Bragg reflector geometries for colorimetric orientation and deformation sensing

Bragg Reflector Geometries for Colorimetric Orientation and Deformation Sensing

by

Christopher Wing

B.S. Chemistry, North Carolina State University, 2004

Submitted to the Department Mechanical Engineering

in Partial Fulfillment of the Requirements for the Degrees of

Naval Engineer and

Master of Science in Mechanical Engineering

at the

Massachusetts Institute of Technology

June 2016

0 2016 Massachusetts Institute of Technology. All rights reserved.

Signature redacted

Signature

of A uthor:...

Christophe

_I

Department of Mechanical Eng' &ring

May 11, 2016

Signature redacted

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rtifie d

b

y

:

...

...

Mathias Kolle

Assistant Professor of Mechanical Engineering

Thesis Supervisor

Signature redacted

Joel Harbour

Professor of the Practice of Naval Construction and Engineering

Thesis Supervisor

Signature redacted.

A ccep ted b y : ...

Rohan Abeyaratne

Professor of Mechanical Engineering

MASSACHUSETTS

INSTITUTE

Chairman, Department Committee on Graduate Studies

OF TECHNOLOGY

Bragg Reflector Geometries for Colorimetric Orientation and Deformation Sensing

by

Christopher Wing

Submitted to the Department Mechanical Engineering in Partial Fulfillment of the Requirements for the Degrees of Naval Engineer and Master of Science in Mechanical Engineering

June 2016

Abstract

Propulsion systems of commercial and naval ships are typically large and expensive. They must be kept well-aligned and free of corrosion to efficiently and reliably transfer torque to a ship's propeller. Early identification of misalignment or surface corrosion is therefore crucial, making an easily deployable, reliable, lightweight system that visually indicates potential alignment and structural integrity issues desirable. This thesis demonstrates the design of a system for visual deformation and orientation indication based on naturally occurring micro-scale surface geometries that show a strong variation in their optical appearance as a function of illumination and observation directions. Specifically, the fabrication of a micro-structured surface covered with appropriately modified mimics of the spherical cavities on Papilio blumei butterfly wings is the first step in developing a low-cost, easy-to-install detection and indication system. For a specific illumination and observation geometry, the cavities' material and structural characteristics define the surface's reflection characteristics and the resulting visual signature for a far-field observer. Here we present an evolution and screening of the cavity design space, including cavity wall height and the combination of conformal and flat Bragg reflectors in order to identify suitable cavity designs. A MATLAB-based simulation environment was created to estimate the surfaces' intensity profile in monochromatic light and color chromaticity under any illumination source and incidence angle as a function of observation angle. The theoretical results are validated through characterization of a succession of physical prototypes

-a m-acro-sc-ale c-avity before -and -after -addition of -a pl-an-ar Br-agg reflector cover -as well -as -a conformally-clad microcavity array. The resulting data provides a basis for identifying the most suitable cavity designs for determination of misalignments, bends, and localized surface pitting in marine propulsion system components. The future development of specific in situ prototypes for the demonstration of the described visual sensing paradigm is facilitated through the results reported in this thesis.

Thesis Supervisor: Mathias Kolle

Title: Assistant Professor of Mechanical Engineering Thesis Supervisor: Joel Harbour

Acknowledgements

Thank you to Bethany Lettiere and Alvin Tan for sputter coating the gold macro-cavity.

Thank you to Maik Scherer for the production of glass cavity array masters.

Thank you to Cecile Chazot for production and imaging of microcavity and Bragg reflector

samples.

Thank you to Mathias Kolle for the explanations of concepts, assistance in geometric derivations,

identification of errors in code, and guidance in writing this thesis.

Table of Contents

Abstract ... 3

Acknow ledgem ents ... 5

Table of C ontents...7

List of Figures ... 9

I. Introduction and Background...11

A . The H igh Costs of Shaft M isalignm ent and Corrosion ... I 1 B. An Introduction to Color ... 14

C. Hypothesis and Course of Action...15

I. Background ... 18

A . Light Sources and Spectra...18

B. Additional Properties of Light and Geom etrical Optics (Raytracing)... 19

C. D istributed Bragg Reflectors (DBRs) ... 21

D . CIE Color Space and RGB Values...22

III. Theoretical Investigation to Determine a Suitable Surface Morphology for Colorimetric Orientation and Deform ation Sensing...25

A. Spherical Mirrors - A Raytracing Analysis for Incident Light with Small Angular Divergence...25

B . Addition of a D istributed Bragg Reflector Cover... 33

C. Addition of a Conform al Bragg Reflector ... 35

IV. Experimental Characterization of Proposed Cavity Reflector Structures... 38

A . Characterization of a Gold M acro-Cavity ... 38

M anufacturing M aterials and M ethods... 38

Gold M acro-Cavity Optical Characterization Process... 38

Gold M acro-Cavity Optical Characterization Data ... 40

B. Characterization of a Gold Macro-Cavity Covered with a Planar Bragg Reflector ... 40

M anufacturing M aterials and M ethods... 40

Covered Gold M acro-Cavity Optical Characterization Process ... 41

Covered Gold M acro-Cavity Optical Characterization Data... 41

C. M icrocavity w ith Conform al Bragg Layer ... 41

M anufacturing M aterials and M ethods... 41

M icrocavity Optical Characterization Results... 42

D. Microcavity Array with Conformal Bragg Layer Covered with a Planar Bragg Reflector...42

M icrocavity Array Prototype Properties...43

Dual Bragg Reflector Microcavity Optical Characterization Process and Results...44

