth summarizing qualitatively the important attributes of microring behavior. First, the variable ϕ determines resonance conditions, and it depends on both λ and n. That means the MRR is wavelength selective; also, its resonance spectrum will change if n is altered by some means. Second, the buildup factor

, a function of ϕ, is the ratio of circulating to incident power. The maximum of can be much greater than 1, with higher heights witnessed for larger values of self-coupling r. Third, the complex input-output transfer of the MRR usually has magnitude near unity, but its complex argument Φ depends strongly on ϕ. This dependence has a slope much greater than one near resonance that becomes close to zero away from resonance.

#### Finesse and Quality Factor

Finesse and quality factor of an MRR are convenient figures of merit that will allow later discussion of scaling laws and performance comparisons. The transmission or reflection spectrum of an optical resonator is a frequency comb as in Fig. 3.11(b). The spacing between transmission peaks or valleys is called the free spectral range (FSR). In terms of single-pass phase, FSR is always equal to 2π, and this converts to frequency and wavelength according to Eq. 3.21. The width of those peaks is typically stated as the full-width at half depth (FWHD). Half depth is simply halfway between the maximum and minimum of the spectrum’s range.

3.27()

The finesse

of the resonator is then defined as the ratio between FSR and FWHD.
3.28()

Finesse is an important figure of merit for optical cavities, as it is in essence the selectivity of that cavity as a filter. In terms of physical significance, it is roughly 2π times the expected value of round trips taken by a photon in the cavity before leaving it. An MRR thus increases the effective optical path experienced by a light wave by a factor proportional to

. When some detuning is applied to the cavity, it is effectively applied many times to the circulating lightwave. As seen in Fig. 3.12, small detunings near resonance are greatly amplified in the overall phase shift with a gain proportional to . This increase in sensitivity can be quite extreme, which explains the wide interest in microrings for biosensing and even single atom sensing [19].

Figure 3.12   The MRR effective phase shift as a function of single-pass phase over a range of finesse values. A completely decoupled ring has finesse of π (straight line). As finesse increases, a smaller range of single-pass detuning is required to span an effective range of π (dotted lines).

The finesse is closely related to the quality (Q) factor of the cavity, which is the sharpness of a resonance relative to its center frequency. Q factor is a standard oscillator concept formally defined as the quotient of stored energy to energy lost per oscillation cycle. This is indeed a close relation to the finesse, 2π times the mean number of round trips made by light in the cavity. The quality factor of an MRR is expressed

3.29()
where m is the harmonic order of the resonance of interest. m is also the number of wavelengths that fit around the ring’s circumference:
3.30()

The definition of quality factor can alternately be stated as the number of wave cycles before the amplitude is depleted to

of its original value. We are concerned with ultrafast capabilities of these devices, so the response bandwidth is an important property.
3.31()
3.32()
where B is bandwidth or bit-rate.

#### Round Trip Loss

The total loss experienced in one round trip of the ring, has several origins such as material absorption, bending loss, and diffuse scattering due to waveguide edge roughness. The distributed loss rate a accounts for all of these types of loss mechanisms.

There is another type of loss in the ring-coupler system caused by coupler nonideality. In a realistic directional coupler, two waveguides are brought from a large, noninteracting separation distance into very close proximity. In this close region, the asymmetric and symmetric normal modes of the system propagate at slightly different phase velocities, which results in power transfer between the two waveguides. If the waves are brought into interacting proximity very slowly, the eigenstates of the coupled oscillator system change adiabatically. This is typically impossible for microresonator coupling because the adiabatic transition takes a lot of unnecessary space. When this change is not adiabatic, some energy is lost from the coupler, which is referred to as mode conversion loss. Mode conversion losses in microresonators are examined in detail by [14] and examined more closely for our purposes in Section 3.1.2. To account for imperfect coupler efficiency

, Eqs. (3.23) and (3.26) can be rewritten
3.33()
3.34()

#### 3.4  Lasers

Lasers have revolutionized science, sensing, and communication since their invention. Many different forms of laser have been created. In general, they consist of an optical gain medium inside of a cavity. The cavity creates feedback that is phase dependent, thus favoring one or a small number of discrete wavelengths. Optical gain takes place through a special light-matter process called stimulated emission, in which electronic energy is converted into light with the same phase, wavelength, and direction as incoming light. If enough gain is present so that the cavity round trip gain exceeds unity, then an extremely narrow set of wavelengths will take over and “lase.”

