Abstract

A new semiconductor laser structure with digitally switchable wavelength is proposed. The device comprises two coupled c.avities with different optical path lengths, which form V-shaped branches with a reflective 2 × 2 half-wave optical coupler at the closed end. The reflective 2 × 2 coupler is designed to have a π-phase difference between cross-coupling and self-coupling so as to produce synchronous power transfer functions. High single-mode selectivity is achieved by optimizing the coupling coefficient. The switchable wavelength range is greatly increased by using Vernier effect. Using deep-etched trenches as partial reflectors, additional waveguide branch structures are used outside the laser cavities to form a complete Mach-Zehnder interferometer, allowing space switching, variable attenuation, or high speed modulation to be realized simultaneously. Detailed design principle and numerical results are presented.

©2008 Optical Society of America

1. Introduction

Wavelength switchable lasers are of great interest for both long-haul and metropolitan optical networks [1]. Besides their use for source sparing with the advantages of reduced inventory and cost, they are important components for reconfigurable optical add-drop multiplexing (ROADM) and open the possibility for new intelligent system architectures with more efficient and more flexible network management. High speed wavelength switching can also be used for wavelength based packet switching and optical CDMA.

Monolithically integrated wavelength switchable or tunable semiconductor lasers offer many advantages over external-cavity tunable lasers assembled from discrete components. They are compact, potentially low-cost, and likely more reliable as they contain no moving parts. A conventional monolithic tunable laser comprises a multi-electrode structure for wavelength tuning. A typical semiconductor tunable laser [2] consists of a distributed Bragg reflector (DBR), an active gain section, and a phase shift region. When the reflectivity peak wavelength of the DBR grating is tuned by injecting current, the phase shift region must be adjusted simultaneously in order to prevent the laser from hopping from one mode to another. Besides, the tuning range of such a laser is limited to a few nanometers due to the limitation of commonly achievable refractive index change in semiconductor materials.

More sophisticated tunable lasers with wider tuning ranges have been developed in the forms of sampled grating distributed Bragg reflector (SGDBR) laser [3–5], superstructure grating (SSG) DBR laser [6], grating-assisted codirectional coupler (GACC) laser [7], modulated grating Y-branch (MG-Y) laser [8] and digital supermode (DS) DBR laser [9]. In addition to the fabrication complexity involving non-uniform (or selective area) gratings and multiple epitaxial growths, they usually use at least three electrodes for wavelength tuning. Complex electronic circuits with multi-dimensional current control algorithms and look-up tables are required. Furthermore, external wavelength locking mechanism is needed in the packaged module to stabilize the wavelength at ITU grid. Such complexities reduce the fabrication yield and operational reliability, and increase the cost. These problems have been the major roadblocks to the wide deployment of tunable lasers in commercial systems.

A widely tunable or wavelength switchable laser can also be realized by using two serially coupled cavities of slightly different lengths. The coupled-cavity laser can be fabricated either by etching a groove inside a cleaved Fabry-Perot laser [10] or by using a cleaved-coupled-cavity structure [11]. While the theory for such a serially coupled cavity laser has been well established [12–15], it is difficult to realize optimal cross-coupling coefficient and phase simultaneously. Consequently, the mode selectivity of such a coupled-cavity laser is very poor, which results in very limited use in practice.

Coupled-cavity lasers have also been previously investigated in the form of a Y-laser [16–19]. The Y-laser has the advantage of fabrication simplicity without requiring gratings or deep trenches. However, the mode selectivity of the Y-laser is similar to serially coupled cavity lasers, which is insufficient for stable single-mode operation with good side mode suppression.

In this paper, we propose a new monolithically integrated half-wave V-coupled cavity semiconductor laser. It has the advantages of compactness and fabrication simplicity, and can achieve significantly improved single-mode selectivity while allowing the lasing wavelength to be switched over a wide range.

2. Device structure and operation principle

Figure 1 shows the schematic diagram of a V-coupled cavity laser placed within a Mach-Zehnder interferometer (MZI). The monolithically integrated semiconductor laser consists of two cavities, each comprising an optical waveguide bounded by deeply etched trenches at both ends. The two waveguides have slightly different optical lengths, and are disposed side-by-side on a substrate to form V-shaped waveguide branches with a predetermined cross-coupling at the closed end and no cross-coupling at the open end. The deep trench and the coupling region at the closed end form a reflective 2 × 2 optical coupler which is designed to be half-wave coupler with synchronous power transfer functions (see Section 4). The length of one of the cavities (i.e. the fixed gain cavity) is chosen so that its resonance frequency interval Δf matches the spacing of the operating frequency grid (e.g. 100GHz spacing as defined by ITU). An electrode is deposited on top of the waveguide to inject a fixed current to produce an optical gain. The second cavity (i.e. the channel selector cavity) has a slightly different optical length and thus slightly different resonance frequency interval Δf′. It has two sets of electrodes deposited on the waveguide, one for injecting a fixed current to produce an optical gain and the other for applying a variable current to change the effective index in order to switch the laser wavelength. The waveguide segment of variable index is disposed away from the coupling region so that its refractive index change does not affect the coupling coefficient between the two cavities.

Using deep-etched trenches as partially reflecting elements, additional waveguide branch structures can be designed outside the laser cavities to form a complete Mach-Zehnder interferometer that servers as a power combiner and space switch. The light emitted from the two waveguide branches at the open end is coupled to an output port by a 2 × 2 multiple mode interference (MMI) coupler (or a directional coupler). An electrode is deposited on the space switching section outside the cavity to adjust the phase and to switch the laser from one output port to another. Alternatively, the switch section can be designed and used as a variable optical attenuator for output power adjustment, or as a high-speed Mach-Zehnder modulator. On the other side of the laser, a tapered combiner can be used for power monitoring.

 figure: Fig. 1.

Fig. 1. Schematic diagram of a V-coupled cavity laser within a MZI

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In order for the device to operate as a digitally wavelength-switchable laser, the length of the fixed gain cavity is chosen so that its resonance frequency interval matches the spacing of the operating frequency grid, an example being the widely used frequency grid defined by ITU (e.g. spaced at 200GHz, 100GHz or 50GHz). The resonance frequency interval is determined by

Δf=c2ngL

where c is the light velocity in vacuum, ng the effective group refractive index of the waveguide, and L the length of the fixed gain cavity.

Similarly, the resonance frequency interval Δf′ of the channel selector cavity is

Δf=c2ngL=c2(naLa+nbLb)

where La and Lb are the lengths of the gain section and wavelength switching section, respectively, na and nb are the corresponding effective group refractive indices. L′=La+Lb is the total length, and n′g=(naLa+nbLb)/L′ is the average effective group refractive index of the channel selector cavity.