V. Prediction of visual sensor performance for indication of specific shaft misalignments and corrosion pitting...45

V I. Sum m ary and C onclusions...46

V II. O utlook...48

List of Figures

Figure 1. Conventional propulsion train components and arrangement ... 11

Figure 2. Effects of variable loads: longitudinal bending moments ... 12

Figure 3. Effects of waves: longitudinal bending moments... 13

Figure 4: A lignm ent conditions ... 13

Figure 5. Ships in drydock: costly maintenance and repairs... 14

Figure 6. Typical spectra for common light sources... 18

Figure 7. Typical source beam profile ... 19

Figure 8. Specular reflection and refraction of light... 20

Figure 9. Optical behavior of distributed Bragg reflectors ... 22

Figure 10. CIE 1932 2' observer tristimulus values and chromaticity diagram ... 24

Figure 11. Spherical cavity geometry ... 25

Figure 12. Half-sphere cavity illuminated by perfectly collimated incident light ... 26

Figure 13. Ring contributions to anti-parallel reflection signal... 27

Figure 14. Partially closed spherical cavity under normally incident light... 28

Figure 15. Plot of reflected intensity vs. cavity closure for normally incident light... 29

Figure 16. Half-sphere cavity under 300 incident light... 30

Figure 17. Plot of reflected intensity as a function of angle of incidence for a collimated beam...30

Figure 18. Incipient and complete masking of rings due to variations in cavity opening radius and light in cidence angle... 3 1 Figure 19. Masking of ring signals -reflected intensity map for collimated light... 32

Figure 20. Simulated intensity response for incident monochromatic light and color response for white light into a go ld cavity ... 33

Figure 21. Simulated chromaticity shifts for a gold cavity with a flat Bragg reflector covering the opening ... 3 4 Figure 22. Angle-dependent reflection characteristics of a conformal Bragg reflector cavity ... 35

Figure 23. Simulated color response for a Bragg reflector-clad cavity with a Bragg reflector cover ... 36

Figure 24. Simulated color response for a Bragg reflector-clad cavity with a Bragg reflector cover after considered m odifications ... 37

Figure 25. Spectrum of lamp used for characterization ... 39

Figure 26. Experiment setup for quantification of cavity reflection... 39

Figure 27. Simulated vs. actual cavity reflection behavior as illumination angle changes...40 Figure 28. Simulated vs actual cavity reflection behavior as illumination angle changes for a macro-cavity

Figure 29. P. blumei mimic and chromaticity simulation results by ring ... 42 Figure 30. Microcavity fabrication process ... 43 Figure 31. Scanning electron micrograph (SEM) of a gold-coated microcavity array ... 44 Figure 32. Dual Bragg reflector microcavity array simulation results and comparison with experimental d ata ... 4 4 Figure 33. Simulated application of dual-Bragg reflector indicating film on a machine shaft...45

I.

Introduction and Background

A. The High Costs of Shaft Misalignment and Corrosion

Mechanical systems throughout the marine industry are sensitive to large loads and environmental forces that produce undesired deformations and changes in alignment internal to and between their major components. In the case of a ship's propulsion system. these major components typically include a prime mover, reduction gears, and shafting (Figure 1). The shafting is supported by various bearings as it spans a significant distance.

THRUST BEARING STRUT MAIN SPRING F1 SHAF S4 EARING PU ___

4

IMAIN

ECODU 1 aIE REACTIVE FORCE

COLERN _- -TTBE - s~uTMOVER GEAR PI EITHRUST)

BEARING STERN TUBE

PROPELLER BEARING

-HULL

Figure 1. Conventional propulsion train components and arrangement. The conventional

marine propulsion train consists of many large, heavy components spread over a significant portion of the ship that require precise alignment. (figure source: propellerpages.com)

To efficiently and reliably rotate the propeller and move the ship, this group of components must be precisely aligned in both the horizontal and vertical planes. An initial shaft alignment during construction accounts for the large static forces inherent in the system, due to the large size and weight of these components, which produce large bending moments.

This initial shaft alignment is completed in an environment when the forces on the ship's structure, which supports the bearings, are not changing, but even then it can be challenging to accomplish due to the large size and weight of the shaft, the large distances separating the various bearings and supporting structures, and the need to penetrate multiple watertight boundaries through small, sealed openings. This group of components must be maintained in nearly perfect alignment thereafter to efficiently and reliably move the ship by rotating the propeller. Attempts to maintain that alignment are made difficult not only by the large forces/torques applied to the system in order to generate the required thrust but also by the changing operating conditions of the vessel.

Variations in loading. winds, waves. and temperatures can all produce undesired hull girder deflections, which in turn continually change the offsets of the shaft-supporting bearings. resulting

in changes in alignment. both internal to and between. the major propulsion components.

To understand two of these effects. consider first that the reason for most ships to transit across the water is to carry either many passengers or a large load of cargo. This, combined with an (initially) large amount of fuel. represents a large (and variable) change in the load distributed along a ship's longitudinal axis. Changes in this loading relative to the ship's buoyancy produce significant shear forces and bending moments that result in undesired deformations and stresses. A representative distribution of these load forces. a typical buoyancy distribution for calm water, and the resulting bending moment is shown in Figure 2.

Figure 2. Effects of variable loads: longitudinal bending moments. Typical distributions of

variable load due to passengers, cargo, and fuel (red) and buoyancy due to the hull displacing water (blue) result in shear forces and bending moments along the length of the ship which are transmitted to the propulsion train supports mounted to the hull. (figure source: hcmut.edu)

Unfortunately. for large ships operating in the oceans, calm water is not guaranteed. Waves result in significant variation in the water level along the hull, continuously shifting the distribution of the buoyant force, thereby inducing a continuously shifting bending moment along the longitudinal axis that can be severe enough to result in obvious hull deformation (Figure 3).

