research article measurements of the characteristics of...

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 598737 7 pageshttpdxdoiorg1011552013598737

Research ArticleMeasurements of the Characteristics ofTransparent Material Using Digital Holography

Ding Yu1 Shang Wenbin2 Yang Hong3 and Yang Yan2

1 Chongqing City Management College Shapingba District Chongqing 401331 China2Mechanical Engineering Chongqing University of Technology No 69 Hongguang Road Banan District Chongqing 400054 China3 Key Laboratory of Advanced Manufacturing Technology for Automobile Parts Ministry of Education No 69 Hongguang RoadBanan District Chongqing 400054 China

Correspondence should be addressed to Yang Yan 40129784qqcom

Received 31 July 2013 Revised 28 September 2013 Accepted 28 September 2013

Academic Editor Xing Chen

Copyright copy 2013 Ding Yu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Digital holography is applied tomeasure the characteristics of transparentmaterial A digital hologram recording system tomeasurethe surface of transparent material was established and the digital holograms of transparent object were obtained in high qualityFor postprocessing of hologram the least-squares phase unwrapping algorithm was used in phase unwrapping and the phasereconstruction image of transparent object was obtainedThe information of material surfaces was measured and the characteristicwas presented in 3D visualization The validation experiment was conducted by NanoMap 500LS system the results of validationexperiment are well satisfied with the measurement by digital holography which proved the feasibility of digital holographictechnology as a good measurement tool for transparent material

1 Introduction

In 1948 Gabor [1 2] proposed holography which can beused for reconstructing amplitude and phase of wave fieldWith the fast development of computer technology recordingand reconstruction are made possible by means of computerSubsequently Goodman and Lawrence [3] developed digitalholography in 1960s

Digital holography is a new image processing technologythat uses a high-resolution charge-coupled device (CCD)camera for hologram recording and image reconstructionwith a numerical method by a computer [4] Comparedwith traditional holography which uses a photographic platesas recording media digital holography has a significantimprovement that is to say it does not need complicatedchemical processing [5] thus adding more flexibility andimproving the efficiency of the holographic process Inthe past few years with modern computer technology andCCD technology developing rapidly digital holography hasbeen established as an important scientific means applied in

metrology [6] surface measurement deformation measure-ment [7] and more recently biological microscopy [8ndash10]

In general the objects in applications of digital holog-raphy are not transparent so the holograms were cap-tured by the reflection of the object [11ndash13] In this paperwe adopt the method of transmission to generate thehologram

Noncontact surface measurement techniques are veryimportant in many fields such as science and engineeringIt is very helpful to control the products quality and makethe appropriate diagnosis Digital holographic imaging asa new image processing technology combined with opticalholography and digital image processing of computer iswidely used in the field of surface measurements of objectsIn addition to classical roughness values 119877

119886 119877119905 and so forth

digital holography can also make full-field measurementsof 3D surface material Digital holographic measurementcan measure the surface of the object with some importantcharacteristics such as no damage high resolution andnoncontact fast processing

2 Advances in Materials Science and Engineering

Reflector

Laser

Lens Collimating lens

Beam splitter prismTesting field

ObjectCCD

Computer

Reflector


Beam splitter prism

Figure 1 Off-line recording of digital hologram

Compared with other image processing and measuringtechniques digital holography has a number of importantadvantages [4] Firstly it does not requiremuch optical equip-ment especially changing the hologram plates Secondly itcaptures the image with noncontact nondestructive andfast height accuracy sensitivity and resolutionThirdly thereis no complex chemical processing in digital holographyso reproducing hologram and reconstructing the hologramare quicker than traditional optical holography Fourthlycomputer image processing allows to process the errors andnoise so as to revise additional optical phase for improvingthe quality of the reconstructed image

Holographic interferometry (HI) is a procedure whichenables static and dynamic displacements of objects withoptically rough surfaces to be measured with optical interfer-ometer accuracy [4 14] Then in digital holography we canaccurately measure the slight deformation of the objects bycalculating the change of the information recorded by digitalholograms [4] In this paper through digital holographyand image processing we can measure the height of thematerial surface Otherwise by recording the hologram thephase diagram can be obtained and the surface of materialcan be measured quickly and accurately making the three-dimensional visualization possible

2 General Principles of Digital Holography

21 Recording of Digital Hologram Digital holographicrecording optical path and the traditional optical holographyare consistent only replace holographic plate with a CCDcamera as the recording medium The concept of outlinedigital hologram recording is shown in Figure 1

