Interferometric characterization of 3D printed optic devices

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Interferometric characterization of 3D printed optic devices Oyemakinwa Kehinde Oladimeji

MSc Thesis June 2019

Department of Physics and Mathematics

University of Eastern Finland

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Oyemakinwa kehinde Oladimeji Interferometric characterization of 3D printed optical devices, 50 pages University of Eastern Finland

Master’s Degree Programme in Photonics

Supervisors Ph.D. Petri Karvinen

Ph.D. Henri Partanen

Abstract

In this thesis, interferometric characterization was conducted on seven lenses with different nominal focal lengths (100 mm to 94 mm) in order to study their surface geometry. The Mach-Zehnder interferometer set-up was used to conduct the experiment. A N-BK7 glass lens of 100 mm focal length from Thorlabs was used as the reference lens. The test lenses used were six 3D printed plastic lenses and a glass lens of the same specification as the reference lens. The glass lens was used as test lens in order to serve as a reference for comparing the other lenses. The glass lens and lens 1 (100 mm 3D printed lens) shows no significant change in their phase profile when used as a test lens. Whereas, the other lenses display an evident change in their phase when used as the test lens. The wavefront curvature of the lenses was also analyzed. The glass lens and lens 1 shows an approximate planar wavefront while the other lenses shows a spherical wavefront similar to that of a Newton’s ring wavefront pattern. We observe that slight changes in the focal lengths of the lenses during design can cause a very significant change in the wavefront curvature. This report takes into consideration the inability to properly align the Mach-Zehnder interferometer and as such will require that further measurements be done to properly arrive at a valid conclusion.

Keywords: Interferometric characterization; Lenses; Mach-Zehnder interferometer;

Radius of curvature;...

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Preface

I will like to give glory to God for the successful completion of this report. I am grateful for the support shown by my family, friends and colleagues. Aprreciation also goes to my supervisors who were very helpful in guiding me during the research work. Also to Markku Pekkarinen (MSc) for helping in the design and fabrication of the lenses.

Joensuu, the 25th of May 2019 Oyemakinwa Kehinde Oladimeji

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Chapter I

Introduction

In the field of photonics and the world at large, optical lens play an important role in the functionality of several optical devices both for industrial and scientific use. Glass was the most handy material used for making lenses in its early stage. However, many optical lenses designed for consumer use are made of plastic in recent times because of its attendant benefits. One of such benefits is the low cost of producing plastic lenses while still having a qualitative degree of use. Depending on the area of application and method of fabrication, the decision on what material to use - glass, plastic, or even semiconductors is very critical [1]. Traditional methods of fabrication used grinding, polishing, and injection molding techniques [2–4].

In recent times, additive mixture or 3D printing techniques have been on the rise because it avails manufacturers the opportunity of customizing, small batch produc- tion and low cost of producing lens [5]. 3D printing has enhanced the possibilities of using high quality technologies in the world of fabrication covering industries like display [6], medical [7], biotechnology [8], aerospace [9], imaging [10], lens fabrication [11], etc. Though 3D printing technique of optical devices offers promising prospects, it is not without its shortcomings especially the problem of high optical purity and uniformity of refractive index across the optical components [12]. There- fore, when optical devices are fabricated using 3D printing, an important task is that they are characterized to determine the extent to which they can be used and errors corrected for. One major area of concern in the investigation of these shortcomings is the surface geometry of the optical device i.e. the shape of the lens as it concerns our study.

There are several techniques that have been used in the characterization of optical

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devices. They include morphological characterization [13], mechanical characterization [14], interferometric characterization [15,16] amongst many others. Amongst the methods mentioned earlier, interferometric characterization appears to be the most preferred method because it offers a non-contact mechanism in its diagnosis.

Four key reasons stand out in choosing interferometric characterization and they are the ability to give full-field data, the advantage of not modifying the condition of the test sample, possible levels of resolution that can be realized, and the link between accuracy of the measurement and the measurement scale used [17] for example the ”industry standard” optical profilometers, most of which are essentially white light interferometers. The last reason is perhaps the greatest key factor in choosing interferometric techniques.

Interferometry is a non-contact optical method of measurement that employs the concept of interference of waves (radio, light) with the use of an interferometer. In- terference results in fringe pattern formation from superposition of light wavefronts traveling from different optical path in the same direction [18]. Optical interferometry is an age-long laboratory method spanning approximately 300 years. Improve- ments have been made over the years on its use with the most remarkable being the advent of the lasers as a source of light. This overcomes the problem of coherence imposed by the light generated from the arc mercury traditional source of light [19].

Also, the growing utilization of photo-detectors and computerized processes for its signal has advanced the use of interferometers. Interferometers are designed to be applied principally in two distinct ways. Based on this distinction, there is the amplitude-splitting interferometer and the wavefront-splitting interferometer. The underlying difference is explained in the way the optical paths are oriented and ar- ranged. When doing interferometric characterization, both principles can be used but we narrow down on the use of amplitude-splitting interferometer in this thesis.

