We present a customized multilayered dielectric stack employed as a broadband phase modulator with 6.3 THz optical bandwidth. The bandpass modulator provides up to a full-cycle of near-uniform phase modulation across a defined signal spectrum with maximized transmission and minimized pulse phase distortion. The modulator offers a compact, lightweight approach to active wavefront phase control for large optical apertures without the use of mechanical actuators. The modulator also provides for rapid signal switching. We contrast the narrowband transmission of a standard Distributed Bragg Reflector (DBR) with the broadband transmission of our optimized bandpass modulator. We explore techniques for implementing rapid phase modulation while maintaining high average signal transmission levels.
© 2000 Optical Society of America
Future space missions will likely use optical signals for efficient power transmission and data conveyance across the extreme distances associated with the space environment. Proposed applications include optical power beaming , interplanetary optical communication links , and extrasolar planetary imaging , all of which require structures or constellations of satellites with requisite large aperture baselines. The primary challenge in implementing these systems is the active maintenance of coherent wavefronts across the optical system’s aggregate aperture area, which for some deep-space astronomical applications, may be up to 1,000 m in baseline diameter .
The spatial segmentation of a large primary aperture into an optimal number of smaller aperture elements  provides a useful approach to addressing the challenge of coherent wavefront maintenance. Each aperture element implements an active correction of local wavefront phase errors in response to measurements taken from a known coherent reference signal. The phase error correction implementation must respond rapidly, with a reaction time shorter than the hosting structure’s dynamic vibrational or thermal perturbations. Such implementation requires a phase modulation range of at least a full optical wavelength cycle, allowing for a modulo 2π translation in axial position. For applications requiring precise transmission of pulsed power or high rate data, the implementation must supply a bandwidth wide enough to fully transmit all contributing frequency components of the primary optical signal with low loss and low distortion. Additionally, the overall mass of the implementation should be as low as possible, preferably much lower than the mass of state-of-the-art mechanical actuators traditionally used to actively correct local figure variations in optical applications of reflective geometry.
As a possible means of providing such capabilities, we describe a local phase front correction implementation that employs multilayered, dielectric stacks as real-time, adjustable, broadband phase modulators for pulsed optical signals. Our implementation utilizes recent advances in optical interference coating design to develop a computationally optimized bandpass configuration with a high average transmission region within which contributing frequency components of a propagating optical signal are similarly modulated in phase with low signal loss and low signal distortion. Because the bandpass configuration is compact, lightweight, rapidly configurable, and provides broadband optical phase modulation, its implementation provides an enabling technology for active phase front control in large segmented aperture systems. The bandpass configuration also provides a means for rapid switching of the optical signal. Shifting the spectral position of the transmission function’s bandpass region relative to the signal’s spectral position causes the signal to transition from a region of high transmissivity to a region of high reflectivity.
To contrast the features of our bandpass configuration against a more familiar optical interference coating configuration, we first review the limitations of the Distributed Bragg Reflector (DBR) configuration as a transmissive phase modulator. We quantify the DBR’s limitation in providing fine, broadband control of transmitted phase values for optical signals. We examine several alternative modulation methods that introduce a shift in the interference coating’s transmission function, thereby imparting a phase modulation to the transmitted optical signal. We conclude with a presentation and analysis of our bandpass modulator configuration’s design and performance. We emphasize the computational optimization approach used to generate a spectrally broad transmission region, the maximization of phase modulation available to an optical pulse defined to occupy a fraction of the bandpass region, and the minimization of phase distortion of the modulated optical signal.