Laser light has several important properties that are very different from thermal light. Firstly, laser radiation is spectrally pure; in other words, it has a narrow spectral linewidth. Secondly, laser radiation can exhibit very high powers. Once the laser threshold is reached, a significant portion of further pumping power will be converted to optical power at the lasing wavelength. This is different than filtered thermal light, which can have a narrow linewidth, but wastes optical power in proportion to the narrowness of the spectrum. Thirdly, laser radiation is directional, which has been experienced by anyone using a laser pointer. In the context of integrated waveguides, directionality does not apply in the same way; however, this property manifests as laser light occupying a single spatial mode. This is a very important property for many integrated devices, for example couplers, where device function relies on inputs and outputs being in a single mode.

In this section, we will give a high level overview of light-matter interaction. Stimulated emission is the key effect for optical gain, but all types of interaction are important to consider when designing lasers. We will then discuss some integrated platforms well suited to electrically pumped lasers and, finally, perform a dynamical system analysis of the most basic laser, which will serve as a warmup for advanced laser dynamics studied in further chapters.

#### 3.4.1  Light-Matter Interaction

Photonic devices can be separated into active and passive categories based on the various mechanisms of how materials interact with light. Insulators, such as bulk SiO2 and silicon nitride, are characterized by fully occupied valence energy levels and act as dielectrics. In a dielectric, an applied electric field induces a material polarization and subsequent net displacement field. The degree of this polarization results in different refractive indices of different materials. Dielectrics can absorb light as heat if the induced material polarization results in atomic displacements that correspond to phononic (i.e., vibrational) modes, for example, microwave radiation absorbed into water molecule vibrations. Unlike insulators, metals have many unoccupied valence levels, allowing the free flow of electrons. The resulting high conductivity means that electric fields do not propagate as waves in metal, except under very special circumstances.

The third classification of solids, semiconductors, are characterized by an insulator-like filled valence band but a relatively nearby unoccupied energy band called the conduction band, their energy difference described by the energy bandgap. Photons with energies less than a bandgap do not interact with the electronic state of the semiconductor, so that the semiconductor behaves as a simple dielectric. Photons with enough energy, on the other hand, can be absorbed as electronic energy. This absorption by charge carrier excitation is different from vibrational absorption because 1) it alters the electronic properties of the material in a way that can be easily measured, and 2) these excited carriers can in turn re-emit their energy as light. In particular, a photon that interacts with an already excited electron can stimulate the emission of a second photon with identical wavelength, direction, and polarization, the process underlying optical amplifiers and lasers. A PN diode within an optically active semiconductor injects external carriers into excited states, allowing the creation of electrically-pumped light emitting diodes (LEDs) and, when inside an optical cavity, laser diodes.

Electrons in atoms are confined to discrete energy levels due to quantum confinement effects. Atomic structure, as well as the structure of organized atoms in solids, gives rise to spatially varying electric potentials which confine electrons to certain energies. These energy levels are generally referred to as bands, each of which can hold a certain number of electrons. All energy bands below the Fermi level are filled with electrons according to the Pauli exclusion principle. We will restrict ourselves to discussing the highest occupied energy level in the atom, the valence band, and the first unoccupied level right above that, the conduction band. These bands are of interest to us because they are the easiest to perturb with external electric fields, and because they are the primary site of the electron-photon interaction that make the generation of light possible. Moreover, the conduction bands of most materials have the capacity to hold many more electrons than can generally be found in equilibrium at normal temperatures.

#### Spontaneous Emission and Absorption

The simplest electron-photon interactions are spontaneous emission and absorption. Spontaneous emission occurs when an electron moves from a more energetic energy level to a less energetic energy level, and in the process releases the energy difference between these levels in the form of a photon. Spontaneous emission is a random process both temporally and in the phase and direction of the emitted radiation. Absorption occurs when a photon incident on an atom causes an electron to move from a less energetic band to a more energetic band. It is important to note that absorption can only occur when the incident photon energy matches the energy difference associated with the two bands the electron moves between.

