Overview of Projection Moiré Profilometry with High-Dynamic Range

Chapter 1 Introduction

1.5 Overview of Projection Moiré Profilometry with High-Dynamic Range

As a type of optical measuring apparatus, the charge-coupled diode (CCD) camera provides the capability of increasing the speed of measurement by inspecting an area with only one shot. However, the CCD camera ’ s high-variation range of reflectivity presents an exceptional challenge for the optical measurement established on the surface. The dissertation presents a method that could enable one to acquire an image with a high-dynamic range in one shot without any reduction in spatial resolution. Because of the sufficient signal-to-noise ratio, the method presented could perform the robustness of the phase-retrieving algorithm, and the surface topography could be measured more accurately.

Chapter 2 Analysis of Thermo-Optic Tunable

Dispersion-Engineered Tapered-Fiber Filter

2.1 Review of Taper-Fiber Filter

Stronger interaction of the optical evanescent field with the environment has been widely utilized in many fiber-based devices such as the fused-tapered-based fiber filters [1], [2] dispersion-engineering applications [3]–[5] and nanowire sensing applications[6], [7]. The dispersion-engineering techniques manipulate the dispersion characteristics of fiber waveguides to alter the optical properties of fiber devices. In the literatures the theoretical modeling of tapered fibers had been widely developed [8]–[12], the fundamental-mode cutoff effects induced by dispersion-engineering techniques with fiber tapering have not been theoretically investigated yet. The main objective here is thus to theoretically analyze the fundamental-mode cutoff effects and to determine the optimal tapered-fiber structures for achieving low-loss and high-cutoff efficiency. These results should be very crucial for applications such as high-gain low-noise S-band fiber amplifiers [13] and widely tunable S-band fiber lasers [14].

This chapter will present experimental characterization and theoretical simulation results on the studied thermo-optic tunable fused-tapered-fiber filters made

by tapering standard single-mode fibers (SMF-28). The tapered fibers are immersed in Cargille liquids for implementing dispersion engineering through the control of material dispersion. The filter becomes cutoff at longer wavelengths due to the reducing refractive index of Cargille liquids below the effective refractive index of the Cargille liquids below the effective refractive index of the fiber waveguide. The propagation loss increases rapidly near the cutoff wavelength, and a sharp short-wavelength-pass optical filtering edge is achieved. To investigate these effects more deeply, the effective index (neff) and the mode field diameter (MFD) of the fundamental mode are calculated and carefully examined in order to understand what determines the filter performance. The numerical beam propagation method (BPM) is then adopted to theoretically simulate the cutoff phenomena and the temperature tuning characteristics of the whole device. Good agreement between the simulation and experimental results has been found. Since the filter performance is also significantly influenced by the whole waveguiding structure, the optimal parameters for the uniform taper diameter, uniform taper length, and taper transition length are investigated by the full BPM simulation for achieving best filter performance. From the simulated spectral responses and the effective dispersion curves, find that the taper-waist diameter greatly affects the MFD, which in turn affects the final dispersion relation and the achievable cutoff slope. The taper length can also affect the cutoff slope, and the transition length has direct impacts on the insertion loss. The obtained theoretical results help to determine the optimal device structures for fabricating efficient short-pass tapered-fiber filters which can be utilized in high performance S-band Er-fiber amplifiers and lasers.

2.2 Fabrication Process and Operation Principle

The high-cutoff efficiency of the studied short-pass fiber filter is obtained by locally modifying the material and waveguide dispersion within the uniform taper region. The schematic of the fused-tapered short-pass filter is shown in Fig. 2-1(a).

One can describe the structure of a tapered-fiber filter by specifying the transition length (denoted as τ), the uniform waist length (denoted as L0) and the waist diameter (denoted as ρ). It consists of a transition zone where the diameter is gradually reduced to ρ over a distance τ and then a uniform waist section with the length of L0. The uniform waist section is immersed within suitable Cargille index-matching liquids.

The optical properties of the used Cargille liquids can be summarized as follows. The refractive index n_D =1.456, thermo-optic coefficient dn_D dT = −3.74 10× ⁻⁴ ⁰C,

and optical transmittance is 88% at 1300nm, and 80% at 1550nm for 1-cm-long length. The whole device is mounted on a thermoelectric (TE) cooler for temperature control. Those tapered-fiber filters are fabricated by a homemade tapering workstation comprising several modules for fiber pulling, heating, and scanning as illustrated in Fig. 2-1(b). A hydrogen flame head with high-accuracy flow control is set on a three axis stepper motor to precisely control the traveling of the flame over the distance of few centimeters. When the tapering process begins, the pulling motors move outward to pull the heated fiber and the flame simultaneously starts to travel back-and-forth for heating the region to be tapered. The pulling mechanism employs a high-precision stepping motor with a right-and-left-threaded screw to drive two V-groove clampers moving outward in a reverse direction. The clampers bilaterally hold the single-mode fiber for providing a precise pulling stress. By controlling the moving speed of the

scanning flame and the pulling clampers, the total elongation length can be varied from 30 to 60 mm, and the corresponding length of the uniform waist is measured to be around 10-25 mm, contingent on the waist diameter. When tapering is finished, the fiber is fixed in a graved U-groove on a quartz substrate and immersed in index-matching liquids. A TE cooler is used to control the liquid temperature to change its refractive index for tuning the cutoff wavelength.