2. Phase Velocity, Relative Phase Modulation, and Group Velocity
We define phase velocity, relative phase modulation, and group velocity in terms of wavelength for use in assessing the optical interference coating configurations presented in later sections. The multilayered stack configurations are dispersive in nature and may be described as a function of wavenumber by a generalized dispersion function, ω(k). We use a Taylor series expansion about a defined central wavenumber value k0 to obtain an expression for the dispersion function,
The first term in the expansion determines the phase velocity for the pulse’s central wavenumber. The phase velocity for any particular free-space wavenumber k, or free-space wavelength λ, is
where Δϕ is interpreted with respect to λ as the absolute change in phase accumulated within some specified amount of time, Δt. The overall effective refractive index of the multilayered dielectric stack through which the wavefront propagates is neff. We define Δd as the path length traveled by the propagating wavefront. We substitute Δd/vp for Δt, rearrange the equation, and obtain the absolute change in transmitted signal phase in terms of the structure’s effective refractive index as,
Modulation of either neff or Δd results in a modulation of the absolute change in signal phase. We obtain the relative phase modulation by taking the difference between the unmodulated absolute change in phase and the modulated absolute change in phase.
If we assume the optical signal to be a pulse with a central wavenumber k0, then the group velocity is
This expression is valid only if the pulse envelope is sharply peaked at k0 and the dispersion function is smoothly varying around k0 . We restrict attention to this case.
We use k=2π/λ, and dk=(-2π/λ2)dλ, to express the group velocity in terms of the phase velocity at the central wavelength as
By designing interference coating configurations that increase the rate with which transmitted signal phase changes with respect to wavelength, we reduce the group velocity. This creates not only a desirable region for phase modulation, but also a mechanism for introducing a known amount of group delay to transmitted signals .
The next higher order term in the dispersion function expansion, d2ω/dk2, describes the group velocity dispersion. Higher order terms within the expansion introduce pulse distortions, which are usually undesirable for the purpose of transmitting optical signals with low envelope shape distortion. For maximum effectiveness in phase modulation, we seek multilayer stack designs that possess regions of maximized transmission, reduced group velocity, minimized group velocity dispersion, and minimized higher order terms in the dispersion expansion.
3. The Distributed Bragg Reflector as a Transmitted Optical Signal Phase Modulator
The narrow, sharply peaked transmission resonance regions found within the DBR’s transmission function restrict the bandwidth available to frequency components of short optical pulses. Additionally, the phase component of the DBR’s transmission function does not vary linearly with wavelength within the transmission resonance regions, causing group velocity dispersion within these regions. We examine these limitations exhibited by the DBR stack configuration for purposes of contrasting them with the capabilities of the bandpass modulator configuration presented in later sections.
The DBR is widely used to provide a high level of reflectivity for a band of wavelengths centered about a selected free-space design wavelength, λ0. In its simplest implementation, this structure comprises periods of alternating layers of high and low refractive index materials and thicknesses described by n1d1=n2d2=λ0/4, where n1,2 are the respective high and low refractive indices of the dielectric materials, d 1,2 are the physical layer thicknesses, and λ0 is the stack’s design wavelength .
The DBR interference coating configuration not only exhibits periodic regions of high reflectivity, but also exhibits resonance regions of high transmissivity. The bandwidth of these transmission resonances is dependent on the values of the layers’ refractive indices and the total number of layers within the stack.
Fig. 1 illustrates a typical transmission function as produced by a 100-layer stack of alternating GaAs and AlAs layers on a GaAs substrate. The stack design may be described by the expression
where S represents the interface to the GaAs stack substrate, L represents an AlAs layer with an optical thickness of λ0/4nL, H represents a GaAs layer with an optical thickness of λ0/4nH, and A represents the air interface. The exponent notation implies a repeating period of layers.
As seen in Fig. 1, the transmitted relative phase function changes rapidly with respect to wavelength within the regions of high transmission. This rapid change is most pronounced within the transmission resonances on either side of the reflective bandgap. Here the ability to modulate the phase of the transmitted signal is maximized while the modulation of group velocity and higher ordered phase distortions is minimized.
Use of the DBR configuration as a phase modulator requires a trade off between maximizing phase modulation and maximizing transmission bandwidth. The transmission resonances in a DBR’s transmission function are narrow in width and rounded near the resonance peak, with rapid transmission fall-off on either side. This narrow resonance shape prevents typical DBR stacks from transmitting wideband, short-pulsed optical signals with low distortion.
