Time efficient auto-focussing algorithms for ultrasonic inspection of dual-layered media using Full Matrix Capture

NDT&E International 47 (2012) 43–50

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Time efficient auto-focussing algorithms for ultrasonic inspection of dual-layered media using Full Matrix Capture Miles Weston a,n, Peter Mudge b, Claire Davis c, Anthony Peyton d a

TWI Ltd, NDT Validation Centre (Wales), Port Talbot SA13 2EZ, UK TWI Ltd, Granta Park, Great Abington, Cambridge CB21 6AL, UK c School of Metallurgy and Materials, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK d School of Electrical and Electronic Engineering, University of Manchester, Sackville Street Building, Manchester M13 9PL, UK b

a r t i c l e i n f o

abstract

Article history: Received 1 March 2011 Received in revised form 17 October 2011 Accepted 21 October 2011 Available online 8 December 2011

This paper describesa number ofmethods for calculating the point of incidence at a planarrefractive interface between dual-layered media for ultrasonic applications in the field of non-destructive testing. It is shown how Snell’s law may be expressed as a quartic polynomial, and solved using analytical or numerical techniques to find the point of incidence at the refractive interface. An array transducer mounted onto a Perspex wedge is used to generate ultrasonic imagery of a double ‘v’ butt weld in a low carbon steel plate, using the Full Matrix Capture technique. Curve-fitting algorithms are also presented that allowautomated focussingthrough the wedge-plate interface. Finally, a description is given on how algorithms may be adapted to allow auto-focussing through dual media with a non-planar interface. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Ultrasonic imaging Non-destructive evaluation Full Matrix Capture Signal processing Snell’s law

1. Introduction The use of linear array transducers in the field of ultrasonic non-destructive testing and evaluation (NDE) has become common place in recent years. Typically, elements in an array are excited in parallel to generate interference patterns in a component. By phasing the relative transmission times of individual elements, focussing and beam steering can be achieved, which allowslarge regions to be inspected at once. This can be favourable in comparison with single crystal transducers, where multiple configurations may be required to inspect a given region. In order to inspect a component for discontinuities, the transducer is usually coupled to the specimen via an intermediary medium to allow transfer of energy into the component. In contact mode, a solid Perspex or Rexolite wedge is commonly used for this purpose. The appropriate wedge angle will depend on the characteristics of the component and the types of flaw which are likely to occur. In immersion mode, both transducer and component are submerged in a tank filled with water, which acts as the couplant; a similar method of water couplant could be used without full immersion with water filled inspection wheels or water spray used with sledges. One advantage of using a purely

n

Corresponding author. Tel.: þ441639873125. E-mail address: [email protected] (M. Weston).

0963-8695/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2011.10.006

liquid intermediary medium is that components with irregular surface profiles will still be fully coupled to the transducer. To ensure an accurate representation of a component’s structural integrity,an essential factor that needs to be considered is the beam path. In a situation where the beam is required to propagate through media of different acoustic velocities, the effects of refraction must be considered to determine the beam pathaccurately. The relationship between the angle of incidence and angle of refraction is given by Eq. (1), and is the well known Snell’s law. In modern day engineering Snell’s law is widely applied across a broad range of industries [1–3]. In the field of ultrasonic NDT Drinkwater and Bowler [4] provide one example where the Full Matrix Capture (FMC) technique was used to inspect wrought iron chain-links for fatigue cracking. Here the array transducer was coupled to the curved surface of the chains via a solid wedge. To account for refraction at the wedge-chain interface an iterative search algorithm, using a minimum time of flight approach, was implemented. The FMC technique has also been implemented in immersion mode, where the incidentpoint along the surface profile of a component was found using a minimum time of flight algorithm [5]. A different approach by Long and Cawley [6] used a flexible membrane attached to an array transducerto inspect components with an irregular surface profile. Here curve-fitting algorithms were used to define the surface profile, and an iterative approach was used to calculate the incidentpoint at the interface.