The resonance frequency interval Δf′ of the channel selector cavity is chosen to be slightly different than Δf so that only one resonant peak coincides with one of the resonant peaks of the fixed gain cavity over the material spectral gain window, as shown in Fig. 2. The distance between two aligned resonant peaks, which corresponds to the free spectral range (FSR) of the combined cavity, is determined by

Δfc=ΔfΔfΔfΔf

In order not to have two wavelengths lasing simultaneously, Δfc should generally be larger than the spectral width of the material gain window.

 figure: Fig. 2.

Fig. 2. Schematic diagram illustrating the relationships between the resonant frequency combs of the two cavities and the material gain spectrum.

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The frequency of the channel selector can be tuned by varying the effective index n′ or nb of the wavelength switching segment. The rate of the tuning is determined by

δff=δnn=δnbLbnbL

Since the laser frequency is determined by the resonant peak of the fixed gain cavity that coincides with a peak of the channel selector cavity, a shift of |Δf-Δf′| in the resonant peaks of the channel selector cavity results in a jump of a channel in the laser frequency. Therefore, the change of the laser frequency with the refractive index variation is amplified by a factor of Δf/|Δf-Δf′|, i.e.

δf=ΔfΔfΔfδf

This is the so-called Vernier effect which is also used in other structures such as SSG or SG DBR lasers. However, since the frequency interval of the reflectivity peaks of the SSG or SG structure is determined by the modulation period in the grating and usually at least 10 periods are required in each of the front and back reflectors, they requires a total device length typically at least 20 times larger than that of the V-coupled cavity laser if the same frequency grid is used. The increased tuning range without long and complex grating structures is one of the advantages of the proposed device. Consider an example in which Δf = 100GHz, and Δf′ = 90GHz, the range of the laser frequency variation is increased by a factor of 10 with respect to what can be achieved by the index variation directly. For this numerical example, assuming the effective group refractive index of the waveguide is 3.215, the lengths of the fixed gain cavity and the channel selector cavity are L = 466.24µm and L′ = 518.31µm, respectively. The device length is therefore comparable to conventional DFB or F-P lasers.

Due to the limitations of device length and the associated loss, a SSG or SG DBR laser typically has Δf larger than 600GHz. For common DWDM applications with ITU channel spacing in the range of 50~200GHz, digital wavelength switching is impossible and an external wavelength locking mechanism is required. This complexity is removed with the V-coupled cavity laser. By using etched trenches as reflecting mirrors, the cavity lengths can be accurately defined by photolithographic method. Deviations from the ITU grid due to material dispersion or fabrication errors can be compensated by slightly adjusting the injection current in the fixed gain cavity (with or without an extra electrode) or by temperature tuning.

It should be pointed out that mode hoping may occur if the bias current is changed directly to adjust the output power. Therefore, the laser is preferably operated at a fixed bias current. The power adjustment or intensity modulation needs to be done by an external variable optical attenuator or modulator (which may be integrated on the same chip as shown in Fig. 1).

3. Analysis of threshold conditions

An important difference between the V-coupled cavity laser of the present paper and the previously investigated Y- laser is that the light is partially coupled from one branch to the other without going through a common waveguide section. As we will see in Section 6, this allows an optimal amount of light to be coupled from one cavity to the other (i. e. cross-coupling), relative to the amount of light coupled back to the same cavity (i. e. self-coupling). As a result, much better single-mode selectivity can be achieved with the V-coupled cavity laser as compared with the Y-laser.

Referring again to Fig. 1, assume the amplitude reflectivities of the deep trenches on the closed end and open end are r1 and r2, respectively (for simplicity, the reflectivities at the open end of the two waveguide braches are assumed to be the same). The coupling between the waveguides occurs at the closed end. We denote the amplitude coupling coefficients from fixed gain cavity to channel selector cavity (cross-coupling), from fixed gain cavity back to fixed gain cavity (self-coupling), from channel selector cavity to fixed gain cavity (cross-coupling), and from channel selector cavity back to channel selector cavity (self-coupling), by C12, C11, C21 and C22, respectively. Note that the reflectivity of the trench is treated separately and is not included in the coupling coefficients.

For simplicity, in the following we will treat the two waveguide segments in the channel selector cavity as a uniform waveguide with an average effective refractive index of n′. To analyze the V-coupled cavity laser, we consider one of the cavities as the main laser cavity and treat the coupling effect of the other cavity as a modifying multiplication factor in the reflectivity of the mirror at the closed end of the main cavity. First, let us consider the fixed gain cavity as the main cavity. The effective reflectivity of the deep trench at the closed end for this cavity can be written as r2e=ηr2, where η is an effective reflection factor (in amplitude) taking into account the coupling effect of the channel selector cavity and is calculated by

η=C11+C21C12r1r2e2(g+ik)L(1+C22r1r2e2(g+ik)L+C222r12r22e4(g+ik)L+...)
=C11+C21C12r1r2e2(g+ik)L1C22r1r2e2(g+ik)L

The threshold condition can therefore be written as

r1ηr2e2(g+ik)L=1

Similarly, we can also consider the channel selector cavity comprising the bend waveguide as the main laser cavity. The effective reflectivity of the deep trench at the close end for this cavity can be written as r′2e=η′r2, where η′ is the effective reflection factor taking into account the coupling effect of the fixed gain cavity and is calculated by

η=C22+C21C12r1r2e2(g+ik)L(1+C11r1r2e2(g+ik)L+C211r12r22e4(g+ik)L+...)
=C22+C21C12r1r2e2(g+ik)L1C22r1r2e2(g+ik)L.

The threshold condition for the laser can then be written as

r1ηr2e2(g+ik)L=1

After some manipulation, it can be shown that Eqs. (7) and (9) are identical. They both lead to the following threshold condition of the V-coupled cavity laser:

C11r1r2e2(g+ik)L+C22r1r2e2(g+ik)L(C11C22C21C12)r12r22e2(g+ik)Le2(g+ik)L=1

This complex equation, which can be separated into two equations corresponding to the real and imaginary parts, determines the wavelengths of the lasing modes as well as their threshold gain coefficients. In the case of uncoupled cavities, i.e., C12 = C21 = 0, C11 = C22 = 1, we have η=η′ = 1, and Eqs. (7), (9), and (10) reduce to the threshold conditions of conventional Fabry-Perot cavities.

4. Design of the reflective optical coupler

 figure: Fig. 3.

