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We present a novel system that can measure absolute distances of up to 300 mm with an uncertainty of the order of one micrometer, within a timeframe of 40 seconds. The proposed system uses a Michelson interferometer, a tunable laser, a wavelength meter and a computer for analysis. The principle of synthetic wave creation is used in a novel way in that the system employs an initial low precision estimate of the distance, obtained using a triangulation, or time-of-flight, laser system, or similar, and then iterates through a sequence of progressively smaller synthetic wavelengths until it reaches micrometer uncertainties in the determination of the distance. A further novel feature of the system is its use of Fourier transform phase analysis techniques to achieve sub-wavelength accuracy. This method has the major advantages of being relatively simple to realize, offering demonstrated high relative precisions better than 5 × 10−5. Finally, the fact that this device does not require a continuous line-of-sight to the target as is the case with other configurations offers significant advantages.
©2012 Optical Society of America
1. Introduction
Absolute length interferometry has traditionally been the province of national standards laboratories, which are charged with the responsibility of maintaining a reproducible and auditable chain of length standards. Absolute length interferometers are seen as being highly specialist, capable of being operated only under strictly controlled environmental conditions and were also very expensive. However recent developments in laser physics, in particular the emergence of relatively low cost, extremely narrow bandwidth laser diodes operating on the external cavity principle and grating stabilized, have led to an upsurge in interest in this area. This is motivated by the possibility of bringing ultra high precision absolute length measurement – typically sub-micrometer – out of the standards laboratory and into wider industrial use. That industry has a pressing need for such a capability is beyond doubt, as sectors such as aerospace are facing current difficulties in maintaining high levels of precision over very large assemblies.
In conventional arrangements interferometry can only yield relative distance measurements – i.e. it can only determine the difference in distance to two points if they are linked by a continuous path. This means that conventional interferometry cannot answer questions such as: ‘what is the distance from an arbitrary datum to a point, or collection of points?’; or ‘what is the difference in distance between two sets of points located on two different surfaces?’ Absolute length interferometry offers us the means to answer questions such as these.
Up to now only a small number of techniques have existed to achieve absolute length interferometry. The oldest method is the so called “excess-fractions technique”, which was the basis of the commercially available NPL Absolute Length Interferometer that is found in standards laboratories around the world. Modern variants on this method have been tried. The work of Ikezawa [1] used a CO2 laser to extend the three wavelength NPL technique to five wavelengths – but CO2 lasers are large and expensive. More recently Towers et al. [2] have constructed a fiber-based interferometer which can implement the excess fractions method, and has the virtues of being lower cost and also much more compact. However they only achieve measurement distances of up to 40mm or so, which does not address the great majority of real industrial problems.
The technique that has achieved the most attention recently is undoubtedly the frequency scanning interferometry (FSI) method – sometimes known as wavelength scanning, or wavelength sweeping. For examples see [3–9]. This technique exploits the increasing availability of tunable lasers which are also very stable, although a lot of work had to be devoted to solving the stability problem before the technique was fully successful – see for example [3, 10]. The method is capable of delivering high accuracy – approaching 1 part in 109 [11] and has been deployed over distances of up to 20m [6]. This method is already starting to find practical applications, particularly in monitoring the positional alignment of large complex assemblies, such as those which are often found in the colliders and accelerators that are used in experimental particle physics [12, 13], for use in photo-lithography applications [14] and also on coordinate measurement machines (CMM’s) [15].
Despite all the work that has been put into it, FSI suffers from two major drawbacks. Firstly it is necessary to maintain a continuous line of sight to the target whilst the laser frequency is being scanned. Any loss of this line of sight – even for an instant – results in complete loss of the measurement. Secondly the scanning can only be achieved at limited speeds without destabilizing the laser. This latter problem limits the device’s dynamic measurement potential.
With these limitations in mind other approaches have been developed. These fall into three main groups. First we have the frequency comb methods [16–18]. While these are certainly an elegant solution and overcome the line of sight problem, they have only been made to work over relatively short distances and still require a laser to be frequency scanned.
The second family of techniques are many variants around frequency mixing and the use of optical feed-back. See for example [19–24]. These usually, but not always, employ a broad-band source, and in one case [24] the author uses a white light source. These offer an extremely fast solution and many do not require a continuous line of sight. But all of them only measure accurately over extremely short distances – from some micrometers to 1-2 mm at most, and when extended to distances of the order of 1m or so, accuracies no better than 0.3mm are reported. Such accuracy levels at practical distances render these techniques of little value in many industrial contexts, as many alternative methods already exist for achieving these levels of precision at much lower levels of complexity and cost.
The third variant, to which our proposed method belongs, employs a concept known as “synthetic wavelengths”. Principal examples of this technique are given in [25–28]. In this method, instead of continually sweeping the laser wavelength, a set of discrete wavelengths are used in a heterodyne fashion to synthesize new, usually larger, virtual wavelengths. In one sense this is a return to the concept of Ikezawa [1], although it does not use excess fractions and with modern laser diodes any discrete wavelengths that are within the tunability range of the laser can be used, as opposed to the 5 fixed wavelengths that Ikezawa had available from his CO2 laser.