The object wave 119874(119909 119910) and the reference wave 119877(119909 119910)

interference occur to form a hologram on the CCD surfaceIn the hologram plane the recorded intensity distribution119868119867(119909 119910) can be printed as follows [4]

119867(119909 119910) = |119874|2

+ |119877|2

+ 119877119874lowast

+ 119877lowast

119874 (1)

where119877119874lowast and119877

lowast

119874 represent the conjugation image and theoriginal image respectively

So the intensity distribution of the recorded hologram byCCD is written as

119867(119898 119899) = 119867 (119909 119910) rect( 119909

119871119909

119910

119871119910

) comb(119909

Δ119909

119910

Δ119910

) (2)

Object plane

Laser beam

Recording

Reconstruction plane

Reconstruction

y

x

d

120585998400120585120588

120578998400120578

CCD

d

120588

Figure 2 Coordinate system for image reconstruction

The light intensity is collected by the data acquisition cardand quantity The 3D information of object is saved by thecomputer and formed as a digital hologram

22 Reconstruction of Digital Hologram In traditional holog-raphy the reconstruction is carried out by means of illumi-nation of the hologram intensity with the reference wave Avirtual image and a real image of the object are reconstructedThe digital hologram simulates the traditional optical repro-duction processes and its numerical reconstruction can begained by computer

The laser beam goes through the object and the part ofthe beam diffracted by objects and reaching the recordingsurface is considered the object beam while the beam fromthe laser arriving without any distortion is considered asthe reference beam Two beams superposition create aninterference pattern on the CCD sensor This diffraction canbe described by the Fresnel-Kirchhoff integral as [4]

119877 (1205851015840

1205781015840

)

=119894

120582∬

infin

minusinfin

ℎ (119909 119910) 119864119877(119909 119910)

exp (minus119894 (2120587120582) 120588)

120588119889119909 119889119910

(3)

with

120588 = radic(1205851015840 minus 119909)2

+ (1205781015840 minus 119910)2

+ 1198892 (4)

The coordinates in (3) and (4) are illustrated in Figure 2119877(1205851015840

1205781015840

) is the wave fields of the reconstruction image andℎ(119909 119910) is the hologram function The 120588 is the distancebetween a point in the hologramplane and the correspondingpoint in the reconstruction plane 119889 is the distance betweentwo adjacent planes and 120582 is the wavelength

Different numerical reconstruction algorithms have beenproposed such as Fresnel approximation algorithm and

Advances in Materials Science and Engineering 3

convolution approach algorithm [4] The resolution of thereconstruction image by convolution approach algorithms ismuch better than that by the Fresnel approximation algo-rithm in [10] Therefore the convolution approach algorithmis adopted in this paper and the mathematical expression ofthe convolution approach algorithm is defined as follows

119877 (1205851015840

1205781015840

)

= 119865minus1

119865 [ℎ (119909 119910)]

sdot 119865 [119894

120582( exp[ minus 119894

2120587

120582

times radic1198892 + (119909 minus 1205851015840)2

+ (119910 minus 1205781015840)2

])

times (radic1198892 + (119909 minus 1205851015840)2

+ (119910 minus 1205781015840)2

)

minus1

]

(5)

where 119865[ ] and 119865minus1

[ ] are the Fourier transform and theinverse Fourier transform respectively

In this paper we use numerical simulation hologramto check the ability of our arithmetic for determination ofthe focal plane For numerical simulation reconstruction ofholograms the Fresnel and convolution arithmetic based onthe reconstruction equations has been used well in practiceThe convolution arithmetic producing good holograms andreconstruction images and high quality of holograms doesnot change with different object distances Therefore convo-lution arithmetic will be importantly explained in this paper

In Figure 2 the left and the right part are the same whichmeans the formula for reconstruction image from a hologramis the same as that for the generation of the hologram from theobject imageThus after change of some parameters in (3) wewill obtain the new hologram function as follows

ℎ (119909 119910) =119894

120582∬

infin

minusinfin

119874 (120585 120578) 119864119877(120585 120578)

exp (minus119894 (2120587120582) 120588)

120588119889120585 119889120578

(6)

Because the form of (6) is the same as that of (3)the convolution algorithm can also be used to obtain thehologram function ℎ(119909 119910) from the object function119874(120585 120578)Then the replacement of (3) and (6) can be rewritten in theform of the superposition integral as follows