Fig. 2-1. (a) Diagram of a tapered-optical-fiber structure with a uniform waist. (b) Schematic diagram of the tapering station used to fabricate the tapered fibers.

When a single-mode optical fiber is tapered down to few tens micrometer in diameter, the evanescent tail of the mode field spreads out of the fiber cladding and reaches the external environment (Cargille liquids). The size of the Ge-doped core in the tapered zone is so reduced that its waveguiding effects are negligible. Therefore, the pure silica cladding plays as the new core, whereas the external medium serves as the new cladding. The material dispersion curves for the original fiber core (GeO2 4.1 mol%), cladding (pure silica), and the Cargille liquids are plotted in Fig. 2-2 to illustrate their relative relation. On the righthand side of the cross point indicated in

Fig. 2-2, the refractive index of the liquids is greater than the index of fiber taper and the total internal reflection of the interface is frustrated. Therefore, the lights cannot be guided in the fiber taper and suffer a great amount of optical loss. On the other hand, the light can be nicely confined in the fiber taper when the wavelength is shorter than the cutoff wavelength. The cutoff wavelength of the short-wavelength-pass filter should be very near the cross point of the two dispersion curves indicated in Fig. 2-2, under the condition that the mode in the tapered region is mainly guided by the original cladding. The temperature shift of the cross point as also indicated in Fig. 2-2 provides an intuitive explanation about the temperature-tuning capability of the tapered-fiber filter.

Fig 2-2. Material dispersion curves for the original fiber core (GeO₂ 4.1 mol%), cladding (pure silica), and the index-matching liquids measured at 25, 26, and 27^oC, respectively.

It should also be noted that a related but different kind of short-wavelength-pass optical filtering effects can be observed when the tapered fiber waist is in the 100 nm range. [15] The tapered fiber is surrounded by the air, and an abrupt change of the mode field diameter as a function of the optical wavelength can be found. This leads to an abrupt change of optical loss accordingly. However, since no dispersion engineering is used to produce true optical cutoff, the filtering slope and rejection

ratio are not as large as the cutoff case studied here.

2.3 Experimental and Simulation Results

Both the experimental and simulated spectral responses of the short-wavelength- pass fiber are displayed in Fig. 2-3 for performance studies and comparison. The solid lines of Fig. 2-3(a)-(c) display the experimental spectral responses of the short-wavelength-pass filter with waist diameters measured to be 20, 26, and 40 μm, respectively. The total elongation lengths are about 30 mm, and the lengths of the uniform waist are measured to be around 18 mm. The larger the waist diameter is, the steeper the cutoff slope can be. Here the cutoff slope (in unit: dB/nm) is defined as the average gradient of the rolloff spectral curve in the linear region from -10 to -30 dB transmission loss. The cutoff slopes are calculated to be -0.54, -1.38, and -2.00 for the cases of ρ = 20µm, ρ = 26µm, and ρ = 40µm, respectively. When ρ = 40µm, the cutoff slope is very sharp, but the rejection efficiency is more limited. The extra loss of less than 2 dB at 1390 nm may result from the absorption of hydroxyl ions which were generated from the hydrogen flame and then diffused into the tapered fiber. The tuning efficiencies of the filter are about 52, 47, and 62 nm/^oC for the cases of ρ = 20µm, ρ = 26µm, and ρ = 40µm, respectively. At the guiding wavelengths, the insertion losses of the filters are below 1 dB, 0.5 dB, and 0.3 dB for ρ = 20µm, ρ = 26µm, and ρ = 40µm, respectively. The losses are mainly from the absorption loss of the Cargille liquids and the optical loss due to fiber tapering. The absorption loss caused by the Cargille liquids is estimated to be below 0.2 dB after taking into account the transmittance of the liquids and the evanescent field overlapping effects.

The fiber-tapering loss depends on the tapering conditions. In principle, when the

tapering transition is slow enough to meet the adiabatic criteria [16], the tapering loss can be made very small. For our cases, the tapering loss is reasonably low but still observable, as can be seen from the above numbers. Even smaller tapering losses should be possible at the cost of increasing the total device length.

