telecommunications component modeling in asap · the methodology presented here serves as a general...

ASAP TECHNICAL PUBLICATION BROPN1152 (MARCH 24, 2008)

Telecommunications Component Modeling in ASAP

This technical publication describes how to compute coupled fields and coupling coefficients in the Advanced Systems Analysis Software (ASAP®) from Breault Research Organization (BRO). The emphasis of this document is on the steps necessary to compute coherent coupled fields and coupling coefficients in ASAP. A brief discussion is also included on some of the important issues in modeling physical optics phenomena that are found in telecommunication devices. The coupling calculation is illustrated in two real-world systems: a ball coupling lens, and an optical feedback system that produces narrow linewidths for dense wavelength division multiplexing (DWDM) applications. The methodology presented here serves as a general roadmap for performing other types of coupling calculations in ASAP.

Figure 1. ASAP beams, traced from a diode laser through a ball lens to study fiber coupling efficiency

For more detailed information regarding general physical optics modeling in ASAP, please see the technical guide, Wave Optics. This and other BRO technical publications referenced in this document may be viewed or downloaded from the BRO Knowledge Base.

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Telecommunications was once exclusively the science and technology of communication using electronic signals transmitted by wire or radio wave. However, the inventions of the solid state laser (laser diode) and fiber optics in the last part of the 20th century heralded an entirely new way to transmit significantly broader bandwidth information with optical signals: light. Now, the general goal in fiber optics telecommunications is to increase the data transmission rate even further, using techniques such as wavelength division multiplexing (WDM). WDM, or frequency division multiplexing as it is known outside of the optics discipline, achieves high transmission data rates. It does this by simultaneously transmitting many signals from many separate coherent laser diodes operating at different wavelengths through the fiber. Other optical devices are changing the telecommunications market as well. Light emitting diodes (LEDs), the incoherent cousin of the laser diode, are used in all types of information display systems, such as the illumination source for a cell phone’s liquid crystal display. Waveguides are used to “pipe” light into regions inaccessible with conventional optics. In both these cases, the ability to simulate sources and telecommunication components to optimize the efficiencies or power transfer between these systems in the “bulk” is an important element of telecommunication component modeling. For example, it is important to maximize the amount of power coupled from the laser into a single-mode fiber to minimize transmission losses. To do so, the mode of a laser diode and coupling system should match the mode of the single-mode fiber it is entering. Calculating coupling efficiencies is not only important in the case of a single-mode fiber but in other components as well, such as spectrally tuned WDM systems. In the case of an LED, you want to get as much light as possible to the light crystal display (LCD), with uniform illumination.

System performance requirements and physical properties The general optical system performance requirements (software capabilities) dictate the optical configuration and obviously the performance metrics. There are many different performance requirements for telecommunication devices. However, an important performance requirement is how much light from your source is coupled into the optical system or, for that matter, back into the source. This requirement is most commonly expressed as the coupling coefficient. For example, when coupling light from a source such as a laser diode into a fiber, you are typically interested in how much power from the source is coupled into the fiber. This calculation is dependent on a number of parameters of the source, the coupling optics, and the fiber. Fortunately there are common threads that bind the calculation of the system’s coupling coefficient together. These common threads are found in fundamental characteristics that completely specify the behavior of electromagnetic (E&M) radiation. In fact, this information is what is needed to compute the coupling coefficient. These characteristics are coherence, polarization, amplitude, and phase. Light and its interaction with optical elements are physically and mathematically distinguished and characterized primarily by coherence, polarization, amplitude, and phase. Given these four characteristics and the optical system behavior, we can compute the coupled-mode behavior and coupling coefficient of a wide variety of telecommunication systems. ASAP, unlike many other analysis codes, can simulate all of them. The source of E&M radiation and how it interacts with your optical system determine the relationships between the optical performance requirements and important physical properties. The physical properties you need to simulate physical optics are derived from the optical performance requirements, and are used as input into the software design tools. Table 1 illustrates how coupled-mode behavior and coupling coefficients are related to physical properties and the required software capabilities.

2 Telecommunications Component Modeling

NOTE The important point to emphasize is that these inputs dictate the type of software tool needed for simulation and not the reverse.