The Mach-Zehnder interferometer (MZI) is an amplitude-splitting interferometer and is widely used in investigating lenses. MZI is widely used in characterization because information encoded in the fringes formed particularly provides information on the internal and external properties of the lens. Information such as optical perfor- mance, surface roughness, form or shape of lens and displacement of moving surfaces can be extracted [16]. This extraction of information are even made easy with the compatibility of MZI with available dedicated softwares. Usually, the sample to be investigated is measured by comparing it in relation to another lens often called the

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reference. The reference can either be the interferometer itself which normally has systematic measurement errors calibrated internally or sample comparison. What- ever be the standard comparison, the achievable result accuracy is not as exact as the accuracy of the reference. This implies that the reference sample is chosen such that it is considerably better than the test sample.

In this study, we examine the surface geometry of a 3D printed plastic lens with a standard glass lens as reference using a MZI. The goal is to determine the actual focal lengths that the test lenses should be designed with as opposed to the nominal focal lengths used in its original design. An elaborate discussion on the principles of interferometry, interference is covered in chapter 2. The experimental set-up is explained in chapter 3. Results from conducted experiments are presented and discussions arising from it are covered in chapter 4. Chapter 5 details the conclusions drawn from the study.

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Chapter II

Theory

In this chapter, theoretical background on concepts and phenomenon useful in the understanding of this work are explained as simple as possible.

2.1 Light wave

Light can be seen as a transverse electromagnetic wave that travels spatially. Since the electric field and magnetic field are related and propagate together, it is enough to consider the electric field only at any given point; this field can be treated as a time-dependent vector perpendicular to the direction of the waves propagation [21].

Assume we have a linearly polarized wave traveling in a vacuum in the z axis, it is possible to define the electric field E at any point as a sinusoidal function of time and distance.

E =acos[2πν(t−z/c)], (2.1)

where a signifies the amplitude, ν is the frequency, and c the propagation speed of the wave. Given that T is the period of vibration and ω is the angular frequency

T = 1/ν = 2π/ω. (2.2)

The wavelength λ is related by Eq. (2.3)

λ=cT =c/ν, (2.3)

with the propagation constant of the wave given as

k = 2π/λ. (2.4)

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In a medium with a refractive index n, where the light wave spreads at a speed

υ =c/n. (2.5)

The wavelength of that radiation would be λ_n =υT

=υλ/c=λ/n. (2.6)

2.2 Complex representation of light waves

A complex exponential representation is perhaps more easy to use when describing mathematical representation of light waves, hence we can rewrite Eq. (2.1) as

E =<{aexp[i2πν(t−z/υ)]}. (2.7) The significance of this is that E takes only the real value of the expression in the curly braces where the value of i= (−1)^1/2 [21]. The advantage of this description is that the right-hand side expression can now be presented as product of factors varying spatially and temporarily such that

E =<{aexp(−i2πνz/υ) exp(i2πνt)}

=<{aexp(−iφ) exp(i2πνt)}, (2.8) where

φ = 2πνz/υ

= 2πnz/λ. (2.9)

The multiplication S=nz is referred to as the optical path between the source and the pointz, andφ is the resulting phase difference [20,21]. Assuming all operations are linear, it becomes easier to use the complex notation

E =aexp(−iφ) exp(i2πνt), (2.10)

and extract the real part after solving. Eq. (2.10) can then be written as

E =Aexp(i2πνt), (2.11)

where

A=aexp(−iφ) (2.12)

is termed the complex amplitude of vibration [21].

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2.3 Intensity of light

It is not possible to directly observe the electric field of a visible light wave due to its very high frequencies (ν ≈ 6×10¹⁴ Hz for λ = 0.5 µm). The only quantities that can be measured is the intensity which is defined as the time average of the amount of energy which passes in unit time a unit area perpendicular to energy flow direction [21]. The intensity is in turn a measure of the time average of the square of the electric field.

hE²i= lim

T→∞

1 2T

Z T

−T

E²dt. (2.13)

From Eqs. (2.1), (2.2), and (2.9), we have hE²i= lim

T→∞

1 2T

Z T

−T

a²cos²(ωt−φ)dt

=a²/2. (2.14)

Since our interest is not in the absolute value of the intensity but in the relative values over a given area, we can neglect the factor of 1/2 and other factors of proportionality and express the optical intensity as

I =a² =|A|². (2.15)

2.3.1 Intensity in interference of two monochromatic waves

Suppose at a point P, two monochromatic waves travelling in the same direction and polarized at the same plane are superimposed, the sum of the electric field at this point is

E =E₁+E₂, (2.16)

where E₁ and E₂ are the electric fields of the individual waves. If both waves propagate with equal frequency, the intensity at this given point is

I =|A₁+A₂|², (2.17)

whereA₁ =a₁exp(−iφ₁) andA₂ =a₂exp(−iφ₂) are the complex amplitude of both waves [19,21]. Correspondingly,

I =A²₁+A²₂+A1A^∗₂+A^∗₁A2.

=I₁+I₂+ 2(I₁I₂)^1/2cos ∆φ, (2.18)

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whereI₁=hA₁i²andI₂=hA₂i²are the intensities of the two waves and 2(I₁I₂)^1/2cos ∆φ is the interference term.

2.3.2 Phase determination

In general, the (real) phases of both waves will not be equal due to the different paths traveled to point P. However, if the experiment was conducted such that equal phase difference ∆φ is introduced accordingly between the components, we have

∆φ = 2π

λ ∆S, (2.19)

where ∆S is the optical path difference for the two waves from their common origin to P, and λ_o is the wavelength of the light. Generally, Eq. (2.18) reveals that the amplitude components and phase difference of both waves is integral to the description of the interference term [22].