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M. Weston et al. / NDT&E International 47 (2012) 43–50

The purpose of the investigation presented in this work is to demonstrate how auto-focussing through dual-layered media can be achieved using analytical or numerical methods – both of which can be computationally more efficient than iterative methods. An efficient algorithm is of particular importance to FMC due to the large amount of signal processing required. In this study, ultrasonic arrays were used to acquire data using the FMC technique [7] on a low carbon steel, double ‘v’ butt welded plate. The ultrasonic array transducer was coupled to the test specimen via an intermediary dielectric medium. Data processing using an in-house algorithm known as Sequential Phased Array (SPA) is then discussed. This paper presents the derivation of an analytical solution for the incident point at a planar interface. It is shown how curve-fitting algorithms can enable auto-focussing through dual-layered media with a planar interface. Results are presented which illustrate how the algorithms can be applied experimentally to inspect a double ‘v’ butt weld in a low carbon steel plate. In addition a description of how the data processing algorithmscould be adapted to allow autofocussing through dual media with a non-planar interface is given.

2. Theoretical background 2.1. Full Matrix Capture and the Sequential Phased Array Algorithm FMC is a data acquisition process which collects time domain signals for every possible transmitter-receiver combination in an array transducer. This data set is termed the Full Matrix of Data. For a linear array containing n elements, this yields a total of n2 A-scans. Initially the first element in an array is excited, while all elements are used as receivers. This method of transmitting on one element and receiving on all is then repeated until every element within an array has been excited. An advantage of this technique, over more conventional parallel transmission methods, is that a fully focussed image can be achieved where every pixel acts as a focal point. However, while coherent noise levels of FMC are identical to that of parallel transmission techniques providedthe system is time-invariant; incoherent, random noise levels will reduce by a factor of On [8]. To process the Full Matrix of Data an imaging algorithm termed Sequential Phased Array (SPA) was used. This algorithm allows a fully focussed, cross-sectional image of the test specimen to be generated by treating every pixel as a focal point. For each focal point there is an amplitude contribution from every transmitter-receiver pair; each contribution is summed to give the pixel intensity. C-scan images can be produced using the SPA algorithm by stacking multiple cross-sectional images using the same methods as conventional phased array techniques [9]. 2.2. Snell’s law The incidentpoint of a beam at aninterface between two media of acoustic velocities v1 and v2 can be calculated using Snell’s law, expressed in its most common form in Eq. (1) and illustrated in Fig. 1, where yi and yR represent the angle of incidence and angle refraction respectively. This equation is valid for both longitudinal and shear wave modes. By setting yR equal to 901 for a refracted longitudinal wave mode, the first critical angle can be determined. Beyond this angle no longitudinal wave mode exists and the shear wave mode will be dominant. For a Perspex-steel interface the first critical angle will be approximately 271. sin ðyi Þ sin ðyR Þ ¼ v1 v2

ð1Þ

It can be shown that Snell’s law is derived from Fermat’s principle [10]; by taking the derivative of the acoustic path length, the point

(X0, Z0) Normal

θi Velocity 1 Velocity 2

(Xb, Zb) θR

(Xs,Zs)

Fig. 1. Refraction of a wave at an interface between dual media of different acoustic velocity for a single source (x0,z0) and target (xs,zs).

of incidence can be determined, which allows the beam path to be calculated. The algorithms developed were based on homogeneous and isotropic bulk media which simplifies the required mathematics. We also assume that dispersion is negligible, so Snell’s law holds for all frequencies across a broad spectrum. In order to find the point of incidence at an interface, when both the target and source are well defined, it is necessary to express Snell’s law in its general form (Eq. (2)). In cartesian coordinates, the position of the transducer wasdefined as (x0,z0), the point of incidenceat the interface as (xb,zb), and the focal point in the specimen as (xs,zs), also shown in Fig. 1.

bðxb x0 Þ2 2

ðzb z0 Þ þ ðxb x0 Þ

2

¼

ðxs xb Þ2 ðzs zb Þ2 þ ðxs xb Þ2

ð2Þ

where b ¼ v22 =v21 . In Eq. (2) all parameters except for xb and zb are known. In this paper we consider the case of a planar interface between dual layered media, whereby this interface represents zero material depth (zb ¼ 0). Since zb is defined as a constant, Snell’s law can be expressed as a quartic polynomial given in Eq. (3). p4 x4b þ p3 x3b þ p2 x2b þ p1 xb þ p0 ¼ 0