Fig. 3. Unfolded reflective 2 × 2 coupler in the V-coupled cavities

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The design of the reflective coupler at the closed end of the V-coupled cavities is very important for achieving high single-mode selectivity of the laser. To analyze the coupler, we unfold the laser cavities with respect to the reflection plane at the closed end. The round trip propagation in the coupled laser cavities can be treated as a 2 × 2 four-port system, as shown in Fig. 3. We denote the initial electric field of the fixed gain cavity and the channel selector cavity at the open end as E1 and E2, respectively. After passing through the coupling section and a round trip propagation including two reflections at the trenches, the electric field E1′ and E2′ can be written as

E1=r1r2(C11E1e(g+ik)L+C21E2e(g+ik)L)e(g+ik)L
E2=r1r2(C12E1e(g+ik)L+C22E2e(g+ik)L)e(g+ik)L

Note that in the above equations we do not distinguish the propagation constants in the coupled and uncoupled regions. The actual differences are taken into account in the phases of the complex coupling coefficients. From the laser threshold conditions E 1′=E 1 and E 2′=E 2, we can obtain

(C11+βC21)r1r2e2(g+ik)L=1
(C22+C12β)r1r2e2(g+ik)L=1

with

β=E2e(g+ik)LE1e(g+ik)L

By eliminating β in Eqs. (13) and (14), we can obtain the same threshold condition as Eq. (10).

Consider the case of a symmetrical structure with C11 = C22 and C12 = C21. It will be shown in Section 5 that the highest single-mode selectivity is reached when the two cavities are symmetrically pumped with equal round trip gain, i.e. gL = gL′ if the cavity mirrors have the same reflectivities. The lowest threshold lasing mode occurs when both of the two cavities are resonant. In this case, from Eqs. (13) and (14), we can derive |β| = 1, i.e., the electric fields from the two waveguide branches have the same amplitudes in the coupling region.

Now let us examine more closely the coupling region of the optical coupler. For simplicity and without losing generality, the self-coupling coefficients C11 and C22 can be assumed to be of real positive values (any non-zero phase can be compensated by lengths L and L’, respectively), i.e. C11 = |C11| and C22 = |C22|. Assume the cross-coupling coefficients have a relative phase φ with respect to the self-coupling coefficients, i.e. C12 = |C12|e and C21 = |C21|eiφ. Consider the case where the powers in the two input waveguides at the entrance of the coupling region are equal with the total power normalized to 1. The output powers in the two waveguides at the exit of the coupling region can then be written as

P1=12[C112+C212+2C11C21cos(φ+ϕ)]
P2=12[C122+C222+2C12C22cos(φϕ)]

where ϕ is the relative phase of the input field at the entrance of the coupling region in the second waveguide with respect to that in the first waveguide.

In a conventional 2 × 2 optical coupler, the cross-coupling coefficient has a relative phase of π/2 with respect to the bar-coupling coefficient, i.e. φ = π/2. Therefore, the two output waveguides have complementary output powers when the relative phase of the two input fields changes. Figure 4(a) shows the typical curves of output power versus relative input phase for a 3-dB directional coupler or a 2 × 2 multi-mode interference (MMI) coupler. For an ideal coupler, the energy conservation rule leads to |C11|2 + |C21|2 = 1 and |C12|2 + |C22|2 = 1. Such a coupler (referred as quarter-wave optical coupler below) is commonly used in waveguide based Mach-Zehnder interferometers and optical switches. It is also used in the MMI coupler of Fig. 1 to realize the switch/combiner function outside the laser cavities.

 figure: Fig. 4.

Fig. 4. Power transfer functions of quarter-wave (a) and half-wave (b) optical couplers.

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For optimal operation of the V-coupled cavity laser, the reflective coupler at the closed end needs to have synchronous power transfer functions, in contrast to the complementary power transfer functions as in conventional quarter-wave couplers. The cross-coupling coefficients have a relative phase φ = (m = 0, ±1, ±2, …) with respect to the self-coupling coefficients. Figure 4(b) shows the ideal power transfer functions of such a coupler for the case m = 1 (which is the easiest to realize). When the input fields have opposite phases, the powers at both output ports reach the maximum simultaneously. When the input fields have the same phase, destructive interference occurs at both output ports and the energy is dissipated into radiative modes out of the waveguides. For such a coupler (referred as half-wave optical coupler hereafter), the energy conservation rule requires that the amplitudes of the coupling coefficients satisfy |C11| + |C21| = 1 and |C12| + |C22| = 1 for the ideal case when there is no excess loss. In the above example, we have used C11 = C22 = 0.755, and C12 = C21 = -0.245. Although we have |C11|2 + |C21|2 = |C12|2 + |C22|2 < 1, no energy is lost when the two input fields have opposite phases, as shown in Fig. 4(b). The half-wave optical coupler can be realized in the form of a three-waveguide coupler, i.e., by adding a third waveguide in the middle of a conventional 2 × 2 directional coupler. The middle waveguide forms a quarter-wave coupler with each of the adjacent input/output waveguides, resulting in a 180° coupling phase between the input/output waveguides. Theoretically, low coupling loss can be achieved with such a half-wave coupler. However, for fabrication simplicity, we can use a compact 2 × 2 coupler with a multimode coupling region as shown in Fig. 3. It can be seen as if the middle waveguide is merged with the adjacent waveguides in the coupling region. An arbitrary cross-coupling coefficient and phase can be achieved by adjusting the length Lc and the gap Wc of the multimode coupling region. However, such a coupler will incur some excess loss. The excess loss ε (in dB) can be calculated by ε = 10 log10(|C 11|2 + |C 21|2 + 2|C 11C 21‖cos(φ)|). From Eq. (13), we can obtain the increased laser threshold gain (in intensity) due to the excess loss ΔG = 2Δg = -ε/(8.686L) when the cross-coupling phase is near 180°. The detailed analysis will be given in Section 5.

5. Numerical results and discussions

Now we use a numerical example to illustrate the characteristics of the V-coupled cavity laser. Consider the previously mentioned example where n=n′ = 3.215, L = 466.24µm (Δf = 100GHz), and L′ = 518.31µm (Δf′ = 90GHz). The two cavities have a common resonance wavelength at 1550.12nm, corresponding to a frequency of 193.4THz. Assume the reflective coupler at the closed end is a perfect half-wave optical coupler with C11 = C22 = 0.755, and C12 = C21=-0.245. We also assume that the reflecting mirrors of the cavities are formed by air trenches with r1=r2 = 0.823 and the two cavities are pumped to produce the same round trip gain, i.e., gL = gL′.

 figure: Fig. 5.

Fig. 5. Effective reflection factors versus wavelength.