Table 1. Relationship between performance requirements and physical properties

Optical Performance Requirements Physical Property/Software Capability (feature) Coupled-mode behavior (coupling coefficient) - Coherence (temporal/spatial)

- Polarization (Fresnel relationships) - Amplitude (Fresnel relationships) - Phase (aberrations - Coupling

The software tool must have capabilities to accurately simulate the physical properties that affect performance. ASAP automatically changes the polarization, amplitude, and phase of light as it interacts with optical components. For example, ASAP changes the polarization and amplitude of light incident on an interface according to Fresnel’s equations. ASAP also adjusts the phase of the light according to the indices of refraction, optical path length (OPL), and aberration of the optical components. ASAP uses this information to compute the coupling coefficient.

Physical Optics in ASAP The technique ASAP uses to simulate diffracting sources like laser diodes, and in general other physical optics phenomena, is called Gaussian beam superposition. This technique is substantially enhanced to include the effects of coherence, polarization, amplitude, and phase, which are fundamental characteristics for properly simulating the physical optics phenomena that influence the coupling calculation. Understanding the method ASAP uses to simulate field propagation is important because actual optical fields are used in an overlap integral to compute the coupling efficiency.

Gaussian beam superposition method The Gaussian beam superposition algorithm in ASAP models arbitrary optical fields as a summation of fundamental Gaussian beams. The superposition algorithm is similar to Babinet’s principal, except that ASAP uses Gaussian beams as the basic functions. Furthermore, the Gaussian beams are each represented by a set of real rays describing the coherence, polarization, amplitude, and phase of the Gaussian, which may be traced through the opto-mechanical system. This set of rays characterizes the near-field and far-field properties of a Gaussian beam.

Telecommunications Component Modeling 3

It is possible to superpose the ray trace information from the individual Gaussian beams to recreate an arbitrary optical field at any position in space. Figure 2 illustrates the concept.

Figure 2. Gaussian beam superposition and propagation theory in ASAP

Advantages of Gaussian beams The advantage of using Gaussian beams is that they are mathematically well understood, well behaved, and their propagation can be represented by a set of geometrical rays. Furthermore, Gaussian beams retain their shape as they propagate through free-space. And, unlike plane waves, which exist everywhere in space, the Gaussian profile (apodization) falls off rapidly with radial distance. If a Gaussian beam interacts with the local curvature of a well-behaved optical element the Gaussian retains its shape after reflection or refraction. The coherence, polarization, amplitude, and phase of each beam are always known and information from the geometrical ray trace can be used to recreate the Gaussian beam at any position in the optical system. The reconstruction and summation of all the Gaussian beams allows you to recreate the optical field at any position in the optical system. This is all done without Fourier or Fresnel transforms or integrals. Propagation of an individual Gaussian beam is done with geometrical ray tracing (see Figure 2). The location and direction of a Gaussian beam is determined by a single ray called the “base ray”. The base ray is just like the ray used in lens design codes to set up point sources from grids of rays. The base ray is accompanied by additional rays, known as parabasal rays. “Para” means around and parabasal rays are rays around the base ray. The first two parabasal rays describe the width and divergence of the Gaussian beam. The parabasal rays are traced just like the ordinary base ray. As their size and divergences change, so does the size and divergence of the Gaussian beam. Two additional parabasal rays are traced in planes orthogonal to the basic parabasal rays. The four parabasal rays allow the Gaussian to acquire a different width and divergence in each local paraxial axis (along the base ray), and in doing so allows the Gaussian beam to become generally astigmatic. The mathematical relationships between the widths, locations, and directions of the base and parabasal rays allows the amplitude and phase of the Gaussian beam to be reconstructed at any point in the optical system even as the base and parabasal rays become completely intertwined. As each individual Gaussian beam propagates through the optical system, its amplitudes and phases are changed respectively by the local diattenuation (polarization selectivity) and aberration of the optical element. In this sense, each Gaussian beam samples a local area of the optical surface, and the summation of


all the Gaussian beams constitutes the polarization changes, power losses and aberrations imparted to the optical field by the surface. Each Gaussian beam, and thus the entire wavefront, is then rigorously propagated through virtually any optical system. The resulting field can be calculated and displayed at any location or skew plane within the optical system and not just near focus. This can be done without any complex matrix techniques, Fourier or Fresnel transforms, or integrals. Furthermore, the effects of multiple beam paths and ghost imaging are simulated in straightforward way, since geometric rays can be easily split at interfaces with non-zero reflection and refraction coefficients.

The way in which the Gaussian beams interact to recreate the optical field is a function of the coherence of the source and the calculation method used to superpose the beams. For example, the Gaussian beams simulating a monochromatic point source are coherent with respect to each other. This means that ASAP adds their electric field amplitudes linearly to calculate the composite optical field. If the individual beams are incoherent with respect to each other, ASAP automatically linearly superposes the squared modulus of the electric fields, which is the energy density, to calculate the optical field.

