WO1999054677A1

WO1999054677A1 - A phase shifting interferometer

Info

Publication number: WO1999054677A1
Application number: PCT/SG1998/000029
Authority: WO
Inventors: Kai Man Hung; Hiroyuki Ueda; Takatoshi Yamada
Original assignee: Sony Precision Engineering Center (Singapore) Pte Ltd; Sony Corporation
Priority date: 1998-04-16
Filing date: 1998-04-16
Publication date: 1999-10-28
Also published as: JP2002507285A; AU7242898A

Abstract

A phase shifting interferometer (1) includes a light source (2), a reference optical path (5), a measurement optical path (4), which includes a mounting device (32) to support an optical element (37) to be tested, and a light splitter (3). The light splitter (3) splits light (18) from the light source (2) to the reference optical path (5) and to the measurement optical path (4) and to recombine light from the reference optical path (5) and the measurement optical path (4) and directs the recombined light (27) to a detector. The light source includes a GaAs-AlGaAs distributed Bragg reflector wavelength tunable laser diode (10).

Description

1 A PHASE SHIFTING INTERFEROMETER

The invention relates to a phase shifting interferometer, and in particular a phase shifting interferometer for determining aberration parameters of optical elements.

It is known to measure lens aberrations using a Twyman-Green interferometer and this type of interferometer has been used to measure aberrations in optical pick-up lenses for optical disc players, such as players for compact discs and digital video discs. Conventionally, aberration parameters in optical pick-up lenses have been measured using a Twyman- Green interferometer by either rotating a flat-block phase object along the reference arm or tuning a piezo-electric device mounted on the reference mirror.

However, these phase shifting interferometers have the disadvantage that the measurement of the aberration in the lenses is not accurate as the wavelength of light emitted by the light source, which is usually a laser, does not correspond to the wavelength of light with which the lenses are actually used. Hence, errors can be introduced as the refractive index of the lens will vary for different wavelengths of light.

In addition, the use of these conventional phase shifting interferometers for detecting aberrations in optical pick-up lenses requires the use of difficult and/or complicated 2 calibration procedures.

With the rotating phase block type of interferometer it is necessary to calibrate the interferometer to ensure that the detector captures the image when the phase block is at the correct position during rotation of the phase block.

Phase shifting interferometers using piezo-electric devices for phase shifting have the disadvantage that the piezo- electric device is non-linear and so a complicated feed-back loop is required in order to ensure that linearity is obtained.

In order to determine the abberations in the optional pick-up lens an automated optical fringe analysis is normally carried out by either phase shifting or the Fourier transform of the fringe patterns [1] . Both techniques, however, involve the use of trigonometric functions in their calculation, so only principal value phase maps, i.e. the phase values lying in between -π and π (or equivalently, between 0 and 2π) , are obtained. Therefore, any phase values lying outside this range will not be represented in the demodulated phase map. Hence, a process called phase unwrapping must be used to remove the 2π phase discontinuities in the phase maps in order to recover the actual phase values of the tested object .

A number of phase unwrapping algorithms have been discussed 3 extensively in the past decade [2-5] (and the references therein) . The aim of these algorithms is to handle noise, discontinuities and below-modulation-threshold pixels in the phase maps, while at the same time minimizing the computational effort for the process [6] . There is a trade off between the robustness of an algorithm and the computational time that is needed to execute the algorithm. Global or path-independent algorithms, which firstly scan through the phase map before any calculations, are generally more robust but take a longer time to unwrap the phase map; whereas, the reverse is true for local or path-independent algorithms, which process the phase map pixel by pixel.

Phase unwrapping by regions [2] , which is a path-independent algorithm itself, has been regarded as one of the useful methods, as commented in Ref. [6-7], that makes a good compromise between robustness and execution time. The algorithm involves, firstly, the identification of regions of the phase map within which the phase is continuous. This is done by comparing the phase between a pixel and its neighbour. If the phase difference is within an adjustable tolerance, then the pixel and its neighbour are allocated to the same region. Secondly, after this sorting operation is completed, different regions are phase shifted with respect to one another to eliminate the phase discontinuities. The advantage of this approach is that noisy data will be isolated and thus have only a minor effect on the overall quality of the unwrapped phase map. 4 However, the original proposed phase unwrapping by regions [2] does not cope well with discontinuities, such as physical edges or holes, and mild level of noise [6,7] .