Fig. 2-3. Experimental and simulated spectral responses of the tunable

short-wavelength-pass fiber filters for (a)ρ = 20μm, (b)ρ = 26μm, and (c)ρ

= 40μm at different temperatures.

To further analyze the fundamental-mode cutoff characteristics, numerical simulation by the BPM is performed to study the optical field propagation within the tapered fibers and to predict the filter performance theoretically. The BPM uses the finite difference method to solve the paraxial approximation of the Helmholtz equation with “transparent” boundary conditions [17]. This approach can automatically include the effects of all the guided and radiation modes as well as the mode-coupling and mode-conversion effects. The fiber-taper transition structure is set to be an exponential shape in our simulation. The dotted lines in Fig. 2-3(a)–(c) show the simulated transmission spectra with the temperatures of 25, 26, and 27 ^oC and the waist diameters of 20, 26, and 40 μ m, respectively. The cutoff wavelengths gradually shift to longer wavelengths when the temperature increases. Good agreement of the changing trends for the cutoff slopes and rejection efficiencies between the simulated and experimental results has been found. The mismatch of the cutoff wavelengths between the experimental data and simulation results should be due to the uncertainties of the fiber-tapering parameters and the temperature-reading errors in the experiment. Most importantly, the simulation results correctly reproduce the experimental observation that the larger diameter cases have larger cutoff slopes, but will eventually have poorer rejection ratios when ρ is too large (> 40μm). The thermo-optic-tuning efficiency of the filter for ρ=26μm in Fig. 2-3(b) is about 48 nm/^oC from BPM simulation, which is also in good agreement with the 47 nm/^oC tuning efficiency from the experimental data.

Based on the facts that the theoretically predicted tuning and cutoff efficiencies agree reasonably with the experimental data, we further utilize the BPM as a reliable simulation tool to investigate the effects of different taper parameters for determining

the optimal taper structures.

(a)

(b)

Fig. 2-4. Field distributions along the tapered-fiber filter. (a) When the wavelength (1250 nm) is shorter than the band edge of the filter, the fields are guided over uniform waist and coupled back to the fundamental mode. (b) When the wavelength (1350 nm) is longer than the band edge of the filter, the fields spread out along the uniform waist.

Fig. 2-4(a) and (b) show the simulated field evolution within a tapered fiber with the cutoff wavelength around 1330 nm. The waist diameter ρ is 26 μm, the

transition length τis 6 mm, the taper transition angle is around 0.5^o , the uniform waist length L0 is 18 mm, and the temperature is set at 25 ^oC. In Fig. 2-4(a), when the propagation wavelength (1250 nm) is shorter than the cutoff wavelength, the fundamental eigenmode of the input SMF is smoothly transformed into the fundamental eigenmode of the uniform tapered waist, propagates through the uniform waist region and then gradually reconverts to the fundamental mode of the output SMF within the second transition distanceτ. The fundamental MFD is larger in the uniform tapered region when compared to that of the SMF-28. On the other hand, Fig.

2-4(b) shows the case when the propagation wavelength (now 1350 nm) is longer than the cutoff wavelength. The optical field quickly disperses away within the first taper transition region as well as the uniform waist region due to the higher refractive index of the surrounding liquids. Only a very small fraction of the optical field can be coupled back to the fundamental mode of the output SMF. In this way, huge optical losses are induced for wavelengths longer than the cutoff wavelength.

14 (a)

(b)

Fig. 2-5. (a) Effective index of the fundamental mode versus wavelength under different waist diameters of tapered fibers surrounded by (a) the air, and (b) the Cargille liquids (nD = 1.456).

The material dispersion of the surrounding medium can significantly modifies the dispersion curve of the propagation mode due to the stronger overlap with the evanescent field spread out of the taper waist. To analyze how the surrounding material affects the dispersion properties of the fused-tapered fiber, the effective index of fundamental mode is calculated under different waist diameters of tapered fibers surrounded by the air and the Cargille liquids(n_D = 1.456) . The results are plotted in Fig. 2-5(a) and (b). The material dispersion curves of the original Ge-doped core, the

cladding (pure silica), and the Cargille liquids are also plotted for easy comparison.

