Explaining diffraction patterns of compact discs using the grating equation

 

Physik  |  Technik

 

Levin Glatz, 2008 | Wettingen, AG

 

The microstructure compact discs (CDs) use to store data acts as a diffraction grating that differs from simple textbook-style gratings. In my investigation I explain the observed diffraction pattern using the grating equation and identify strengths and limitations of this approach. An experiment with a laser and a C++ simulation of the constructed model showed that the transparent polycarbonate layer creates additional visible diffraction features through the total internal reflection of exiting light. Two-dimensional Fourier analysis of microscope images of CDs was used as an alternative way to model and simulate CD diffraction. Comparisons showed that the 2D Fourier transforms can capture the data-dependent diffraction pattern in the second dimension along the grooves that gets lost when CDs are modeled through the grating equation. The differences between the two considered CD types, CD-DA and CD-R, are not yet fully understood.

Introduction

My research question was: “To what extent can the diffraction patterns of compact disc surfaces be explained using the grating equation for one-dimensional periodic structures, considering the effects of internal reflections in the polycarbonate layer and the circular geometry?” Two types of CDs were investigated: CD-DA (Digital Audio) and CD-R (Recordable).

Methods

As a first step I did research to see what resources already exist on my topic. Next, I developed a first theoretical model of the phenomenon. This first model ignored the transparent polycarbonate (PC) layer of CDs. I designed an experiment with lasers to test the model. The setup consisted of a laser mounted to a tripod, a CD-R, and a wall that the diffracted laser rays would hit. I measured the laser spots on the wall for different positions of the laser.

The first experiment showed differences between the model predictions and measurements. The theoretical model was extended by considering the PC layer that leads to refraction described through Snell’s law. In a second “experiment” photographs of CDs with known positions of a white light source, the CD, and the camera were compared to a C++ ray tracing simulation of the theoretical model. The simulation uses the grating equation for its calculations and the OpenGL Utility Toolkit (GLUT) to display the result.

The limitations of the theoretical model motivated a third “experiment”. CDs were photographed with an optical microscope, and the images were analyzed with the 2D Discrete Fourier Transform. The 2D transforms were used as an alternative way to simulate the appearance of CDs based on the Fourier optics description of diffraction. The differences between the results from this simulation (based on Fourier optics) and the results from the second experiment (ray tracing simulation based on the grating equation) were analyzed to finally answer the research question.

Results

The first experiment with a laser showed that the grating equation does hold for main diffraction features. However, additional features were observed that could not be explained with the grating equation on its own.

The simulations of CDs based on the grating equation and Snell’s law for refraction and total internal reflection showed high resemblance to the photographs. They also showed that the PC layer has an observable effect on the appearance of CDs. It leads to total internal reflection of high diffraction orders, which can then undergo further diffraction upon hitting the grating surface a second time. The visible features of such total internal reflections are more intense on CD-R than CD-DA. The extended theoretical model still failed to explain all visible features, with the CD-DA producing unexplained features along the data grooves, perpendicular to the other features.

The third experiment showed that the missing feature can be simulated based on the 2D Fourier transforms of microscope images. This suggests that the appearance of CDs is partly dependent on the data written to them.

Discussion

The accuracy of the simulations based on the grating equation presents it as a powerful tool in modeling the optics of CDs; however, the method of comparison to photographs provides no quantitative measure for the accuracy.

The diffraction features created by the semi-periodic patterns in the data cannot be modeled accurately with the grating equation and reveal its limitations compared to the Fourier optics approach.

Conclusions

As a next step the grating equation approach could be combined with the Fourier optics approach to modeling the CDs. Alternatively, the difference between CD-DAs and CD-Rs could be investigated.

 

 

Würdigung durch den Experten

Prof. Dr. Lukas Gallmann

Levin Glatz lotet mit einem umfassenden wissenschaftlichen Ansatz aus Experiment, Theorie und Simulation die Leistungsfähigkeit und Grenzen der eindimensionalen Gitterbeugungsgleichung als Modell zur Beschreibung der Optik einer Compact Disc (CD) aus. Trotz der Einfachheit des Modells wird unter Berücksichtigung interner Reflexionen in der Polycarbonatschicht der CD eine ausgezeichnete Übereinstimmung mit der Realität erreicht. Mit hohem Verständnis für Konzepte und technischem Geschick zeigt Levin Glatz, wie verbliebene intrinsische Lücken des Modells beseitigt werden können.

Prädikat:

Gold

Sonderpreis «Regeneron International Science and Engineering Fair (ISEF)» gestiftet von der Gebauer Stiftung

 

 

 

Kantonsschule Wettingen
Lehrerin: Mathilde Rüfenacht