Konstantinos Falaggis

Method of excess fractions with application to absolute distance metrology: theoretical analysis

The method of excess fractions (EF) is well established to resolve the fringe order ambiguity generated in interferometric detection. Despite this background, multiwavelength interferometric absolute long distance measurements have only been reported with varying degrees of success. In this paper we present a theoretical model that can predict the unambiguous measurement range in EF based on the selected measurement wavelengths and phase noise. It is shown that beat wavelength solutions are a subset of this theoretical model. The performance of EF, for a given phase noise, is shown to be equivalent to beat techniques but offers many alternative sets of measurement wavelengths and therefore EF offer significantly greater flexibility in experimental design.

Multiwavelength interferometry: extended range metrology.

We present an optimized method for multiwavelength interferometry that allows measurements beyond the largest beat wavelength. The approach exploits wavelength coincidence between two beat wavelengths in order to measure unambiguously over an extended range. Performance of the approach has been validated both through simulations and experimentally by means of a fiber interferometer for four measurement wavelengths. Initial results have demonstrated 1/200th of a fringe phase resolution, giving absolute metrology over 18.16 mm, or a dynamic range of 1 part in 2.4x10(6). With improved phase resolution the method has the potential to range over >100 m using femtosecond laser frequency comb sources.

Method of excess fractions with application to absolute distance metrology: analytical solution

Multiwavelength interferometry provides a solution to a number of applications in metrology for the measurement of optical path differences longer than the source wavelength. To this day, the method of excess fractions (EF) has proved to provide very long, unambiguous measurement ranges with the highest reliability for a given set of wavelengths and level of phase noise. This is achieved because EF combines the individual phase values in an equivalent least-square problem and evaluates the correspondence for all possible solutions. However, this procedure can be slow for a number of applications. In this paper, an analytical solution for EF is presented that allows the direct calculation of the unknown integer fringe order. It is shown that this solution is consistent with the other phase unwrapping approaches as beat wavelength or Chinese remainder theorem-based solutions, but moreover, it can be understood as a unified representation and solution of the fringe order problem.

Method of excess fractions with application to absolute distance metrology: wavelength selection and the effects of common error sources.

Multiwavelength interferometry (MWI) is a well established technique in the field of optical metrology. Previously, we have reported a theoretical analysis of the method of excess fractions that describes the mutual dependence of unambiguous measurement range, reliability, and the measurement wavelengths. In this paper wavelength, selection strategies are introduced that are built on the theoretical description and maximize the reliability in the calculated fringe order for a given measurement range, number of wavelengths, and level of phase noise. Practical implementation issues for an MWI interferometer are analyzed theoretically. It is shown that dispersion compensation is best implemented by use of reference measurements around absolute zero in the interferometer. Furthermore, the effects of wavelength uncertainty allow the ultimate performance of an MWI interferometer to be estimated.

Optimum wavelength selection for the method of excess fractions

In this paper, we present an understanding of the failure modes of excess fractions solutions to multi-wavelength interferometry. From this basis, an approach to select optimum measurement wavelengths has been introduced. A practical fiber optic sensor has been constructed for simultaneous detection of the intensity at four measurement wavelengths. The system has been demonstrated using two wavelength selections that are very near the optimal configuration and the data analyzed using an excess fractions solver. Initial results have shown a measurement range of 17 mm with reliable and robust absolute metrology from a system with a phase noise of 1/200th of a fringe. This corresponds to an overall dynamic range of 1 part in 2×106.

Unified theory of phase unwrapping approaches in multiwavelength interferometry

Multi-wavelength interferometry (MWI) has a long tradition and provides a solution to a number of applications in the field of optical metrology. In MWI phase unwrapping procedures are usually based on beat wavelength approaches, Chinese Remainder Theorem (CRT) techniques, or the method of Excess Fractions (EF). Each of these unwrapping approaches has distinct advantages making it suitable for a given application. Beat wavelength and CRT based approaches offer a direct calculation of integer fringe order, however, the unambiguous measurement range (UMR) is limited by the available measurement wavelengths. On the other hand, EF offers many alternative sets of wavelengths to achieve a large UMR with high reliability; however, the calculation of the integer fringe order involves a large number of computational steps. In this work, a unified theory of beat wavelength, EF and CRT approaches is reported. It is shown that the calculation of the integer fringe order requires a low computational effort, which hitherto had only been possible for CRT and beat wavelength approaches, whilst offering flexibility in choosing the measurement wavelengths for a given UMR, which had only been the case for EF. As the model can be used in a predictive way to determine the UMR and measurement reliability it is possible to define optimization criteria that are based on parameters which are dependent on the choice of the measurement wavelengths.