For example, a polychromatic source in ASAP is a series of spectrally apodized monochromatic sources. The Gaussian beams constituting the monochromatic source are coherent with respect to each other, but are incoherent at different wavelengths. ASAP linearly adds the electric field amplitudes of the coherent Gaussian beams, while linearly adding their energy densities at different wavelengths.


Combining Ray Tracing with Physical Optics Calculations Combining ray tracing with a physical optics calculation offers many advantages over other techniques, and ASAP has the capability to simulate it all. 1. Ray tracing is an accurate and efficient way to simulate the propagation of light in optical systems with

simple or complex geometrical surfaces. You now need to construct only a single geometrical model. As the rays describing each Gaussian beam propagate through the optical system, their amplitudes and phases are changed respectively by the local diattenuation and aberration of the optical element. In this sense, each Gaussian beam samples a local area of the optical surface, and the summation of all the Gaussian beams constitutes the polarization changes, power losses, and aberrations imparted to the optical field by the surface. Each Gaussian beam, and thus the entire wavefront, is then rigorously propagated through the optical system. The resulting field can be calculated and displayed at any location/skew plane within the optical system (including on the surface of grazing incidence optics), and not just near focus. This can be done without any complex matrix techniques, Fourier or Fresnel transforms, or integrals.

2. The effects of multiple beam paths and ghost imaging are straightforwardly simulated since geometric rays can be easily split at interfaces with non-zero reflection and refraction coefficients.

3. Rays can be non-sequentially traced through optical systems resulting in light propagation along physically realizable paths.

You can simulate complicated telecommunication-component optical and mechanical geometries including ball lenses, aspheric coupling lenses, diffraction gratings, apertures, and their optical properties (indices of refraction including gain and absorption, gradient index, coatings, scatter). The Gaussian beam superposition algorithm in ASAP is extremely powerful; however, it does have some limitations. When you simulate aperture dimensions (or object spatial frequencies) that are near to or below the wavelength of light, you can still simulate diffraction gratings, small apertures, or calculate coupling efficiencies in fibers. Diffraction gratings are handled macroscopically with the grating equation and not by constructing the small structure of the grating. In a similar sense, apertures are handled with a special decomposition technique instead of simply “clipping” Gaussian beams. This limitation generally implies that ASAP cannot simulate physical optics phenomena that do not have a ray equivalent model. Propagation in a single-mode fiber is an example. This is purely a diffractive phenomenon with no equivalent ray trace model. However, you can still simulate such optical systems in ASAP. ASAP provides “hooks” to input and output arbitrary fields that can be decomposed into Gaussian beams. So, for example, even if ASAP cannot simulate propagation in a single-mode fiber, you could still model the optical system up to the fiber, calculate the coupled-mode behavior, and output the complex field at the fiber face to an external simulation software that simulates single-mode fibers. You could then decompose the results from the external software back into an ASAP optical field to continue the simulation in ASAP.


Coupling Calculation in ASAP The coupling coefficient is a mathematical representation of how the mode of the incident light “couples” or is “accepted” into an optical system, such as a fiber, in terms of its inherent mode structure. The coupling coefficient is the fraction of the power of the incident optical field mode that is coupled into the propagating mode of the optical system. In our special case, this is a single-mode fiber.

Light propagating in single mode fibers and, in general, very small waveguides is a purely diffractive phenomena. Light propagation in the fiber can be described in the form of transverse modes. The superposition of the modes yields the resulting field. This behavior is mathematically similar to the Gaussian beam superposition algorithm in ASAP. Complex optical fields are represented by the superposition of a “mode”, in this case a fundamental Gaussian beam, whose propagation is described by a set of real rays. The fundamental Gaussian beam is one mode of the family of Hermite-Gaussian beams.

The fiber will support the propagation of only certain mode types. This is dependent on the geometry and optical properties of the fiber. Furthermore, the modes that are excited in the fiber are dependent on the incident light. Recall that incident light can be represented in terms of a summation of modes. Depending on how the modes in the incident light match those that are supported by the fiber determines how much light is coupled into the fiber and will propagate through the fiber. The incident light is sometimes said to excite a certain mode behavior in the fiber. Single mode fibers allow only one type of mode excitation, which is approximately a Gaussian in shape. Multimode fibers can be excited by multiple modes as opposed to a single mode.