The main aim of using the synthetic wavelength approach is to extend the ambiguity range beyond the wavelength of the optical source that is used, typically a laser. Lu and Lee used a stable tunable laser source to produce a 50 mm synthetic wavelength [27]. This enables them to measure heights up to 25 mm with an accuracy of 80 nm. Bourdet and Orszag used a number of synthetic wavelengths (N) simultaneously to measure a distance of up to 0.5 m with an accuracy of 0.1 μm. They constructed N equations with N + 1 unknowns and they claim that they can solve these equations since there are N unknown integers [29]. Yu et al. used multiple reference heights to extend the ambiguity range beyond half a synthetic wavelength. The shortest reference height that is used should be shorter than half the longest synthetic wavelength that is used in their system. This algorithm does not require the use of wavelength meters. Their system is capable of measuring heights up to 100 mm with an accuracy of 1 μm [30, 31]. Marron and Gleichman used N (2≤N≤63) synthetic wavelengths simultaneously to measure 3D shape and distances of up to 100 mm [32]. A noticeable advance in extending the ambiguity range was proposed by Tan et al. They used a number of synthetic wavelengths simultaneously to measure distances of up to 20 m, with an accuracy of 60 μm. They developed a new algorithm to eliminate the half synthetic wavelength error for multi wavelength absolute distance measurement [33]. A noticeable advance in improving the accuracy of absolute distance measurement was proposed by Schödel [34]. Their system uses three very stable laser sources and a 16 bit camera. The resultant fringe patterns are analyzed using phase stepping algorithm. Their system is capable of measuring a distance of 400 mm with an accuracy of 0.1 nm. The ability to perform absolute distance measurements across distances larger than 10 m has been investigated by the European long distance project, using femto second laser technology and synthetic wavelength interferometry [35]. The aim of this project is to measure distances up to 1 km with a relative accuracy of 10−7.
In this paper, we propose a new method to extend the ambiguity range and eliminate the half synthetic wavelength error that occurs during the measurement of distances longer than the longest half synthetic wavelength that is used in our system. In contrast to Ref [33], here we have used a commercially available tunable laser with a coherence length of 40 m and produce the different synthetic wavelengths sequentially. The proposed algorithm commences with a low precision starting guess at the distance to be measured, which may be obtained in any of a variety of low cost ways, is used alongside a convergence algorithm and Fourier transform fringe analysis to perform absolute length interferometry. This set-up uses an experimental arrangement which is relatively simple, when compared to those that are found in many other comparable implementations of absolute length interferometry. This system has so far been shown to deliver accuracies of the order of ± 2µm over distances of up to 300mm. In common with other synthetic wavelength methods it does not require a continuous line of sight to the target – as only two time-discrete sightings are required.
The proposed system has a number of advantages. All the used equipments are available commercially and are fairly low in price. For example, the stability requirement for the tunable laser source is not stringent. This makes the use of low price laser sources viable. Also, the building and the programming of the proposed system is relatively simple compared with other systems explained in the literature. Additionally, the system is built in a lab where the temperature is not controlled. This makes building such system within the reach of most metrology labs. This makes this system suitable of industrial applications.
This paper is organized as follows. The next section explains the system setup that was used to implement the proposed algorithm. In section 3, we explain the principle of measuring an absolute distance using the Michelson interferometer. Also, we explain a method for measuring the phase shift between two fringe patterns. In section 4, we explain the proposed algorithm in addition to some aspects that must be taken in consideration when practically implementing the proposed algorithm. In section 5, the system performance is evaluated using computer simulation. Experimental results are presented in section 6. Finally, in section 7 conclusions are drawn for the work that has been presented in this paper.
2. System description
The proposed absolute distance measurement system consists of two main parts; namely the system hardware and software. The hardware consists of a Michelson interferometer, a tunable laser, a wavelength meter, a CCD camera, a screen, a motorized stage and a computer, as shown in Figs. 1 and 2 . The Michelson interferometer comprises a reference arm mirror Mr, a target arm mirror Mt, and a beam splitter. The system software is written in the interactive data language IDL and the C programming language, so as to achieve a fully automated measurement system.
A New Focus 6300 LN external cavity tunable diode laser is used as a light source to drive the Michelson interferometer. The wavelength of this laser can be tuned either manually, or by automatic PC control. The laser can be set over a tuning range of wavelengths varying between 680 nm and 690 nm, with a repeatability (uncertainty) of 0.1 nm. Stably setting the laser to the required wavelength using automatic computer control takes approximately three seconds. This includes the time that is required by the tunable laser to stabilize and also the time overhead for communication between the PC and the laser controller via the serial port [36].
A Bristol Instruments laser wavelength meter, model 621A, determines a more precise measurement of the wavelength of the tunable laser, with a relative uncertainty of λre = 3 × 10−8 [37]. Suppose that the wavelength of the laser is λ = 680 nm. Then the uncertainty in measuring the wavelength is determined as λe = λ × λre = 680 × 3 × 10−8 = 2.04 × 10−5 nm. The USB port is used for communication between the PC and the wavelength meter. When a measurement of the wavelength is required, the PC commands the wavelength meter, which in turn measures the wavelength of the laser beam and then returns the result to the PC. The time overhead for this wavelength measurement operation is 0.1 seconds.
A Prosilica GE 1380 CCD camera is employed, which has a resolution of 1360 × 1024 pixels and can capture images at a rate of 20 frames per second. The pixel depth for this camera is 12 bits. This monochrome camera is connected to the PC via a Gbit Ethernet port.
For the purpose of evaluating the system’s performance, the target mirror is fixed upon a Newport ‘ultra-precision’ XMS160 motorized stage. The target mirror can therefore be translated forwards, or backwards, over distances of up to 160 mm, with a repeatability of 50 nm and a resolution of 1 nm. This stage also comes pre-calibrated with a manufacturer’s guarantee of accuracy better than 1µm. This accurate motion is utilized for testing the resolution of the proposed algorithm. In order to move the target mirror, the PC sends its instructions to the motion controller, which in turn controls the motorized stage to achieve the required movement.