Generalized theory of phase unwrapping: approaches and optimal wavelength selection strategies for multiwavelength interferometric techniques

Multi-wavelength interferometry (MWI) has a long tradition in the field of optical metrology and is used as a solution to a number of applications. MWI phase unwrapping procedures are usually based on beat wavelength approaches, Chinese Remainder Theorem (CRT) techniques, or the method of Excess Fractions (EF). Each of these unwrapping approaches have a distinct advantage for a given application: Beat wavelength and CRT based approaches offer a direct calculation of integer fringe order, and EF offers many alternative sets of wavelengths to achieve a large unambiguous measurement range (UMR) with high reliability. Nevertheless, a drawback of Beat wavelength and CRT based approaches is that they have a limited UMR due to the available measurement wavelengths, and the alternative approach the EF is often impractical in practice, because the calculation of the integer fringe order involves a large number of computational steps. Recently, we have reported a unified theory of beat wavelength, EF and CRT approaches, which enables the derivation of phase unwrapping approaches with low computational effort, which hitherto had only been possible for CRT and beat wavelength approaches, whilst offering flexibility in choosing the measurement wavelengths for a given UMR, which had previously only been the case for EF. In this work, we briefly summarize the previous developed framework that determines the UMR and measurement reliability and derive optimization criteria that are based on parameters, which are dependent on the choice of the measurement wavelengths. The developed optimum wavelength selection strategies maximize the dynamic range of interferometer for a given value of phase noise the dynamic range of interferometer.

A hybrid technique for ultra-high dynamic range interferometry

In this paper, we explore the optimization and implementation of multi-wavelength interferometers such that measurements beyond the largest beat wavelength can be achieved reliably. A hybrid beat wavelength approach is presented that also exploits wavelength coincidence between two beat wavelengths in order to measure unambiguously over an extended range. The performance of the approach has been explored both through simulations and experimental validation has been obtained using a fiber interferometer with 4 measurement wavelengths. The initial results have demonstrated 1/200th of a fringe phase resolution giving absolute metrology over 18.16 mm, or a dynamic range of 1 part in 2.4×106.

Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation

The phase information of a complex field is routinely obtained using coherent measurement techniques as, e.g., interferometry or holography. The obtained measurement result is subject to a 2π ambiguity and is often referred to as wrapped phase. Phase-unwrapping algorithms (PUAs) are commonly employed to remove this ambiguity and, hence, obtain the absolute phase. However, implementing PUAs can be computationally intensive, and the accuracy of those algorithms may be low. Recently, the transport of intensity equation (TIE) has been proposed as a simple and practical alternative for obtaining the absolute phase map. Nevertheless, an efficient implementation of this technique has not yet been made. In this work, we propose an accurate solution for the TIE-based PUA that does not require the use of wave-propagation techniques, as previously reported TIE-based approaches. The proposed method calculates directly the axial derivative of the intensity from the wrapped phase when considering the correct propagation method. This is done in order to bypass the time-consuming wave-propagation techniques employed in similar methods. The analytical evaluation of this parameter allows obtaining an accurate solution when unwrapping the phase map with low computational effort. This work further introduces the use of the iterative TIE-PUA that, in a few steps, improves significantly the accuracy of the final absolute phase map, even in the presence of noise or aliasing of the wrapped data. The high accuracy and utility of the developed TIE-PUA technique is proven by both numerical simulations and experiments for various objects.

Optical encryption with protection against Dirac delta and plain signal attacks.

This Letter proposes an optical encryption technique that disguises the information with modular arithmetic concepts and time-varying noise components that are unknown to the receiver. Optical encryption systems that use these techniques produce a nondeterministic system response, as well as noise like image data that can easily be generated with ordinary spatial light modulators. The principle of this technique is demonstrated for the double random phase encoding (DRPE) method. The conventional DRPE method has major vulnerabilities for Dirac signal and plain signal attacks, making them impractical for secure encryption. It is shown that the proposed encryption technique provides a robustness against these types of attacks, allowing optical DRPE to be employed in secure encryptions. Moreover, applications of this Letter are not limited to DRPE alone but can also be adopted by other optical encryption techniques such as fractional Fourier transform and Fresnel-transform-based techniques.