Amanda D. Encallado, Fritz Randulf S. DioricoDepartment of Physics, University of San CarlosTalamban, Cebu City, Philippines 6000a_encallado@yahoo.com, fdiorico@yahoo.comAdvisers: Albert James Licup and Claude Ceniza

PRESENTATION OUTLINE

Introduction

Theory

Methodology

Results

Conclusions

INTRODUCTION

Fizeau interferometer

InSAR

INTRODUCTION

Michelson interferometer

Case 1: Flat Case 1: Flat surfacesurface

Case 2: Flat surface + round Case 2: Flat surface + round peripheryperiphery

Case 3: Convex surface + round Case 3: Convex surface + round peripheryperipheryCase 4: Concave surface + round Case 4: Concave surface + round peripheryperiphery

INTRODUCTION

INTERFEROGRAM CORRESPONDENCE OF LENS SURFACE

To construct a Fizeau interferometer

To be able to automate the three-dimensional reconstruction of the lens surface.

To obtain a three dimensional perspective of lenses

OBJECTIVES

INTRODUCTION

THEORY

4 ( , )( , ) airn d x yx y

1-D Interference

Fourier Transform

Inverse Fourier Transform

Phase Unwrapping

THEORY

*( , ) ( , ) ( , ) ( , )i x y a x y c x y c x y *( , ) ( , ) ( , ) ( , )I u v A u v C u v C u v

( , )1( , ) ( , )

2j x yc x y b x y e

1 Im ( , )( , ) tan

Re ( , )

c x yx y

c x y

THEORY

Phase Sign Ambiguity Correction

Interferogram Frequency Spectrum Wrapped Phase Distribution

Different possible orientations for filters for the Frequency Spectrum

Phase Sign Ambiguity Correction

Interferogram Frequency Spectrum Multi-regional filters

Resulting Phase Distributions Phase Masks

Phase Sign Ambiguity Correction

Phase Sign Ambiguity Correction

Sign corrected phase distribution

THEORY

Phase Unwrapping

4

1yx xy i

i

q

0.1 0.2 0.3

-0.1 -0.2 -0.4

-0.2 -0.2 -0.3

0.1 0.2 0.3

-0.1 -0.2 -0.4

-0.2 -0.2 -0.3

0.1 0.2 0.3

-0.1 -0.2 -0.4

-0.2 -0.2 -0.3

0.1 0.2 0.3

-0.1 -0.2 -0.4

-0.2 -0.2 -0.3

0.1 0.2 0.3

-0.1 -0.2 -0.4

-0.2 -0.2 -0.3

1234

THEORY

Phase Unwrapping

+

-

+

-

THEORY

Phase Unwrapping

Residue Distribution Branch Cuts

Phase Unwrapping

THEORY

Modified version of the Fizeau Interferometer

Beam Expander

Beam Splitter 50-50

Screen

Optical Flat or Reference FlatLens under

observation

Interferogram to be captured by

CCD camera and Analyze using

MATLAB®

Collimated Beam

C

CD

METHODOLOGY

If Complex

Multi-regional filtering

Fourier Transform

Pre-processed interferogram from LabVIEW

Phase-Unwrapping using Goldstein’s Branch Cut Algorithm

Pizza-slice masks append

View 3-D Lens Surface

Inverse FT

If Non-complex

Select best Phase Derivative Variance

METHODOLOGY

Information Flow Diagram

Interferogram

RESULTS

Log magnitude spectrum

RESULTS

( , ) *( , )C u v or C u v

Multi-regional filters

RESULTS

Filtered frequency domain

Wrapped

Phase

Distribution

Phase

Derivative

Variance

RESULTS

Wrapped Phase Distribution of highest quality

Phase residues

Branch cuts

RESULTS

Wrapped phase distribution

Unwrapped phase

RESULTS

Unwrapped Phase

Unwrapping Error

1, , ,( 1, ) xx i j i j i jx y

, 1 , ,( , 1) yy i j i j i jx y

x y (mean: 0.008, max: 12.57 radians)

RESULTS

RESULTS

Displacement plot

Displacement error(mean: 0.001,

max: 0.649 μm)

RESULTS

Interferogram

Wrapped Phase Distribution

RESULTS

Without mean filtering

Mean= 0.011 radians

With mean filteringMean=0.008

radians

RESULTS

Phase residues

Branch cuts

Unwrapped Phase

RESULTS

RESULTS

Displacement plot

Displacement errormean: 0.001, max: 1.343 μm)

RESULTS

Precision and Accuracy ±1.343 μm (accuracy based on unwrapping error) with mean =

0.001 μm Precision:

4 4precisiond z

2precision m

where m is the number of samples per fringe4 ( , )

( , ) airn d x yx y

0.1584

d m

RESULTS

2

4 4

Nd z

m

RESULTS

Conclusions Phase Measurement using the Fourier

transform method Multi-regional filtering for sign ambiguity

correction Goldstein’s branch cut algorithm,

obtained minimum PU error, 0.001 radians.