3. Theoretical background
A Michelson interferometer lies at the heart of this absolute distance measurement system. The Michelson interferometer operates as follows. The tunable laser source acts as an illumination source for the Michelson interferometer with a wavelength of λ. The laser beam strikes the beam splitter, which amplitude divides the laser into two discrete beams. The first beam travels towards the reference mirror Mr, and then reflects back towards the beam splitter. Similarly, the second beam travels towards the target mirror Mt, and then reflects back towards the beam splitter. The beam splitter transmits both reflected beams towards the screen, where they interfere and produce a fringe pattern [38].
The reference mirror is adjusted so that it is slightly tilted. This ensures that the fringe pattern which is projected onto the screen contains a spatial fringe frequency, as shown in Fig. 3(a) [38]. This tilt is introduced in order to enable us to analyze the fringe pattern more easily using Fourier fringe analysis [39, 40].
Fig. 3 Fringe patterns produced by the Michelson interferometer which operates using a single wavelength.(a) ∆L = 0, (b) ∆L < λ/2.
The absolute distance measurement system has two modes of operation. In the first mode, the system operates using a single wavelength λ for the laser source. In the second mode, the system operates using two discrete wavelengths λa and λb.
3.1 The operation of the Michelson interferometer system using a single wavelength
For the first mode of operation, consider the case when the length of the reference arm (Lr) is equal to the length of the target arm (Lt). The term Lr represents the distance between the beam-splitter and the reference mirror. Similarly, Lt represents the distance between the beam-splitter and the target mirror. The tunable laser source illuminates the Michelson interferometer with a laser beam that has a wavelength of λ. The two laser beams interfere together and produce a fringe pattern. This fringe pattern is captured using the CCD camera and a typical image is shown in Fig. 3(a). Let us then vary the length of the target arm by changing the location of the target mirror by using the motorized stage. The value of ΔL is assumed in this case to be smaller than λ/2. The term ΔL that is used here represents the optical path difference (OPD). At this new position a second fringe pattern is now captured and this is shown in Fig. 3(b). The change in the length of the target arm introduces a phase shift Δϕ between the two fringe patterns. The phase shift Δϕ is measured here for a specific pixel in both fringe pattern images. The following equation relates the value of the phase shift Δϕ to the optical path difference ΔL.
Also, it must be remembered that in a Michelson interferometer the optical path difference ΔL is actually double the difference in length between the two arms of the interferometer, as the light beam traverses the distance twice, both in the forward and reverse directions, after being reflected from the mirror, as stated in Eq. (1b).For the case when ΔL is larger than or equal to λ/2, the value of the phase shift is given by;
In order to measure ΔL, we require a knowledge of the values of Δφ = Δϕ + 2πn, and λ respectively. The term Δϕ can be determined by finding the phase shift between the two fringe patterns that were captured, which can be accomplished using Fourier transform fringe analysis [39]. The value of λ can be precisely determined from the wavelength meter. The value of n could be found by counting the number of fringes passing a specific pixel on the CCD camera during the process of changing the length of the target arm. However this method has a serious drawback, in that the system must maintain a continuous line of sight during the movement of the target mirror. If sight of the target is lost during this process then the number of fringes being counted (n) as they pass a particular point cannot be determined correctly. Therefore, an alternative method is proposed here to determine a value for n, which eliminates the need for such fringe counting, by using the iterative synthetic wavelength principle.
The optical path difference ∆L can be determined by using either Eq. (1a) or Eq. (2a). The value of λ is known, since it can be measured using the wavelength meter with an uncertainty of λe = λ × λre nm. Suppose that λ = 680 nm. Then λe = 680 × 3 × 10−8 = 2.04 × 10−5 nm.
Suppose that the value of ∆φ can be measured with an uncertainty of ∆φe = 0.036 radians by using the 1D Fourier transform method, as will be shown in section 3.2. The value of ∆L can be determined with an uncertainty that is determined by Eq. (4).
Where ∆Le is the uncertainty in measuring ∆L. Suppose that λ = 680 nm, then ∆L can be measured with an uncertainty of [41]The term Δφ is set to a value of π in Eq. (5), since this the largest value of phase difference that can be extracted using the Fourier transform method, as will be explained in section 3.2 (n = 0). In the case where the measured distance is longer than the synthetic wavelength, then the error increases. This is because Δφ will have a value larger than π.
3.2 Determining the phase shift between two fringe patterns using the Fourier transform method
The phase shift Δϕ between the two fringe patterns that are shown in Figs. 3(a) and 3(b) may be determined in the following manner. Initially the fringe pattern that is shown in Fig. 3(a) is analyzed using the Fourier transform method [39, 40]. This produces a wrapped phase map φ1(x,y). The second fringe pattern, that was acquired after moving the target mirror and which is shown in Fig. 3(b), is then also analyzed using the Fourier transform method and this produces another wrapped phase map φ2(x,y). The phase shift Δϕ is then determined by using Eqs. (6) & (7).