2.3.3 Interference order

From Eq. (2.19), the optical path difference ∆s is expressed as

∆S= (λ/2πc)∆φ, (2.20)

or a time delay

τ = ∆S/c= (λ/2πc)∆φ. (2.21)

The order of interference is given by

N = ∆φ/2π = ∆S/λ=ντ, (2.22)

If the phase difference ∆φ of the beams changes across the field of view linearly, there is a corresponding cosinusoidal change in intensity leading to alternating light and dark regions known as interference fringes [22]. These fringes reflect the effect of the constant optical path difference or phase difference.

2.3.4 Visibility of Interference fringes

Assuming fully coherent light, the total intensity in an interference pattern is given by Eq. (2.18) and consequently will have its maxima of intensity at

I_max=I₁+I₂+ 2(I₁I₂)^1/2, (2.23)

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when ∆φ = 2mπ, or ∆S = mλ, where m is an integer and its minima of intensity at

Imin =I1+I2−2(I1I2)^1/2, (2.24) when ∆φ = (2m+ 1)π , or ∆S = (2m+ 1)λ/2. The intensity changes between a maximum value ofImax = 4I1 and a minimum value of Imin = 0 in the special case (standing waves) whenI1 =I2 [22].

The visibility V of the interference fringes is therefore defined in terms of the relation between Imax and Imin as

V= I_max−I_min

I_max+I_min, (2.25)

where 06V61. From Eqs. (2.23) and (2.24), we have V= 2(I₁I₂)^1/2

I1+I2

. (2.26)

2.4 Two beam interference

Thermal sources of light are not monochromatic. They exhibit random variations in their amplitudes and phase coupled with the uncorrelated fluctuations when the beam’s source are different. Therefore, it is impossible to observe interference effects from two thermal sources. In order to observe constant interference pattern, it is imperative that the two interfering beams maintain constant phase with respect to time and propagate with the same frequency [21]. To achieve this, both beams must emanate from the same source. With this approach, it is sufficient to describe interference between two beams using the same theory expounded for monochromatic light in previous section. The resulting phenomena can be categorized based on the approach used in obtaining these beams.

2.4.1 Wavefront division

The double-slit experiment carried out by Young is perhaps the earliest attempt made in explaining interference of light which applied the use of wavefront division.

The configuration was such that two secondary beams were isolated from different parts of the primary wavefront. Fig. 2.1 shows the experiment conducted by Young in explaining the wave nature of light. The two pin holes are referred to as

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the secondary monochromatic light sources which are equidistant from the primary source and in phase. In the region beyond it, the two beams combine forming an interference pattern. A good application of the wavefront division is in the Rayleigh interferometer [21].

opaque screen

Figure 2.1: Schematic diagram of the Young’s double slit experiment showing the theory of wavefront division interferometers.

2.4.2 Amplitude division

The alternative way to get two beams from one source is to apply the amplitude division method across same portions of the wavefront. Principles of reflection and refraction come into play when using this method because a part of the light is transmitted and the other reflected through the use of appropriate optical devices.

The most common device used is the beam splitter (a transparent plate coated partly with a reflecting film transmitting one part of the beam and reflecting the other) [20]. The diffraction grating has also been used; producing in addition to the transmitted beams a number of diffracted beams. A cube comprising two right- angle prisms whose hypotenuse faces are glued together with a partly reflecting thin film incorporated into it can also be used. Yet another way is to make a polarizing beam splitter (PBS). A PBS is a beam splitting cube with a multilayered film in it transmitting one polarization and reflecting the other. However, for the

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two beams to recombine, the electric vectors must be directed back to the same plane by means of another polarizer. These optical devices are shown in Fig. 2.2.

Examples of amplitude-splitting devices are Michelson interferometer, Fabry-Perot interferometer, Sagnac interferometer, and Mach-Zehnder interferometer which is our area of interest for this study [21].

beamsplitter

beamcombiner

Figure 2.2: Schematic diagram of the theory of amplitude splitting interferometers.

2.5 Interference in a plane parallel plate

Suppose a point source of monochromatic light is incident on a transparent plane- parallel plate as shown in Fig. 2.3. Regardless of the position, two sets of rays arrive at a point K on the same side of the plate as the source with approximately equal amplitude - one reflected at the upper surface of the plate while the other at the lower surface of the plate - causing the formation of non-localized fringes [22].

Symmetrically, we observe that the interference fringes in the planes parallel to the plates are circles with it’s center at O⁰. O⁰ being the point where K intersects the axis SO, the normal to the plate. The visibility of the fringes start to reduce when the source is extended in the planes parallel to SOK with an exception when point K is at infinity [22].

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Plane of observation

Plane-parallel plate

O'

O

K

S

Figure 2.3: Interference fringe formation in a plane parallel plate by reflection.

2.5.1 Fringes of equal inclination

In this situation, the fringes are viewed from the back focal plane of an objective as shown in Fig. 2.4. Here, two interfering rays BB⁰ andCC⁰ emanates from the same incident ray and are parallel. Let d be the thickness of the plate with refractive index n₂ whilen₁ is the refractive index of the surrounding medium.