ð3Þ

Where the coefficients are defined as p4 ¼ b1 p3 ¼ 2x0 2bx0 þ 2xs 2bxs p2 ¼ x20 þ bx20 4x0 xs þ 4bx0 xs x2s þ bx2s ðz0 þ zb Þ2 þ bðzb þzs Þ2 p1 ¼ 2x20 xs 2bx20 xs þ 2x0 x2s 2bx0 x2s þ 2xs ðz0 þzb Þ2 2bx0 ðzb þzs Þ2 p0 ¼ x20 x2s þ bx20 x2s x2s ðz0 þzb Þ2 þ bx20 ðzb þzs Þ2 The roots of Eq. (3) can be found analytically using Ferrari’s method [11], given in Appendix A. The point of incidence can then be determined by finding the path of minimum time. Snell’s law can also be used to find the point of incidence at a non-planar interface, provided such an interface is well defined. The equation which governs this scenario is not expressed here, but is presented in Appendix B. Since this equation is nonpolynomial and the exponent of the variable exceeds 4th order, solutions must be found using numerical or iterative methods. For a planar interface the point of incidence will always be that which yields the minimum time of flight. However, in the case of a nonplanar interface, it is possible for the point of incidence to be located at one or more local minima or maxima. Fig. 2a shows the time of flight of a wave propagating from a transmitter to a receiver through a quadratic interface as a function of the point of incidence. The simulated beam path in Fig. 2b shows the wave to be incident at the point along the interface which corresponds to a local maximum time of flight. This satisfies Fermat’s principle

M. Weston et al. / NDT&E International 47 (2012) 43–50

45

40

16

interface beam path Transmitter

20

12

z (mm)

Time (μs)

14

10

0

8 Receiver 6

-20 -6

-4

-2

0 2 x (mm)

4

6

-6

-4

-2

0 x (mm)

2

4

6

Fig. 2. Simulations of (a) the time of flight and (b) the beam path, of an ultrasonic wave through dual media with a quadratic interface.

which requires the acoustic path length to be stationary, but not necessarily a minimum [10]. In order to check that the calculated point of incidence is correct, the known coordinates of the source, target and point of incidence can be used to find the angle of incidence and refraction. These values can then be substituted into Eq. (1) to solve Snell’s law in its common form.

Computer

Micropulse Array Controller

2.3. Auto-focussing through dual media To find the point of incidence between dual media using Snell’s law, a mathematical expression describing the interface between the two media must be defined. However, for many applications in industry a known equation is unlikely to exist. One solution to this problem is to map the surface of the component from data collected using the FMC technique. Curve-fitting methods can be applied to the Full Matrix of Data to generate a polynomial expression which defines the interface. Snell’s law can then be solved without the need for the operator to define the surface profile between the dual media. In effect, curve-fitting to the interface allows for automatic focussing through dual layered media. In this work algorithms were developed which automatically generate, fully focussed cross-sectional images through dual media using linear arrays. To mapfully the interface between dual media in three dimensions, a two dimensional array would be required. Adaptations would also need to be made in all algorithms to account for the extra dimension.

Wedge

Probe

Weld test piece Fig. 3. A schematic diagram of the experimental setup.

as the size and locations of the flaws contained within it are shown in Fig. 4. In the section that follows, a comparison has been made between the processing times for iterative, numerical and analytical approaches. It is also shown how curve-fitting techniques can be used to detect and characterise the interface between the wedge and plateautomatically to enable auto-focussing in the weld region.