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The wavelength switching function of the V-coupled cavity laser can be understood from the effective reflection factors η and η′, which are wavelength dependent functions with sharp resonant peaks. Figure 5 shows the squared modulus |η|2 (dotted line) and |η′|2 (solid line), which are the effective reflection factors in intensity, as a function of the wavelength when the laser is at the threshold. The periodic peaks of the effective reflection factor |η|2 occur at the resonant wavelengths of the wavelength selector cavity. The effective reflection factor |η|2 effectively modifies the reflectivity of one of the mirrors of the fixed gain cavity, producing a comb of reflectivity peaks. Consequently, a resonant mode of the fixed gain cavity that coincides with one of the peaks of the effective reflection factor |η|2 is selected as the lasing mode. Since the periodic peaks of the effective reflection factor |η′|2 correspond to the resonant wavelengths of the fixed gain cavity, the lasing wavelength occurs at the position where a peak of |η|2 overlaps with a peak of |η′|2.

Figure 6 shows the small signal gain spectra of the laser near its threshold when the signal is transmitted through the cavities for a cavity length difference of 10% (a) and 5% (b). There are multiple peaks separated by a FSR as determined by Eq. (3). The variation of the material gain is not considered in this calculation. In practice, the laser is limited to single mode operation due to the material gain variation. The FSR is about 7.2 nm (900 GHz) and 15.2nm (1900GHz), respectively, for the two examples. The tuning range as determined by the FSR can be further increased by decreasing the cavity length difference. Thanks to the Vernier effect, there is no tuning range limitation due to achievable refractive index variation, in contrast to the case of a DBR laser. Instead, the increase of tuning range will compromise the mode selectivity, as will be shown later (Fig. 9). These features of the proposed device are not specific to any material system.

 figure: Fig. 6.

Fig. 6. Small signal gain spectra of the laser near its threshold when the signal is transmitted through the fixed gain cavity (dotted line) and through the wavelength selector cavity (solid line) for a cavity length difference of 10% (a) and 5% (b).

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 figure: Fig. 7.

Fig. 7. Effective reflection factors corresponding to χ = 0.1 (solid line) and χ = 0.5 (dotted line).

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The mode selectivity, which relates to side-mode suppression ratio (SMSR), is an important consideration in the design of the laser. The SMSR is directly proportional to the threshold difference between the side mode and the main mode [20]. The mode selectivity can be optimized by appropriately choosing the cross-coupling coefficient. We define the normalized cross-coupling coefficient (in intensity) as

χ=C212C112+C212=C122C122+C222

To illustrate the effect of the cross-coupling coefficient, we calculate the effective reflection factor |η|2 as a function of wavelength for different χ values. Figure 7 compares the spectra of the effective reflection factor |η|2 for χ = 0.1 (solid line) and χ = 0.5 (dotted line), when the laser is pumped at the lasing threshold. We can see that when the cross-coupling coefficient decreases, the peaks of the effective reflection factor |η|2, which is proportional to the effective reflectivity of the mirror at the closed end, become narrower while the contrast decreases.

Since the discrimination of side modes is based on the misalignment of resonant modes between the fixed gain cavity and the channel selector cavity, the narrower the effective reflectivity peaks, the better the mode selectivity between adjacent modes. Quantitatively, the mode selectivity can be characterized by threshold difference between the side modes and the main mode. Figure 8 shows the lasing threshold gain G (in intensity, G = 2g) of the cavity modes for two different normalized cross-coupling coefficients χ = 0.1 (circles) and χ = 0.5 (crosses). For the lowest threshold mode at the common resonance wavelength of 1550.12nm, the threshold difference between the main mode and the next lowest threshold mode is about 7.2 cm−1 for χ = 0.1, but is only about 1.2 cm−1 for χ = 0.5. Although the threshold gain of the next matched mode is almost the same as the main mode, the limitation of the material gain window (or the spectral gain variation) would in practice prevent the adjacent matched modes from lasing, as schematically shown in Fig. 2.

 figure: Fig. 8.

Fig. 8. Lasing threshold of the cavity modes for χ = 0.1 (circles) and χ = 0.5 (crosses).

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For a perfect symmetric half-wave coupler without any excess loss, we have C11 = C22 = |C11|, C12 = C21=-|C21|, and |C11| + |C21| = 1. When the two cavities are pumped with equal round-trip gains, from Eqs. (13) and (14) we can derive β = -1 for the common resonant mode with the lowest threshold and this threshold is the same as that of the uncoupled Fabry-Perot laser.

In Fig. 9, we show the threshold gain difference between the lowest threshold mode and the next lowest threshold mode, and the corresponding side-mode suppression ratio (SMSR) as a function of the normalized cross-coupling coefficient χ for a cavity length difference of 10% and 5%. The SMSR is calculated for 1mW output power according to the simple model of Ref. [20] without considering the spatial hole-burning effect. The threshold of the main mode is independent of χ and is the same as in the case of an uncoupled Fabry-Perot cavity (χ = 0). For the case of 10% cavity length difference, the largest threshold difference occurs around χ = 0.096. The threshold difference increases as χ decreases from 1 to 0.096, because the peak width of the effective reflection factor |η|2 decreases, resulting in an increased selectivity between the main mode and its adjacent modes. As χ further decreases to below 0.096, the threshold difference decreases. This is because the peak width of the effective reflection factor |η|2 becomes narrower than the mode spacing and it no longer affects the threshold difference. Instead, the threshold difference is determined by the contrast in the effective reflection factor |η|2 which decreases with the decreasing cross-coupling coefficient. When the cavity length difference decreases, the largest threshold difference and SMSR decrease and the cross-coupling coefficient at which the largest threshold difference (and SMSR) occurs also decreases.

 figure: Fig. 9.

Fig. 9. Threshold gain difference between the lowest threshold mode and the next lowest threshold mode (a) and the corresponding side-mode suppression ratio (b) as a function of the normalized cross-coupling coefficient χ for a cavity length difference of 10% (solid line) and 5% (dashed line).

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 figure: Fig. 10.

Fig. 10. Threshold gain difference between the lowest threshold mode and the next lowest threshold mode as a function of the normalized cross-coupling coefficient χ for two different pumping conditions corresponding to gL = g′L′ (solid line) and g = g′ (dotted line). The cavity length difference is 20%.

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By increasing the length difference between the fixed gain cavity and the channel selector cavity, the threshold difference between the lowest threshold mode and the next lowest threshold mode can be increased, at the expense of reduced free spectral range as determined by Eq. (3). It is also found that the maximum threshold difference is achieved when the simple round trip gains in the two cavities are equal, i.e. gL = g′L′ (if the cavity mirrors have the same reflectivities) or r1r2e2gL = r1′r2′e2g′L′. Consider the case L = 466.24µm (Δf = 100GHz) and L′ = 582.68µm (Δf′ = 80GHz) with other parameters the same as in the previous example. Figure 10 shows the threshold difference between the lowest threshold mode and the next lowest threshold mode as a function of the cross-coupling coefficient for two different pumping conditions corresponding to gL = g′L′ (solid line) and g = g′ (dotted line). The cavity length difference is about 20%. Compared to the case of 10% cavity length difference, the maximum achievable threshold difference is increased from 7.2cm−1 to 16cm−1 (for the case gL = g′L′), and the optimal cross-coupling coefficient at which the maximum threshold difference is achieved is increased from 0.096 to 0.29.