In accordance with a first aspect of the present invention, a phase shifting interferometer for testing an optical pickup lens comprises a light source, a reference optical path, a measurement optical path and a light splitting device to split light emitted from the light source to the reference optical path and to the measurement optical path and to recombine light from the reference optical path and the measurement optical path and direct the recombined light to a detector; the measurement optical path comprising a mounting device adapted to have an optical pick-up lens to be tested mounted thereon, a spreading lens adapted to spread incident light in the measurement optical path across the surface of an optical pick-up lens to be tested, and a concave spherical reflecting surface adapted to reflect the incident light back through an optical pick-up lens element to be tested and the spreading lens to the light splitting device; and the light source comprising a wavelength tunable laser.

The term "light" as used herein includes electromagnetic radiation in the ultraviolet, visible and infra-red regions of the electromagnetic spectrum.

An advantage of the first aspect of the invention is that by 5 using a wavelength tunable laser diode, it is possible to perform phase shifting interferometry on an optical pick-up lens, at the same wavelength with which the lens will be used.

Preferably, the laser is a laser diode. Typically, the laser diode comprises a distributed Bragg reflector (DBR) region and may be wavelength tuned by varying the injection current to the DBR region.

Typically, there is an optical path difference between the reference and measurement optical paths and the length of reference optical path is greater than the length of the measurement optical path. Typically, the optical path difference is at least 10 mm, preferably at least 30 mm, more preferably at least 40 mm and most preferably, in the region of 40 mm to 70 mm and may be for example, 50 mm.

Preferably, the light source further comprises two anamorphic prisms arranged to receive the output light beam from the laser diode and to emit a light beam having a substantially circular cross-section.

Preferably, the light source further comprises a spatial filter. Typically, the spatial filter comprises a focusing lens and an aperture through which the focusing lens focuses the output light from the laser diode. 6

Preferably, the light source also comprises a collimator, such as a focusing lens, which collimates the light to produce a collimated light beam from the light source. Typically, two collimators are provided, one collimator adjacent the laser diode which receives the light beam emitted by the laser diode and a second collimator which receives the light beam after the light beam has passed through the anamorphic prisms and the spatial filter. Typically, the anamorphic prisms and the spatial filter are located between the two collimators.

Typically, the light source also comprises an isolator, preferably located between the anamorphic prisms and the spatial filter. The advantage of the isolator is that it minimises the risk of the laser diode being exposed to retroreflected light from the beam splitter.

Preferably, the spreading lens comprises a converging lens which typically has a numerical aperture greater than or equal to the numerical aperture of the optical pick-up lens being tested. Preferably, the distance from the focus of the converging lens to the optical pick-up lens is substantially equal to the distance from an optical light source, s\ιch as a laser diode, to the optical pick-up lens in use, for example in an optical disc player.

Preferably, the detector may be a recording device, for example an image recording device, such as a camera. 7 Typically, the image recording device is a digital image recording device, such as a CCD array.

Preferably, the reference optical path comprises' a plane reflecting surface which reflects light transmitted by the beam splitter along the reference optical path back towards the beam splitter.

In accordance with a second aspect of the present invention, a method of calibrating a phase shifting interferometer having a light source comprising a laser, comprises the steps of detecting the intensity of light in a region of the image plane of the interferometer and monitoring the change in intensity of the region as a function of a phase shifting parameter of the interferometer to determine the change in the phase shifting parameter required to cause the intensity at the monitored region to change through a minimum and a maximum and return to the initial intensity, the required change in phase shifting parameter being indicative of a 2π phase shift between the reference and the measurement optical paths of the interferometer.