Δφ is then wrapped using the equation
The terms and are the phases of the first and the second fringe patterns determined using the Fourier transform method respectively.The uncertainty in the 1D Fourier transform method was evaluated experimentally as follows. Firstly 100 fringe patterns were captured, each with dimensions of 512 × 512 pixels, using the CCD camera. The images are captured in approximately 30 seconds. During this time, the mirrors remained stationary. Then the wrapped phase was determined for all of the fringe patterns using a 1D Fourier transform approach. This produces 100 different phase maps. The phase shift between each wrapped phase map and the subsequent map is then determined using Eqs. (6) and (7). This therefore produces 99 phase difference images. The phase shift values for the pixels along a central vertical line are plotted for the 99 phase difference images, as shown in Fig. 4(a) . The standard deviation for these 99 values is found to be ∆ϕe = 0.036 radians (equivalent to 0.0056 of a fringe). We shall therefore consider this number to be the uncertainty of the 1D Fourier transform approach to phase measurement.
The above procedure was repeated for the 2D case in order to evaluate the uncertainty of the 2D Fourier transform method. The corresponding phase shift values are plotted in Fig. 4(b). The standard deviation was found to be 0.034 radians (equivalent to 0.0054 of a fringe) for the 2D Fourier transform method. We shall consider this number to be the uncertainty of the 2D Fourier transform approach to phase measurement.
The performance of both the 1D and 2D Fourier transform algorithms is very similar. We have used the 1D Fourier transform algorithm within this system because it offers faster execution times when implemented using digital computers.
3.3 The operation of the Michelson interferometer system using two wavelengths and Lr = Lt
For the second mode of operation, consider the case when Lr = Lt and the tunable laser source acts as an illumination source for the Michelson interferometer with a wavelength of λa. The two laser beams interfere and produce a fringe pattern, the image of which is shown in Fig. 5(a) . Let us then tune the laser source to another wavelength λb. The resultant fringe pattern image at this second wavelength is shown in Fig. 5(b). Both fringe patterns are very similar and the phase shift between the fringes in both images is close to zero. This phase shift is introduced mainly by errors produced by the system, such as vibration and speckle noise.
Fig. 5 Fringe patterns produced by the Michelson interferometer which operates using two wavelengths and ∆L = 0. Fringe patterns produced when the Michelson interferometer operates using the wavelengths (a) λa, (b) λb.(a) ∆L = 0, (b) ∆L < λ/2
3.4 The operation of the Michelson interferometer system using two wavelengths and Lr ≠ Lt
Let us consider the case when Lr ≠ Lt and the tunable laser source acts as an illumination source for the Michelson interferometer with a wavelength of λa. The two laser beams interfere and produce a fringe pattern, the image of which is shown in Fig. 6(a) . We then measure the phase shift between the two fringe patterns Δϕa which is shown in Figs. 5(a) and 6(a) for a specific pixel in both images. This phase shift is determined using the Fourier transform method, as explained in section 3.2 and is given by the equation
Fig. 6 Fringe patterns produced by the Michelson interferometer which operates using two wavelengths and ∆L ≠ 0. Fringe patterns produced when the Michelson interferometer operates using the wavelengths (a) λa, (b) λb.
The Michelson interferometer is then tuned to a second wavelength λb. The resultant fringe pattern is shown in Fig. 6(b). We then measure the phase shift Δϕb between the fringe patterns that are shown in Figs. 5(b) and 6(b) for a specific pixel in both images using the Fourier transform method. The phase shift is given by the equation
Subtracting Eq. (8) from Eq. (9) gives the phase difference δϕ as shown in the following equation.
Where λs is defined as the synthetic wavelength. In the proposed algorithm, the phase difference δϕ is computed by directly finding the phase shift between the two fringe pattern images that are shown in Figs. 6(a) and 6(b). It must be noted that the operation of the proposed algorithm does not require the use of the images that are shown in Figs. 5(a) and 5(b), which are presented here solely for the purposes of simplify the understanding the principle of using the synthetic wavelength.The synthetic wavelength λs is given by the equation
Where δλ = λb - λa. Suppose that λa = 680 nm and λb = 690 nm. By using Eq. (11), the synthetic wavelength that is produced by these two wavelengths λs = 46920 nm. Suppose that ∆L is smaller than λs/2, then ∆L can be determined using the equationFor the case when ΔL is larger than λs/2, the value of the phase shift is given by
Measuring the value of ΔL requires a knowledge of the values of δϕ, λs and n. The phase difference δϕ can be determined by finding the phase shift between the two captured fringe patterns using Fourier fringe analysis [39, 40]. The value of n could be determined by counting the number fringes passing a specific pixel on the CCD camera during the process of changing the length of the target arm. It is worth stressing once again here that we do not use fringe counting to measure ∆L in the proposed algorithm, and this fact is one of the main advantages of the proposed method.
The uncertainty in measuring the synthetic wavelength λse depends to a great degree upon the specific values that are chosen for λa and λb. This behavior is because of the denominator of Eq. (11). On one hand, suppose that we use the full range of the laser, i.e. λa = 680 nm and λb = 690 nm. By using Eq. (11), the corresponding synthetic wavelength λs = 46,920 nm. The uncertainty in measuring the synthetic wavelength λse is given below in Eq. (14) as being 0.2 nm. Therefore, the relative uncertainty that is present in measuring the synthetic wavelength λsre = 0.2/46920 = 4.3 × 10−6.
On the other hand, suppose that only a very small amount of the total possible tuning range is used, for example λa = 680 nm and λb = 680.1 nm. For this case the size of the synthetic wavelength λs = 4,624,680 nm. The uncertainty in measuring the synthetic wavelength is λse = 1886 nm. Therefore, the relative uncertainty in measuring the synthetic wavelength λsre = 1886 /4624680 = 4.1 × 10−4.