Vri1h

Load

Buoyancy

Hogging Sagging

W>B

B>W

Figure 3. Effects of waves: longitudinal bending moments. Waves at the water surface can

result in significantly varying buoyancy forces along the longitudinal axis of a ship - the same axis along which the propulsion train is situated - producing significant changes in the shear forces and bending moments. Maximum effects are seen for wavelengths equal to the ship's length and can result in noticeable deformations in the hull shape referred to as hogging or sagging.

When misalignment results from these hull deflections. it is due to parallel or angular misalignment (Figure 4(a) and 4(b)), or even a combination of both. The consequences of this misalignment can be serious, including elevated friction and therefore higher fuel consumption, premature bearing and seal failure. and increased lubricant leakage (Motion System Design 2010). Side effects such as failure of coupling and foundation bolts and increased noise and vibration levels are also a possibility.

(a) Offset misalignment (b) Angular misalignment (c) Correct Alignment

Figure 4: Alignment conditions. Shaft misalignments (a, b) account for nearly half of rotating machinery repair costs. Maintaining correct alignment (c) can reduce repair costs and machinery downtime. (Motion System Design 2010)

Since the entire value of a naval vessel is dependent upon its ability to effectively maneuver, care must be taken to avoid damaging the propulsion train by operating outside of established alignment limits. These large, expensive propulsion components require continual condition monitoring and periodic maintenance and repair as a result. An installed system that can provide real-time indications of excessive misalignments could allow for immediate identification and quick correction of any problems before serious damage occurs, saving time and money on repairs. Because these repairs often require the ship to be put into a drydock (Figure 5). the associated costs

are very high, making early identification of problems via deformation sensing with high spatial resolution extremely important.

Figure 5. Ships in drydock: costly maintenance and repairs. Accessing and repairing large

components outside of the hull can require placing a ship into a drydock - combining a costly service with the base repairs and the inability to operate the ship.

Unfortunately, the necessary space, weight, funding. and manning for adding equipment to a ship can be very hard to come by. An easily deployable, lightweight, thin film-based, colorimetric indicator that visualizes deformations with high local resolution could potentially be applied to the shaft surface to clearly indicate upon sight that a predetermined threshold has been exceeded without any other associated components required (power/data cables. etc.). To attempt coupling a visual, easily perceivable response to mechanical deformations in a physical surface, we must understand the ways in which color can be produced and altered.

B. An Inooduction to Color

We perceive color constantly. Every time we look at an object our eyes and brain interpret color(s) that we associate with that object as an inherent characteristic. Usually our interpretation of this color is a result of the atoms or molecules within the object (partially) absorbing some wavelengths of the visible light incident upon it while reflecting the rest. However, these individual pigments (atoms and molecules) are not the only source of the colors that we perceive.

The microscopic structural and geometric properties of an object can also produce color from this same incident light through scattering or diffraction, resulting in what is commonly referred to as

siruclural color(tijon) (Johnsen 2011). Materials without a high degree of order (those with significant variations in dimensions and spacing) do this through Rayleigh or Tyndall scattering. usually resulting in apparent blue hues because the lower wavelengths are scattered more

efficiently by small particles, as seen when observing the sky (Johnsen 2011). Objects with a high

degree of order (very consistent dimensions and spacing relative to the visual wavelength scale)

in their structure can behave as a diffraction grating. Light that encounters regularly repeating

micro- or nano-structures along one or more axes undergoes diffraction, leading to constructive

interference of light scattered from different parts of the structure along different reflection angles

simultaneously for different wavelengths of incident light (Born and Wolf 1980). An example of

this can be seen when observing the back of a CD due to the highly consistent radial groove spacing

of 1600nm (Brewer 1987).

In solid materials, a special type of pattern can exist and function not on the object's surface but

along the linear path of incident light travelling into and within the object. Specifically, if the bulk

object contains alternating sections of different optical indices, they can produce color through

constructive interference of the light reflected at each successive internal transition. This is

commonly referred to as a Bragg grating and the effect will be strongest for a certain wavelength

(based on the exact materials and spacing). A characteristic color will result if the incident light

spectrum contains that wavelength.

Because the orientation and dimensions of this periodic structure determine the optical appearance

of the objects in which they reside, the following question arises: Can changes in the bulk

material's mechanical state (dimensions and alignment) due to external forces drive a shift in the

object's apparent color that can provide useful knowledge concerning those changes to an

interested observer?

C. Hypothesis and Course ofAction

Based on the desired qualities of the system that we have set (easily deployable, lightweight, thin

film-based, colorimetric indication), our proposed method should have the following effects on

the visual signal:

* Redistribute the reflected light over a wide angular range;

* Vary the reflected light predictably in color and/or intensity with a predetermined variation

in local surface orientation;

" Accomplish the previous at very high spatial resolution, providing a continuous-appearing

response for all local points on the surface.

Since Bragg reflectors exhibit large angle-dependent shifts in their reflected color, a phenomenon called iridescence, and spherical mirrors scatter light over a wide angular range, a combination of these two components might provide the desired response. In fact, naturally occurring micro-scale structural geometries that reflect light in this manner could provide a template for accomplishing this with appropriately high resolution. A potential candidate is the structure that has been discovered on the wings of the green swallowtail butterfly (Papilio blumei). They feature micro-cavity arrays with a conformal Bragg reflector surfaces. The spherical geometry of these cavities results in a bright and widely-distributed reflection. The presence of easily-defined geometrically-separate reflection paths within them allows the conformal Bragg reflector surface to vary the combined response of these individual reflection paths differently, controlling the visual signature for a far-field observer.