ℎ (119909 119910) = ∬

infin

minusinfin

119874 (120585 120578) 119892 (120585 120578 119909 119910) 119889120585 119889120578 (7)

with

119892 (120585 120578 119909 119910)

=119894

120582

exp [minus119894 (2120587120582)radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

]

radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

(8)

This superposition integral can be considered as a convolu-tion So (7) will become

ℎ (119909 119910)

= 119865minus1

119865 [119874 (120585 120578)]

sdot 119865 [119894

120582(exp [minus119894

2120587

120582

radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

])


+ (120578 minus 119910)2

)

minus1

]

(9)

Equation (9) is used for numerical simulation of hologramsby the convolution algorithm Based on this equation we canget the hologram function ℎ(119909 119910) equivalent to some objectimage function 119874(120585 120578) [4]

3 Theories of Surface Measurements byDigital Holography

31 General Principles As the material surfaces are unevenwhen irradiated by the laser there will appear a differentphase on the material surfaces In Section 2 after recordingand reconstruction of digital hologram the intensity andphase distribution of the reconstructed image can be obtainedby [4]

119868 (120585 120578) =1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816

2

= Re2 1003816100381610038161003816119877 (120585 120578)1003816100381610038161003816 + Im2 1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816 (10)

Φ(120585 120578) = arctanIm 1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816

Re 1003816100381610038161003816119877 (120585 120578)1003816100381610038161003816

(11)

where Re |119877(120585 120578)| and Im |119877(120585 120578)| denote the real and imag-inary parts of the object complex amplitude respectively

32 Phase Unwrapping Phase distribution value obtainedby (11) is limited in the range of (minus120587 +120587) for the theoryof the arctan function so also through phase unwrappingthe accurate phase information can be obtained [4] Phaseunwrapping is a method applied to wrapped phase imagesto remove the 2120587 incoherence embedded within the phasediagram It detects a 2120587 phase jump and adds or subtracts aninteger offset of 2120587 to adjoining pixels following that phasejump based on a tolerance mechanism thus retrieving thecontinuous form of the phase map The simulation of phaseunwrapping process is illustrated in Figures 3(a) and 3(b)

Phase unwrapping is a technique which can generate acontinuously phase distribution It constitutes essential partsof optical metrology by heterodyne techniques In surfacemeasurement by digital holographic the phase unwrappingis a key technique Numerous phase unwrapping algorithmshave been proposed in the past several years [15 16]Two types of strategy have been developed to solve the


The wrapped phase

Sample index

Wra

pped

pha

se in

radi

ans

4

3

2

1

0

minus1

minus2

minus3

minus40 100 200 300 400 500 600

(a)

The unwrapped phase

Sample index

Unw

rapp

ed p

hase

in ra

dian

s

6

4

2

0

minus2

minus4

minus60 100 200 300 400 500 600

(b)

Figure 3 Phase unwrapping (a) wrapped phase and (b) continuous phase

phase unwrapping problem path-following and minimum-norm methods Path-following methods include Goldsteinrsquosbranch cut algorithm [17] and minimum spanning tree algo-rithm [18] Minimum-norm methods contain least-squaresphase unwrapping algorithm [19] minimum Lp-norm phaseunwrapping algorithm [16] and so on

In this paper the continuous phase image is acquired byleast-squares phase unwrapping algorithmwithout weighingThe discrete cosine transforms (DCT) is used to solve thediscrete Poisson equation in this method The least-squaresolution of the unwrapped phase and the expanded phase isobtained

33 The Height Distribution of Surface As mentioned aboveafter the phase unwrapping with computer the phase distri-butionΦ(120585 120578) can be simply converted into the height distri-bution ℎ(120585 120578) on the material surface Then the continuousphase distribution can be used for measurements of materialsurface So the height of the material surface ℎ(120585 120578) can becalculated by [4]

ℎ (120585 120578) =120582

4120587Φ (120585 120578) (12)

where 120582 is the wavelengthThe surface of homogenous optical properties was mea-

sured in the light of the reconstructed phase-contrast image[19] which showed the phase changes corresponding tooptical path was less than 10 nm In this paper the target wechoose is much bigger than 10 nm

4 Experiments and Results

41 Optical Experiment Since the height of the target surfaceis uniformly distributed in this experiment a test target isused as the target surface The optical experimental setupfor recording test target (USFA 1950) holograms is shown in