The index difference of the pure silica and the air is so large that the optical field is strongly confined when the tapered fiber is surrounded by the air. Thus, the dispersion curves of the fundamental mode have almost the same slope (_௘௙௙⁄ 1.2 10^ିହ /nm) with that of the pure silica when ρ>20μm. The effective mode index is lower than those of the pure silica and Ge-doped core because parts of the optical mode field are now in the air. When the waist diameter gets smaller than 5 μm, the dispersion curve of the fundamental mode becomes more wavelength dependent. This is because now a larger fraction of the mode field is in the air. The slopes of the dispersion curves in Fig. 2-5(b) are calculated to be -0.55×10^-5, -0.87×10^-5, -0.94×10^-5, -1.07×10^-5, and -1.14×10^-5 for ρ=10μm, ρ=20μm, ρ=26μm, ρ=33μm, and ρ=40μm, respectively. In contrast, the slopes of the modal dispersion curves for tapered fibers immersed in the Cargille liquids are slightly flatter than those in the air, which indicates that the effective mode index of the tapered fiber is indeed modified by the surrounding dispersive liquids. The insect of Fig. 2-5(b) shows the detailed curves near the cutoff wavelength when the tapered fibers are immersed in the Cargille liquids. The cross points between the dispersion curves of the tapered fibers and the surrounding liquids exactly determine the cutoff wavelength, which will shift to longer wavelengths as the waist diameter increases. When the optical wavelength is near the cross point, the optical field quickly spreads out of the tapered fibers and large optical losses are induced. Intuitively the cross angle between the two intersected dispersion curves determines the cutoff slope. A suitable surrounding material which can produce a larger intersection angle is thus a key for achieving high-cutoff efficiency.

16 (a)

(b)

Fig. 2-6. MFD/waveguide width versus wavelength under different waist diameters of tapered fibers surrounded by (a) the air, and (b) Cargille liquids (nD

= 1.456).

Fig. 2-6(a) and (b) plot the ratio of the 1/e MFD to the waveguide width (waist diameter of the tapered fiber) as a function of the optical wavelength. When the tapered fiber is surrounded by the air, the optical field is strongly confined inside the tapered fiber and thus the width ratio is almost flat with respect to the optical wavelength for the waist diameters considered here. In contrast from Fig. 2-6(b) it is obvious that a larger fraction of the optical field spreads out of the tapered fiber when it is surrounded by the Cargille liquids due to the small index difference. As the waist

diameter is getting smaller, the width ratio gets larger. When the waist diameter is smaller than 10 µm, the optical fields will largely spread out of the tapered waveguide region and will experience higher losses even at guided wavelengths. When the waist diameter is between 20 µm and 26 µm, the width ratio curves exhibit a significant turning point between the guided region and the unguided region, as shown in Fig.

2-6(b). When the waist diameter is larger than 40 µm, the cutoff wavelength is shifted toward the longer wavelengths, and the optical field is more strongly confined in the waveguide. The dispersion-engineered effects can also be seen from the wavelength dependence of the MFD. Fig. 2-7(a) and (b) show the mode field distribution of the fundamental mode in the tapered waist under different taper diameters. Two optical wavelengths at 1250 nm (guiding) and 1350 nm (near cutoff) are used as the examples to illustrate the difference. The field is strongly confined in the tapered region for the guiding wavelengths when ρ > 20µm. As the waist diameter gets smaller than 10 µm, the field spreads out of the tapered fiber region and larger losses at the guiding wavelengths are produced. Near the cutoff wavelength in Fig. 2-7(b), the field extends more widely into the Cargille liquids due to the weak guiding condition. Thus, the loss becomes huge and the short-wavelength-pass band edge is formed. Moreover, the larger taper diameters (ρ > 33µm) can confine the fields more tightly than the smaller ones even near the cutoff wavelength.

Fig. 2-7. Fundamental-mode-field distribution in the tapered waist under different taper diameters at (a) 1250 nm (guiding wavelength) and (b) 1350 nm (near cutoff wavelength).

Fig. 2-8. Simulation results of transmission spectra at different waist diameters of 5, 10, 20, 26, 33, and 40μm, respectively.

Fig. 2-8 shows the simulated transmission spectra of the fiber taper with the waist diameter of 5, 10, 20, 26, 33, 45, and 60 µm, respectively. The fiber transition length τ is 6 mm, the uniform waist length L₀ is 18 mm, and the temperature is at 25^oC. The spectral cutoff responses are not the same for different waist diameters.

Note that as the waist diameter increases, the band edge is steeper. The cutoff slopes are calculated to be -0.38, -0.79, -1.14, -1.68, and -2.01 for ρ = 10µm, ρ = 20µm, ρ = 26µm, ρ = 33µm, and ρ = 40µm, respectively. The band-edge shifts to the longer wavelengths as the diameter increases. The cutoff characteristics in Fig. 2-8 can be described by the waveguide dispersion behavior with different waist diameters. The optical field intensely spreads out into the Cargille liquids when the waist diameter is getting small. The ratio of the optical field distributed in the waveguide with respect to that in the Cargille liquids is strongly decisive to the effective index of the mode field. When the diameter is less than 10µm, the waist is too thin to confine the optical

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