An advantage of the second aspect of the invention is that it provides a relatively straightforward and fast calibration of the tuning parameter of the laser to the phase shift between the reference and the measurement optical path of the interferometer . 8

Preferably, the calibration method further comprises plotting the light intensity of the region as a function of the phase shifting parameter of the interferometer laser. Typically, the variation in intensity as a function of the phase shifting parameter of the interferometer approximately describes a cosine relationship and the amount of change required in the phase shifting parameter to cause a 2π phase shift between the reference and the measurement optical path of the interferometer is equal to one cosine cycle of the phase shifting parameter versus intensity.

Typically, the laser may be a wavelength tunable laser and the phase shifting parameter is a wavelength tuning parameter of the laser.

The wavelength tunable laser may be a wavelength tunable laser diode, such as a distributed Bragg reflector laser diode, and the tuning parameter may be the injection current to the Bragg region of the diode. The wavelength tunable laser diode may be a GaAs-AlGaAs distributed Bragg reflector wavelength tunable laser diode, such as that produced by Yokokawa Electric Company of Tokyo, Japan, and may be wavelength tunable from an output wavelength of 780nm to 790nm.

Alternatively, the interferometer may comprise a piezo-electric phase shifting device, typically coupled to the reference mirror, and the phase shifting parameter is a 9 phase shifting parameter of the piezo-electric phase shifting device .

Preferably, the phase shifting interferometer calibrated by the method according to the second aspect is a phase shifting interferometer in accordance with the first aspect of the invention.

Preferably, the phase shifting interferometer in the first and the second aspects of the invention is a Twyman-Green interferometer. Alternatively, the phase shifting interferometer may be a Mach-Zehnder or a Fizeau interferometer .

In accordance with a third aspect of the invention, a method of phase unwrapping a phase map of a fringe pattern obtained from a phase shifting interferometer comprises the steps of:

(a) dividing the phase map into a number of regions ... , R_J-_JR_J, ... such that any two pixels within a region have a phase difference of less that 2π;

(b) assigning a class variable ... , μ_k, μ_x , . . . to each region ... , R_k, R_1# ... ;

(c) applying the following equation to each of the the boundary pixels (i,j), (m,n) between a pair of adjacent regions R^Rj_.:

μ_k ~ Mi φ(i, j) - Φ(m,n)

2π (1) 10 where φ(i,j) is the phase of pixel (i,j), φ(m,n) is the phase of pixel (m,n) and [ ] denotes a rounding operation to the nearest integer;

(d) summing up the equations (1) for the boundary pixels to obtain:

ω(μ* - Mi) Σ φ(i, j) - φ(m,n)

(i, j) ER_k 2π (2)

(m,n) 6R_λ where ω is the number of pair-pixels along the boundary;

(e) transforming the resulting equations (2) into a matrix equation

AΦ = P (3) where A is a (KxL) matrix with K equal to the number of regions and is the total number of combination by which other regions are in contact with a particular region; and Φ is a vector:

. . .

Φ = μ_k μi

I . ^' )

(f) calculating the rank of the matrix A;

(g) solving the matrix equation (3) as a subject of a least square problem;

(h) rounding off the solutions to integers; and, (i) adding the solutions in the unit of 2π to the phase value of each region to obtain the unwrapped phase map . 11

An example of a phase shifting interferometer in accordance with the invention will now be described with reference to the accompanying drawings, in which: -

Figure 1 is a schematic diagram showing a Twyman-Green interferometer;

Figure 2 is a schematic diagram showing components of a light source for the Twyman-Green interferometer shown in Figure 1 ;

Figure 3 is a schematic block diagram of a GaAs-AlGaAs distributed Bragg reflector wavelength tunable laser diode ;

Figure 4 is a schematic diagram showing components of a measurement arm of the Twyman-Green interferometer shown in Figure 1 ;

Figure 5 is a graph showing the intensity variation of a patch of pixels versus injection current, and fitted with a nonlinear cosine curve;

Figure 6 is a schematic diagram illustrating phase unwrapping by regions ;

Figure 7 is an example of an original phase map;

Figure 8 is a wrapped phase map of the original phase map shown in Figure 7 ;

Figure 9 is an unwrapped phase map of the wrapped phase map shown in Figure 8 ;

Figure 10 is a wrapped phase map of an optical pick-up lens with a spherical aberration; and,

Figure 11 is an unwrapped phase map of the wrapped phase 12 map shown in Figure 10.