3. Results and discussion The acquisition system used for this work was a Micropulse5PA 128/128 array controller. Data were acquired using the FMC techniqueand processed by the SPA algorithm using a desktop computer with two quadcore 3 GHz CPUs and 48 GB of RAM. The received data was sampled at 50 MHz and stored in 8 bit format. The maximum data transfer rate was approximately 7 MB/s; appropriate gates were used to minimise the size of data transferred across the Ethernet. The transducer used was an Olympus, 64 element, linear array probe with 0.6 mm pitch and 5 MHz centre frequency. The probe was mounted onto a 361 Perspex wedge; inducing a refracted shear wave mode into the test specimen. A diagram of the setup is shown in Fig. 3. A low carbon steel plate containing a double ‘v’ butt weld with known flaws was inspected using the FMC technique. The types of flaws contained in the plate included lack of root penetration (LORP), porosity and a toe crack. The geometry of the plate as well

3.1. Comparison of techniques to determine the point of incidence for a planar interface Table 1 shows the time taken for different computational approaches to calculate the point of incidenceat a refractive interface for a single source and target,as shown in Fig. 1. The simulated positions of the source, target and interface were indicative of the experimental setup; the source was simulated to represent the position of the first element in the transducer array. The beam angle in the wedge was set to 361 and the depth of inspection (zs) to 50 mm, which represented the maximum depth of inspection at full skip. The point of incidence was found analytically using Ferrari’s method to solve Eq. (3), while the MATLAB function roots.m was used to find the numerical solution. This function employed numerical methods based on computing the eigenvalues l of the companion matrix M given in Eq. (4), where x is the

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M. Weston et al. / NDT&E International 47 (2012) 43–50

20

88

62

30

1

72

30

2

78

25

3

Plate thickness 25mm 1

2

3

Defect Type 1) Lackof root penetration 2) Porosity 3) Toecrack

Fig. 4. A low carbon steel plate containing a double ‘v’ butt weld with 3 known flaws: (1) lack of root penetration, (2) porosity and (3) toe crack.

Table 1 A comparison of the different computational approaches used to solve Snell’s law. Computational approach

Processing time (ms)

Step size Dxb (mm)

Dxs (mm)

Numerical Analytical Iterative

1.2 2.5 960 380 190 99 40 20

N/A

N/A

0.001 0.0025 0.005 0.01 0.025 0.05

0.02 0.04 0.09 0.18 0.44 0.89

(X0, Z0)

Beam accuracy

(Xb, Zb)

ΔXb

(Xb+ΔXb, Zb)

Velocity 1 Velocity 2

ΔXs (Xs, Zs)

eigenvector of the companion matrix. Mx ¼ lx

(Xs+ΔXs, Zs)

ð4Þ

Finally an iterative method based on Fermat’s principle was implemented. This approach calculated the time of flight of a sound field with respect to position along the interface. The point of incidence was then determined from the stationary point which lay between the source and target along the interface (x-axis). A fundamental limit of ultrasonic imaging is that artefacts cannot be resolved if they are separated by less than half the insonifying wavelength [12]. For a 5 MHz shear wave mode in low carbon steel, the half wavelength is approximately 0.3 mm. Using basic trigonometry it can be shown that to achieve a resolution of 0.3 mm in the test specimen, the incident point at the interface must be known with considerably more accuracy, this is illustrated in Fig. 5. To achievea theoreticallateral beam errorof 0.3 mm for all beam paths, a minimum step size of 0.017 mm wasrequired. With this step size the iterative algorithm took 74 ms to process (Fig. 6). It is interesting to note however that when increasing the step size up to as much as 1 mm, the image quality remained unchanged. This was because the lateral beam error stated in this work is in reference to the maximum beam path error; consequently all other beam paths will experience less error. Also the predicted point of

Fig. 5. Refraction through dual layered media, illustrating how small changes in beam position at the interface (Dxb) can cause significant inaccuracy to the beam path (Dxs).

incidencealong the interface had an equal likelihood of having a positive or negative error. Therefore image quality was maintained through virtue of cancelation of errors (analogous with random noise cancelation via averaging). To minimise the processing time of the iterative method, the constraint that the point of incidence was between the source and target was imposed. It is impossible to be totally conclusive on the optimum method to calculate the point of incidence, as processing times will depend on many factors includingprocessor power, programming language and quality of code. In this work all algorithms were implemented in MATLAB. This language is designed for efficient matrix multiplication and is subsequently poor at looping. Had a lower level language such as C been used, then it is possible that the numerical and iterative methods would have seen a greater performance increasethan the analytical approach.