Displacement error through PU error Displacement resolution > 3-D Reconstruction process, successfully

implemented

0.158 m

References

[1] . K. Rastogi, Holographic Interferometry (Springer-Verlag Berlin Heidelberg,Berlin, Germany, 1994).[2] E. Hecht, Optics, 2nd ed. (Addison-Wesley Publishing Company Inc., New York, USA, 1990).[3] D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping; Theory, Algorithms, and Software (John Wiley and Sons Inc., USA,

1998).[4] T. Kreis and W. J¨uptner, “Fourier-transform evaluation of interference patterns: demodulation and sign ambiguity,” SPIE 1553 Laser

lnterferometry IV: Computer-Aided lnterferometry, 263–273 (1991).[5] T. Kreis, “Digital holographic interference-phase measurement using Fouriertransform method,” J. Opt. Soc. America 3, 847–855

(1986).[6] C. Gorecki, “Interferogram analysis using Fourier transform method for automatic 3D surface measurement,” Pure Applied Optics 1,

IOP 1, 103–110 (1992).[7] F. T. S. Yu and X. Yang, Introduction to Optical Engineering (Cambridge University Press, Cambridge, CB2, United Kingdom, 1997). 50

BIBLIOGRAPHY 51[8] M. C. P.F. Almoro and M. Daza, “Measurement of 3-D deformation by Phaseshifting Holographic Interferometry,” 19th SPP Philippine

Physics congress proceedings pp. 57–60 (2001).[9] B. O. S. of America, Handbook of Optics: Volume II (McGraw-Hill Professional, USA, 1994).[10] R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley Pulishing Company, New York, USA, 1992).[11] N. J. Giordano, Computational Physics (Prentice Hall, Upper Saddle River, New Jersey, 1997).[12] P. M. A. Nabeel Shirazi and A. L. Abbott, “Implementation of a 2-D Fast Fourier Transform a FPGA-Based Custom Computing

Machine,” presented at the 5th International Workshop on Field Programmable Logic and Applications, FPL 1995 (Oxford, UK, Sep 1995.).

[13] P. L. Devries, A First Course in Computational Physics (John Wiley and Sons, Inc., New York, USA, 1994).[14] E. Weisstein, “Fourier Transform,” http://mathworld.wolfram.com/ FourierTransform.html (Accessed November 18, 2007).[15] A. D. Marshall, “The Fast Fourier Transform Algorithm,” http://homepages.inf. ed.ac.uk/rbf/CVonline/LOCAL

COPIES/MARSHALL/node20.html (Accessed Novembe 18, 2007).[16] Matlab, “Four Quadrant Inverse Tangent,” http://www.mathworks.com/access/ helpdesk/help/techdoc/ref/atan2.html (December 1,

2007). BIBLIOGRAPHY 52[17] Wikipedia, “Matrix Multiplication,” http://en.wikipedia.org/wiki/Matrix multiplication (December 10, 2007).[18] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall Signal Processing Series, Englewood Cliff, New

Jersey 07632, USA, 1989).[19] Y. H. B. W. J. K. J.J. Akerson, Y.E. Yang, “Automatic Phase Unwrapping Algorithms in Synthetic Aperture Radar (SAR) Interferometry,”

IEICE TRANS. ELECTRON. E83-C, 1896–1904 (2000).[20] J. Bioucas-Dias, “Phase Unwrapping via Graph Cuts,” IEEE Transactions on Image processing 16, 698–709 (2007).[21] Matlab, “Phase Unwrapping,” http://www.mathworks.com/access/helpdesk/ help/toolbox/dspblks/ref/unwrap.html (April 20, 2007).[22] M. A. Schofield and Y. Zhu, “Fast phase unwrapping algorithm for interferometric applications,” OPTICS LETTERS, Optical Society of

America 28, 1194–1196 (2003).[23] R. L. Smith, “The Velocities of Light,” Am. J. Phys. 38, 978–983 (1970).[24] T. S. H.D. Young, R. Freedman and A. Ford, University Physics, 10th ed.[25] Wikipedia, “Interferometry,” http://en.wikipedia.org/wiki/Interferometry (April 15, 2007).