Taking θ₁ and θ₂ as the angles of incidence and refraction at the upper surface respectively, we get

AB=BC =d/cosθ₂, (2.27)

AC = 2dtanθ2, (2.28)

and

AD=ACsinθ1 = 2dtanθ2sinθ1, (2.29) Correspondingly, the optical path difference between the two rays should be

∆S =n2(AB+BC)−n1AD

= 2n₂dcosθ₂. (2.30)

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d

n1

n2

n

1

B

B'

C

C'

D

1

2

P S'

S

Figure 2.4: Fringes of equal inclination formation in a plane parallel plate by reflection.

It is important to also take into consideration a phase shift ofπat one of the surfaces caused by reflection. The interfering wavefront consequently have a resultant optical path difference of

∆S = 2n₂dcosθ₂±λ/2. (2.31)

Consequently, a bright fringe will satisfy the condition

2n₂dcosθ₂±λ/2 =mλ, (2.32) where m is an integer while a dark fringe accordingly satisfies the condition [22]

2n₂dcosθ₂±λ/2 = (2m+ 1)λ/2. (2.33) Clearly, we see from Eq. (2.31) that phase difference between the wavefronts is dependent only on angleθ₂ for a given value ofd. This allows us to replace the point

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source with an extended monochromatic source without any problems. Since this is correct for only one specific plane of observation, the fringes are termed localized (localized at infinity in this instance) [22]. Visibility is unaffected because fringes formed at the back focal plane of the lens by any other point S⁰ of the extended source are similar to the one formed by the original point S. Hence, they are called fringes of equal inclination (Haidinger fringes) because they are circles centered to the normal of the plate [22]. In the case of transmission likewise, the beam transmitted directly interferes with the beam generated by two internal reflections. The optical path difference between the beams is given by

∆S = 2n₂dcosθ₂, (2.34)

since the net phase shift as a result of reflections at the two plate surfaces is either zero or 2π. The fringes thus harmonizes with the ones observed due to reflection though the visibility is low because of the difference in the relative amplitude of the beams.

2.5.2 Newton’s ring

In optical metrology, different complex forms of fringe pattern can be realized from interferometric measurements. However, all these complex fringes can be reduced to or represented by two basic fringe patterns: straight equally spaced fringe pattern or quadratic shaped fringe pattern otherwise called Newton rings [23,24]. Newton’s ring are traditional examples of interference fringes often produced when two spherical wavefronts of different curvatures interfere. The rings are composed of circular coaxial fringes with relative spaces. The spaces become thinner as the distance increases from the center of the pattern. Analysis of Newton’s ring are helpful in the determination of radius of curvature of optical devices, wavelength measurements, displacement detection, etc [25].

2.5.3 Radius of Curvature

There are several techniques that have been used in the examination of Newton’s ring for the measurement of physical parameters. One common technique used is to find the center of the circular ring and compute the radius of curvature. Other methods involve examining the structure of the fringes in the interferogram. The method described in [26] is a classical technique based on the physical consideration

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of the interferogram to estimate the strongest and weakest points of the fringes where the value of the intensity of the fringes is maximum or minimum. There are other methods used such as the spin filtering method, wavelet method, windowed fourier transform method, etc. in estimating the center of the circular fringes [25].

The intensity distribution of Newton’s ring can be described by, I(x, y) = I₀+I₁cos(∇_ϕ)

=I₀+I₁cos 2π

λ₀δ

=I₀+I₁cos 2π

λ₀Rr²+π

, (2.35)

whereI₀ and I₁ are the average intensity and the amplitude of sine variation of the fringes respectively, ∇_ϕ is the phase difference given by ϕ₁(x, y)−ϕ₂(x, y), where ϕ₁(x, y) and ϕ₂(x, y) represent the phases of the reflected wavefront of the two lenses. δ is the path difference of the two reflected beams and by simple analysis, δ = r²/R+λ₀/2 and λ₀ is the wavelength of the incident light, r is the radii of the Newton’s ring and R is the radius of curvature of the lens under investigation.

if we assume that coordinates (x₀,y₀) represents the center of the Newton’s ring, replacing r² with (x−x₀)²+ (y−y₀)² and adopting 2π/λ₀R =πK, Eq. (2.35) can be rewritten as

I(x, y) = I₀+I₁cos[πK(x−x₀)²+πK(y−y₀)²+π]. (2.36)

2.6 Mach-Zehnder Interferometer (MZI)

The MZI designed over 100 years ago ranks as one of the most used optical set-up in capturing interferograms. It is an amplitude division interferometer and comprises two sets of fully reflecting mirror and another two identical sets of partially reflecting mirrors otherwise called beam splitters. These mirrors are used to split the beams into separate optical paths and recombined afterwards with the interference pattern acquired by a detector. Differences in optical path length can be introduced by tilting one of the beam splitter or putting a test sample along the path of one of the beams.

A degree of control is achievable when using the MZI such as control over the fringe spacing by changing the angle between the beams coming from the interferometer, control over the location of the intersection points of a pair of rays coming from the

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same point on the source by separating the beams laterally, etc [20]. By substituting with an extended source, possibilities of fringe localization at any desired plane becomes possible. Some interesting features of the MZI are that only one transverse mode of a laser is allowed in the measurement path even when a multiple mode laser is used (e.g pulsed Ruby laser), splitting of beams can be made as large as possible, and the test sample can be made to match the area of fringe localization permitting an extended source of high power to be used. The major drawback of the MZI is the difficulty encountered in its alignment. All these interesting features combine to make the MZI an important tool in the studies of density variation in gas-flow patterns in research chambers, heat transfer, and temperature spread in plasmas, etc [21].