The coupling calculation in ASAP is different from that typically found in most lens design codes. Many of those calculations are based upon the paper by R. E. Wagner and W. J. Tomlinson, "Coupling efficiency of optics in single-mode fiber components," Appl. Opt. V. 21, n. 15, p. 2671 (1982). Wagner and Tomlinsons’ method requires computing the overlap integral of the far-field system amplitude distribution and the fiber amplitude distribution. This result is then multiplied by the coherent optical transfer function (OTF) of the coupling system.

The method used by ASAP does calculate an overlap integral to compute the coupling efficiency. However, ASAP does this calculation by directly operating on the amplitude distribution from the system and a mathematical distribution describing the fiber mode. The diattenuation and phase change, normally represented in the OTF of the system, are inherently present in the system amplitude distribution through the individual Gaussian beams. The individual Gaussian beams discrete sampling of the power losses and aberrations, as a function of field and pupil, each contribute to the superposed system amplitude.

The mathematically normalized coupling calculation in ASAP is operationally defined as follows:

∫∫∫∫∫∫

ψψψψ

ψψ==η

dxdy)y,x()y,x(dxdy)y,x()y,x(

dxdy)y,x()y,x(c

*fiberfiber

*syssys

2*fibersys2

Here ψsys is the optical field of the input light and ψfiber is the fiber optical field mode.

ASAP ball lens coupling example ASAP has a recommended analysis methodology that is built directly into its interface, which is demonstrated in this example. This analysis methodology consists of three basic steps: 1. Constructing and verifying the optical system model 2. Constructing and verifying the source model 3. Ray tracing and analysis


In particular, we examine the coupling behavior between a laser diode and a fiber using a ball lens as the coupling optical system using this methodology.

Constructing and Verifying the Optical System Model ASAP accurately models most optical/mechanical systems and their optical properties. ASAP is similar to 3D solids modeling software in that it utilizes a powerful geometrical modeling approach that permits a nearly limitless variety of systems to be constructed in a straightforward manner. Geometries can be constructed in its Builder user interface (UI), translated from an IGES file, translated from a lens design file, or in the command language. As opposed to sequential ray tracing algorithms found in typical lens design codes, all surfaces may interact with rays, and are referenced by default to a single global coordinate system. Smooth, continuous object surfaces can be represented by a sequence of simple conicoids or a general 286-term polynomial. Therefore, anything from a simple plane to an arbitrarily oriented elliptical toroid can be modeled precisely. Further, ASAP can simulate parametric mesh surfaces (NURBS) that are common in Computer-Aided Design (CAD) tools. In fact, polynomial and parametric entities can be used to clip each other. ASAP allows you to create optical properties databases that describe real and complex refractive indices with absorption or gain, complex multilayered coatings, uniaxial crystals, and scattering surfaces. The standard Fresnel equations are used not only to calculate (as a function of incident angle and polarization) transmission losses at interfaces between two dielectric media, but also reflection losses at any dielectric/conductor interfaces. ASAP is capable of splitting any ray into transmitted, reflected, diffracted, near specular, diffuse, and back-scattered components. The basic optical system and the coupling calculation can be simulated with the UI in ASAP, or through the flexible command language. We will illustrate the ball lens coupling geometry with both the UI and command language in ASAP. The command language is used for the remainder of the example. While script files (files containing ASAP commands) can be complex, they bring out the power of the application. They are often the most efficient and compact way to set up ASAP geometry, sources, and commands for analyses. Furthermore, some of the commands and syntaxes in ASAP are not fully implemented in the UI. The UI is complete enough to be a primary tool for many users, or an aid in learning ASAP before moving into the ASAP command language. Our ball lens coupler is probably the simplest optical coupling system outside of direct or butt coupling. In direct or butt coupling, the laser is placed directly against the fiber. The ball lens is symmetric so the orientation of the lens is irrelevant. However, other factors such as fiber positioning are relevant and affect the coupling efficiency, which is demonstrated at the end of this example. A symmetric ball optical system has many advantages. However, it imparts a large amount of spherical aberration to the laser diode wavefront. Spherical aberration is the only significant aberration present because the extremely small fiber cores limit off-axis aberrations. The ball lens couplers are typically used at large magnifications, with small diameters and large refractive indices, to minimize the effects of spherical aberration. This aberration changes the modal behavior of the light from the laser diode, and therefore the coupling of the laser diode light into the fiber. In the ball lens example, the system is set up for a –4x magnification. The focal length of the ball lens is slightly less than the ball radius and is therefore inside of the ball lens. This magnifies the laser diode waist of 1.5 microns to approximately 6 microns at the fiber. The laser diode is placed at the appropriate distance in front of the ball lens. The detector is placed beyond the –4x Gaussian image plane to show the geometric caustic primarily due to spherical aberration in the ray trace plot.