4. Modulation of the Distributed Bragg Reflector
Modulation of the optical signal as transmitted by the DBR configuration can be accomplished by actively shifting the DBR’s complex transmission function relative to the transmitted signal. We assess the DBR stack configuration’s capability for modulating narrow band signals by computationally modeling the result of electrorefractive changes within the stack.
One method of modulating the position of the DBR’s transmission function with respect to the transmitted signal wavelength is through the inclusion of one or more quantum wells within each layer of GaAs. This quantum confinement within the GaAs layers allows the formation of an excitonic absorption peak near the GaAs absorption band edge and provides stability for the excitons against ionization in the presence of an electric field. When applied perpendicular to the stack’s layers, an electric field broadens and shifts the excitonic absorption peak to longer wavelengths. This effect, known as the Quantum Confined Stark Effect (QCSE) , modulates absorption of wavelengths near the absorption peak. Changes in absorption translate into changes in refractive index as described by Kramers-Kronig relations.
The phase modulation applied to the transmitted optical signal depends on the location of the excitonic absorption peak in relation to the optical signal’s center wavelength. The absorption peak must be near enough to the transmitted optical signal’s center wavelength to benefit from the field-induced refractive index changes, yet not so near as to incur undesirable absorption.
Fig. 2 shows a detailed view of the transmission resonance nearest the long wavelength side of the DBR’s transmission stop-band region as it is modulated using a 100 kV/cm electric field applied perpendicular to the 100-layer DBR stack.
The results depicted in Fig. 2 show a maximum phase modulation of 2.8803 radians for a center wavelength of 1055.1 nm. This level of modulation is obtained at a significant cost in spectral variation of the transmitted signal amplitude. The transmitted amplitude ranges from 65% of the initial input signal when no modulation is applied, through 92% near the mid-modulation point, then to 43% at full modulation. For most applications, this large range of amplitude variation is not acceptable.
To obtain this data, we use a matrix method  for calculating the transfer function of the 100-layer stack, from which we obtain the wavelength-dependent, complex transmission function. We refer to modeling data obtained by Nelson et al.  as a reference for the change in refractive index of a single 80 Å In0.2Ga0.8As quantum well embedded within a layer of GaAs. Their results indicate that the described quantum well enhances an excitonic absorption peak at 1011 nm and, through the QCSE, red-shifts the peak in the presence of a 100 kV/cm applied electric field to ~1030 nm. In response to the change in absorption, the difference in refractive index within a single quantum well at a wavelength of 1061 nm is Δn=0.0041.
We note that the operational wavelength of this particular DBR modulator is near 1055.1 nm, not the 1061 nm target wavelength used by Nelson. This difference in operational wavelength placement with respect to the excitonic absorption peak results in a slightly higher level of refractive index modulation accompanied by a slightly higher level of absorption. Our computational models conservatively account for this small difference in refractive index modulation by adjusting the change in refractive index for a single quantum well from Δn=0.0041 at 1055.1 nm toΔn=0.0043 at 1061 nm.
We model three quantum wells per GaAs layer in an effort to increase the range of phase modulation obtained. We utilize a multiplicative factor of 2.5 rather than 3 to account for the multiple quantum wells per GaAs layer due to the geometrical dependence of well placement within the layer and their disproportionate strength of interaction with the quasi-standing electric field of the propagating optical signal .
Previous modeling efforts show that the range of phase modulation obtainable from a multilayered dielectric stack may be further improved by using computationally optimized stack designs derived from initial DBR configurations to narrow the operational transmission resonance . This solution increases the rate with which phase changes with respect to wavelength, but does so by a further reduction in bandwidth to the already narrow transmission resonance used for optical signal transmission.
As described, we apply three performance improvement factors to the initial DBR configuration; inclusion of up to three quantum wells per GaAs layer rather than only one, careful selection of the transmitted signal’s operational center wavelength with respect to the exciton absorption peak induced by the quantum wells, and optimization of the overall stack design to narrow the operational transmission resonance thereby increasing the rate with which the phase changes with respect to wavelength. After examining the DBR configuration’s response to these performance improvement factors, we conclude that the DBR configuration is insufficient for transmitting phase modulated optical signals with broad signal spectrums. The narrow transmission resonance bandwidth and the transmitted amplitude variations occurring during signal modulation prevent the DBR configuration from transmitting broadband optical signals with low distortion. We therefore seek alternative multilayered stack configurations that exhibit broadband signal transmission regions with a corresponding phase-to-wavelength function capable of supporting full-cycle phase modulation for a select band of wavelengths.