As it can also be seen in Fig. 10, the maximum achievable threshold difference decreases as the pumping condition deviates away from the optimal condition of gL = g′L′ (or more generally, equal round trip gains). Therefore, it is preferable that gain variations be avoided when the refractive index of the channel selector cavity is changed to switch the laser wavelength. This can be realized by using a separate tuning section in the channel selector cavity which is substantially passive (with little gain or loss), as shown in Fig. 1. This also allows flexible output power control independent of the wavelength switching.

 figure: Fig. 11.

Fig. 11. Threshold gain difference versus the normalized cross-coupling coefficient for different cross-coupling phases.

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 figure: Fig. 12.

Fig. 12. Maximal threshold gain difference and the corresponding χ opt versus cross-coupling phase.

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Ideally the reflective coupler at the closed end of the V-coupled cavities is a half-wave optical coupler. As the cross-coupling phase deviates away from half-wave (i.e. 180°), the maximal threshold gain difference decreases. Figure 11 shows the threshold gain difference versus normalized cross-coupling coefficient χ for different cross-coupling phases. When the cross-coupling phase deviates from 180°, the peak decreases and becomes less pointed, similar to the case when the gain coefficients deviate away from the balanced pumping condition. The optimal cross-coupling coefficient χ opt at which the maximal threshold gain difference occurs also decreases. For conventional directional couplers with 90° coupling phase, the threshold difference becomes zero, which means there is no mode selectivity.Figure 12 shows the variations of the maximal threshold gain difference and the χ opt value when the cross-coupling phase changes.

 figure: Fig. 13.

Fig. 13. Variations of the cross-coupling phase, cross-coupling coefficient and excess loss versus the width and length of the coupling region.

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Now let us consider the 2 × 2 optical coupler with a multimode coupling region as shown in Fig. 3. As mentioned in the previous section, an arbitrary cross-coupling coefficient and phase can be achieved by adjusting the length Lc and the gap Wc of the multimode coupling region, with a certain amount of excess loss. Assume the effective indices of the waveguide and the cladding regions are 3.220 and 3.189, respectively. The width of the waveguide is 3.0µm. Using two-dimensional Beam Propagation Method (BPM), we obtain the variations of the normalized cross-coupling coefficient, phase, and the excess loss as a function of the gap Wc as illustrated in Figs. 13(a), 13(b), and 13(c), respectively, for a fixed coupling length Lc = 34µm. The variations of the normalized cross-coupling coefficient, phase, and the excess loss with the coupling length Lc are shown in Figs. 13(d), 13(e) and 13(f), respectively, for a fixed gap Wc = 2.6µm. We can see that the cross-coupling coefficient increases with Lc and decreases with Wc. The coupling phase mainly depends on the gap Wc, although the coupling length Lc also slightly affects the coupling phase. Therefore, although the cross-coupling coefficient and phase are interrelated and cannot be adjusted independently, any target values of these two parameters can be achieved by adjusting the two design variables Lc and Wc simultaneously. To achieve optimal cross-coupling coefficient and phase in our numerical example, we choose Wc = 2.6µm and Lc = 34µm. The coupling phase approximately equals to 180° and we have C11 = C22 = 0.689, and C12 = C21=-0.217. The normalized cross-coupling coefficient is about 0.09. The excess loss is -0.86dB, which results in an increase of 2.1 cm−1 in the threshold gain. The excess loss does not affect the threshold gain difference which only depends on the normalized cross-coupling coefficient χ and the cross-coupling phase φ.

6. Comparison of mode selectivity with Y-coupled cavity laser

Now let us examine the difference in mode selectivity between the V-coupled cavity laser of the present paper and the previously investigated Y-coupled cavity laser [12–15]. Consider a Y-coupled cavity laser as shown in Fig. 14 with two waveguide branches A, B and a common waveguide section C of length LA, LB and LC, respectively. The electrodes on the three waveguide sections are separated by shallow isolation trenches. The amplitude coupling coefficients from the common waveguide section C to waveguide A and B are denoted as C1 and C2, respectively. The amplitude coupling coefficients from waveguide A and B to the common waveguide section C are denoted as C′1 and C′2, respectively. From the reciprocity of light wave propagation, we have C1 = C′1, and C2 = C′2. Assuming there is no coupling loss, we have | C1|2+| C2|2 = 1.

 figure: Fig. 14.

Fig. 14. Schematic diagram of a Y-coupled cavity laser

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Similar to the previous analysis for the V-coupled cavity laser, we denote the initial electric field at the open end of the A and B branches as E1 and E2, respectively. After a round trip propagation including two reflections at the cavity mirrors, the electric field E1′ and E2′ can be written as

E1=r1r2(C1E1e(g+ik)L+C2E2e(g+ik)L)C1e(g+ik)L
E2=r1r2(C1E1e(g+ik)L+C2E2e(g+ik)L)C2e(g+ik)L

where L = LA + LC and L′ = LB + LC are the total cavity lengths of the branch A and branch B, respectively; g = (gALA + gCLC)/L and g′ = (gBLB + gCLC)/L′ are the average gain coefficients; and k = (kALA + kCLC)/L and k′ = (kBLB + kCLC)/L′ are the average propagation constants of the two cavities. From Eqs. (19) and (20), we can derive the threshold condition for the Y-coupled cavity laser:

C1C1r1r2e2(g+ik)L+C2C2r1r2e2(g+ik)L=1

By comparing Eqs. (19) and (20) with Eqs. (11) and (12), we can obtain the following equivalent relationships: C11 = C1C′1, C22 = C2C′2, C12 = C′1C2, and C21 = C′2C1. Substituting these equations into Eq. (10), one can obtain Eq. (21). One can also see that, no matter what is the splitting ratio of the Y-branch coupler, the equation C11C22-C12C21 = 0 always holds. Therefore, in terms of the threshold condition, the Y-coupled cavity laser is equivalent to a special case of the V-coupled cavity laser when C11C22-C12C21 = 0. This means that for the Y-coupled cavity laser, the amplitude of the cross-coupling coefficient is either the same as that of the self-coupling coefficient when the Y-coupler is an equal power splitter, or is between the two different self-coupling coefficients when the Y-coupler is an unequal power splitter. This limitation is removed for the V-coupled cavity proposed in this paper. The V-coupled cavity laser allows the optimized cross-coupling coefficient to be smaller than the self-coupling coefficients of both cavities, resulting in a much larger side-mode threshold difference, i.e. much higher single-mode selectivity.