The relative uncertainty for the wavelength meter is λre = 3 × 10−8. Whereas the relative uncertainty in measuring a synthetic wavelength λsre may vary from 4.3 × 10−6 to 4.1 × 10−4, depending upon the specific operating position within the tuning range of the laser. Such large relative uncertainty values in measuring the wavelength of the synthetic wave put stringent requirements upon the specifications of the wavelength meter, as will be explained later in section 5.
Suppose that the value of δϕ can be measured using the Fourier transform method with an uncertainty of δϕe = 0.036 radians, as was experimentally proven to be a reasonable estimate in section 3.2. Also, suppose that λa = 680 nm and λb = 690nm. The value of ∆L can be measured with an uncertainty that is determined by Eq. (16).
For our system, ∆L can therefore be measured with a theoretical uncertainty of
The term δϕ is set to a value of π in Eq. (17) since this the largest value of phase that can be extracted using the Fourier transform method as was explained in section 3.2.
The entire tuning range for the laser source is 680 nm to 690 nm. The smallest synthetic wavelength that can be produced using the absolute distance measurement system is achieved by using the full tuning range. That is with λa = 680 nm and λb = 690 nm, hence producing a synthetic wavelength size of λs = 46920 nm. Therefore, the best theoretical measurement uncertainty that this system can produce using the synthetic wavelength principle is given in Eq. (17) as being 268 nm.
The tunable laser source has an uncertainty of 0.1 nm. The largest synthetic wavelength that can be achieved using the system that has been produced is limited by this uncertainty. Suppose that λa = 680 nm and λb = 680.1 nm, then λs = 4,624,680 nm. Therefore, the largest distance ∆L which can be measured with this system, as calculated by using Eq. (12), may be given by λs/2 = 2,312,340 nm, or approximately 2.3 mm.
4. The proposed algorithm
The operation of the iterative synthetic wavelength algorithm can be explained as follows. Initially, ∆Linitial is estimated with an absolute accuracy that is better than ε = 2.3 mm. This relatively low precision measurement can be carried out by using, for example; a time-of-flight laser system, or even a simple ruler, or caliper. The tunable laser is set to the lowest wavelength in the tuning range of λa = 680 nm, and a fringe pattern ga(x,y) is captured using the CCD camera.
In the first iteration, ∆L1 and ε1 are set to values of ∆Linitial, and ε respectively. The suffix number “1” here refers to the first iteration of the proposed algorithm. The wavelength of the synthetic wave λs1 must be set to be equal to, or larger than, 2ε1. Since λa and λs1 are known, we can then calculate λb1 such that both wavelengths λa and λb1 form the synthetic wavelength λs1 as shown in Eqs. (18) and (19). Both of these equations can easily be derived from Eq. (11).
The tunable laser is then set to the new wavelength of λb1. A second fringe pattern gb1(x,y) is captured. The phase difference δϕ1 between fringe patterns, ga(x,y) and gb1(x,y), is then calculated using the Fourier transform method.
The term ∆L1 can be thought of as the number of synthetic wavelengths that traverse this distance, multiplied by the synthetic wavelength, as given by the equation
Where N1 is an integer number, and its value is large and unknown. To find N1, we define two integer numbers NU1 and NL1 and they represent an upper and lower limit for N1 respectively. Both numbers can be calculated using Eqs. (21) and (22). Where ⌊.⌋ represents the operation of neglecting the fractional part of a real number. For example, ⌊4.6⌋ = 4, and ⌊-4.6⌋ = −4. Both integer numbers NU1 and NL1 satisfy the equationSince the difference between NU1 and NL1 is equal to “1”, then the number of whole integer waves that make up the optical path distance N1 must be equal to either one, or the other, of these two numbers. We then calculate an upper and lower limit for ∆L1 using Eqs. (24) and (25).
As mentioned above, we know the absolute length of ∆L1 with an absolute accuracy of ε1. Only one of the following two equations is satisfied
If Eq. (26) is satisfied, we set N1 to NL1, and ∆L1 is set to ∆LL1. On the other hand, if Eq. (27) is satisfied, we set N1 to NU1, and ∆L1 is set to ∆LU1. Both cases are illustrated in Fig. 7 . The optical path difference ∆L1 is now known to an improved precision. This concludes the first iteration.
Fig. 7 (a) Finding ∆L for the case when Eq. (26) is satisfied. (b) Finding ∆L for the case when Eq. (27) is satisfied.
In the second iteration, the absolute accuracy parameter for the measurement ε2 is now set to a new smaller value of ε1/η. Where η is the convergence factor and which has a value that is greater than 1. The parameter ∆L2 is now set to the new value of ∆L1 that was produced as a result of the first iteration. The size of the synthetic wavelength λs2 is reduced, being set to a value of twice the size of the new uncertainty, i.e. λs2 = 2ε2. Then we calculate a new second wavelength λb2, which when used in conjunction with the wavelength λa will produce this new desired synthetic wavelength λs2. The laser is then re-tuned to the new wavelength λb2 and another fringe pattern is acquired by the camera. Then the phase difference between this fringe pattern and the one that was acquired previously at the wavelength λa is measured to produce a relative phase shift value δϕ2. Subsequently we can calculate values for the upper and lower limits NU2, NL2, ∆LL2, ∆LU2, and hence produce a new and better estimate ∆L2 for the optical path difference. This second iteration estimate for the value of ∆L is more precise than that which was obtained during the first iteration.