2.6.1 Working principle of a Mach-Zehnder interferometer

In this section, a brief non-mathematical approach of how the MZI works is described based on the simple set-up shown in Fig. 2.5. Few things to note before description is that when light ray arrives on a surface and the material has a higher refractive index on its other side, then a phase shift of exactly λ/2 is experienced by the reflected light ray. The refractive index of a perfect mirror can be imagined as infinite, so light reflected by a mirror has its phase changed by λ/2. When a light ray arrives on a surface and the refractive index of the material on the other side is lower, then there is no phase change. When a light ray travels from one medium into another, effects of refraction causes its direction to change without a phase change occuring at the surfaces of the two mediums. When a light ray propagates through a medium, its phase will be shifted by a value dependent on the refractive index of the medium and the path length of the light ray through the medium.

There are two detectors D1 and D2 each showing the effect of a beam traveling through different optical path as labeled in A and B. The detection of light on both detectors is a sum contribution of the effect of phase shifts on both optical path. If we consider light arriving at D1 along path A, light is first reflected by the front of the BS1 causing a phase change of half wavelength (λ/2), then reflected by M1 with an additional λ/2 phase change and transmitted through the BS2 with a constant phase change. Along path B, light arriving at D1 is first transmitted through BS1 with constant phase change, then reflected by the front of M2 introducing a λ/2 phase change, and finally reflected by the front of BS2 with an additionalλ/2 phase

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Figure 2.5: Schematic diagram of a Mach-Zehnder interferometer.

change. Summing the contributions from both optical paths, we observe that the light arriving are both in phase because same phase change has been introduced thereby producing constructive interference.

Turning to D2 and following the same principle, we observe along path A that light arriving at D2 is first reflected by the front of BS1 with a λ/2 phase change, then reflected again by M1 causing a further phase change ofλ/2, transmits through BS2 with a constant phase change, reflected by the inner surface of BS2 with no phase change and finally transmitted through BS2 again with additional constant phase change. Along path B, light is first transmitted through BS1 with a constant phase change, followed by a reflection at the front of M2 with a phase change ofλ/2 and finally transmitted through BS2 with another constant phase change. Summing the phase shifts up, we observe that light through path A has gone through an additional phase change of λ/2 causing a differential in the phases. Therefore, at D2, there is no detection of light proving the essence of the MZI that regardless of the wavelength, light is detected at only one detector in this case D1.

However, when a test sample and reference is placed between the beams along these two optical paths as is the case for the interferometric characterization in this thesis, the intensity of light seen at the detectors changes enabling the analysis of phase shift introduced by the test sample in determining its surface geometry.

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2.6.2 Laser as a light source

As earlier mentioned, the invention of the laser as an alternative source to the traditional light sources (e.g mercury arc) opened up the world of interferometry to lots of possibilities. Lasers have therefore replaced other forms of light sources in the setup of MZI. A list of commonly used lasers are shown in Table. 2.1. The Helium- Neon (He-Ne) laser is mostly used in the optical laboratory. They usually function at a wavelength of 633 nm but versions with other visible and infrared wavelengths that gives relevant outputs are available. Diodes is another common type of laser used for example in CD, DVD, blu-ray, etc. [27].

Table 2.1

Laser types used for interferometry

Laser type Wavelength (µm) Output

He-Ne 3.39, 1.15, 0.63, 0.61, 0.54 0.5-25 mW

Ar⁺ 0.51, 0.49, 0.35 0.5 W-a few W

Diode 1.064, 0.780, 0.660, 0.635, 0.594, 5-50 mW 0.532, 0.475, 0.405

Dye 1.08-0.41 10-100 mW

CO₂ ∼10.6,∼9.0 few W-few kW

Ruby 0.69 0.6-1o J

Nd-YAG 1.06 0.1-0.15 J

2.6.3 Beam splitter

Beam splitters are optical components used to divide incident light in a specific ratio into two different beams which may or may not have equal optical power [28].

Alternatively, they are also used as a means of recombining two split beams into one beam. They are mostly grouped as either a cube or plate based on their design.

Plate beam splitter

If it is a plate, the separating face is made to have a partially reflective coating and reflects the light accordingly while the other face is made not to reflect light. To do this, a multi-layer anti-reflective coating is applied on it, or the plate is positioned

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Figure 2.6: Diagram of a plate beamsplitter.

at brewster angle and a p polarized light source is used allowing it to pass through without reflection [29]. One other way is to introduce a wedge angle on the plate enabling the unwanted reflected light to go out of the system. In this beam splitter, we see a laterally parallel shift in the optical axis of the transmitted section of the beam as shown in Fig. 2.6. This shift is defined by Eq. (2.37)

d =tsinθ

1− cosθ (n²−sin²θ)^1/2

, (2.37)

whereθ is the angle of incidence,t is the thickness of the plate andn is the refractive index.

Beam splitter cube

In a beam splitter cube (polarizing or non polarizing), the splitting happens at an interface inside the cube. they are often constructed by gluing or cementing two triangular glass prisms. Crystalline materials may also be used which are birefringent hence allowing for several types of polarizing beam splitter to be produced [30]. Fig.

2.7 shows examples of cube beam splitters.

