Advances in Complex Analysis and Applications

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Introduction
Many coherence interferometers systems find practical applications for the physical parameter measurement, such as are temperature, strain, humidity, pressure, level, current, voltage, and vibration. Physical implementation and signal demodulation are real important for the good measurement. Many implementations are based on the Bragg gratings, fiber optics, vacuum, mirrors, crystals, polarizer, and their combinations ; whereas in the signal demodulation, has been applied commonly the Fourier transform. This transform permits us to know all frequency components of any interference pattern, doing possible the signal demodulation for the interferometer systems.
The Laplace transform has many practical applications in topics such as control systems, electronic circuit analysis, mechanic systems, electric circuit system, pure mathematics, and communications. The linear transformation permits us to transform any time function into a complex function whose variable is s=iω+σ , where i is the complex operator, ω is the angular frequency, and σ is a real value. The complex function can represent in the complex s-plane, where their axes represent the real and imaginary parts of the complex variable s. This complex plane does feasible the study of dynamic systems, and some applications are the tuning closed-loop, stability, mathematical methods, fault detection, optimization, and filter design. In addition, the s-plane permits graphical methods such as pole-zero map, Bode diagrams, root locus, polar plots, gain margin and phase margin, Nichols charts, and N circles .
In dynamic system analysis, pole-zero map and Bode diagrams are 2 graphical methods which have many practical applications. Both methods require a complex function, where the frequency response plays a real important role. In the pole-zero map, poles and zeros have been calculated from the complex function, and so, their locations are represented on the complex s-plane. It is usual to mark a zero location by a circle ⋄ and a pole location a cross ×. In the Bode diagram, the magnitude and phase are calculated from the complex function, and so, both parameters are graphed. The graphic is logarithmic, and it shows the frequency response of our system under study.
Under our knowledge, the coherence interferometer system was not studied on the s-plane, and as a consequence, its interference pattern was not represented over the pole-zero map or Bode diagrams. In this piece of work, the complex s-plane was used to represent the output signal of an interferometer system. Applying 2 graphical methods, such as pole-zero plot and Bode plot, the optical signal was represented. Numerically was verified that the pole location and the zero location depend directly on the optical path difference, piece a Bode diagram shows the stability/instability of the interferometer.