One version of rigid optical micro-cavity arrays that mimic this natural structure have already been produced which generate iridescent reflections similar to P. blumei wings. This project examines the potential for designing and embedding customized versions of these structures within a thin film for use as a passive sensor for visual indication of excessive deformations and/or misalignments in sensitive mechanical equipment. This thesis presents a concept for a visual indicator of surface deformation, the development of a MATLAB-based raytracer for modeling the behavior of a system of spherical cavities in combination with Bragg reflectors to identify a suitable system, and validates the simulation results through inspection of a macroscopic cavity and the fabrication and characterization of microscale analogue for high-resolution deformation identification. It focuses on rigid versions due to the additional complexities of fabricating flexible materials with consistent and distinct values. If the methods and tools develop prove useful, they can then be applied to that challenge. First, we provide additional background information on the specific principles to be employed and how they determine the combined response of the two components of interest, spherical mirrors and distributed Bragg reflectors. With the necessary theory in mind, we then propose a system of these components that interact with light in a manner that can provide our desired visual signals in the presence of surface deformations or misalignments by shifting the relative distribution of wavelengths in the reflected light from those surfaces as their physical orientation changes. A custom MATLAB-based simulation environment is presented as a way of predicting the response of such a system for any chosen construction parameters. Simulated results for this theoretical system are presented and then compared to data obtained by constructing and characterizing corresponding physical samples on the macro and micro-scale in order to validate

the results of the simulations. The response of the photonic surface applied to a propulsion system

shaft is then modeled for different defects, including misalignment, bending, and localized pitting

in addition due to its effects on surface geometry in much more local fashion. The Summary and

Conclusions section presents the key insights of this thesis and an Outlook serves to discuss further

milestones that need to be reached in order to advance this concept towards demonstration in the

intended applications.

II.

Background

In order to investigate possible means of producing the desired visual signal, we will first review some fundamental characteristics of light and principles defining its transmission and interaction with matter. including human observers, so that we can identify and understand methods and tools that may be useful for designing and fabricating a surface sensitive to deformation and changes in orientation.

A. Light Sources aind Spectra

Visible light is electromagnetic radiation to which the human eye responds, most strongly within the wavelength (X) range of 400 to 700 nm (Buser and Imbert 1992). which produces all of the approximately 10 million colors that we can perceive - either as the response to an individual wavelength or a combination of some relative proportions. This light can come from a variety of sources, such as the sun, an incandescent light bulb, or an LED. A light source may produce only photons of a single wavelength or innumerable photons over a range of wavelengths in any variety of proportions (relative intensities), which is defined as the source's spectrum. Typical emission spectra for these sources are shown in Figure 6 to illustrate the extent to which they can differ and why they must be considered to accurately simulate a possible system's response. While polarization is another fundamental property of light, and is important in determining how light interacts with matter. this analysis will assume that light emitted by the sources relevant to this work is non-polarized. Sunlight LED Incandescent Fluorescent Wavelength (nm)

Figure 6. Typical spectra for common light sources. Sunlight, LEDs, incandescent bulbs, and fluorescent bulbs are four of the most common light sources that we encounter. The colors along the spectra represent the color of light for the wavelength at that point on the x-axis.

B. Additional Properties of Light and Geometrical Optics (Ratiracing)

The emission of light from any real light source is subject to angular spreading, resulting in a variation in illuminated area on a receiving surface as a function of travel distance and a variation in intensity per unit area as a function of the angular deviation from the center of the beam. A typical emission profile is shown in Figure 7.

Figure 7. Typical source beam profile. The cone of light emission of a source depends inversely on its size. A point source emits light in a broad angular range, while an extended area source has a narrow emission cone, i.e. a small divergence angle, a, as shown.

When light is propagating within one medium with refractive index n, and encounters a flat interface with another medium with refractive index n2 # n, (e.g. when incident upon an object).

a fraction of the light will be reflected and a fraction will be transmitted. This reflection of light at a flat interface between two distinct media satisfies the relationship

02 = 7 - 01

where 01 is the angle of light incidence measured from the interface normal and 02 is the angle of reflection (Figure 8(a)). This phenomenon is often called specular reflection, as opposed to the diffuse reflection of light that occurs when it encounters rough surfaces. The light transmitted from

Source

a

Angular divergence

one optical medium to another undergoes refraction at the surface (Figure 8(b)). where the transmitted light's new direction is governed by Snell's law (Born and Wolf 1980):

n1cosO1 = n2cos02.

Refraction

Glass (n ~ 1.50)

(a) _{Reflecting surface (b) Air (n = 1.00)}

Figure 8. Specular reflection and refraction of light. (a) For specular reflection, the incident and

reflected angles are the same magnitude with respect to the surface normal. (b) Light travelling from one medium to another undergoes refraction in accordance with Snell's Law.

The refractive indices of non-absorbing materials are real numbers, while materials that absorb some light within a specific spectral range have a complex refractive index in that region with the imaginary part representing the material's absorption coefficient (Born and Wolf 1980). While most materials considered here are transparent and only have a real refractive index, gold is also considered as a substrate material and its complex refractive index is taken into account.

The fraction of the incident light that is reflected is given by the Fresnel equations, which provide the reflection coefficients, i.e. the ratios of the electric field amplitudes E and Er of incident and reflected light, as a function of the refractive indices, nj and n2. of the two materials forming the

optical interface and also the angle of light incidence, 01. The Fresnel equations take different forms depending on whether the incident light's electric field vector is oriented parallel or perpendicular to the plane of incidence, as defined by the light's propagation path and the surface normal at the point of incidence. These two fundamental states of light polarization are termed "parallel polarized" and "perpendicularly polarized", respectively. The Fresnel coefficients, rp and r, for parallel and perpendicularly polarized light, respectively are given by:

Ep,r _ n₂

cosO

and

Es,r

n

cos6

Esj n1cos61 + n2cos62

where the angle

is given as previously described by Snell's Law:

= asin(I .sin6

). The

n₂

reflectivities of parallel and perpendicularly polarized light, RP and Rs

,

are then found by squaring

the reflection coefficients:

2

and the intensity of transmitted light, TP,,, is given by the relationship:

TS =

1

provided the materials are transparent. For non-polarized incident light and a detector lacking

polarization sensitivity, the reflectivity and transmission fractions are given by the averages of

these values,

respectively (Born and Wolf 1980).