M. Weston et al. / NDT&E International 47 (2012) 43–50

However, with the setup used in this paper, the analytical and numerical approaches, and the iterative approach with a 1 mm step size were comparable in performance, with the numerical approach to solving Snell’s law being the fastest (Table 1). 3.2. Curve-fitting to a planar interface Fig. 7a shows the reflection from the wedge-plate interface, generated from the Full Matrix of Data using the SPA algorithm. Fig. 7b displays the line of best fit, used to express the interface between the wedge and plate as a polynomial equation. Using simple trigonometry, the coefficients of this polynomial were used to estimate the wedge angle and the perpendicular distance from the centre of the transducer to the wedge-plate interface. Table 2 compares the predicted wedge parameters from the curve-fitting algorithm with the actual measured parameters. It can be seen that the estimated wedge parameters match very closely with actual values. Comparing images generated using actual and predicted wedge parameters showed little difference between images in terms of flaw location and image quality. This is illustrated in Fig. 8 where a 3 mm side drilled hole at a depth of 50 mm in low carbon steelwas imaged using boththe actual wedge parameters and the parameters predicted from the curve-fitting algorithm. The image shows that, when using the

parameters predicted by the curve-fitting algorithm, the 3 mm side drilled hole was misaligned by a distance of 1.11 mm and 1.03 mm in the x-axis and z-axis, respectively. The illustration also shows a very strong correlation between the size and shape of both indications, which is important when characterising defects. The accuracy with which the wedge parameters were predicted was largely due to the wedge surface, which acted as an ideal planar reflector. In immersion mode where the couplant is water, the interface would be calculated from reflections directly from the surface of the test specimen. Here a greater error in surface mapping would be expected due to greater surface roughness and localised surface imperfections. Using the FMC technique all information required to curve-fit to the interface was captured in the same Full Matrix of Data used by the SPA algorithmfor imaging of the components sub-surface. Consequently no additional transmissions were required to map the interface between the wedge and plate whenusing the FMC technique.

Table 2 A comparison of the predicted wedge parameters using the curve-fitting algorithm against measured parameters.

1

Wedge parameters

Actual wedge parameters

Predicted wedge parameters

Angle (deg.) Distance (mm)

36.0 27.5

35.5 27.7

0.8

-45 -47

0.6 z (mm)

processing time (s)

47

0.4

-49 Indication 2

-51

Indication 1 0.2

-53

0

-55 80 0

0.2

0.4 0.6 step size (mm)

0.8

Fig. 6. A graph showing the step size used to calculate the point of incidence at a refractive interface against processing time and lateral beam accuracy, at a depth of 50 mm for a single source and target, using the iterative method.

88

90

10

20

25

15 z (mm)

wedge-plate interface

15 z (mm)

84 86 x (mm)

Fig. 8.  6 dB contour plot of a 3 mm side drilled hole at a depth of 50 mm in low carbon steel. The plot compares the imaging accuracy of the SPA algorithm using wedge parameters predicted by the curve-fitting algorithm (indication 1) against the SPA algorithm using the actual wedge parameters (indication 2).

10

-25

82

1

20

25 -20

-15 -10 x (mm)

-5

-25

-20

-15 -10 x (mm)

-5

Fig. 7. (a) An image of the interface between the wedge and low carbon steel plate, generated using the SPA algorithm. (b) A line of best-fit generated using the polynomial curve fitting algorithm to calculate the interface between the wedge and plate.

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M. Weston et al. / NDT&E International 47 (2012) 43–50

One potential limitation of the curve-fitting algorithm is that the technique can only map regions of the surface profile where transmission signals are reflected directly back to the transducer. A potential solution to this problem would be to use a larger aperture to increase the amount of data capture. This could be implemented either by synthesising a large array transducer using a scanning array [13], or by simply using a transducer with a larger aperture.