We start by building and verifying the system geometry. Figure 3 illustrates the ball lens geometry and a simple circular detector plane built in the ASAP Builder. Figure 3 contains the ASAP main window, the Builder, the 3D Viewer, and the Command Output window.

Figure 3. ASAP Builder, 3D Viewer, and Command Output windows

The Builder is a spreadsheet-like interface with menus for defining the geometry, properties, and sources, of the system, as well as performing analyses. ASAP is “object”-based software in the sense that we first construct databases describing the optical properties and system geometries, and then we associate items in the database to objects, which are the parts ray traced in ASAP. In addition to defining the databases, you can also define variables and mathematical functions for use in ASAP commands within the spreadsheet. A graphical verification of the system geometry is shown in the ASAP 3D Viewer. Abundant numerical information is also available to verify that your optical system is constructed correctly. In our example, we do not need to construct the fiber geometry, because we require only its mathematical model description to perform the coupling calculation.


Figure 4 illustrates the geometry set up directly in the ASAP command language. Since the command language is really an optical programming language, you are free to write in your own style within the syntax. In fact, the resulting Builder information is eventually converted into a similar command script before it is processed in ASAP.

Figure 4. ASAP command language

Constructing and verifying the source model Sources and their interactions with optical elements are physically and mathematically distinguished and characterized primarily by the four basic parameters:

1. Coherence 2. Polarization 3. Amplitude 4. Phase

Given these four characteristics, we can completely describe the behavior of light within an optical system and calculate the coupling coefficient. All these parameters for many sources and optical elements can easily be set up with the ASAP UI or through the command language. We will simulate a simple, unpolarized laser diode in the script language as an example of how to set up a source in ASAP. In our example, we set up the laser diode source as a command macro that can be called at any time during an ASAP analysis. ASAP requires the fiber mode to be mathematically defined for the coupling calculation, so it is also created as “source” macro. Macro programming is a powerful feature that allows multiple commands and procedures to be combined into a common set that is run with a single command.


Coherence When we speak of the coherence of a source of radiation, we are usually referring to its amount of spatial and temporal coherence. Spatial coherence has to do with the extent or size of the source. In other words, how well do the point emitters of the extended source, emitting radiation at the same wavelength, emit in spatial phase with respect to each other? A point source is by definition spatially coherent. Temporal coherence has to do with the monochromaticity or bandwidth of the source. In other words, how well matched in phase is the radiation from a point source or points of an extended source, emitting at different wavelengths? Although a monochromatic source is temporally coherent and completely polarized, it is a physically nonexistent source. Coherent sources typically are “quasi” monochromatic point sources, such as lasers that emit radiation over a very narrow spectral band. Incoherent sources typically are extended sources such as filaments that emit radiation over a broad spectral band. NOTE ASAP can simulate optical fields that exhibit total or partial temporal coherence. ASAP can also perform total or partial spatial coherence calculations with appropriate combinations of spatially distributed coherent sources. Source coherence in ASAP is easily defined by telling ASAP whether the source is coherent or incoherent. This can be done in the UI or in the Command Input window. Figure 5 illustrates coherence definition in the ASAP Editor window, via a split screen. .

Figure 5. Defining the source coherence, wavelength, parabasal rays, amplitude, and phase

The BEAMs command sets up the spatial coherence of the source. To give the source a temporal component, we simply assign a wavelength to the source. Appropriately spaced, individual, monochromatic sources are used to define a bandwidth for a polychromatic source. Each monochromatic source can be flux weighted to simulate spectral apodizations or spectral power distributions. In this example, we are also defining the source in a macro called LASER_DIODE, delimited by the {}. This macro can be called at any time with the single command, $LASER_DIODE. This causes the contents of this macro to be run in ASAP. A similar macro is created for the fiber mode, called FIBER_MODE, which is also described by a set of rays in ASAP. After the ray trace, ASAP uses the information from the complex


amplitude distribution of this ray set, with a similar output from the laser diode and ball lens in the overlap integral of the coupling efficiency calculation. The next part of our source definition is to assign parabasal rays to the source. Parabasal rays are geometrical rays that simulate a Gaussian beam and its propagation through an optical system. The superposition of these individual Gaussian beams yields the resulting optical field. To add parabasal rays to the command file, you set the PARABASAL and WIDTHS commands to the recommended parameters. This is illustrated in Figure 5. Defining the source coherence, wavelength, parabasal rays, amplitude, and phase.