Understanding these principles of light reflection and refraction at optical interfaces now allows

us to consider the more complex case involving a series of alternating optical interfaces within a

distributed Bragg reflector (DBR).

C. Distributed Bragg Reflectors (DBRs)

A Bragg reflector comprises a stack of alternating thin layers with distinct refractive indices, ni

and n2, and thicknesses, di and d

. Light incident on the Bragg reflector will undergo multiple

events of simultaneous refraction and reflection. In other words, at each optical boundary within

the Bragg reflector, a portion of the incident light is reflected while the rest is transmitted into the

next layer as shown in Figure 9(a).

The Bragg reflector's specific thickness and refractive indices (materials) can be chosen to result

in all reflected rays of a certain wavelength within the visible range having the same phase upon

leaving the Bragg reflector's surface

-

regardless of the individual paths taken

-

for a certain angle

of incidence according to the equation:

where m is an integer number and no and O are the refractive index and the angle of light incidence in the medium surrounding the Bragg reflector, respectively (Heavens 1955).

Spectra of a typical Bragg reflector for two different incidence angles are shown in Figure 9(b). Figure 9(c) presents the spectrally resolved reflectivity of a Bragg reflector as a function of light incidence angle. This data was created using a matrix transfer algorithm analogous to the one described in Heavens, Chapter 4.

Incident Reflected light 100 100

light 1q 1 q-30' 90 - - - 80 700 80 650 70 60 60

--

600

(a) Transmitted ight (b) A (nm) W N (deg)

Figure 9. Optical behavior of distributed Bragg reflectors. (a) Alternating thin layers with different refractive indices can provide strong reflectivity in a specific spectral range. (b) The location of a DBR's reflection peak varies with light incidence angle. The shift can be seen for incidence angles of 8 = 0 and 30'. (c) Theoretical reflectivity of a Bragg reflector composed of alternating layers of thicknesses d1l = 55 nm and d2 = 81 nm and refractive indices n1 = 2.5 and n2 = 1.7. The angle dependence of a

Bragg reflector's reflection spectrum can produce a distinct color change.

Due to this angle-dependent spectral reflection distribution, Bragg reflectors composed of thin layers (~ 100 nm thick) represent lightweight, customizable optical elements that could be used for colorimetric visualization of variation in surface orientation and localized deformations.

D. (E Color Space and RGB Values

The spectral reflection response of an object illuminated by a specific light source is given by the product of the object's reflectivity spectrum and the light source spectrum. Once the object's spectral response is known, we can use the CIE color matching functions to determine the object's apparent color as described below (Fairinan, Brill and H-emmendinger 1997).

The CIE color matching functions (Figure 10(a)) are an experimentally-derived set of curves that estimate the spectral sensitivity of human light receptors and help to predict the perceived color of a typical human observer when exposed to light of a specific spectral intensity distribution in the

visible range. In particular, the color matching functions map the combined response of the three

types of cones in the human eye to a set of three numerical functions, ;(A),

y()

and Z(A), such

that the experimental result of a human observer's perception of any object's spectral signature,

S(A), is equivalent to that mapped onto the chromaticity diagram of a standard observer shown in

Figure 10(b) by the following process (Fairman, Brill and Hemmendinger 1997). The combined

response is calculated as the tristimulus values:

X

Y

LS(A)-

J4Ai),

Z =

LXS(i) zQAi),

where Ai represent the set of wavelengths at which spectral data S( ) was acquired. The

normalizations

X Y

X

-

and y =

X+Y+Z

an

M+

provide the color chromaticity coordinates x, and y, while Y is the sample's luminosity.

Electronic display devices utilize a different system for defining color, typically the RGB

specification, which defines how much red, green, and blue light is emitted by each of the display's

pixels. Using an appropriate mathematical transform (Fairman, Brill and Hemmendinger 1997),

the CIE chromaticity values can be converted to RGB coordinates and the appropriate color can

be displayed on a screen. Note that the possible RGB values do not actually encompass all possible

chromaticity coordinates. Specifically, RGB does not include colors near the edges of the color

space but the deviations are usually not very significant to a human observer. Therefore, given any

object's reflection spectrum we can use the CIE color space and the CIE XYZ to RGB conversion

to represent the object's apparent color.

x(x) y(X) -- (X) 2.0-1.5 1.0 0.5 -M / 400 500 X4(nm) (0) 700 0W9 520 0.8 0.7 0.6 5 04 0.3 0.2 480) 0.1 470 0.01 40Q 0.0 0. 1 58~0 N00 620 0(.2 0.3 0.4 0.5 0.6 0.7 0.8

x

Figure 10. CIE 1932 2* observer tristimulus values and chromaticity diagram. (a) The

tristimulus curves represent the empirical spectral sensitivities of the light receptors of human observers.

(b) The CIE chromaticity diagram that represents the observed color as a function of an objects chromaticity values, which derive from the correlation of its reflection spectrum with the tristimulus curves. (figure source: wikipedia.org)

/

0.0

(a)

III.

Theoretical Investigation to Determine a Suitable Surface Morphology

for Colorimetric Orientation and Deformation Sensing

A surface that allows for visual indication of localized deformations or global surface rotation needs to satisfy several different criteria simultaneously. First, it should scatter light in a broad angular range to ensure visibility from many different directions. Second, the light reflected by the surface should change significantly as a function of the orientation of surface normal to the directions of light incidence and observation. In the following, potentially suitable surface structures with these desired properties are identified and characterized through optical modeling. The key ingredients are concave surface structures that scatter light in a broad angular range in combination with iridescent Bragg reflectors that provide the orientation dependent variation of the surface's optical signature. The aim of this section is to identify an optimum set of parameters

that define cavity and Bragg reflector geometry within a large parameter space.