Figure 1 shows a Twyman-Green interferometer 1 which comprises a light source 2, a beam splitter 3, a measurement arm 4, a reference arm 5 and an output path 22. The reference arm 5 includes a reflecting surface 6 and a reference mirror 7. The output path 22 includes a reflecting surface 8 and a zoom lens 9 which direct a recombined light beam 27 to a CCD camera and frame grabber (not shown) .

The light source 2 is shown in more detail in Figure 2 and includes a gallium arsenide-aluminium gallium arsenide (GaAs- AlGaAs) distributed Bragg reflector wavelength tunable laser diode 10, a first collimator 11, two anamorphic prisms 12, 13, an optical isolator 14, a lens 15, an aperture 16 and a second collimator 17.

A schematic block diagram representing the laser diode 10 is shown in Figure 3. The laser diode 10 comprises three regions, an active region 19, a phase control region 20 and a Bragg selector region 21. Each of the three regions 19, 20, 21 are controlled by an injection current i_LD, i_PH and i_DBR, respectively. The population inversion occurs in the active region 19, the phase control region 20 is used to control the phase of the laser by injection of charge carriers and the Bragg selector region 21 is essentially a waveguide corrugation formed to select the modes of the laser diode 10 which are Bragg matched. The Bragg wavelength can be changed 13 by varying the refractive index of the layers near the Bragg selector region 21 through the thermal effect or by the band filling effect by the injection of current i_DBR. The characteristics of GaAs-AlGaAs wavelength tunable laser diodes with distributed Bragg reflectors and phase control sections are described in more detail in the article by T Hirata et al entitled "Fabrication and Characteristics of GaAs-AlGaAs Tunable Laser Diodes with DBR and Phase-Control Sections Integrated by Compositional Disordering of a Quantum Well" in IEEE Journal of Quantum Electronics, Vol. 27, No. 6, June 1991, pages 1609-1615.

The laser diode 10 has a lasing wavelength of approximately 780nm to 790nm.

The purpose of the optical arrangement shown in Figure 2 is to produce a collimated, circular "clean" beam of output light from the light source 2. It is well known that the cross-sectional shape of an output light beam from a semiconductor laser is elliptical due to asymmetric radiation caused by diffraction at the exit from the diode laser cavity. Therefore, two anamorphic prisms 12, 13 are used to correct the cross-sectional shape of the output beam to a circular cross-section. However, before the output beam from the laser diode 10 enters the prisms 12, 13, the beam is first collimated by the first collimator 11 to minimise astigmatism resulting from the prisms 12, 13. The output beam from the anamorphic prism 13 then passes through an 14 optical isolator 14. The optical isolator 14 protects the laser diode 10 from being exposed to retroreflected light from the reflecting surfaces of the interferometer 1. The optical isolator 14 may, for example, use the Faraday effect to isolate the laser diode 10 from retroreflected light.

After exiting through the optical isolator 14, the light is "cleaned" by a spatial filter which comprises the lens 15 and the aperture 16. The lens 15 is arranged to focus the collimated output beam from the optical isolator 14 through the aperture 16. The output beam from the aperture 16 is then recollimated using a second collimator 17 to produce an output light beam 18.

The output light beam 18 from the light source 2 is incident on a beam splitter 3 which splits the beam 18 into a measurement path beam 25 and a reference path beam 26.

As shown in Figure 1, the measurement arm 4 is orientated perpendicularly to the other sections of the interferometer 1. Typically, the beam 18, reference path beam 26 and the output path beam 27 are orientated in a generally horizontal plane and the measurement arm 4 is orientated in a generally vertical plane.