In the third iteration, the parameter ε3 is set to an even smaller value of ε2/η, and so on, each time producing an ever more precise value for ∆L. The iterative process terminates when the value that is calculated for the new required wavelength λbi falls outside the tuning range of the tunable laser source. Where i refers to the iteration number.
4.1 Practical considerations
The wavelength meter has an uncertainty in measuring the wavelength of the laser beam of λe = 2 × 10−5 nm. Whereas the tunable laser source itself has an uncertainty of 0.1 nm. For example, suppose that the PC instructs the laser source to generate a beam with a wavelength of 685nm, in this case the laser may actually generate a beam with a wavelength that lies in the range of 684.9 nm to 685.1 nm. In order to improve the accuracy of the proposed system, we have relied here solely upon using wavelength values as measured by the more precise wavelength meter. This is indicated in the flow chart that is shown in Fig. 8 . This flow chart summarizes the steps that are required to implement the proposed algorithm using the equipment that was outlined in section 2.
Fig. 8 A flow chart that indicates the procedures required to practically implement the proposed algorithm. The subscript “i” refers to the iteration number. The subscript “m” refers to a measured value.
The convergence factor should be chosen with a value between 1 and . The value of is given by Eq. (28) in order the system to work with 6σ reliability [42]
5. Computer simulation
The operation of the iterative synthetic wavelength algorithm was simulated using the Matlab program that is shown in Fig. 9 . In this program, the operation of the wavelength meter is simulated using the get_lambda function. The uncertainty for the wavelength meter is set to a value of 3 × 10−8 × wavelength in nm. The operation of the tunable laser was simulated using the set_laser function. The uncertainty for the tunable laser was set to a value of 0.1 nm. The phase difference δϕ was simulated using the function get_phase. The uncertainty in extracting the phase was set to 0.0056 of a fringe period which was experimentally validated in section 3.2 above. We have used the rand function in Matlab to generate uniformly distributed random numbers in order to obtain these simulation results.
Fig. 9 A Matlab program simulates the operation of the absolute distance measurement using the iterative synthetic wavelength algorithm.
Here we will use computer simulation to study the effect of each hardware component on the overall performance of the proposed absolute distance measurement system. Firstly, we will study the effects that any uncertainty in correctly reading the wavelength of the laser beam using the wavelength meter has on the overall performance of the system. In the Matlab program, we set the uncertainty for both the tunable laser and the phase difference measurement to zero. The uncertainty of the wavelength meter is then varied over the range 0 to 1 pm in one thousand uniformly spaced steps. For each step, the simulation program is repeated over ten thousand iterations. In these iterations, we count the number of times that the algorithm converges to a “correct” answer and we use this number to calculate the probability of convergence for the iterative synthetic wavelength algorithm. Because we are performing a simulation here, the answer may be considered to be “correct” when the absolute value of the mathematical difference between the simulated known length (deltaL_Actual) and the estimated length (deltaL_estimated) is less than 1 μm as indicated in the Matlab program shown in Fig. 9. As shown in Fig. 10(a) , the probability of convergence is 1 for wavelength meter uncertainties that fall in the range from 0 to 0.0004 nm. Also, we shall study the effects of uncertainties in the wavelength meter upon the overall measurement accuracy. As we can see from Fig. 10(b), the measurement accuracy begins to deteriorate when the uncertainty for the wavelength meter exceeds a value of 0.00037 nm.
Fig. 10 (a) The effect of uncertainty in the wavelength meter on the probability of convergence for the proposed algorithm. (b) The effect of uncertainty in the wavelength meter on the accuracy of the proposed algorithm. (c) The effect of uncertainty in measuring the phase shift on the probability of convergence. (d) The effect of uncertainty in measuring the phase shift on the accuracy of the proposed algorithm. (e) The effect of uncertainty in setting the wavelength of the tunable laser on the probability of convergence. (f) The effect of uncertainty in setting the wavelength of the tunable laser on the accuracy of the proposed algorithm.
Secondly, we move on to study the effect of any uncertainty in calculating the phase difference δϕ upon the overall performance of the system. Here we set the uncertainty for both the wavelength meter and the tunable laser to zero. The uncertainty in determining δϕ is then varied over a range of 0 to 5 radians. The probability of convergence and the measurement accuracy are both determined over this range. As we can see from Figs. 10(c) & 10(d), the probability of convergence and the measurement accuracy begin to deteriorate when the uncertainty in calculating the phase difference δϕ exceeds a value of 2.1 radians – which is much larger than the size of the uncertainty we see in practice.
Lastly, we study the effect of any uncertainty in setting the wavelength of the tunable laser upon the overall performance of the system. We therefore set the uncertainty for both the wavelength meter and for calculating δϕ to zero. The uncertainty of the tunable laser is then varied over a range of 0 to 0.5 nm. The probability of convergence and the measurement accuracy are determined over this range. As we can see from Figs. 10(e) & 10(f), the probability of convergence and the measurement accuracy begin to deteriorate when the uncertainty for setting the wavelength of the tunable laser exceeds a value of 0.2 nm.