A. Spherical Mirrors - A Raytracing Analysis

fbr

Spherical surface cavities are an interesting structural component for colorimetric orientation and deformation sensors because they scatter light in a broad angular range. An illustration of this geometry and the parameters of interest is shown in Figure 11.

z

r

Figure 11. Spherical cavity geometry. The cavity geometry parameters that will be discussed are visually represented along with the conventions for discussing departure from the standard case of a half-sphere cavity (Az = 0). The values Az and rz discussed in the following sections are normalized by the cavity radius, r.

Under collimated illumination a concave half-spherical reflector will reflect part of the light back into the direction of light incidence, an effect called retro-reflection. Retro-reflected light can only originate from specific areas within the cavity. Light hitting the cavity surface parallel to its surface normal will directly reflect into the direction of light incidence. Light hitting the cavity walls at a

450 angle will undergo a double reflection before propagating anti-parallel to the incident light. Light hitting the cavity at 600 will reflect three times before propagating at 180' to the incident

.. Ir n n-1 7T

light. In general, if the local light incidence angle 6, satisfies the condition 0" = 2 - -2n - ( 1)2

the light will be reflected by the cavity n times and ultimately leave the cavity in a direction anti-parallel to the incident radiation. For a cavity of unit radius, only the light that is incident in a

ring of radius,

rn =

sin(

)

=

sin

)

-

in the plane perpendicular to the propagation path

will satisfy this condition.

We will focus only the center point and the first three rings outside of it, referring to them as R 1, R2, R3, and R4 (Figure 12(c)). We neglect the rings that experience a number of bounces higher

than four since they occur at the edge of the cavity with minimal surface area coverage and thus contribute very little to the overall cavity reflection. Because of the ease with which we can geometrically define the behavior of this subset of light, we will investigate it with the assumption that we are attempting only to detect its return signal in the vicinity of the incident light source (i.e. observation angle illumination angle).

C R3

0

(a)

(b)

(c)

Figure 12. Half-sphere cavity illuminated by perfectly collimated incident light. In this

condition, Az = 0, all light that enters the cavity through a point on one of the circular rings, Ri, is effectively

reflected 1800 and returns towards the source. This occurs either through a single reflection at an incidence angle of 00 for R, (black), or a succession of reflections, each at the same angle of 450 for R2 (red), 600 for R3 (green), or 67.50 for R4 (blue).

In general., a light beam incident on the cavity will have some angular divergence as discussed in section II.B. This angular divergence results in an effective area that produces a return signal for each ring. The effective area of each ring determines the proportions of each ring's contribution to the total signal reflected by the cavity. For an angular divergence of a, we can use the following equations to determine the effective area of each ring:

All

(n 2 2) n 2 A

with the second term being set to 0 for the case of n = 1. i.e. the center ring.

For a =1 00 and after normalizing to the area of the center ring (n = 1), the relative areas are:

A1 = 1.0, A2 = 21.9. A3 = 19.0. and A4 = 15.5 as shown in Figure 13.

25 0 c20

U-~15

(a)

Ar

_(b)

₀

Figure 13. Ring contributions to anti-parallel reflection signal. (a) Different effective areas for R1 (the center) and R2 for incident light with a given angular divergence (b) Relative area sizes for R1 to R4 with a source divergence angle of 100. These relative ring areas must be estimated in order to accurately calculate each ring's contribution to the return signal for intensity and color estimation.

The weighting of the intensity of light reflected by each of these rings becomes necessary, when the cavities are shrunk down to a size where the rings cannot be resolved by the observer. In this case, a human observer will see the color associated with the spectrum that is the weighted sum of the contributions of each of the rings. Scaling the cavities to smaller sizes will enable the visualization of deformations with higher spatial resolution and sensitivity. Appropriate accounting for the contributions of the individual reflection rings by weighing them with the above

discussed effective areas permits the estimation of a cavity's overall reflection spectrum that can then through the CIE color space algorithm and CIE XYZ to RGB color conversion be displayed as the corresponding color enabling simulation of the response of the surface as it physically rotates or deforms.

It is important to consider the specific geometry of the concave reflectors in order to predict their reflection characteristics. Assume a cavity with a spherical surface curvature with radius, r. A parameter that will have a substantial impact on the cavity's reflection is its height, z, which is defined as z = 0 at the center of the spherical cavity (i.e. in the x-y plane that bisects the sphere). By changing the cavity height (closure amount), the cavity opening radius, r,. decreases according to the relationship:

rz2 = r 2 - Az2 .

which upon nondimensionalization with the cavity radius, r, becomes: rz = 1 - Az2

This causes each ring to be masked from entering/returning as the radius of the cavity opening, rz, becomes smaller than the ring's radius. r (Figures 14 and 15).

(a)

(b)

(c)

Figure 14. Partially closed spherical cavity under normally incident light. (a) Representation

of a partially-closed cavity of radius, r, with opening radius r, < r. (b) As r, decreases, the area of the

cavity's opening decreases and light that would have entered the cavity through a point on the outer ring (blue) in the x-y plane will be blocked from entering. This rapidly removes one ring's entire contribution to the return signal, reducing the intensity and possibly shifting the spectrum. Subsequent rings would be masked in the same fashion at larger closure fractions. (c) This physical cause of this is also shown in shown in the x-z plane.

Figure 15. Plot of reflected intensity vs. cavity closure for normally incident light. For

increasing cavity closure, Az = Ir - zi, light from one ring after the other is blocked from entering the cavity. This effect begins with the ring of largest radius, and continues inward.