Using the same parameters as those in the example of 10% cavity length difference of Fig. 9, i.e., L = 466.24µm, and L′ = 518.31µm, also assuming that the two cavities are pumped to produce the same round trip gain, i.e., gL = g′L′, Fig. 15 shows the threshold gain coefficient G (= 2g) of the lowest threshold mode (solid line) and the next lowest threshold mode (dotted line) as a function of the Y-coupling coefficient |C2|2 (or |C1|2). The threshold of the main lasing mode is G = 2g = 8.5 cm−1 which is independent of the coupling coefficient and is the same as the uncoupled cavity of the same length L. It is also the same as the threshold of the V-coupled cavity. However, the largest threshold difference between the lowest threshold mode and the next lowest threshold mode is only 1.2 cm−1 and it occurs at |C1|2 = |C2|2 = 0.5, i.e., when the Y-branch is an equal power splitter. This compares to a maximal threshold difference of 7.2 cm−1 for the V-coupled cavity laser of the same cavity lengths with a cross-coupling coefficient of 0.096. Note that the value of the maximal threshold difference of the Y-laser is consistent with the value of the threshold difference corresponding to χ = 0.5 in V-coupled cavity laser for which the condition C11C22-C12C21 = 0 is also satisfied. The six-time improvement in threshold difference for the V-coupled cavity laser translates into 7.8dB improvement in SMSR (see Fig. 9). This improvement in mode selectivity is even more significant when the length difference between the two cavities is smaller (i.e. when a large FSR is needed to accommodate for a large number of wavelength channels), as can be seen from the comparisons between the two cavity length differences in Fig. 9.

 figure: Fig. 15.

Fig. 15. Threshold gain coefficient of the lowest threshold mode (solid line) and the next lowest threshold mode (dotted line) as a function of the Y-coupling coefficient of the Y-laser.

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7. Conclusions

In conclusion, a V-coupled cavity structure with a reflective 2 × 2 half-wave optical coupler is proposed for semiconductor lasers to achieve wideband digital wavelength switching with high single mode selectivity. Detailed design principle and numerical results are presented. While the switchable wavelength range is greatly increased by using the well-known Vernier effect, the device offers significant advantage of fabrication simplicity and compactness compared to existing tunable lasers based on complex grating structures. No phase shift region is needed and the digital wavelength switching only requires a single electrode control. Compared to previously investigated Y-lasers and serially-coupled-cavity lasers, the V-coupled cavity laser can achieve much higher single-mode selectivity due to the fact that by appropriately designing the reflective 2 × 2 coupler it allows the optimized cross-coupling coefficient to be much smaller than the self-coupling coefficients of both cavities while maintaining a half-wave relative phase. Using deep-etched trenches as partially reflecting elements, additional waveguide branch structures can be designed outside the laser cavities to form a complete Mach-Zehnder interferometer, allowing monolithic and efficient integration of additional functionalities such as space switching, variable optical attenuation, or high speed modulation.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under grant No. 60788403 and the Open Project Foundation of the State Key Laboratory of Modern Optical Instrumentation (No. LMOI-0602).

References and links

1. J. W. Raring and L. A. Coldren, “40Gb/s Widely-Tunable Transceivers,” invited paper, IEEE J. Sel. Top. Quantum Electron. 13, 3–14 (2007). [CrossRef]  

2. N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990). [CrossRef]  

3. V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993). [CrossRef]  

4. L. A. Coldren, “Monolithic Tunable Diode Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 988 (2000). [CrossRef]  

5. P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008). [CrossRef]  

6. Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993). [CrossRef]  

7. R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992). [CrossRef]  

8. J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference, Washington DC, paper TuE2 (2004).

9. D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

10. L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981). [CrossRef]  

11. W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimetals , 22, 257 (1985). [CrossRef]  

12. L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659–682 (1984). [CrossRef]  

13. T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 742–754 (1985). [CrossRef]  

14. D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223–226 (1986). [CrossRef]  

15. R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66–72 (1988). [CrossRef]  

16. O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993). [CrossRef]  

17. M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992). [CrossRef]  

18. M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992). [CrossRef]  

19. M. Kuznetsov, “Design of widely tunable semiconductor three-branch lasers,” J. Lightwave Technol. 12, 2100–2106 (1994). [CrossRef]  

20. T. L. Koch and U. Koren, “Semiconductor Lasers for Coherent Optical Fiber Communications”, J. Lightwave Technol. 8, 274–293 (1990). [CrossRef]  

References

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  • |

  1. J. W. Raring and L. A. Coldren, “40Gb/s Widely-Tunable Transceivers,” invited paper, IEEE J. Sel. Top. Quantum Electron. 13, 3–14 (2007).
    [Crossref]
  2. N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
    [Crossref]
  3. V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
    [Crossref]
  4. L. A. Coldren, “Monolithic Tunable Diode Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 988 (2000).
    [Crossref]
  5. P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008).
    [Crossref]
  6. Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993).
    [Crossref]
  7. R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992).
    [Crossref]
  8. J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference, Washington DC, paper TuE2 (2004).
  9. D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).
  10. L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
    [Crossref]
  11. W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimetals,  22, 257 (1985).
    [Crossref]
  12. L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659–682 (1984).
    [Crossref]
  13. T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 742–754 (1985).
    [Crossref]
  14. D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223–226 (1986).
    [Crossref]
  15. R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66–72 (1988).
    [Crossref]
  16. O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993).
    [Crossref]
  17. M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992).
    [Crossref]
  18. M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992).
    [Crossref]
  19. M. Kuznetsov, “Design of widely tunable semiconductor three-branch lasers,” J. Lightwave Technol. 12, 2100–2106 (1994).
    [Crossref]
  20. T. L. Koch and U. Koren, “Semiconductor Lasers for Coherent Optical Fiber Communications”, J. Lightwave Technol. 8, 274–293 (1990).
    [Crossref]

2008 (1)

P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008).
[Crossref]

2007 (1)

J. W. Raring and L. A. Coldren, “40Gb/s Widely-Tunable Transceivers,” invited paper, IEEE J. Sel. Top. Quantum Electron. 13, 3–14 (2007).
[Crossref]

2000 (1)

L. A. Coldren, “Monolithic Tunable Diode Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 988 (2000).
[Crossref]

1994 (1)

M. Kuznetsov, “Design of widely tunable semiconductor three-branch lasers,” J. Lightwave Technol. 12, 2100–2106 (1994).
[Crossref]

1993 (3)

O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993).
[Crossref]

V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[Crossref]

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993).
[Crossref]

1992 (3)

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992).
[Crossref]

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992).
[Crossref]

M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992).
[Crossref]

1990 (2)

T. L. Koch and U. Koren, “Semiconductor Lasers for Coherent Optical Fiber Communications”, J. Lightwave Technol. 8, 274–293 (1990).
[Crossref]

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

1988 (1)

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66–72 (1988).
[Crossref]

1986 (1)

D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223–226 (1986).
[Crossref]

1985 (2)

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 742–754 (1985).
[Crossref]

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimetals,  22, 257 (1985).
[Crossref]

1984 (1)

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659–682 (1984).
[Crossref]

1981 (1)

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
[Crossref]

Alferness, R. C.