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(a) (b)

Incident beam

Transmitted Reflected

p

p s

%

Cube size Beam splitter

coating

s

Incident beam

Transmitted Reflected

p

p s s

%

Cube size Beam splitter

coating

s p

Figure 2.7: Diagram of a cube beamsplitter: (a) Polarizing beamsplitter and (b) Non polarizing beamsplitter.

2.6.4 Mirrors

Mirrors are optical devices that function on the basis of law of reflection which states that light rays incident on a reflective surface at an angle will be reflected at that same angle. This law is based on the Fermat’s principle. Fermat’s principle stipulates that light traveling between two points takes the path that requires the least time compared to other nearby paths. The law of reflection can be derived from this principle as shown in Fig. 2.8. Suppose we want light to move from point P to point Q bouncing through the mirror. Light takes the path shown in Fig. 2.8 then the path length is given as,

L=√

a²+x²+p

b²+ (d−x)², (2.38)

since speed of light is constant everywhere across all possible paths, the shortest path takes the minimum time path. Therefore, taking the derivative ofL with respect to x and equating it to zero, we get the shortest path.

dL

dx = 2x

2√

a²+x² − 2(d−x) 2p

b²+ (d−x)² = 0, (2.39)

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Figure 2.8: Light path from P to Q describing Fermat’s principle.

the equation reduces to

√ x

a²+x² = (d−x)

pb²+ (d−x)², (2.40)

From Fig. 2.8, we see that Eq. (2.40) is equivalent to sinθ_i = sinθ_r. Hence, we can conclude that θ_i =θ_r thus fulfilling the law of reflection. This principle can also be extended to concave and convex mirrors. The mirrors used in the MZI fully agrees with this principle reflecting light from the angle at which it is incident on it. It is possible to control this angle by adjusting the mechanism used in fitting the mirror in the interferometer. This is part of the initial precautions that must be taken in aligning the interferometer.

2.7 Detectors/solid-state sensors

In most interferometric systems in use today, solid-state detector arrays are used to acquire the needed intensity shapes or frames. These detectors are preferred because the linearity of the pixels with intensity shows a very good compatibility, no image delay in between measured frames and no geometric error caused by sensor in the acquired interference pattern [31]. Moreover, the measured intensity can be digitized and stored in computers for further processing. Examples of these detectors include, photodetectors, photomultipliers, photodiode arrays, charge coupled devices (CCDs), charge injection devices (CIDs). They are generally grouped based on

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their geometry as area arrays or linear arrays. The most commonly used ones for interferometry are the area arrays because they offer the advantage of measuring two- dimensional section of a wavefront or surface [32]. The number of pixels available in each dimension of the sensor determines its resolution. Earliest type of sensors have about 500 by 500 pixels similar to that of a television. Subsequent ones range from 1000 by 1000 pixels to 4000 by 4000 pixels and are used for machine vision applications. Though the linearly arrayed sensor take a one-dimensional section of the surface, it compensates for it in its resolution usually up to 7000 pixels enabling measurements with very high spatial resolutions [33]. Nowadays, sensors available can handle up to 40 megapixels. In practise, any camera designed with these sensors can be used for interferometric purposes. The only difference seen is in its resolution, response to light, range coverage and flexibility, and the data output arrangement.

2.8 Phase unwrapping

When the test sample used in the interferometric diagnosis is measured as a 3D spatial distribution, the angle of incidence of the sample beam is varied between 0 ≤ θ < π range. The wavefront transmitted is a measure of different rotation angles of θ. Several phase shift methods are available for analyzing the phase of the wavefronts. The end result is a wrapped phase in an interval of −π to π. A reconstruction of this wavefront is needed to eliminate the 2π discontinuities that is embedded in the raw phase data. The reconstruction process is what is termed phase unwrapping or phase continuity. The modulo 2πphase information is transformed to a continuous wavefront display of the test sample. On observing a large discontinuity in the reconstruction, 2π or multiples of 2π are added to the neighboring data to eliminate the discontinuity [34]. Several algorithms have been deployed in phase unwrapping ranging from its 1D data to its 3D data. Some of the algorithms used have been reported in [35] for 1D, [36] for 2D and [37] for 3D.

2.9 Optical profilometry

Optical profilometry is a method of investigating the surface properties of an optical device. The properties investigated which include the surface roughness, surface morphology and step heights are determined from the topography of the surface [38]. There are two basic methods used in profilometry - the contact method and

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non-contact method. The contact method otherwise called the stylus profilometer uses a probe to physically scan the surface of the optical device to extract data.

However, the non-contact method uses light in place of a physical probe. The important thing is to ensure that light is focused on the sample in a way that it can detect the surface. Several techniques such as confocal microscopy, low coherence interferometry, etc. are some of the methods that have been employed [39]. The vertical scanning interferometry technique was used in this report.

2.10 Vertical scanning interferometry (VSI)

Vertical scanning interferometry is a vital, non-intrusive tool that has been utilized effectively too evaluate the surface topography of solids i.e glasses, ceramics, metals, with a high degree of precision. It makes up for the constraints encountered while using phase shifting interferometry where phase uncertainty issues limits the height variations between two adjacent data points to the wavelength of the light used [40].

It capitalizes on the low coherence feature of white light. Interference is achieved only when the delays in path length coincide within the coherence time of the light source.