Another important aspect that needs to be considered is the cavities' response to light entering the cavity at non-normal angles. Recall that we are only considering an observer in the vicinity of the light source. In this case, the intersection of the rings of light reflected back at the observer with the plane of the cavity opening form an elliptical shape, with the ellipse's major axis increasing as a function of incidence angle and its minor axis remaining constant. This causes some of a ring's perimeter to be masked from entering/returning as the illumination angle departs from the normal of the cavity's opening (0 > 00) (Figures 16 and 17).

29

I1

(b)

\ \.

(a)

(c)

Figure 16. Half-sphere cavity under 300 incident light. (a) Reflection of light from a spherical cavity at non-normal incidence for a cavity with height z = r. (b), (c) As the angle of illumination increases, some fraction of the light that would have entered the cavity on the outer ring (blue) is blocked from entering the cavity. This is due to the rays' intersection points with the horizontal plane through the cavity center stretching out into an ellipse along the direction of the incoming light's deviation from the surface normal projected into the same plane.

Figure 17. Plot of reflected intensity as a function of angle of incidence, 6, for a collimated beam. The decrease in reflected intensity as a function of change in illumination angle, 0, can be shown by tracking the fraction of the rings' circumferences that are masked by the cavity walls at each angle.

While either of these changes alone can be enough to mask some/all of a ring that previously had all of its light enter the cavity, a combination of these two changes allows for an even more rapid masking of rings with a change in illumination angle. Even the center point can be masked at some

60--- R1 R2 50 R -- R4 40 Total U, C 30 C 20 10 0 0 20 40 60 80 100 0 (deg) I

closure/angle combinations, providing an additional shift in the return spectrum. These characteristics should allow for significant control over each ring's level of contribution to the return signal for a given change in illumination angle through careful choice of cavity closure. Investigating these trends mathematically in order to visualize this impact of this possible control parameter, we derive the following equation for the fractional intensity, It, for a ring of radius = r:

sin

1- z

_{= zz* tan 0+ r*}

_{I V}

COS 0 + r * cos

I -f))2 2

and we find that for any given ring, the geometrical limits (closure/incident angle) for the beginning and completion of a ring's masking (Figure 18) can be simplified as:

inception: 1 - Z2 z * tan 0 + r, COS 0 completion: 1 - z 2 = (z

*

0.5

-0.5

-1

-90

-60

-30

0

30

60

90

0

(deg)

Figure 18. Incipient and complete masking of rings due to variations in cavity opening radius and light incidence angle. The combined conditions that start (dotted lines) and complete (solid lines) masking of any ring can be determined geometrically and mapped simultaneously. The symmetry of the plots shows that only the magnitudes of the change in closure fraction and illumination angle affect ring masking.

Recognizing the symmetry in the onset and completion of a ring's masking as a function of cavity opening radius and light incidence angle. we find that we only need to consider one quadrant to evaluate the cavity's response. We can then define a ring's full intensity as that returned from a half-sphere cavity with normal incidence and plot within these limits the expected fraction of this full intensity that remains unmasked at all possible combinations of illumination angle and closure amount for each of the rings using the same equation. Figure 19(a) presents these values visually for the center point and the first three rings while Figure 19(b) shows the combined response.

50 a 222 _{40 C} 05 (a) 03 20 10 0 1 0 0 20 40 60 0 20 40 60 80 0 20 40 60 80

(a) 0 (deg) 0 (deg) (b) 0 (deg)

Figure 19. Masking of ring signals - reflected intensity map for collimated light. When each

individual rings' contribution is weighted by its initial relative area and then reduced based on the previous masking rates, the expected total intensity of the return signal per ring (a) and as a combination from all rings (b) can be calculated for all possible conditions.

We see that there is a rapid initial decline that continuously slows until complete masking occurs for each ring. Accounting for the rings' relative areas, combined with these masking effects. a final set of weighting factors can be determined for any wavelength to set each ring's initial -full" intensity spectrum contribution. We can sum the signals and then see the plot for any given cavity geometry (Az) to determine what range of closure fractions might best provide the desired visual shift. The results suggest that a closure fraction between 0.3 and 0.7 would provide an immediate shift for a small variation in the incidence angle 0, as intensity drops due to masking of the fourth

ring.

We now have the information necessary to simulate a base real scenario - reflection of an actual light source from an actual material - a lamp shining a collimated beam of light into a gold cavity.

For a real application, we need to consider the variations in each reflection ring's signal as a function of wavelength and local incidence angle. Calculating the intensity response of a gold cavity to monochromatic light of 470 nm wavelength and its color response to white light using the complex refractive index of gold nQ() produces the results shown in Figures 20(a) and 20(b).

1 11

UC 00.

I.25

0-25

0 30 60 90 0 30 60 90

(a) 0 (deg) (b) 0 (deg)

Figure 20. Simulated intensity response for incident monochromatic light and color response for white light into a gold cavity. (a) Accounting for the reflectivity of the cavity material

at the angle of incidence appropriate for each ring allows the response to a single wavelength (470 nm) to be estimated in order to simulate the appearance of a microcavity array under monochromatic light. (b) It also allows the color of the returned light to be estimated in order to simulate the appearance of a microcavity array under any conditions - in this case white light.

The variation in intensity and color achieved in a bare cavity is not adequate for our proposed deformation sensing application. To accentuate the color shift, we can investigate the effects of the other proposed components. First we apply a Distributed Bragg Reflector (DBR) that will filter the spectrum transmitted into the cavity in an angle dependent manner.