The lay out and components of the measurement arm 4 are shown in more detail in Figure 4. The measurement path beam 25 exits the beam splitter 3 horizontally and is directed 15 vertically upwards by a folding mirror 30. The primary components of the measurement arm 4 include a spherical reflecting surface 31, a lens stage 32, an objective lens 33 mounted on an objective lens stage 34 and a cover glass 35. The objective lens 33 focuses the measurement path beam 25 (which is collimated) to a focus 36 after which the light diverges to spread across a test lens 37 mounted on the lens stage 32. After passing through the test lens 37, the light is focused by the test lens 37 to a focus 38 before diverging through the cover glass 37 to the reflecting surface 31. The light is then reflected from the reflective surface 31 and reflected back along the same path to output from the measurement arm 4 to the beam splitter 3. As the radius of curvature of the reflecting surface 31 is equal to the distance from the focal point 38 to the surface 31, the reflected light follows the same path as the light incident on the reflecting surface 31.

As shown in Figure 1, the reference path beam 26 is reflected by the reflecting surface 6 to the reference mirror 7 which reflects the reference beam 26 back along the path of the incident light from the reflecting surface 6 and beam splitter 3 towards the beam splitter 3.

The reflected light from the reference mirror 7 and the spherical reflecting surface 31 is then recombined at the beam splitter 3 into the output beam 27 which is reflected by the reflecting surface 8 through the zoom lenses 9 to form an 16 image on the CCD camera.

It should be noted that where the laser diode 10 is a GaAs-AlGaAs wavelength tunable laser diode with distributed Bragg reflector, the reference optical path length (i.e. the optical distance from the beam splitter 3 to the reference mirror 7) should be approximately 50 mm longer than the length of the measurement optical path (i.e. the optical distance from the beam splitter 3 to the spherical reflecting surface 31) . This is important as it helps to prevent mode hopping occurring in the diode 10 if one wants to obtain one period of fringe shift.

After the interferometer has been set up and aligned correctly, as explained above and as shown in Figures 1, 2 and 4, it is necessary to calibrate the tuning of the laser to the wavelength shift of the interferometer. In this example with a GaAs-AlGaAs distributed Bragg reflector wavelength tunable laser diode it is necessary to calibrate the change in injection current i_DBR to the wavelength shift. In order to obtain the calibration, the intensity of a region of pixels obtained from the CCD camera is recorded as the injection current i_DBR is increased. The current i_DBR is increased in very small increments and is plotted in points, as shown in Figure 5, which shows the injection current i_DBR (in units of digital to analog steps) versus intensity of the region of pixels being monitored. 17 Ideally, the points plotted should define a cosine-like waveform. However, as the relationship between the injection current i_DBR and the wavelength shift is not completely linear, and the experimental environment is not perfectly stable, the points plotted have minor deviations from the ideal waveform. Hence, in order to evaluate the change in injection current i_DBR which causes a 2π phase shift between the reference and measurement paths, the points plotted are fitted nonlinearly with a cosine curve 40, as shown in Figure 5. From the cosine curve 40 shown in Figure 5 it is possible to determine the change in injection current i_DBR which is necessary to obtain a 2π phase shift from the graph. For example, the change in injection current i_DBR between points 41 and 42 on the curve 40 represent a 2π phase shift. Therefore, for example, if a four step fringe scanning method is used, the injection current differences at which it is necessary to grab an image from the CCD camera for every 2π/4 phase shift can be determined.

To enhance the accuracy of the calibration, it is preferred that only pixels which have an intensity modulation above a threshold value are chosen for the calibration.

Hence, this method of calibration provides a relatively straightforward and quick calibration of the injection current i_DBR to the fringe scanning.

After calibration, the test lens 37 may be tested for 18 aberrations by recording images from the CCD camera at 2π/4 phase shifts and then demodulating and phase unwrapping the images using suitable algorithms to determine aberration values for the lens 27. In addition, the values can be normalised by the root-mean-square values by dividing the coefficient of, for example, the third-order coma term (3p³-2p) sinθ by f "f (3p-2p) sinøpdpdø.