Unlike other interferometry techniques that monitor the shape of the fringes, VSI focuses on fringe contrast i.e. fringes are seen only around the best focus position.

Due to low coherence of white light, optical path difference (OPD) matching becomes tedious because the contrast changes as the path length changes [41]. However, this shortcoming provides a useful method to evaluate the measured fringes from the changing contrast. Fig. 2.9 shows how a typical VSI is set up in the laboratory.

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CCD sensor

Frame grabber

Computer

PZT controller PZT

Sample surface Beam

splitter Magnification selector White light

source

Microscope objective Reflectance mirror Plate beamsplitter Mireau objective

Figure 2.9: Schematic diagram of a vertical scanning interferometer.

The wave property of light is used to compare OPD between a reference surface and a test surface inside the interferometer. The beam splitter splits the light passing through the test surface reflecting half of it through the focal plane microscope objective while the other half is reflected from the reference mirror. When an equal distance from the beam splitter to the reference mirror and to the test surface is achieved, and the beams are recombined, interference fringes are formed wherever there is variation in the length of the light beams. The OPD is as a result of variations of height on the test surfaces. The interference beam is directed to a digital camera which interprets constructive interference as light bands and destruc- tive interference as dark bands. Each transition from light to dark represents λ/2 difference between the test path and reference path.

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Chapter III

Experimental set-up and measurements

In this chapter, a detailed description of how the Mach-Zehnder interferometer was assembled and steps taken in aligning it is given. The sample lenses under investigation are 3D printed plastic lens fabricated using an inkjet-based method specially designed for printing of optical components [42]. The reference lens is a glass lens of 100 mm focal length. Also, procedures taken in getting the captured fringes are explained in details.

3.1 Device set-up

The set-up of the experiment is as described in Fig. 2.5 with few other optical components added to it to enhance the quality of beam propagation. Fig. 3.1shows the full experimental set-up with the additional optical components. In this set-up, a Siemens He-Ne laser with a wavelength of 543 nm and power of 5 mW was used as the laser beam source. An expansion of the beam was done by the beam expander lens from a 4 mm Raspberry PI v2 camera objective and a lens of 75 mm focal length. The beam is made to pass through a neutral density (ND2) filter to control the intensity of the light. The beam is then directed towards the non-polarizing beam splitter (NPBS1) by means of two mirrors (M1 and M2) properly aligned.

The NPBS1 is a CCM1-BS013M model from Thorlabs working with a wavelength range of 400−700 nm. The NPBS1 was used to split the beam in a 50 : 50 ratio into the two arms of the interferometer; one serving as a reference beam and the other as the test beam. The test wave propagates straight while the reference wave deviates at a right angle. The test wave is reflected at right angle by another mirror (M4) towards the sample lens while the reference wave is also reflected at right angle by

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another mirror (M3) towards the reference lens. An aperture of diameter 15 mm was placed after both lens to control the amount of light that passes through the lens before they are then combined at another non-polarizing beam splitter (NPBS2) of the same model as NPBS1. The combined beam passes through a ND3 filter and is detected by a Canon FD 50 mm f/1.4 objective and Thorlabs DCC1545M camera.

The detected interference fringes are fed into a computer system for processing and displayed on the computer screen.

He-Ne laser Beam expander

ND 2 filter M1

M2

M3

M4 NPBS 1

NPBS 2 reference

lens reference lens

test lens

ND 3 filter

Detector

computer system

Aperture

Shutter arm control Shutter

arm control

Figure 3.1: Schematic diagram of the experimental set-up used for measurements.

Figure 3.2: Experimental set-up of the interferometer in the Laboratory used for measurements.

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3.2 Alignment of the system

As stated in section 2.6, aligning the MZI often proves to be a difficult task. The set-up used in this thesis was not an exception. The beam splitters (placed on mi- crometer slides) and mirrors in the MZI ideally should give room for tilt adjustments about the vertical and horizontal axes. However, in our set-up, the beam splitters are fixed about their axes leaving the mirrors as the only adjustable optical components thereby making alignment even more difficult than usual. The beam from the He-Ne laser is directed through the center of the NPBS1 and is incident on the center of M3. Mirrors M1 and M2 are adjusted such that the reflected beam is centered at mirror M3. This was monitored by the use of an aperture placed on the path of the propagated beams. Mirrors M3 and M4 are adjusted so that the beams reference wave and test wave are parallel and overlap at the center of NPBS2. These procedures were repeated several times until the beams are made to coincide at an accepted range as perfect overlap could not be achieved. At the region of overlap, interference fringes are formed and are adjusted until vertical fringes are clearly seen.

The mirrors M3 and M4 are adjusted until the zeroth order white fringe appears in the field. The success of coincidence of the beam is shown in Fig. 3.3 from the screen shot on the computer.

Figure 3.3: Screen shot of the degree of alignment recorded from the interferometer set-up as opposed to a perfect alignment of the two beams ideally.

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3.3 Measurement procedures

As stated in this set-up, there are two interferometer arms (I₁ and I₂); one arm for the glass reference lens (I₁) and the other arm for the 3D printed test lens (I₂). Seven test lenses were printed with different focal length and each one was measured. In taking the measurements, the Mach-Zehnder setup incorporates servo motor shutters controlled with Arduino board. This enables us to automatically block interferometer arms when necessary with Matlab commands using serial port communication. As a precaution, the reference lens and the test lens are kept at the same optical path length to minimize the circular fringes on the camera. The reference lens is then shifted in the x-direction to introduce narrow linear fringes before measurements are taken. For each lens, the process of shifting the reference lens in the x-direction is repeated to detect the narrow linear fringes. Measurements were taken in this order: first, both arms are set to open by default at initial set-up.