3.3. Inspection of a double ‘v’ weld in low carbon steel using an auto-focussing algorithm Angled beam inspection was used to interrogate the weld region from both sides of the weld and weld plate using a fixed stand-off of 54 mm as shown in Fig. 9. The wedge angle and perpendicular distance from the centre of the transducer to the

A

B

Fig. 9. Cross-sectional view of the weld region displaying the four locations from which the weld was interrogated. Path ‘A’ represents a direct beam path to the region of interest, path ‘B’ shows the beam path skipping off the backwall to reach the region of interest.

interface were calculated automatically using the curve-fitting algorithm. This information was used in the focal law calculation which allowed for auto-focussing in the weld region using the SPA algorithm. In this paper the analytical method was implemented into the SPA algorithm to find the point of incidence between dual media. This method was shown to be computationally efficient, and offers an exact solution to the linear interface problem. Fig. 10a shows a volume corrected, unencoded C-scan of the weld region, with a 54 mm stand-off, generated using the SPA algorithm. Three dotted lines, parallel to the y-axis, have been overlaid on the image which represent the centre and edges of the weld body. Identified in the image are the three flaws, which are clearly visible above the background noise level. Since the scan was unencoded, the dimensions of the flaws in the y-axis are only approximate. To accurately determine flaw sizes in this axis, an encoder would need to be incorporated into the scanning system. Other artefacts which are present in the C-scan were identified as geometric or as being located outside the weld region through B-scan interpretation and can be ignored. Fig. 10(b)–(d) show the B-scan images of the flaws; LORP, porosity and toe crack respectively. For each image a weld overlay was generated for clarity. The dashed horizontal line represents the plate backwall. Any indications imaged above this line were in direct line of sight of the probe. Below the horizontal line is a mirror image of the weld; any indications plotted in this region represent the beam reflecting off the backwall before insonifying the weld region. Comparing B-scan images to master drawings (Fig. 4), showed that all flaws were accurately positioned in the weld in the plane perpendicular to the weld run (i.e. x–z plane). In the y-direction both the length and positioning of the flaws were inconsistent with master drawings, though this was expected because scanning was unencoded.

Toe Crack

-10

70

LORP

Porosity

z (mm)

x (mm)

-20

60 LORP 50

-30 -40

40

-50

100

200 y (mm)

300

40

50 60 x (mm)

70

-10

-10 Porosity

-20 z (mm)

z (mm)

-20 -30

-30

-40

-40

-50

-50 40

50 60 x (mm)

70

Toe Crack

40

50 60 x (mm)

70

Fig. 10. Images of the carbon steel double ‘v’ weld plate showing (a) an unencoded, volume corrected C-scan of the entire weld region, (b) B-scan of lack of root penetration, (c) B-scan of porosity, (d) B-scan of a toe crack.

M. Weston et al. / NDT&E International 47 (2012) 43–50

49

4. Conclusion Algorithms to process data acquired using the FMC technique were developed in the Matlab environment. It has been shown that for a planar interface an iterative, numerical or analytical approach can be used to find the incident point between dual media. For a non-planar interface however, the point of incidence could only be found using numerical or iterative methods, as the equation used to describe the system exceeds 4th order. It was found that the time taken to find the point of incidence at a planarinterfacefor the analytical, numerical and iterative methods were similar for the set-up investigated. Curve fitting algorithms were also developed to allow autofocussing through a planar interface. Experimental results from a double ‘v’ butt welded plate demonstrated how auto-focussing through dual-layered media with a planar interface can be implemented using an analytical approach to detect and image real defects in industrial components. Fig. B1. A diagram of refraction at a non-planar interface between dual layered media.