The algebraic procedures for this can be summarized as follows :

i. Dividing the effective phase domain into regions. For example, in Figure 6, if the phase difference between the two pixels labelled (g,h) and (i,j) is smaller than 2π, then they will be assigned to the same region, say R_k . However, if the phase difference between pixels (i,j) and (m,n) is bigger than 2π, then they will be allocated in two different regions .

ii . Assigning class variables μ to different regions. For example, the region R_k can be assigned a class variable μ_k .

iii. Applying the following equation to the boundary points between regions :

_k φ(i, j ) - ,φ(m,n)

2π (1)

where φ(i,j) is the phase of pixel (i,j) and [ ] denotes a 19 rounding operation to the nearest integer.

iv. Summing up the equations for the boundary points such as (i,j) and (m,n) to obtain:

(ύ (μ_k - μ_λ) = Σ φ(i, j) - φ(m,n)

( i , j ) eR_k 2π (2)

(m,n) eRj where ω is the number of pair-points along the boundary.

v. Transforming the resulting equations (2) into a matrix equation

AΦ = P (3) where A is a (KxL) matrix with K equals to the number of regions and L is the total number of combination by which other regions are in contact with a particular region; and Φ is a vector:

. . )

_k

I . J

vi . Calculating the rank of the matrix A.

vii. Solving the matrix equation as a subject of a least square problem (e.g. QR decomposition can be used) .

viii .Rounding off the solutions to integers. 20 ix. Adding the solutions in the unit of 2π to the phase value of each region.

A program has been developed and tested for phase unwrapping based on the algorithm outlined above. In order to illustrate the robustness of the algorithm against noise and physical discontinuities, the program was applied to a wrapped phase map shown in Figure 8 which is obtained from demodulating four intensity maps (after the four-step phase shifting interferometry) that encode the original phase map depicted in Figure 7. The unwrapped phase map is displayed in Figure 9 in which 494 linear equations and 559 variables were generated and solved by the algorithm. It took less than 10 seconds to obtain the phase map when the program was executed in WINDOWS 95 using Mathematical software on a 266MHz Pentium PC. However, it should be noted that the run time depends significantly on the complexity of the phase map, i.e. the number of equations generated, as well as the speed of the computer. From the unwrapping results it can be verified that the unwrapping algorithm performs well against spikes of noise and physical discontinuities. The robustness of the phase unwrapping by regions is significantly improved.

There are many applications in which the new algorithm developed in this paper can be applied. One of them is to unwrap the phase maps of lenses obtained by phase shifting interferometry . Figure 10 shows a wrapped phase map of a lens with a spherical aberration. One may observe in Figure 21

10 that the phase values are bound between -π and π which gives rise to phase discontinuities. Figure 11 displays the unwrapped phase map of Figure 10 after the map was unwrapped by the algorithm. It took less than 2 seconds to finish the unwrapping operation. Once again, the results show that the algorithm is robust against noise and missing data and that: it is computational efficient to implement the algorithm.

It should be noted that the injection current i_DBR should not be chosen near the mode hopping points of the laser and that the change in the injection current i_DBR should not be too large in order to minimise changes in the average intensity and the modulation ratio of the fringe pattern when the laser diode 10 is tuned. However, it should also be noted that the change in injection current i_DBR should be sufficiently high to maintain accuracy when tuning the laser diode 10 using the injection current i_DBR- In addition, for the particular laser diode used in the arrangement described above, that is a GaAs-AlGaAs DBR laser diode manufactured by Yokokawa Electric Company of Tokyo, Japan, the injection current i_LD to the active region 19 should be sufficiently high to minimise fluctuations in the modulation ratio of the fringe pattern and that the optical path difference between the measurement arm 4 and the reference arm 5 should be in the region of approximately 40 to 70 mm (i.e. the length of the reference arm 5 should be 40 to 70 mm longer than that optical path difference of the measurement arm 4) . 22 Hence, the invention has the advantage that it is possible to test optical pick up lenses for aberrations at the wavelength at which they will actually be used without requiring mechanical phase shifting devices for the interferometer, such as piezoelectric devices. In addition, the invention permits a relatively straightforward calibration method to be used for the laser diode and interferometer.