5. The Bandpass Configuration as a Transmitted Optical Signal Phase Modulator
We now present an alternative multilayer dielectric stack design for an interference coating that addresses many of the limitations intrinsic to the DBR configuration. Our unique bandpass modulator configuration exhibits a near-uniform phase modulation across a much wider bandwidth than the DBR configuration. Our bandpass modulator also maintains a high average transmission across the bandpass region, whereas the DBR configuration exhibits rapid transmission fall off on either side of the resonance peak. Using the same notation introduced by Eq. (6), an initial design for a 79-layer bandpass configuration  is described by the expression
where 9H represents a GaAs “defect” layer that is nine times the thickness of a regular quarter-wave layer. Unlike the periodic quarter-wave layer configuration of the DBR represented in Eq. (6), the initial bandpass configuration of Eq. (7) includes several thick defect layers interspersed at regular intervals within the stack construction. These defect layers act to create a family of new transmission resonances within the reflective region of the DBR where transmission was previously forbidden. These resonances blend to form a transmitting bandpass region centered about the design wavelength. As with the DBR example, we model the bandpass configuration using a GaAs substrate with layers of GaAs as the high refractive index material and AlAs as low refractive index material.
This initial bandpass configuration possesses a transmission region with the transmission ranging between 93.00 and 100.00 percent. The rippling peak-to-valley variations of the transmission region may be reduced further through computational optimization. We employ an optimization technique, referred to as “needling,”  to strategically insert 10 nm thick layers of AlAs within specific layers of GaAs at specific locations such that the average transmission level across the width of the bandpass is improved. This incremental process results in a much smoother transmission function across the bandpass configuration’s bandwidth ranging between 96.85 and 100.00 percent. Our optimization process converted the initial configuration of 79 layers into a final configuration of 91 layers by splitting six layers of GaAs with thin layers of AlAs. Fig. 3 compares the details of the initial filter design’s transmission curve to the details of the improved filter design’s transmission curve. We emphasize that this improved design is still not globally optimal, but illustrates the value of using the bandpass configuration as a transmission phase modulator.
At normal incidence, the bandpass configuration exhibits a bandpass region of ~21.0 nm, or ~6.3 THz, in bandwidth, and has an edge-to-edge phase change of 2.08 full cycles, slightly greater than 4π radians. This large edge-to-edge relative phase difference allows access to essentially any level of relative phase modulation.
For the sake of example, we define an optical signal with a gaussian amplitude envelope, a center wavelength of 993.3 nm, and a root-mean-square (rms) width of 5 nm, corresponding to a pulse length of 52 fs. Selection of these parameters intentionally positions the signal’s rms bandwidth within the shorter wavelength half of the bandpass region. We select the signal’s center wavelength of 993.3 nm to correspond with the peak of the second transmission resonance ripple within the bandpass.
By defining the optical signal bandwidth to be less than half of the transmission bandwidth, we provide access to a full cycle of phase modulation without adversely affecting signal transmission levels. Appropriate modulation of the stack’s optical characteristics causes the transmission function to shift in spectral position relative to the optical signal. This shift translates the transmission function across the signal’s spectral bandwidth, effectively causing the signal’s spectral position to migrate from one half of the full transmission bandpass region to the other half. This translation of the transmission function with respect to the optical signal’s bandwidth brings all signal frequency components through a complete cycle of phase modulation. Important to the successful implementation of this phase modulation approach is the requirement that the technique used to shift the transmission function in wavelength space does so in a uniform fashion.
