Lock-in amplifiers are key devices in numerous instruments used in the optical sciences or in optical equipment in industry. In many experimental configurations, they represent the means to reliably detect and record small or weak signals. The purpose of this text is to provide a step-by-step introduction to this technique. The first part explains how modulation is used to extract a signal from noise and describes different types of modulation. The focus shifts from electronics to optics in the second part, which covers lock-in amplifier applications in optical instruments. The book is intended for readers who want to better understand instruments and experiments based on lock-in detection and/or to design (and perform) new experiments in which lock-in amplifiers are applied.
The tolerancing of optical devices is of importance in many industrial applications. Illumination devices that produce a
cut-off line in the light distribution are examples where demanding requirements have to be met.
The approaches that will be presented are used as a prerequisite to make these illumination devices more robust, i.e.
relatively insensitive to position tolerances and to adverse conditions under which they have to be operated. Emphasis
will be laid on LED optics that feature high optical efficiency.
The method of optical transfer matrices represents a convenient tool to study optical devices in a paraxial approximation.
The results of such calculations also provide a guideline on how to plan the numerical simulations. This is helpful,
because the simulations that have to be performed in this branch of illumination engineering are quite time-consuming.
To sum up the simulation results abstract representations will be used. They are considered helpful to keep the overview
in a wealth of data gained by automated simulations and to communicate them in a concise way.
Tolerancing within tight requirements prescribed by legislation for safety reasons, as it is typical for frontlighting
applications, will be addressed.
While building and adjusting interferometric sensors it is useful to have a clear picture of how different adjustment errors
may affect optical properties of the device. While operating the interferometer it is helpful to know how different
sensitivities of the instrument influence the measurement. The well-known matrix method provides a convenient means
to study these effects systematically in a first order of approximation.
It can be applied to calculate the influence of axial misadjustment errors, the influence of tilts as well as the effect of
decentering. It will be illustrated that a relatively limited set of mathematical approaches suffices to treat these
sensitivities and to study their interplay. It forms a kind of toolbox that can be used to analyze interferometer layouts in a
The emphasis will be laid on two-beam interferometers that are described after unfolding the reference and the
measurement arm, respectively. The method is advantageous to check compensation schemes and it does allow for a
first-order balancing of the arms of two-beam interferometers. Extension to anamorphic optical systems, i.e. optical
arrangements that feature a low symmetry, is possible and the influence of a rotation error with respect to the optical axis
can be investigated as well.
An advantage of the matrix approach can be seen in the fact that it also facilitates to communicate essential features of
the interferometric device in a concise way. The attempt is made to present it in a way that it might be applied to a
variety of interferometric sensors.
Optical phase conjugation designates the "reflection" of a ray of light back into the direction from where it came. This
physical effect can be realized by sophisticated non-linear optical devices. Examples are phase conjugating mirrors based
on stimulated Brillouin scattering. Phase-conjugated mirrors are advantageous in instrument design, because they can
make an optical instrument less sensitive to misalignments or a laser resonator less sensitive to optical aberrations caused
by a phase distorting medium.
As quasi-phase-conjugated devices we might designate devices that show this special feature in an approximated way.
They can be realized with less technical and scientific effort and still offer an improvement in system stability and
instrument insensitivity. An established way to make quasi-phase-conjugated devices is by implementing arrays that are
made up of tiny retro-reflectors. These retro-reflectors may be realized using micro-lenses or tiny corner cube reflectors.
Quasi-phase-conjugated arrays have applications in optical sensors and instruments as well as in illumination
Such retrodirective arrays are analyzed from the point of view of the sensitivity of their retro-reflective properties.
Emphasis is put on considerations of geometrical device tailoring, namely on the question how the geometry of a unit
cell that is repeated in the array arrangement influences the retro-reflection properties of the optical arrangement.
A parameter study is provided that is targeting at robust retrodirective arrangements.
Electrostatic stresses represent a phenomenon that is frequently encountered when dielectric media are exposed to an
electric field. Depending on the application, they might be considered a spurious effect or an effect that should be
enhanced to improve performance. While measuring quadratic electrostriction with an optical interferometer they
superimpose on the measurement signal that stems from electrostriction. Designing Maxwell-stress actuators they
represent the effect that has to be maximized for best performance of the device.
For the purpose of optimization, it is of importance to understand how the electrostatic stresses depend on the elastic and
dielectric properties of the materials that are used in an actuator or in an optical instrument. These stresses are also
functions of the orientation of the surface of the dielectric material with respect to the electric field and depend on the
anisotropy of the material.
Using a phenomenological description, it will be shown how they can be predicted for the purpose of improved
instrument design. The method of analysis can be represented as an algorithm. It will be discussed what pitfalls should
be avoided while deriving the results. The results can also be applied to materials that feature high anisotropy.
The phase space beam analyzer is a measurement instrument that is applied in laser technology to perform analyses of
the spatial and angular distribution of rays. We are interested in this instrument as a means to characterize non-coherent
light sources. In this context, a closer look at the tolerances of this optical instrument was considered useful.
Having a so-called quadrupole lens as a key element, the phase space beam analyzer is a device that features anamorphic
optical properties. To describe these anamorphic properties, recurrence was made to a description by extended ray-transfer
matrices. This formalism allows for an analysis of the alignment tolerances of the phase space beam analyzer
and facilitates a study of the sensitivities of the instrument. The analysis is complemented using numerical ray tracing.