References

[1] J.E. Greivenkamp and J.H. Bruning, "Phase Shifting Interferometry" , Chap. 14 in Optical Shop Testing, D. Malacara, Eds pp. 501-598, 2nd edition, John Wiley & Sons, Inc. (1992)

[2] J.J. Gierloff, "Phase Unwrapping by Regions", in Current Development in Optical Engineering II (Proc. SPIE) , R.E. Fischer and W.J. Smith, Eds, Vol. 818, 2-9(1987)

[3] D.C. Ghiglia, G.A. Mastin and L.A. Romero, "Cellular-Automata Method for Phase Unwrapping", J. Opt . Soc . Am. A, 4(1), 267-280 (1987)

[4] H.O. Saldner and J.M. Huntley, "Temporal Phase Unwrapping: Application to Surface Profiling of Discontinuous Objects", Applied Optics, 36(13), 2770-2775(1997)

[5] K. Creath, " Phase-Measurement Interferometry Techniques" , in Progress in Optics, E. Wolf, Ed. (Elsevier, Amsterdam), pp.349-393 (1988)

[6] For an overview see [7] and P.G. Charette and I.W. Hunter, "Robust Phase-Unwrapping Method for Phase Images with High Noise Content" , Applied Optics, 35(19), 3506-3513(1996) 23

[7] P. Stephenson, D.R. Burton and M.J. Lalor, "Data Validation Techniques in a Tiled Phase Unwrapping Algorithm", Opt. Eng. , 33(11), 3703-3708(1994)

[8] H. Takajo and T. Takahashi, "Noniterative Methods for Obtaining the Exact Solution for the Normal Equation in Least-Squares Phase Estimation from Phase Difference", J^". Opt. Soc . Am . A, Vol. 5, p. 1818-1827 (1988).

[9] D.C. Ghiglia and L. A. Romero, "Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping that Uses Fast Transforms and Iterative Methods", J. Opt . Soc . Am . A, Vol. 11, 107-119(1994)

Claims

24 Claims

1. A phase shifting interferometer for testing an optical pick-up lens comprising a light source, a reference optical path, a measurement optical path and a light splitting device to split light emitted from the light source to the reference optical path and to the measurement optical path and to recombine light from the reference optical path and the measurement optical path and direct the recombined light to a detector; the measurement optical path comprising a mounting device adapted to have an optical pick-up lens to be tested mounted thereon, a spreading lens adapted to spread incident light in the measurement optical path across the surface of an optical pick-up lens to be tested, and a concave spherical reflecting surface adapted to reflect the incident light back through an optical pick-up lens element to be tested and the spreading lens to the light splitting device; and the light source comprising a wavelength tunable laser.

2. A phase shifting interferometer according to claim 1, wherein the laser is a laser diode.

3. A phase shifting interferometer according to claim 2, wherein the laser diode comprises a distributed Bragg reflector.

4. A phase shifting interferometer according to any of the 25 preceding claims, wherein the reference optical path and the measurement optical path have an optical path difference which is at least 10 mm.

5. A phase shifting interferometer according to claim 4, wherein the optical path difference is at least 30 mm.

6. A phase shifting interferometer according to claim 5, wherein the optical path difference is at least 40 mm.

7. A phase shifting interferometer according to claim 6, wherein the optical path difference is in the region of 40 mm to 70 mm.

8. A phase shifting interferometer according to claim 7, wherein the optical path difference is approximately 50 mm.

9. A phase shifting interferometer according to any of the preceding claims, wherein the spreading lens is a converging lens and distance between the converging lens and the optical pick-up lens is greater than the focal length of the converging lens .

10. A phase shifting interferometer according to any of the preceding claims, wherein the spreading lens has a numerical aperture which is greater than or equal to the numerical aperture of the optical pick-up lens. 26

11. A phase shifting interferometer according to any of the preceding claims, wherein the light source further comprises two anamorphic prisms arranged to receive the output light beam from the laser diode and to emit a light beam having a substantially circular cross-section.

12. A phase shifting interferometer according to any of the preceding claims, wherein the light source further comprises a spatial filter.

13. A phase shifting interferometer according to claim 12, wherein the spatial filter comprises a focusing lens and an aperture through which the focusing lens focuses the output light from the laser diode.