Polarization The vector description of the electric field of the source is the polarization of the source. A polarized electric field exhibits a preferential behavior in the direction of the vibration of the transverse E&M wave. For example, a linearly polarized wave’s electric field vibrates in a plane, which also contains the propagation vector (direction). In natural or unpolarized light, the atomic dipoles radiate randomly so that any single resultant polarization state is unrecognizable. ASAP allows you to create and simulate polarized light and its interaction with polarizing components that polarize by reflection/transmission (diattenuation), dichroism (selective absorption), and birefringence (phase retardance). ASAP sources can be polarized in a variety of handedness, orientation, and ellipticity. This is done using the POLARIZ command. The POLARIZ command input variables may be interpreted as Jones vector coefficients, if the source model is set up in a specific orientation with respect to the global coordinate system. The axis parameter of the POLARIZ command designates the first polarization component direction; the second direction is chosen to be perpendicular to both the first polarization component direction and the direction of propagation. Adjustments are sometimes required for rays that are skewed relative to these components. In our laser diode example we assume unpolarized light. For an example of a polarized source, see the technical guide, Wave Optics. This and any other BRO technical publication referenced in this document may be viewed or downloaded from the BRO Knowledge Base, http://www.breault.com/k-base.php.

Amplitude The amplitude of a source is a manifest part of its oscillatory behavior. Many phenomena in nature exhibit oscillatory behavior. Sources of radiation, specifically their electric fields, are no different and are conveniently described mathematically by harmonic or periodic functions such as sines and cosines containing amplitude terms. The amplitude is the maximum extent of vibration of the E&M wave. The squared modulus of the electric field is the energy density. It is the quantity that detectors “detect”. NOTE ASAP allows you to create user-defined, arbitrarily apodized (spatial or angular flux weighting) sources. ASAP has many common, pre-defined point sources that you can use in your simulations. These point sources usually have uniform apodizations. Apodization is a term that mathematically describes how a source’s amplitude changes as a function of position or propagation direction. It is just a flux weighting of the power in the rays making up the source as a function of the ray’s position or propagation direction. Not all sources have uniform apodizations, certainly not our laser diode example. Fortunately, ASAP has a very flexible way for you to put in your own user-defined apodizations as mathematical functions or actual measured data. This is done in general with the USERAPOD and APODIZE commands. However, in our example we use a special GAUSSIAN command, which allows us to model both the amplitude and phase of laser-like sources in a single command. We create our laser diode source with only one ray and then decompose it into an angular spectrum of Gaussian beams to model the actual source rays propagating through the ball lens. This is done to create a set of Gaussian beams and their associated rays


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that sample the amplitude and phase (aberrations) changes of the laser diode light as it propagates through the ball lens. A single ray would not contain this information because it samples the lens at only one point. However, the single ray of the GAUSSIAN command is a convenient and easy way to define the laser diode’s basic emission properties. The decomposition technique is another way to create a coherent source from an existing ray set. It actually creates a set of Gaussian beams from a special complex matrix file and is the general “hook” to input and output arbitrary fields that can be decomposed into Gaussian beams. Decompose is commonly used when the parabasal rays from simple two-dimensional ray grids cannot be properly propagated through the optical system or when individual Gaussians are clipped by apertures, as in the case of spatial filters.

Phase The phase of a source is the fraction of an oscillatory cycle of a wave of the source, measured from a specific reference or fixed origin. It is usually represented as an optical path length of a ray in geometrical ray trace codes. NOTE With ASAP, you can create user-defined wavefront functions (phase functions), and even wavefront Eikonals (the complete mathematical description of the wavefront). You can assign many pre-defined phases to your grids.

The phase in the present context is the optical path length (OPL) of the rays at their starting points. The ASAP GRID command sets the spatial positions of the rays, and the SOURCE command sets the phase and propagation direction of the rays in the grid. However, the GAUSSIAN command we are currently using does both with one command. Keep in mind that we are setting up point sources and not extended sources. The wavefront from a point source is the mathematical surface over which the optical lengths of the rays are constant. Rays are everywhere perpendicular to the wavefront. Perfect wavefronts are spherical and a spherical wavefront at infinity is planar. Wavefronts that are not diffraction limited have aberration and are not spherical. Laser diodes are perfect examples. Some laser diode wavefronts are astigmatic, meaning that a fan of rays along one axis focuses at a different position than a fan of rays in the orthogonal axis. Its optical path difference map is shaped like a saddle. ASAP allows you to simulate virtually any wavefront that can be described by a 10th-order polynomial equation. See the technical guide, Wave Optics. The GAUSSIAN command simplifies sources that have a Hermite-Gaussian form.