Not all electron energy changes are dependent on interactions with photons. Electron-phonon processes involve the transfer of energy from an electron to an acoustic or thermal vibration of an atom or molecule. Nonradiative re-combination is a common electron-phonon process that, in semiconductors, generally involves defect states. These localized energy bands originate from electric field perturbations caused by defects in semiconductor lattices, and allow for an electron’s kinetic energy to be dissipated into a thermal or vibrational state associated with the defect. This is referred to as thermalization.

#### Stimulated Emission

Stimulated emission occurs when an incident photon causes an excited electron to drop to a less energetic band, emitting a photon in the process. Stimulated emission is important to an understanding of laser dynamics because it results in the generation of a photon with identical phase, frequency, and direction to the incident photon. As in electron-photon absorption, the electromagnetic field of the incident photon resonates with the energy difference between two possible bands that an electron can occupy. This resonance causes the electron to vibrate with the same frequency as the incident radiation. Stimulated emission occurs when the electron which the photon interacts with is already in an excited state. This electron is perturbed by the incident electric field, causing it to transition between the energy bands. Since the downward transition of the electron does not draw energy from the incident electric field, the incident photon is unchanged by the interaction, and the extra energy associated with the downward band transition adds to the overall amplitude of the electric field in the form of an additional photon. Since the recombining electron has been perturbed by the incident electric field, the generated electromagnetic field oscillates with the same frequency, phase, polarization, and direction as the incident photon!

Figure 3.13   Light-matter interaction with valence (Ev ) and conduction (Ec ) electron energy levels. (a) Absorption: an incident photon raises electron energy. (b) Spontaneous emission: an excited electron falls to the valence energy and emits a photon. (c) Stimulated emission: an incident photon triggers an excited electron to release a second photon with identical phase, frequency, and momentum.

#### Direct and Indirect Bandgap Semiconductors

All semiconductors are subject to spontaneous emission, absorption, and non-radiative recombination, but only special types are capable of providing optical gain. Indirect bandgap semiconductors have a valence band maximum energy and conduction band minimum energy appearing at different momentum states (Fig. 3.14). Both energy and momentum must be conserved in a transition, and photons carry relatively little momentum. This momentum must come from another particle, a phonon, in indirect gap semiconductors. The three particle electron-photon-phonon interaction is less likely than the two particle electron-photon interaction, thereby making spontaneous and stimulated emission much stronger in direct gap materials (relative to competing nonradiative recombinations). Indirect bandgap materials, including silicon and germanium, are therefore unsuitable for producing practical optical amplifiers or lasers.

Figure 3.14   Band diagrams of (a) silicon and (b) gallium arsenide, showing differences between indirect and direct bandgaps. Black lines represent edges of allowed electron and hole states in terms of energy, E, and momentum, k. Gray regions are band gaps, energies where no states are present. Vertical red arrows represent the minimum energy of the conduction to valence transition, and the horizontal blue dashed line is the momentum of the transition for indirect bandgap materials. Adapted and reproduced from Semiconductors: Group IV Elements and III-V Compounds (1991) by O. Madelung Ref. [20]. With permission of Springer-Verlag.

With these things in mind, optical devices can be classified as active if they exchange or convert photonic energy with electronic energy, and passive otherwise, although passive devices always have dielectric properties and nonideal absorptive processes. Naturally, a combination of active and passive devices is desirable for integration of complete photonic systems, consisting of active transmitters, passive routing waveguides, and active receivers.

#### Population Inversion

We have seen that photons incident on direct bandgap materials result either in stimulated emission or absorption, depending on whether the interacting electron is in an excited state or not. As stimulated emission requires an electron to drop from the conduction to the valence band, the likelihood, or rate, at which stimulated emission occurs is proportional to the conduction band electron density. Inversely, absorption requires an electron start in the valence band, and is therefore proportional to the valence band electron density. In a bulk semiconductor with many electrons, this means the net emission/absorption effect depends on the ratio of excited electrons to total electrons. This is often referred to as the carrier density, since only conduction electrons can flow freely to carry charged current.