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992).
[Crossref]

Anandarajah, P. M.

P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008).
[Crossref]

Anis, G. D.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Barry, L. P.

P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008).
[Crossref]

Barton, E.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Baums, D.

O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993).
[Crossref]

Bava, G. P.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Buhl, L. L.

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992).
[Crossref]

Busico, G.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Caponio, N. P.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Coldren, L. A

V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[Crossref]

Coldren, L. A.

J. W. Raring and L. A. Coldren, “40Gb/s Widely-Tunable Transceivers,” invited paper, IEEE J. Sel. Top. Quantum Electron. 13, 3–14 (2007).
[Crossref]

L. A. Coldren, “Monolithic Tunable Diode Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 988 (2000).
[Crossref]

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 742–754 (1985).
[Crossref]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659–682 (1984).
[Crossref]

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
[Crossref]

Dentai, A. G.

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992).
[Crossref]

Diitting, K.

M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992).
[Crossref]

Duck, J. P.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Goano, M.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Griffin, R. A.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Hammerfeldt, S.

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference, Washington DC, paper TuE2 (2004).

Hildebrand, O.

O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993).
[Crossref]

Idler, W.

M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992).
[Crossref]

Iga, K.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
[Crossref]

Ishii, H.

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993).
[Crossref]

Jayaraman, V.

V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[Crossref]

Koch, T. L.

T. L. Koch and U. Koren, “Semiconductor Lasers for Coherent Optical Fiber Communications”, J. Lightwave Technol. 8, 274–293 (1990).
[Crossref]

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 742–754 (1985).
[Crossref]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659–682 (1984).
[Crossref]

Koren, U.

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992).
[Crossref]

T. L. Koch and U. Koren, “Semiconductor Lasers for Coherent Optical Fiber Communications”, J. Lightwave Technol. 8, 274–293 (1990).
[Crossref]

Kuznetsov, M.

M. Kuznetsov, “Design of widely tunable semiconductor three-branch lasers,” J. Lightwave Technol. 12, 2100–2106 (1994).
[Crossref]

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992).
[Crossref]

Lang, R. J.

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66–72 (1988).
[Crossref]

Maher, R.

P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008).
[Crossref]

Maio, I.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Marcuse, D.

D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223–226 (1986).
[Crossref]

Mathur, A.

V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[Crossref]

Meliga, M.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Miller, B. I.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
[Crossref]

Montrosset, I.

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

Ponnampalam, L.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Raring, J. W.

J. W. Raring and L. A. Coldren, “40Gb/s Widely-Tunable Transceivers,” invited paper, IEEE J. Sel. Top. Quantum Electron. 13, 3–14 (2007).
[Crossref]

Reid, D. C. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Rentschler, J. A.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
[Crossref]

Robbins, D. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Sarlet, G.

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference, Washington DC, paper TuE2 (2004).

Schilling, M.

O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993).
[Crossref]

M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992).
[Crossref]

Tohmori, Y.

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993).
[Crossref]

Tsang, W. T.

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimetals,  22, 257 (1985).
[Crossref]

Verlangieri, P.

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992).
[Crossref]

Ward, A. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Wesström, J.-O.

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference, Washington DC, paper TuE2 (2004).

Whitbread, N. D.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Williams, P. J.

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

Yariv, A.

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66–72 (1988).
[Crossref]

Yoshikuni, Y.

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993).
[Crossref]

Appl. Phys. Lett. (2)

R. C. Alferness, U. Koren, and L. L. Buhl, “Broadly tunable InGaAsP/InP laser based on a vertical coupler filter with 57-nm tuning range,” Appl. Phys. Lett. 60, 3209–3211 (1992).
[Crossref]

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315~317 (1981).
[Crossref]

Electron. Lett. (1)

M. Schilling, K. Diitting, and W. Idler, “Asymmetrical Y laser with simple single current tuning response,” Electron. Lett. 28, 1698–1699 (1992).
[Crossref]

IEEE J. Quantum Electron. (5)

Y. Tohmori, Y. Yoshikuni, and H. Ishii, “Broad-range wavelength-tunable superstructure grating (SSG) DBR lasers,” IEEE J. Quantum Electron. 29, 1817–1823 (1993).
[Crossref]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers,” IEEE J. Quantum Electron. 20, 659–682 (1984).
[Crossref]

D. Marcuse, “Coupling coefficients of coupled laser cavities,” IEEE J. Quantum Electron. 22, 223–226 (1986).
[Crossref]

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66–72 (1988).
[Crossref]

V. Jayaraman, A. Mathur, and L. A Coldren, “Theory, design, and performance of extended tuning range semiconductor lasers with sampled gratings,” IEEE J. Quantum Electron. 29, 1824–1834 (1993).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

N. P. Caponio, M. Goano, I. Maio, M. Meliga, G. P. Bava, G. D. Anis, and I. Montrosset, “Analysis and Design criteria of Three-Section DBR Tunable Lasers,” IEEE J. Sel. Areas Commun. 8, 1203–1213 (1990).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

J. W. Raring and L. A. Coldren, “40Gb/s Widely-Tunable Transceivers,” invited paper, IEEE J. Sel. Top. Quantum Electron. 13, 3–14 (2007).
[Crossref]

L. A. Coldren, “Monolithic Tunable Diode Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 988 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (2)

P. M. Anandarajah, R. Maher, and L. P. Barry, et al., “Characterization of frequency drift of sampled-grating DBR Laser module under direct modulation”, IEEE Photon. Technol. Lett. 20, 239–241 (2008).
[Crossref]

M. Kuznetsov, P. Verlangieri, and A. G. Dentai, “Asymmetric Y-Branch Tunable Semiconductor Laser with 1.0 THz Tuning Range,” IEEE Photon. Technol. Lett. 4, 1093–1095 (1992).
[Crossref]

J. Appl. Phys. (1)

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 742–754 (1985).
[Crossref]

J. Lightwave Technol. (3)

O. Hildebrand, M. Schilling, and D. Baums, et al., “The Y-laser: A Multifunctional Device for Optical Communication Systems and Switching Networks,” J. Lightwave Technol. 11, 2066–2074 (1993).
[Crossref]

M. Kuznetsov, “Design of widely tunable semiconductor three-branch lasers,” J. Lightwave Technol. 12, 2100–2106 (1994).
[Crossref]

T. L. Koch and U. Koren, “Semiconductor Lasers for Coherent Optical Fiber Communications”, J. Lightwave Technol. 8, 274–293 (1990).
[Crossref]

Semicond. Semimetals (1)

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” Semicond. Semimetals,  22, 257 (1985).
[Crossref]

Other (2)

J.-O. Wesström, G. Sarlet, and S. Hammerfeldt, “State of the art performance of widely tunable modulated grating Y-branch lasers,” Optical Fiber Communication Conference, Washington DC, paper TuE2 (2004).