B. Addition of a Distribu/ed Bragg Re flec/or Cover

A planar Bragg reflector will specularly reflect a portion of the incident light's spectrum, mainly in the area of its peak wavelength (section II.C.). Adding a Bragg reflector on top of a spherical cavity should result in a reflected signal that consists of a superposition of the light reflected by the cavity and the Bragg reflector for illumination angles near 0'. For an increasing angle of light incidence, the light that is specularly reflected from the Bragg reflector is not captured for an observation direction that is parallel to the illumination angle. However. the transmitted light will enter the cavity and some of it will still be reflected back out towards such an observer, with its intensity dropping as the illumination angle increases as shown in Figure 19. This results in a signal

for that observer that is at first characterized mainly by a color associated with the Bragg reflector's main spectral peak. followed quickly by a signal depleted of any wavelengths near the Bragg reflector's main peak and instead characterized by a weakening sum of the remaining portions of the source spectrum as reflected by the cavity surface. Because the Bragg reflector's main peak will also shift down in wavelength as the illumination angle continues to deviate from 00, the spectral composition. i.e. color, of this return signal should continue to vary as its intensity is diminished. Expected results for a 23-layer SiN/SiO2 DBR with a peak wavelength of 620 nm are

shown in Figure 21.

0.75

0.5

0.25

0

0

30

60

90

0

(deg)

Figure 21. Simulated chromaticity shifts for a gold cavity with a flat Bragg reflector covering the opening. With a peak reflectivity wavelength of 620nm, the simulated Bragg reflector (23 layers, film thicknesses of d1l 74 nm and d2 = 104 nm, and refractive indices of n1 2.1 and n2 = 1.5) shown here initially prevents most of the red part of the source's spectrum from being returned after a small initial angle. The reflection peak shifts with the incident angle and by 400, the return signal's color has shifted significantly as the red portion of the spectrum becomes available for intra-cavity reflection.

For normal light incidence. the results suggest a rapid shift in observed color over the first few degrees away from the surface normal due to the loss of specularly reflected light from the cover in the observer's direction. For non-normal incidence the magnitude of color change per unit angle variation is much smaller. To enhance the response, we add a conformal multilayer to the cavity in order to increase the variation of perceived color as function of incidence angle.

C. Addilion of a Conformal Bragg Reflector

A half-spherical cavity provides an opportunity to employ the Bragg Reflector principles in another way to amplify the changes in its reflected spectrum based on angle of light incidence. Light reflected within each of the rings will hit the cavity surface at a different incident angle (00, 45', 60', and 67.5'). Because the light that is reflected back towards the source will only interact with the cavity at this set of very specific angles and a DBR's reflectance spectrum is angle-dependent, each ring's contribution to the return signal will feature a different peak wavelength as shown in Figure 22. These peak positions will remain fixed for varying incidence angles due to the spherical cavity walls maintaining constant incidence angles with the incoming light. However, due to varying masking rates (described in section Il.A), the peak intensity will decay from each peak's weighted maximum values at different rates, which will lead to additional variations in the perceived color as a function of illumination angle. This color variation can be tailored through careful selection of the DBR materials and layer number and thicknesses of individual layers. center point (0')

T

/

Figure 22. Angle-dependent reflection characteristics of a conformal Bragg reflector

cavity. Angular reflection spectrum (left) and reflection spectra for individual angles that contribute in the composition of the reflected light.

A simulation for a conformal DBR with matching parameters of the planar DBR previously discussed provides the estimated color map shown in Figure 23. In this case, the combination does not appear to dramatically affect the results. The most obvious change is that the first ring shifts

some of the initial blue region further towards a green color for low to moderate deviations from the surface normal but the required angular change for a shift after the first few degrees remains at roughly the same position although it is a little more rapid and accentuated for cavity closure below 0.7.

1

0.7.

<10.5

0.25

0

0

30

60

90

0 (deg)

Figure 23. Simulated color response for a Bragg reflector-clad cavity with a Bragg reflector

cover. Adding a conformal Bragg reflector strengthens the perceived shift in color below 300. Further enhancement of the perceived color shift could result from individually tuning the band gap of the flat and conformal Bragg reflectors. The simulation parameters for both Bragg reflectors are 23 layers, film thicknesses of d1 = 84 nm and d2 = 103 nm, and refractive indices of n1 = 1.83 and n2 = 1.5 on an epoxy substrate of nsub = 1.47.

However, the conformal DBR can be physically different (materials/dimensions) than the flat DBR cover, which provides the opportunity to thoughtfully engineer the response based on the principles previously discussed. For example. shifting the conformal DBR's reflection peak wavelength to 550 nm with the flat DBR's peak center wavelength remaining at 620 nm leads to a more substantial contribution from the center ring by allowing it to reflect more of the light transmitted through the cover at small angles from the surface normal. Additionally, reducing the difference in the refractive indices by using a material with n = 1.8 instead of 1.5 for the lower value narrows the spectral width of the reflection peaks. providing more distinction in the reflection spectra for the rings. Furthermore. reducing the cover DBR's number of layers can allow

more light into the cavity to be separated into these distinct signal components and increasing the difference in the refractive indices will flatten the cover's reflectivity spectrum. resulting in the simulated color map shown in Figure 24. Validation of this simulation method would justify further exploration to improve the rate of color shift with a change in illumination angle.

1

0.75

~0.5

0.25

0

0

30

60

90

6 (deg)

Figure 24. Simulated color response for a Bragg reflector-clad cavity with a Bragg reflector cover after considered modifications. Different choices of materials and layer geometries for the two Bragg reflectors can dramatically change the simulated color map and improve the response. The simulation parameters for the flat DBR are 9 layers, film thicknesses of dl = 73 nm and d2 = 128 nm and refractive indices of n1 = 2.1 and n2 = 1.2. For the conformal DBR they are 23 layers, film thicknesses of dl = 65 nm and d2 = 76 nm, and refractive indices of n1 = 2.1 and n2 = 1.8 on a gold substrate.