Population inversion refers to the situation where more than half of the outer electrons are in an excited state and is a necessary condition for a bulk material to amplify incoming light through net stimulated emission. At first, it seems such a state can be achieved by irradiating the material with photons which will be absorbed, raising the carrier concentration. However, since increasing the number of conduction band carriers decreases the rate of absorption, it is impossible to excite electrons in the material farther than the point where it becomes transparent (i.e., equal density of conduction and valence electrons). In order to achieve optical gain, the semiconductor must be pumped by some other effect. Many types of lasers are described by three-level systems, optically pumped by a shorter wavelength (higher energy photons). Electrons pumped to the third energy level quickly relax to the second, creating a buildup of carriers that will provide gain to the lasing wavelength. In most photonic integration contexts, however, it is advantageous to use electrical pumping. Population inversion can be induced with electronic energy in a PN semiconductor junction.

#### Semiconductor Laser Diodes

While optical pumping of media is a simple way to alter the carrier concentrations, it is generally a cumbersome method to use in a real world device. Instead, semiconductor junctions are often used to affect the carrier concentrations of optical material through electrical pumping. Electrons can be injected in one side of the junction while removing electrons from the other side, without affecting the junction directly. Constantly pumping the junctions this way creates a permanent population inversion at the junction. Such a pumping scheme requires direct bandgap semiconductors in order to facilitate direct stimulated emission. Electronically pumped devices often use multi-quantum well structures to maximize the overlap between excited electrons’ locations and the optical mode confinement. Electrical power input to these devices directly causes carrier transitions which lead to optical gain as described earlier. Chapter 6 will display different examples of electrically pumped lasers. One of the interesting advantages of semiconductor lasers is that they can be fabricated in a standard way on active substrates that will be in the near future compatible with current technology platforms. In the next section, we discuss one of these active platforms.

#### 3.4.2  III-V Platforms

Semiconductors consisting of combinations of group III (including Al, Ga, In) and group V (including P, As, Sb) elements are very important for integrated photonics, largely because of the flexibility by which they can be fabricated. This makes them an ideal material platform for the exploration of integrated lasers, which require simultaneous engineering of energy bandgaps, refractive index guiding, and electronic PN junctions. High quality III-V crystals can be grown epitaxially by vaporizing the elements and exposing this vapor to a growth substrate. Binary compounds (GaAs, InP, AlAs, and GaP) each have fixed properties; however, ternary (e.g., AlxGa1-xAs) and quaternary (e.g., InxGa1_xAsyP1_y) alloys can realize a large variety of material properties, such as bandgap and refractive index. Figure 3.15 shows the range of bandgaps and lattice constants possible with common alloys. Lattice matching is an important consideration during design, since mismatch can induce a strain or even growth defects. Growers obtain multiple layers with different energy bandgaps (thereby creating quantum wells), while maintaining the same lattice constant (e.g. AlxGa1-xAs in the horizontal line in Fig. 3.15). Epitaxial fabrication techniques offer precise control over these mole ratios x and y over many layers of growth with nanometer accuracy in layer thickness.

Figure 3.15   Lattice constants and bandgaps of III-V semiconductors. Lines correspond to ternary compounds: solid for direct bandgap, dashed for indirect bandgap. Areas correspond to quaternary compounds; for example, the gray area indicates possible alloys of InxGa1-xAsyP1-y. Group IV elements Ge and Si are shown as points. Republished with permission of John Wiley and Sons, Inc., from Fundamental of Photonics (1991) 1st ed. by B. E. A. Saleh and M. C. Teich Ref. [1]; permission conveyed through Copyright Clearance Center, Inc.

Often referred to as bandgap engineering, a III-V stack can be designed to be absorbing in some layers and transparent in others. Thin layers sandwiched between larger bandgap layers, called quantum wells, are also a key feature of III-V platforms for photonic applications. By trapping excited carriers in a localized energy potential dip, quantum well and multiquantum well (MQW) regions have exceedingly favorable optoelectronic gain and efficiency over bulk crystals. Additional elements from groups II and IV can also be introduced as dopants in the growth for making PN junctions and, thereby, electrically pumped lasers.

The substantial flexibility of III-V platforms make them a powerful research tool. Early demonstrations of PICs with active devices were in InP [21]. Refer to [22] for a more recent overview. III-V wafers can also be designed to make high-contrast, low-loss waveguides for high-performance passive devices [2325]. The downside of III-V platforms is that, despite their flexibility, epitaxial stacks are grown uniformly over a wafer, making it difficult to integrate both active devices and high-performance passives on a single chip. The optimal epitaxial stack design varies even among active devices. Nevertheless, some semiconductor foundries for III-V have created standardized PIC platforms [26, 27]. The passive waveguides on these platforms are weakly-guiding in order to interface more readily with active sections, but this also means that waveguide bends, couplers, and filters are very large and lossy compared to competing platforms.