The laser diode output from the GAUSSIAN command is shown in Figure 6. These various calculations show the effects of coherence, polarization, amplitude, and phase from our laser diode source.

Figure 6. Laser diode input perpendicular to (top, right) and along the propagation axis (bottom)

NOTE The calculations of the amplitude (irradiance) need not be done in planes orthogonal to the propagation direction. They can be calculated in any skew plane.

In our example, we are looking at the divergence of the source along its propagation axis. This source and its output were originally created with a single ray.


Ray Trace and Analysis Once you are certain that the system and source models are correct, you can then trace the actual source rays through the optical system. A ray race through the optical system is illustrated in Figure 7.

Figure 7. Ray trace through the ball lens

An optical field exists everywhere in between the output of the ball lens and the detector. We can calculate the complex optical field characteristics anywhere in between these two points. So even though the detector plane is not at the plane of –4x magnification, we can still calculate the optical field at this point. In fact, we can calculate the optical field along the axis to see the diffraction envelope or caustic. The diffraction and geometric envelopes are illustrated in Figure 8. Diffractive and geometric caustics. The complex optical field computed at the plane of –4x magnification is used in the overlap integral. Its irradiance pattern is illustrated in Figure 9. Field at fiber face (left) and fiber mode and fiber mode (center, right). The overlap integral is computed using this field and the fiber-mode field computed at the same location. The fiber mode is illustrated in the plot on the right of Figure 9. The resulting coupled fields are illustrated in Figure 10. Coupled fields. The output dialog is shown in Figure 11. User-defined output for ASAP fiber coupling calculation.

Figure 8. Diffractive and geometric caustics


Figure 9. Field at fiber face (left) and fiber mode and fiber mode (center, right)

Figure 10. Coupled fields


Figure 11. User-defined output for ASAP fiber coupling calculations

The macro language in ASAP allows you to customize input and output screens to pass and capture data to and from ASAP. The capability has been used to define menu environments that isolate a user from the ASAP command language. In this way, you can create special applications that your colleagues can use without ever seeing any ASAP commands. The coupling efficiency for our –4x magnification is about 37%. A fundamental question confronting all these types of systems is where to place the fiber for maximum coupling efficiency. This position determines where the maximum amount of light from the source is coupled into the fiber, and it is usually not in the plane of the best geometric focus. You need a physical optics diffraction theory to properly compute the plane of optimum coupling efficiency. We can find this position by using the powerful Gaussian beam superposition algorithm, its coupling efficiency calculation, and a parametric iteration technique, all in ASAP. The ASAP macro language allows us to automatically change the fiber position while re-computing the coupling coefficient at each new fiber position. The coupling efficiency can then be plotted as a function of fiber position. This is a general feature in ASAP that allows you to define virtually any independent variable and compute a resulting dependent variable’s behavior. This can be extended to source alignments, divergences, optical system parameters (radii, thickness, and rectilinear positions), fiber parameters, and even ghost reflections off the fiber. For example, you could calculate and plot the coupling efficiency as a function of the ball’s X and Y positions.


A plot of coupling efficiency versus fiber axial position is shown in Figure 12. Note that the maximum coupling efficiency is not at the Gaussian image plane but rather inside of this position.