Our studies using computer-simulation reveal that the uncertainty in the wavelength meter plays a major role in determining the overall performance of the proposed system. To show this using computer simulation, we set the uncertainty for the tunable laser to 0.1 nm, and the uncertainty in determining δϕ to a value of 0.0056 of a fringe. These are similar to the realistic uncertainly values for the New Focus 6300 LN external cavity tunable diode laser and the practical uncertainty that is inherent to the 1D Fourier transform fringe analysis algorithm as determined experimentally. The uncertainty for the wavelength meter is varied over a range of 0 to 0.001 nm. An inspection of Fig. 11(a) reveals that the simulated system converges to a correct answer as long as the uncertainty in the wavelength meter is less than a value of 0.00025 nm. The laser wavelength meter that is used in our system (Bristol instruments, model Bristol 621A) has a measurement uncertainty of λe ≈2 × 10−5 nm. This explains the reasons behind the fact that our system always appears to converge in practical experiments. Computer simulations show that the accuracy of the proposed system could be improved by using a wavelength meter with a better measurement uncertainty, as shown in Fig. 11(b). For example, the measurement accuracy of the proposed system could be improved to 220 nm by using a different wavelength meter with a measurement uncertainty of λe = 1 × 10−5 nm. Using the system’s model 621A Bristol Instruments laser wavelength meter, with its associated uncertainty as stated above, gives us a theoretical uncertainty for the overall absolute distance measurement of 0.5 μm, as shown in Fig. 11(c). This theoretical simulated measurement uncertainty is similar to that which is practically achieved in our experimental results.
Fig. 11 (a) The effect of uncertainty in the wavelength meter on the probability of convergence for the proposed algorithm. (b) The effect of uncertainty in the wavelength meter on the accuracy of the proposed algorithm. (c) Zoom in for Fig. 11(b). (d) The effect of uncertainty in measuring the phase difference δϕ on the accuracy of the proposed algorithm. (e) The effect of uncertainty in setting the wavelength of the tunable laser on the accuracy of the proposed algorithm. (f) The effect of η value on the system performance.
Computer simulation for our system shows that improving the uncertainty in measuring the phase difference δϕ does not have a significant effect in terms of improving the overall accuracy of the proposed system, as is shown in Fig. 11(d). Here, we have set the uncertainties for the wavelength meter and the tunable laser to values of 3 × 10−8 × 680 nm and 0.1 nm respectively.
Similarly improving the uncertainty in setting the wavelength of the tunable laser also does not improve the overall accuracy of the proposed system, as shown in Fig. 11(e). Here, we set the relative uncertainties for the wavelength meter and for measuring the phase difference δϕ to values of 3 × 10−8 and 0.0056 of a fringe, respectively.
In all the computer simulations that have been described above, we have set the convergence factor η to a value of 2.
The system performance was also evaluated using different values of η varying in a range from 1 to 10, as is shown in Fig. 11(f). Here, we set the uncertainty for the wavelength meter to typical experimental values of 3 × 10−8 × 680 nm, the uncertainty in measuring δϕ to 0.0056 of a fringe, and the uncertainty in setting the wavelength of the tunable laser to a value of 0.1 nm respectively. Computer simulation shows that the value for the convergence factor η that is used in the system should be determined very carefully. For example, by inspecting Fig. 11(f) we can see that setting the value of η to 5 may not be a wise choice, since this may deteriorate the measurement accuracy.
6. Experimental results
Initially, the length of the reference arm Lr was set such that it was close to the length of the target arm Lt within an accuracy of two millimeters. This was achieved by moving the target mirror via the Newport motorized stage and measuring both distances (Lr and Lt) using a simple ruler. Then the optical path difference ΔL = 2(Lr - Lt) was measured using the proposed system and this measurement was repeated at this distance ten times. The mathematical average of these ten measurements was then calculated. Then the target mirror was moved by a distance equal to this average OPD value in a manner so as to act to cancel out the OPD. This process was repeated such that the location of the target mirror was adjusted using the motorized stage until the average of the ten consecutive measurements at the same distance reached a value that was less than one micrometer. This prepares the system in order to test its operation.
In the first experimental test, we set ΔLinitial = 0.5 mm as an initial starting guess for ΔL. We know in advance that ΔL is approximately 1μm, because this condition was achieved during the system preparation stage that was described in the paragraph above. Also, we set the absolute accuracy ε to a value of 2.3 mm as indicated in the flowchart that is shown in Fig. 8. Then we used the system to measure the optical path difference ΔL. The system converged after six iterations and returned a measurement value for ΔL of 2.48089 μm. The result of this measurement procedure is shown in Fig. 12(a) using a solid line. The above test was then repeated using a value of ΔLinitial = 0 mm. The system converged after five iterations and measured ΔL as 4.41 μm. The result of this measurement procedure is shown in Fig. 12(a) using a dashed line. The above test was then repeated once again, for a third time, in this case using ΔLinitial = −0.5 mm. The system converged after six iterations and measured ΔL as −2.57 μm. The result of this measurement procedure is shown in Fig. 12(a) using a dotted line.
Fig. 12 (a) & (b) Measuring the optical path difference ΔL using three different values of ΔLinitial. (c) Measuring ΔL five-hundred times. (d) The histogram for these five-hundred measurements. (e) Uncertainty for different values of η. (f) Number of iterations required by the system to converge for different values of η.
In the second experimental test, the target mirror was moved away from the beam splitter by a distance of 20 mm using the Newport motorized stage. Hence the optical path difference was now equal to 40 mm with one micrometer accuracy. Then we set ΔLinitial = 40.5 mm and used the system to measure the optical path difference ΔL. The system converged after six iterations and measured ΔL as 39.9984 mm. The result of this measurement procedure is shown in Fig. 12(b) using a solid line. The above procedure was repeated using values for the initial estimate of ΔLinitial = 39.5 mm and 40 mm and the results are shown in Fig. 12(b) using dashed and dotted lines respectively. Both measurements produced respective values for ΔL of 39.999 mm and 39.994 mm.