The optimization of mirrors with respect to efficiency is a common task of illumination engineering. To solve this
optimization task various numerical methods and methods that make use of partial or ordinary differential equations are
described in the technical literature. Applying these methods one is often faced with the problem of finding a good
starting configuration that can then be refined. A method has been developed that allows to determine a good starting
configuration in a systematic way. The analytical approach can be used as a guideline through "parameter space" and
gives inside into the parameter dependencies. It has first been applied to special geometries.
In this contribution, it will be shown how the method can be generalized and applied to mirrors that feature different
geometries. To this end, the analytical description will be put on a slightly broader basis. Special emphasis will be laid
on showing how to describe optical reflectors with a relatively low symmetry, because they are frequently encountered in
engineering practice. The analytical method leads to an integral expression that features the parameter dependencies.
In a way, this approach might be considered analogous to describing an imaging optical system in the paraxial
approximation to understand its key feature before starting aspherization of the optical surfaces.
This book is intended to familiarize the reader with the method of Gaussian matrices and some related tools of optical design. The matrix method provides a means to study an optical system in the paraxial approximation. This text contains new results such as theorems on the design of variable optics, on integrating rods, on the optical layout of prism devices, etc. The results are derived in a step-by-step way so that the reader might apply the methods presented here to resolve design problems with ease.
The problem of generating a cut-off line with a carefully calculated reflector contour has been treated in detail by
Spencer et al. for the case of a cylindrical source of light mounted perpendicular to the optic axis.
Because this geometry does not properly represent the geometry in which standard light sources are used in the
illumination systems which we study, the attempt was made to extend this theory to anisotropic light sources. This case
of lower symmetry is closer to the geometry of light sources encountered in headlamp design. Spencer et al. were able to
obtain an implicit algebraic equation for the problem of high symmetry that they analyzed. After adopting their method
to the problem under investigation, the method of analysis used was different insofar as an algebraic equation was not
obtained and the corresponding ordinary differential equation and the corresponding initial-value problem were solved
instead and the solutions are visualized with the aid of a computer-algebra system.
In this context, the concept of a so-called polar line or surface proved helpful. This describes a set of points that connect
the tangent lines that link a given point of the reflector contour to a given extended lightsource of low symmetry. The
extension of the lightsource is assumed to be elliptical in the plane that contains the optic axis and the plane
perpendicular to the cut-off line.
The analysis extended to the anisotropic case gave some insight into the underlying scaling laws and geometrical
Efficiency considerations have direct implications on the performance of a projection headlamp and on its implementation into a car. Therefore, an attempt was made to work out a theoretical basis to gain a better understanding of the underlying parameter dependencies that determine the efficiency of a projection system.
Designing a modern headlamp, it seems indispensable to use numerical algorithms to obtain the final design. The reason why this analytical study on the parameter dependence of the relative efficiency has been undertaken is that it provides some insight into the structure of the multidimensional solution space of this multi-parameter problem and gives a straightforward guideline to identify optimum solutions. Another motivation stems from the fact that understanding the influence of systems parameters is of key importance for the design of adaptive illumination systems.
The method of analysis combines geometrical considerations, analytical ray tracing and calculus to find an expression for a system consisting of an open tri-axial ellipsoidal reflector and a projection lens. The new integral expression allows, for example, to study the influence of the transversal anisotropy of the tri-axial reflector on the relative efficiency of the system. In addition, the effect of mechanical boundary conditions imposed on the reflector of low symmetry and/or the lens can be considered in detail. From the theoretical point of view, it is advantageous that the integral expression, which has been found, is useful to elucidate the underlying scaling laws.
The lighting systems of a car provide a variety of challenges from the point of view of illumination science and technology. Engineering work in this field has to deal both with reflector and lens design as well as with opto-mechanical design and sensor technology. It has direct implications on traffic safety and the efficiency in which energy is used. Therefore, these systems are continuously improved and optimized. In this context, adaptive systems that we investigate for automotive applications gain increasing importance. The properties of the light distribution in the vicinity of the cut-off line are of key importance for the safe and efficient operation of automotive headlamps. An alternative approach is proposed to refine the description of these properties in an attempt to make it more quantitative. This description is intended to facilitate intercomparison between different systems and/or to study environmental influences on the cut-off line of a system under investigation. Designing projection systems it is necessary to take a delicate trade-off between efficiency, light-distribution characteristics, mechanical boundary conditions, and legal requirements into account. Considerations and results on optical properties of three-axial reflectors in dependence of layout parameters will be given. They can serve as a guideline for the optical workshop and for free-form optimization.
SC911: Optical Layout and Analysis Using the Matrix Approach
In optical design, the matrix method is used to find a solution to a given optical task, which can then be refined by optical-design software or analytical methods of aberration balancing. In some cases, the method can be helpful to demonstrate that there is no solution possible under the given boundary conditions. Quite often it is of practical importance and theoretical interest to get an overview on the "solution space" of a problem. The paraxial approach might then serve as a guideline during optimization in a similar way as a map does in an unknown landscape.
The course familiarizes attendees with the application of the method of transfer matrices and related techniques to a variety of optical engineering problems. After an introduction to the method, it describes applications to imaging optics as well as to illumination systems. The course concentrates on devices of practical importance as zoom systems, interferometric devices, and laser resonators. Emphasis is also on providing a toolbox for first-order tolerancing and sensitivity analysis. The course comprises the analysis of anamorphic optical devices, because of their growing market penetration.