14. A phase shifting interferometer according to any of the preceding claims, wherein the light source further comprises a collimator, such as a focusing lens, which collimates the light to produce a collimated light beam from the light source .

15. A phase shifting interferometer according to claim 14 when dependent on claims 12 and 13, wherein two collimators are provided, one collimator adjacent the laser diode and which receives the light beam emitted by the laser diode and a second collimator which receives the light beam after the light beam has passed through the anamorphic prisms and the spatial filter. 27

16. A phase shifting interferometer according to any of the preceding claims, wherein the interferometer is a Twyman-Green interferometer.

17. A method of calibrating a phase shifting interferometer having a light source comprising a laser, the method comprising the steps of detecting the intensity of light in a region of the image plane of the interferometer and monitoring the change in intensity of the region as a function of a phase shifting parameter of the interferometer to determine the change in the phase shifting parameter required to cause the intensity at the monitored region to change through a minimum and a maximum and return to the initial intensity, the required change in phase shifting parameter being indicative of a 2╧Ç phase shift between the reference and the measurement optical paths of the interferometer .

18. A method according to claim 17, further comprising plotting the light intensity of the region as a function of the phase shifting parameter of the interferometer.

19. A method according to claim 17 or claim 18, wherein the laser is a wavelength tunable laser and the phase shifting parameter is a wavelength tuning parameter of the laser.

20. A method according to any of claims 17 to 19, wherein the wavelength tunable laser is a laser diode. 28

21. A method according to claim 20, wherein the tuning parameter is an injection current applied to the laser diode.

22. A method according to claim 21, wherein the laser diode includes a distributed Bragg reflector region and the injection current is applied to the distributed Bragg reflector region.

23. A method according to claim 17 or claim 18, wherein the interferometer comprises a piezo-electric phase shifting device, and the phase shifting parameter is a phase shifting parameter of the piezo-electric phase shifting device.

24. A method according to any of claims 17 to 23, wherein the phase shifting interferometer is a Twyman-Green interferometer .

25. A method according to any of claims 17 to 23, wherein the phase shifting interferometer is a Mach-Zehnder interferometer .

26. A method according to any of claims 17 to 23, wherein the phase shifting interferometer is a Fizeau interferometer.

27. A method of phase unwrapping a phase map of a fringe pattern obtained from a phase shifting interferometer, the method comprising the steps of:

(a) dividing the phase map into a number of regions 29

... ,R_k,R₁, ... such that any two pixels within a region have a phase difference of less that 2╧Ç;

(b) assigning a class variable ... , ╬╝_k, ╬╝_{l t} . . . to each region ... , R_k, R_1; ... ;

(c) applying the following equation to each of the the boundary pixels (i,j), (m,n) between a pair of adjacent regions R_k. j_.:

╬╝ i = ╬ª(i> j ╧å(m,n)

2╧Ç (1)

where ╧å(i,j) is the phase of pixel (i,j), ╧å(m,n) is the phase of pixel (m,n) and [ ] denotes a rounding operation to the nearest integer;

(d) summing up the equations (1) for the boundary pixels to obtain:

(╧ì ( ╬╝_k - ╬╝_╬╗) ╬ú ╧å(i,j) - ╧å(m,n)

⁽i.j) eR_k 2╧Ç (2)

(m,n) Γé¼Rj where ╧ë is the number of pair-pixels along the boundary; (e) transforming the resulting equations (2) into a matrix equation

A╬ª = P (3) where A is a (KxL) matrix with K equal to the number of regions and L is the total number of combination by which other regions are in contact with a particular region; and ╬ª is a vector: 30

. . )

╬ª = ╬▓_k i

i ; J

(f) calculating the rank of the matrix A;

(g) solving the matrix equation (3) as a subject of a least square problem;

(h) rounding off the solutions to integers; and, (i) adding the solutions in the unit of 2╧Ç to the phase value of each region to obtain the unwrapped phase map.

28. A method according to claim 27, wherein the phase map is a phase map of an optical pick-up lens.