6. Modulation of the Bandpass Configuration
We find through simulation that a 1.3 percent reduction in refractive index values for all GaAs and AlAs layers access a full cycle phase modulation. Increasing the refractive indices by a similar amount shifts the transmission function bandpass region to longer wavelengths, moving the transmission bandpass region away from the signal’s spectral bandwidth rather than across it. This shift translates the transmission function such that the signal’s spectral components experience a transition from a region of high transmissivity to a region of high reflectivity. This change in transmission level has the effect of providing an optical switch configuration for the optical signal. We discuss methods of obtaining the requisite change in refractive index for the constituent materials and the challenges associated with each approach.
6.1 Modulation Techniques
Free carrier injection provides a direct method for changing the refractive index of semiconductor materials. Adjusting the flow of free carriers through the stack varies the semiconductor refractive index, resulting in a shift of the transmission function’s position with respect to wavelength. This modulation technique is a leading candidate for the implementation of a non-mechanically induced change in the optical characteristics of the multilayer stack’s constituent dielectric materials. The injection of free carriers into a forward biased stack causes semiconductor bandfilling, band-gap shrinkage, and free-carrier absorption within the semiconductors; all of which contribute cumulatively to a change in refractive index. At a photon energy of 1.2 eV (1033 nm), GaAs experiences a negative change in refractive index of about one percent in response to the introduction of a free carrier concentration of ~2×1018/cm3 . A similar change is expected for AlAs. Because the absorption band edge of AlAs occurs at a shorter wavelength than that of GaAs, the refractive index of AlAs will not change at the same rate or level as that of GaAs in response to a specific free carrier concentration. Care should be exercised in using this method of modulation to keep the rate and level of refractive index change in both semiconductors as similar as possible. Departures from a similar level of refractive index change contribute to transmission function figure distortion, causing an exaggeration of the ripples found within the transmission region and resulting distortion of the transmitted signal.
We previously discussed the effect of the field-induced QCSE as applied to the modulation of an optical signal transmitted by the DBR configuration. This approach to signal modulation produces localized changes in the refractive index of GaAs by slightly increasing its value within the quantum confined region. Estimates of switching times for field-effect, multilayered devices similar to the modulator configuration described here are on the order of several hundred picoseconds , with this speed being limited primarily by the resistance capacitance (RC) time constant of the individual element. This electrorefractive approach provides, at best, only a few nanometers of shift in the transmission function, resulting in only a fraction of the desired full-cycle phase modulation. Also, because the quantum wells are local to only the GaAs layers, the QCSE tends to modify the stack’s optical characteristics in a nonsymmetrical fashion, deforming the transmission function’s carefully optimized figure and causing distortion in the transmitted optical signal. Asymmetric quantum well construction provides improved refractive index modulation , but is still a phenomenon localized to only the GaAs layers. We therefore conclude that the field-induced QCSE is not the most effective method available for modulating the wide bandpass region of the bandpass design.
Doping the semiconductor materials with a medium that exhibits a refractive index nonlinearity in response to optical intensity is another approach to providing the requisite refractive index change. This technique allows the refractive index of the stack’s layers to be modulated through the combination of an optical pump beam and the propagating optical signal. Implementation of this modulation method requires a similar nonlinearity in both materials within the stack. The doping of only one of the two materials within a DBR configuration to produce an optical limiting switch near the reflective bandgap has been suggested  for use in the implementation of an optical switch, but does not maintain the required balance of refractive index change in both materials. An imbalance of refractive index change results in the deformation of the transmission function figure and distortion of the transmitted signal.
6.2 Modulation Results
For purposes of discussion, we examine the relationship of the transmission function to the angle of incidence of the transmitted optical signal. As the angle of incidence increases, the optical thickness of the layers as experienced by the propagating signal increases correspondingly. An increase in the angle of incidence provides an approximation to the effects resultant from a uniform change in refractive index throughout the stack’s layers. At normal incidence, both p and s polarizations of the optical signal exhibit the same transmission spectrum. Off normal incidence assessments must be made separately for both the p and s polarization states. Figs. 4 and 5 illustrate the p and s polarization cases respectively.