Interferometer arm (I₁) is blocked while interferometer arm (I₂) is open and the fringe information taken, the procedure is interchanged with I₁ now open and I₂ closed. Next, the two interferometer arms (I₁ and I₂) are closed to obtain a dark frame and subsequently both (I₁ andI₂) are opened to obtain the fringe information when they interfere (I). The significance of obtaining dark fringes and blocking the arms of the interferometer is to reduce the effect of un-even illumination and in the normalization of the fringes as shown in Eq. (3.1).

I_norm = I−I₁ −I₂ 2√

I1I2

. (3.1)

3.4 Optical profilometer measurements

Optical profilometer measurements were taken for three of the lenses used. The reference lens and two of the printed lenses (100 mm and 97 mm focal length). The measurement procedures were taken as described in Section 2.10. The objective used has a 10×magnification (based on Mirau interferometer). The interferometer carried out 132 measurements for each lens which were stitched by the device software to the final result. Filtering, tilt removal and fitting adjustments were done for optimal spherical shape.

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Chapter IV

Results and discussion

This chapter is intended to present the results that were obtained from the measurements carried out in the laboratory. It also includes discussions on the results, points of observation as it relates to the different lenses measured, etc.

4.1 Results

As stated in this set-up, there are two interferometer arms (I₁ and I₂); one arm for the glass reference lens (I₁) and the other arm for the 3D printed test lens (I₂) kept at same optical path length to minimize the circular fringes on the camera. The reference lens was a N −BK7 plano-convex glass lens manufactured by Thorlabs with refractive index, n = 1.517, focal length of 100 mm and radius of curvature of 51.5 mm. Seven lenses in total were measured as the test lens. One of the test lenses is a glass lens having the same specifications as the reference lens (same refractive index and focal length). The other six lenses are 3D plastic lenses with equal refractive indices,n = 1.533 but different nominal focal lengths. The reason we have a glass lens as one of the test lenses is to observe what happens with the phase information when lenses of the same specifications are used in both interferometer arms. This will enable us to compare results of lenses with different materials and different specifications relative to the glass lens. Ideally, if the lenses are of the same material and specifications given no abberations, there should be no phase change.

However, if the lenses are not of same materials and specifications, we expect to see phase changes or variations in the phase information. In our measurements however, there is a slight phase change even while using the glass lens of the same specification with the reference lens as shown in Fig. 4.1.

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(a) both arms open

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Figure 4.1: Measurement of the phase profile difference from interferogram of reference lens F = 100 mm, (a)I1 and I2 open. (b) I1 open, (c) I2 open, (d)intensity normalized interferogram Inorm, (e) Fourier transform of Inorm, (f) cropped area, (g) inverse Fourier transform of (d), (h) wrapped phase, and (i) un-wrapped phase

Fig. 4.1(a),(b), and (c) shows the fringe information recorded when both (I₁ and I₂) are open, I₁ is open and I₂ closed, and I₂ is open and I₁ is closed respectively.

The dark frame is also captured however not shown in the figure. Fig. 4.1(d) shows the normalized fringe data from applying Eq. (3.1) which is used in finding the phase profile. In getting the phase profile, a 2D Fourier transform ofI_norm is taken, The result is shown in Fig. 4.1(e). One of the two peaks of the Fourier transform is cropped and its maximum value shifted to the center of the Fourier coordinates as shown in Fig. 4.1(f). The inverse Fourier transform enables us to extract the wrapped phase shown in Fig. 4.1(h). The unwrapped phase is also calculated and shown in Fig. 4.1(i). The detailed process and calculations of how this images are calculated and extracted are documented in [26].

Similarly, the measurement procedures are repeated for the other six lenses and adjustments on the x-axis to get narrow linear fringes are made for each lenses. The

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results for each lenses are shown in Figs. 4.2-4.7.

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Figure 4.2: Measurement of the phase profile difference from interferogram of 3D printed lens 7,F = 100 mm, (a)I1andI2open. (b)I1open, (c)I2 open, (d)intensity normalized interferogram Inorm, (e) Fourier transform of Inorm, (f) cropped area, (g) inverse Fourier transform of (d), (h) wrapped phase, and (i) un-wrapped phase

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Figure 4.3: Measurement of the phase profile difference from interferogram of 3D printed lens 2,F = 99 mm, (a)I1 andI2open. (b)I1open, (c) I2 open, (d)intensity normalized interferogram Inorm, (e) Fourier transform of Inorm, (f) cropped area, (g) inverse Fourier transform of (d), (h) wrapped phase, and (i) un-wrapped phase

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Figure 4.4: Measurement of the phase profile difference from interferogram of 3D printed lens 3,F = 98 mm, (a)I1 andI2open. (b)I1open, (c) I2 open, (d)intensity normalized interferogram Inorm, (e) Fourier transform of Inorm, (f) cropped area, (g) inverse Fourier transform of (d), (h) wrapped phase, and (i) un-wrapped phase

Interferometric characterization of 3D printed optic devices