#### 3.4.3  Laser Dynamics

To analyze the behavior of a simple laser, we start with a single cavity mode case with a single gain section. While there are many device and physical parameters that affect the relation between the gain carrier density and optical mode intensity, we find it is most instructive to start with an undimensionalized set of equations to first elucidate qualitative behaviors. A reduced, undimensionalized version of the Yamada equations [28, 29] describing a laser can be stated:

3.35()
3.36()
where G(t) models the gain carrier density, I(t) is the laser intensity, and A is the gain pumping parameter. τ G is the gain medium carrier lifetime, normalized to the cavity photon lifetime. The photon leakage term, normalized to 1 · I, accounts for light coupling out of the cavity, spontaneous emission, and scattering. The magnitude of I and G are normalized to simplify the set of equations, but are still proportional to photon quanta—the terms G(t)I(t) have to cancel each other when the two differential equations are added.

#### Fixed Point Analysis

Equations (3.35) and (3.36) have two dynamical variables and a single bias parameter: the gain pumping, A. They take the form of a coupled ordinary differential equation where

3.37()

Dynamical analysis begins by identifying fixed points or steady states. Fixed points are points in phase space,

, that do not change as time t increases. In other words, fixed points occur when . See a brief introduction to dynamical systems in Section 2.4 and footnote therein. We find the following set of steady state equations:
3.38()
3.39()

This system of equations has two solutions for all values of A, corresponding to the roots of the second equation. The first solution is ISS = 0, GSS = A, where there is no optical intensity. The other solution is GSS = 1, ISS = A – 1. Although solutions with negative ISS exist mathematically, they are non-physical because optical intensity cannot take on negative values.

#### Stability Analysis

Fixed points can come in different varieties depending on whether nearby states evolve towards or away from them. This is referred to as stability. For example, a pencil balancing on its point can be thought of as a dynamical system with a fixed point, but since this fixed point is unstable, any small perturbation will make the pencil fall down. To assess stability mathematically, the system is linearized around fixed points to consider only small perturbations. The first partial derivatives of

with respect to is called the Jacobian matrix, J .
3.40()

The eigenvalues of the Jacobian determine whether a fixed point is stable (i.e., attracting) or unstable (i.e., repulsive). To be stable, the real part of all eigenvalues must be less than zero. Substituting steady state values and solving for eigenvalues, λ,

3.41()
and
3.42()
3.43()

There is a stability change of both solutions at A = 1. This is obvious in the first solution, and can be seen in the second by setting A to 1, which results in a zero eigenvalue. This describes a transcritical bifurcation, which occurs at a parameter where two fixed points coincide and exchange stability.

The steady state behavior of G and I are shown in Fig. 3.16. There is a key change in behavior where the transcritical bifurcation occurs. This is called the lasing threshold, which is an important quantity for every laser. Below threshold, no optical intensity builds up through stimulated emission (although spontaneous emission was neglected in this analysis). The pump energy proportionally energizes the gain medium. Above threshold, the gain carrier concentration locks to its threshold value, and increases in pump power are 100% converted into optical intensity. This analysis was a basic starting point to enter into the rich area of laser dynamics, which will be a central topic of Chapters 5 and 6.

Figure 3.16   Steady state solutions of a simple laser vs. pumping parameter. Blue lines show gain variable, and red lines show intensity variable. Solid lines indicate stable fixed points, while dashed lines indicate unstable. Non-physical solutions < 0 are not shown.

The threshold of a real laser is determined by the rate of pumping required to counteract cavity round trip losses. One important consequence of the existence of a threshold is that traditional lasers can never approach perfect efficiency in converting electrical power into optical power, since they must continuously dissipate enough power to keep the gain population inverted in an excited state. The slope of the output optical power solution above threshold is referred to as the slope efficiency, usually stated in terms of Watts/Amp or a quantum efficiency (photons per electrons). Ref. [30] contains a thorough reference on the physical analysis and design of integrated lasers.

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