The uncertainty of our experimental system was measured as follows. The optical path difference was measured five-hundred times. This measurement process was carried out within a timeframe of approximately five hours. During this time, the system parameters such as Lr and Lt were not changed. The values of ΔL that were produced by the system are shown in Fig. 12(c). The mathematical standard deviation for these measurements was calculated and is found to be 1.6 μm. Hence, the uncertainty in measuring the distance Lr - Lt is half this value, i.e. 0.8 μm. The histogram for these five-hundred measurements is shown in Fig. 12(d) as a bar plot. The shape of the histogram appears to be similar to a Gaussian normal distribution function.
The experimental uncertainty that was achieved for the system, i.e. 1.6 μm, is larger than the theoretical 0.5 μm uncertainty that was determined using computer simulation. This discrepancy may be explained as follows. Firstly, the commercial tunable laser source that is used in this system must be considered to be an imperfectly stable laser source. A wavelength drift has been observed for this laser during its experimental operation with a range of up to 0.1 nm. This has adverse effects on the operation of the system. As explained previously, during the measurement process the tunable laser source was set to a specific wavelength and was then allowed a three-second settling period in order for the laser to stabilize. The wavelength of the laser was then measured using the wavelength meter. The CCD camera then captures an image of the resulting fringe pattern. Reading the wavelength of the laser and capturing the image requires a time overhead of 0.4 seconds. During this period, there is no guarantee that the wavelength of the laser has not changed. Secondly, external environmental conditions, such as the presence of vibration, may also disturb the operation of the system.
We have attempted to determine the optimal convergence factor η to use when practically operating this experimental system. To perform this task, the parameter η was initially set to a value of 1.25 and then the uncertainty for the system was measured as explained above. This procedure was repeated for values of η = 1.5, 1.75, 2, 2.25,…,10. For each different value of the convergence factor η, the uncertainty for the system was determined. The results of this experiment are shown in Fig. 12(e). Inspection of this figure reveals that by setting the value of η to 2, this gives the lowest uncertainty value for our system. Values of η in the range of 3 to 7 should be avoided as they produce large measurement errors.
The measurement time for this system is mainly dependent upon the number of iterations that are required for the system to reach convergence, which in turn depends on the value of the convergence factor η. Setting η to a low value, e.g. 1.25, ensures that the system converges stably without overshooting in its response and produces measurements with low levels of uncertainty, but with such a low convergence factor it requires 15 iterations to converge and hence the associated measurement time is relatively large. On the other hand, by setting η to a large value, e.g. 9, this reduces the number of iterations that are required from 15 to only two and hence makes the measurement process significantly faster, but it also increases the measurement uncertainty within the system and may run the risk of encountering instability in convergence. The number of iterations required by the system to converge for the values of η = 1.25, 1.5, 1.75, 2, 2.25,…,10 have been measured and these are plotted in Fig. 12(f). Experimentally, we have found that a compromise of setting η to a value of 2 gives optimal results, as the system converges stably and requires only a fairly moderate six iterations in order to do so, meaning that the measurement process can be completed in less than 40 seconds and with a measurement uncertainty of 1µm.
7. Parameters affect measurement accuracy
The accuracy of absolute distance measurement is mainly affected by four error sources. The first source is caused by unknown and unpredictable changes in the external environmental factors, such as mechanical vibrations and thermal expansions that affect the target position during consecutive measurements. The second source is related to changes of the refractive index of air during consecutive measurements, due to air temperature and humidity variations across the optical path. The third error source is due to potential misalignment of the laboratory setup for the two laser beams in the Michelson interferometer. The fourth source of error occurs due to systematic errors in experimental observations, which are usually caused by the individual components of the measuring system. These types of errors may severely influence the measurement accuracy. The measurement errors that result directly from the laser source are mainly connected to the stability of the tunable laser source. It is obvious that variations in temperature can cause length changes of the resonator, which in turn has direct influence on the laser wavelength that is produced [43]. Also, the value of the measurement error increases with the increase in the measured length.
The stability of the tunable laser improves by increasing the length of the laser’s warm up time. During the first fifteen minutes, the wavelength stability of the laser system is relatively poor. The standard deviation of the laser wavelength during the first fifteen minutes is 0.059 nm. After a warm up time of 30 minutes, the standard deviation drops to 0.006 nm. Wavelength standard deviation records of 0.00099 nm and 0.000086 nm after warm-up times of 60 and 80 minutes respectively, have been measured experimentally by the authors. The stability of the tunable laser is worse at the extremes of the tuning range than the center. Figure 13 shows the wavelength stability performance of the laser system at upper extremity of the tunability range, 690.1 nm, after 120 minutes warming up time. Figure 14 shows the wavelength stability of the laser system at lower extremity of the tunability range, 680.3 nm, after 120 minutes warming up time.
Fig. 13 Stability of wavelength of laser system at the upper extremity of its tunability range, 690.1 nm, after 120 minutes warming up time.
Fig. 14 Stability of wavelength of laser system at the lower extremity of its tunability range, 680.3 nm, after 120 minutes warming up time.
8. Conclusions
In this paper we have proposed a new system that has the tested to measure absolute distances of up to 300 mm with an uncertainty that is better than one micrometer. This system does not require a continuous line-of-sight to the measured target and can perform the measurement by using only two discrete sightings of the target in a timeframe of less than 40 seconds.
The proposed system is built using off-the-shelf commercially available equipment with a cost of approximately $30,000, and this reduces the system cost considerably when compared to that of many typical absolute length measurement systems that use specifically designed, custom-built equipment.
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