Acknowledgements The authors wish to acknowledge TWI Ltd, the University of Manchester and the University of Birmingham for their facilities and technical support and the EPSRC for funding the studentship at TWI Ltd/University of Manchester/University of Birmingham. The authors also wish to acknowledge Dr P.I. Nicholson for producing the drawings presented in this paper.

p4 x4b þ p3 x3b þ p2 x2b þp1 xb þp0 ¼ 0

A quartic polynomial of the form 3

2

p4 xb þp3 xb þ p2 xb þ p1 xb þ p0 ¼ 0 can be solved using Ferrari’s method, which proceeds as follows:

a¼

3p23 p2 þ 8p24 p4

p3 p p p b ¼ 33  3 22 þ 1 p4 8p4 2p4

g¼

Q ¼ R¼

12

ðB3Þ

Where p4 will be a constant and p3, p2, p1, p0 may be constants or variables and take the following form p4 ¼ b1

ðB4Þ

p3 ¼ 2x0 þ 2xt2 2bxs 2bxt1

ðB5Þ

p1 ¼ 2x20 xt2 þ 2x0 x2t2 þ2x0 z2b 4x0 zb zt2 þ2x0 z2t2 2bx2s xt1 2bxs x2t1

g

a3

ðB2Þ

p2 ¼ bx2s þ bx2t1 þ2bz2b þ bz2s þ bz2t1 x20 x2t2 z20 2z2b z2t2 4x0 xt2 þ2z0 zb þ 2zb zt2 þ4bxs xt1 2bzb zs 2bzb zt1 ðB6Þ

3p43 p p2 p p p þ 2 3 3 1þ 0 p4 256p44 16p34 4p24

a2

dzb ¼ mn nxn1 þ mn1 ðn1Þxn2 þ    þ2m2 xb þ m1 b b dxb

The coordinates (xt1, zt1, xt2, zt2), which lie on the tangent, can then be calculated. Therefore Snell’s law can be expressed in the form

Appendix A

4

parameters which are determined relative to the normal, then the tangent at every point along the interface must be known to determine the point of incidence. To calculate the tangent, it is necessary to differentiate Eq. (B1).

2bxs z2b þ 4bxs zb zt1 2bxs z2t1 2bxt1 z2b þ4bxt1 zb zs 2bxt1 z2s þ2xt2 z20 4xt2 z0 zb þ 2xt2 z2b

ag b2

þ  108 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 8ffi R R2 Q 3 þ S¼ 7 2 4 27 p ffiffiffi 3 U¼ S 5 Q V ¼  a þ U 6 3U pffiffiffiffiffiffiffiffiffiffiffiffiffiffi W ¼ a þ 2V

p0 ¼ bz4b þ 2z0 z3b þ 2z3b zt2 z4b x20 x2t2 x20 z2b x20 z2t2 x2t2 z20 x2t2 z2b z20 z2b z20 z2t2 z2b z2t2 þ bx2s x2t1 þ bx2s z2b þ bx2t1 z2b þ bx2s z2t1 þ bx2t1 z2s þ bz2b z2s þ bz2b z2t1 þ bz2s z2t1 2bz3b zs 2bz3b zt1 þ2x20 zb zt2 þ2x2t2 z0 zb þ2z0 zb z2t2 4z0 z2b zt2 þ 2z20 zb zt2 2bx2s zb zt1 2bx2t1 zb zs 2bzb zs z2t1 2bzb z2s zt1 þ 4bz2b zs zt1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 7 1 W 8 2 ð3a þ2V 7 1 ð2b=WÞÞ xb ¼  3 þ 2 4p4

where xt1 ¼

z0 zb þ ðdzb =dxb Þxb x0 þ ðdzb =dxb Þ þ ð1=ðdzb =dxb ÞÞ ðdzb =dxb Þ2 þ1

Appendix B

zt1 ¼ 

In the general case an arbitrary refractive interface can be defined by the following polynomial equation (Fig. B1)

xt2 ¼

zb ðxb Þ ¼ mn xnb þ mn1 xn1 þ    þm2 x2b þ m1 xb þ m0 b

ðB7Þ

zs zb þ ðdzb =dxb Þxb xs þ ðdzb =dxb Þ þ ð1=ðdzb =dxb ÞÞ ðdzb =dxb Þ2 þ1

ðB1Þ

Since the angle of incidence and angle of refraction are both

xt1 x0 þz0 þ ðdzb =dxb Þ ðdzb =dxb Þ

zt2 ¼ 

xt2 xs þzs þ ðdzb =dxb Þ ðdzb =dxb Þ

ðB8Þ

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