D. J. Robbins, G. Busico, L. Ponnampalam, J. P. Duck, P. J. Williams, R. A. Griffin, A. J. Ward, D. C. J. Reid, N. D. Whitbread, and E. Barton, “A high power, broadband tuneable laser module based on a DS-DBR laser with integrated SOA,” Optical Fiber Communication Conference, Washington DC, Paper TuE3 (2004).

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Figures (15)

Fig. 1.
Fig. 1. Schematic diagram of a V-coupled cavity laser within a MZI
Fig. 2.
Fig. 2. Schematic diagram illustrating the relationships between the resonant frequency combs of the two cavities and the material gain spectrum.
Fig. 3.
Fig. 3. Unfolded reflective 2 × 2 coupler in the V-coupled cavities
Fig. 4.
Fig. 4. Power transfer functions of quarter-wave (a) and half-wave (b) optical couplers.
Fig. 5.
Fig. 5. Effective reflection factors versus wavelength.
Fig. 6.
Fig. 6. Small signal gain spectra of the laser near its threshold when the signal is transmitted through the fixed gain cavity (dotted line) and through the wavelength selector cavity (solid line) for a cavity length difference of 10% (a) and 5% (b).
Fig. 7.
Fig. 7. Effective reflection factors corresponding to χ = 0.1 (solid line) and χ = 0.5 (dotted line).
Fig. 8.
Fig. 8. Lasing threshold of the cavity modes for χ = 0.1 (circles) and χ = 0.5 (crosses).
Fig. 9.
Fig. 9. Threshold gain difference between the lowest threshold mode and the next lowest threshold mode (a) and the corresponding side-mode suppression ratio (b) as a function of the normalized cross-coupling coefficient χ for a cavity length difference of 10% (solid line) and 5% (dashed line).
Fig. 10.
Fig. 10. Threshold gain difference between the lowest threshold mode and the next lowest threshold mode as a function of the normalized cross-coupling coefficient χ for two different pumping conditions corresponding to gL = g′L′ (solid line) and g = g′ (dotted line). The cavity length difference is 20%.
Fig. 11.
Fig. 11. Threshold gain difference versus the normalized cross-coupling coefficient for different cross-coupling phases.
Fig. 12.
Fig. 12. Maximal threshold gain difference and the corresponding χ opt versus cross-coupling phase.
Fig. 13.
Fig. 13. Variations of the cross-coupling phase, cross-coupling coefficient and excess loss versus the width and length of the coupling region.
Fig. 14.
Fig. 14. Schematic diagram of a Y-coupled cavity laser
Fig. 15.
Fig. 15. Threshold gain coefficient of the lowest threshold mode (solid line) and the next lowest threshold mode (dotted line) as a function of the Y-coupling coefficient of the Y-laser.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

Δ f = c 2 n g L
Δ f = c 2 n g L = c 2 ( n a L a + n b L b )
Δ f c = Δ f Δ f Δ f Δ f
δ f f = δ n n = δ n b L b n b L
δ f = Δ f Δ f Δ f δ f
η = C 11 + C 21 C 12 r 1 r 2 e 2 ( g + ik ) L ( 1 + C 22 r 1 r 2 e 2 ( g + ik ) L + C 22 2 r 1 2 r 2 2 e 4 ( g + ik ) L + . . . )
= C 11 + C 21 C 12 r 1 r 2 e 2 ( g + ik ) L 1 C 22 r 1 r 2 e 2 ( g + ik ) L
r 1 η r 2 e 2 ( g + ik ) L = 1
η = C 22 + C 21 C 12 r 1 r 2 e 2 ( g + ik ) L ( 1 + C 11 r 1 r 2 e 2 ( g + ik ) L + C 2 11 r 1 2 r 2 2 e 4 ( g + ik ) L + . . . )
= C 22 + C 21 C 12 r 1 r 2 e 2 ( g + ik ) L 1 C 22 r 1 r 2 e 2 ( g + ik ) L .
r 1 η r 2 e 2 ( g + ik ) L = 1
C 11 r 1 r 2 e 2 ( g + ik ) L + C 22 r 1 r 2 e 2 ( g + ik ) L ( C 11 C 22 C 21 C 12 ) r 1 2 r 2 2 e 2 ( g + ik ) L e 2 ( g + ik ) L = 1
E 1 = r 1 r 2 ( C 11 E 1 e ( g + ik ) L + C 21 E 2 e ( g + ik ) L ) e ( g + ik ) L
E 2 = r 1 r 2 ( C 12 E 1 e ( g + ik ) L + C 22 E 2 e ( g + ik ) L ) e ( g + ik ) L
( C 11 + β C 21 ) r 1 r 2 e 2 ( g + ik ) L = 1
( C 22 + C 12 β ) r 1 r 2 e 2 ( g + ik ) L = 1
β = E 2 e ( g + ik ) L E 1 e ( g + ik ) L
P 1 = 1 2 [ C 11 2 + C 21 2 + 2 C 11 C 21 cos ( φ + ϕ ) ]
P 2 = 1 2 [ C 12 2 + C 22 2 + 2 C 12 C 22 cos ( φ ϕ ) ]
χ = C 21 2 C 11 2 + C 21 2 = C 12 2 C 12 2 + C 22 2
E 1 = r 1 r 2 ( C 1 E 1 e ( g + ik ) L + C 2 E 2 e ( g + ik ) L ) C 1 e ( g + ik ) L
E 2 = r 1 r 2 ( C 1 E 1 e ( g + ik ) L + C 2 E 2 e ( g + ik ) L ) C 2 e ( g + ik ) L
C 1 C 1 r 1 r 2 e 2 ( g + ik ) L + C 2 C 2 r 1 r 2 e 2 ( g + ik ) L = 1

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