By interrogating the structure with an optical signal of p-type polarization, we find that the signal’s center wavelength is modulated by π radians at a 22.60 degree angle of incidence. The signal’s center wavelength is modulated by a full 2π radian cycle at a 33.36 degree angle of incidence. Likewise, by interrogating the structure with an optical signal of s-type polarization, we find that the signal’s center wavelength is modulated by π radians at a 22.66 degree angle of incidence. The signal’s center wavelength is modulated by a full 2π radian cycle at a 33.00 degree angle of incidence.
Wavelength components on either side of the signal’s center wavelength are modulated by slightly different amounts as compared to the amount of modulation imparted to the signal’s center wavelength, resulting in the “saddle” shape of the phase modulation plots in Figs. 4 and 5. This wavelength-dependent variation in modulation level reveals the existence of some residual amounts of group velocity dispersion and other higher order phase distortions near the signal’s bandwidth region. Though the process is more computationally demanding, these residual amounts of pulse phase distortion may be attenuated through further computational optimization of the bandpass configuration’s stack design. The same optimization techniques used to create the bandpass region of high average transmission may also be used to develop bandpass configuration designs with reduced levels of phase distortion. Optimization routines generally rely on the evaluation of a defined merit function to determine when design is approaching some local minimum. By using both transmission level and phase distortion as inputs to the optimization routine’s merit function, the optimizer can generate stack design solutions that not only minimize the variations found in the filter’s bandpass region, but that also minimize higher order phase distortions. Further minimization of these undesirable phase distortion terms will flatten out the saddle shape, causing a more uniform phase modulation across the entire signal bandwidth.
The predicted performance of our bandpass configuration as a broadband phase modulator compares favorably to commercially available phase modulating products. Operating frequencies of commercial electro-optic phase modulators vary from the MHz range for broadband modulators to the GHz range for resonant, narrowband modulators. Depending on the modulation technique employed, and the final dimensions of the device, our bandpass modulator could provide broadband modulation rates at least as high as the GHz range. As discussed previously, the bandpass configuration features the additional capability of signal switching, a process that is implemented at the same operating rate as phase modulation.
The interaction length of the optical signal with the bandpass configuration’s dielectric stack is on the order of micrometers as compared to the longer interaction length required of the electro-optic bulk crystal phase modulator, which is on the order of millimeters. Also, because the fabrication techniques used in creating dielectric stacks are compatible with those used to pattern and fabricate microelectronic devices, these broadband phase modulators have potential application as individually controlled elements in densely packed optical arrays . Each modulator within the element array may be configured to impart a specific amount of phase modulation to a transmitted broadband optical signal, providing the means for producing directional optical beam steering.
6. Summary and Conclusions
We have introduced an optical interference coating design that exhibits a wide bandwidth region of high average transmission, capable of imparting a near-uniform phase modulation to all contributing frequency components of an incident optical signal. The phase modulation has a full-cycle range, causes minimal loss, minimal reflection, and minimal distortion. For purposes of contrast, we examined the common DBR interference coating configuration as an optical signal phase modulator and found it to be deficient in transmitting broadband signals through its narrow transmission resonances. As the DBR is modulated, severe variations in transmission level occur, as well as a distortion of the transmitted signal.
We initially presented a 79-layer specification for the bandpass modulator. To improve the transmissive qualities of this configuration, we employed a computational optimization known as “needling” to strategically split specific layers of the modulator’s high index dielectric with a thin inserted layer of the other constituent dielectric. The optimization resulted in a 91-layer structure that exhibits a transmission bandpass region with a higher average transmission level than that of the initial design.
Though more computationally challenging, further bandpass configuration performance enhancements are possible if the optimization routine calculates its merit function using not only the configuration’s transmission characteristics as input, but also data describing the amount of phase distortion present in the configuration’s transmitted signal phase function. We intend to investigate this method of bandpass modulator design improvement through further enhancements to our optimization merit functions.
Large segmented aperture systems requiring a lightweight solution to rapid, active, wavefront correction could benefit from our technology development presented here. We further suggest the applicability of our compact phase modulation technology to other uses, such as optical switching, and optical beam steering applications.
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