Figure 12. Coupling efficiency as a function of fiber position

ASAP WDM Example Telecommunication engineers have sought to increase bandwidth through higher transmission data rates. This can be done by increasing the source transmission data rate, but only up to a point that is physically achievable. Wavelength division multiplexing (WDM) accomplishes this by using multiple laser diode sources operating at separate wavelengths. Each wavelength, in essence, carries a small part of the total message and each diode constitutes a “channel”. Different parts of the information are transmitted simultaneously at different wavelengths resulting in an increase in the total data transmission rate. The light from many laser diodes closely spaced in wavelength can then be simultaneously propagated (for example, through a fiber), resulting in a higher transmission data rate through channel multiplexing. Even higher transmission data rates can be achieved by decreasing the wavelength spacing between laser diode sources while decreasing the laser diode linewidths. If the linewidths are not decreased, the channels will overlap and become indistinct. Laser diodes have finite bandwidths, and this eventually limits channel spacing. In other words, they emit light not at one single wavelength, but over a range of wavelengths. The bandwidth is caused by changes in the cavity spacing due to things such as thermal and mechanical variations. One way to decrease the laser diode linewidths is through optical feedback. If light of a certain wavelength is re-injected back into the laser it will spectrally tune the subsequent laser output to that “seed” wavelength. Wavelengths different from the tuned wavelength will couple poorly back into the laser diode, which is in fact a waveguide, resulting in a narrower emission bandwidth. One way to spectrally tune the laser diode through optical feedback is with a grating and a mirror, which is a Littman-Metcalf configuration. Light from the laser diode propagates towards a diffraction grating that is oriented near grazing incidence. The diffraction grating is designed to out-couple the zeroth order while directing the minus first order to the mirror. Remember that the laser diode light has a bandwidth and the minus first orders of all these wavelengths are diffracting towards the mirror. The mirror, named the tuning mirror, redirects the minus first orders onto the diffraction grating. Spectral tuning is accomplished by rotating the tuning mirror about the grating center. By the principle of reversibility, only a small range of wavelengths, determined by the tuning mirror angle, propagate toward the


grating, diffract along the minus first-order path, and couple into the laser diode cavity. Wavelengths not reflected along this path couple poorly into the laser diode cavity. The laser diode emission wavelength is “pulled” towards the tuned center wavelength resulting in a narrowed emission linewidth at the tuned, center wavelength. The ASAP model consists of a laser diode source similar to that discussed in the ball lens example, but now with a finite bandwidth and a collimating lens from the ASAP library of stock vendor lenses. The finite bandwidth is obtained by spectrally apodizing a series of monochromatic laser diode sources to a Gaussian spectral distribution centered at 830nm. The spacing between adjacent wavelengths is 0.0025nm. Eleven sources are initially created. The original bandwidth is shown in the first curve of the graph in Figure 13.

Figure 13. Littman-Metcalf system (top), and a spectral graph illustrating resulting line narrowing (bottom)

The laser diode light is collimated and projected onto a grating, with a grating period of 1.0 microns at 80º from the normal. Approximately 5cm separates the planar tuning mirror and grating. The optical system is shown in Figure 13. Littman-Metcalf system (top), and a spectral graph illustrating resulting line narrowing (bottom). The coupling coefficient and the coupled spatial and angular distributions of each individual source as a function of wavelength are computed with exactly the same technique used in the ball lens example. The coupling efficiency is calculated as an overlap integral and the entire computation is parametrically iterated in wavelength. The results for a single round trip through the cavity are illustrated in the graph of Figure 13. The second curve is the coupling coefficient, and the third curve is the resulting laser spectrum. The laser bandwidth is reduced by about a factor of two after one round trip.


Summary The Gaussian beam superposition algorithm in ASAP, with its ability to simulate the coherence, polarization, amplitude, and phase of an optical field, is a powerful and flexible technique that allows you to simulate the physical optics behavior of a wide variety of telecommunication optical systems in the bulk. The Gaussian beam superposition algorithm combines geometrical with physical optics by joining the unique capability and usefulness of geometrical ray tracing with the physical effects of diffraction, interference, and polarization. The change in the coherence, polarization, amplitude, and phase of an optical field is automatically computed as the rays of the individual Gaussian beams propagate through the optical system. ASAP performs coupling efficiency calculations by directly operating on the amplitude superposition of these Gaussian beams and a mathematical distribution describing the fiber mode. The diattenuation and phase changes normally represented in the optical transfer function (OTF) of the system are inherently present in the system amplitude distribution through the individual Gaussian beams. Each Gaussian beam discretely samples the power losses and aberrations as a function of field and pupil coordinates, and contributes to the superposed system amplitude and coupling efficiency. Because of its ray trace approach and similarity to 3D solids modeling software, ASAP permits a nearly limitless variety of optical systems, and allows the effects of physical optics to be simulated in a straightforward manner. Common telecommunication components, such as aspheric coupling lenses, diffraction gratings, and gradient index optical components (GRINS), can be simulated in ASAP. These capabilities, coupled with the ability in ASAP to split rays into reflected, transmitted, diffracted, and scattered light, make ASAP the most realistic commercially available simulation tool for telecommunications component modeling in the bulk.


telecommunications component modeling in asap · the methodology presented here serves as a general...

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