CCDMirror

Beam splitterObjectBeam expander Laser source

MirrorBeam splitter

Figure 4 Experimental setup for recording test target holograms

Figure 4 The experimental conditions are listed as followsthe pixel number119873 is equal to 1024 the pixel size of the CCDcamera is 52120583m the wavelength 120582 is equal to 6328 nm Thelaser with a maximum output power of 6W is adopted

As shown in Figure 4 the input He-Ne laser is dividedinto two parts by a beam splitter (BS) one beam goes throughthe test target as the object beam and another beam isexpanded as the reference beam The object and referencebeams have an interference with BS the hologram is recordedby a CCD detector and then the image information is sent tothe computer by a collection

Figure 5 shows the recording of digital hologram bycomputer The recording distance between the object andthe CCD sensors is set to be 78mm Figure 6 shows thereconstruction intensity image of the test target Then theheight distribution ℎ(120585 120578) can be obtained from (11) and (12)Through the powerful drawing function of MATLAB theheight distribution of the material surface can be displayedin the form of 3D it was shown in Figure 7 1002 points onmaterial surfaceweremeasured in experiment and the heightdistribution information is illustrated in Figure 8 and Table 1

42 Validation Experiment The verification experimentalequipment uses NanoMap 500LS which is produced by AEP


Figure 5 Recording of digital hologram

Figure 6 Reconstruction of digital hologram

technology and the composition of the equipment is the sameas the one in Figure 9 [20] NanoMap has many features(a) seamless integration of conventional contact profilemeterand scanning probe microscope (SPM) technology (b) dualmode operation (tip scan and stage scan) optimized forsmall areas (c) 3D mapping as well as long range profilingand 3D long scan range up to 150mm times 150mm (d)stage scan by using high grade optical reference flat (e)wide vertical range with high accuracy as a result of thedual optical and up to 01 nm vertical resolution with finesensor (f) constant contact forces suitable by software andautomatic sample positioning with 119883119884 motorized stageThe NanoMap 500LS measurement flow chart is shown inFigure 10

After loading the test target onto the stage and setting upscan parameters we can get the scan curve shown in Figure 11We process the data and curve fitting using MATLAB andobtain the result shown in Figure 12 The height distributioninformation is illustrated in Figure 13 and the height dates inTable 2 From Figures 12 and 13 we can see so many morenumbers whose values are smaller than zero the reason forwhich is that the protruding part is less than the hollow onein the part we scan In Figure 13 and Table 2 the averagemeasurement result of the target surface by NanoMap 500LSis 699 nm

05

0

1

08

06

04

02

0

Hei

ght (

mm

)

1

0

02

04

06

08

1

Figure 7 The reconstructed height distribution

Poin

t num

el

Height (nm)

1000

900

800

700

600

500

400

300

200

100

072 73 74 75 76 77 78 79 80

Figure 8 Height distribution

Table 1 Data of heights

Height Min Max AverageValue (nm) 727 791 788



43 Experimental Results Tables 1 and 2 show the mea-surement results of the target surface by digital holographyand NanoMap500Ls The comparison of two methods isshown in Table 3 The relative error can be obtain as follows89699 = 127 The digital holography can serve as aneffective measurement method used in engineering


Table 3 Comparison of the measurement results by two methods

Methods Digital holography (nm) NanoMap 500Ls (nm) Difference (nm) Relative error ()Value 788 699 89 127

Figure 9 NanoMap 500LS

FlowLoad sample onto XY stage and move to the XY stage to center

Lower the Z stage to sample

Move the XY stage to locate themeasurement location

Run scan

Save and analyze data

Unload sample

Figure 10 Flow chart for taking a profile measurement

5 Conclusions

In this paper we have presented the digital holography formeasuring the material surface as an important techniqueThe principle of surface measurement by digital holographyis analyzed In this paper we use a target as a test targetBy recording and reconstruction hologram we obtain thephase and height distribution of the object surface by noisephase unwrapping and so forth Then we use NanoMap500LS 3D profilometer as validation experiment Comprised

100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100minus1500 minus1000 minus500 0 500 1000 1500 2000

60

50

40

30

20

10

y(n

m)

x (120583m)

Figure 11 The scan curve

60

40

20

0

minus20

minus40

minus60

minus1500 minus1000 minus500 0 500 1000 1500 2000

y(n

m)

x(120583m)

minus80

Figure 12 The fitting curve

of contact profilometer the results of two measurementmethods do not make much difference However digitalholography measurement method has the larger advantageDigital holographic measurement can obtain the surface ofobjects with no damage high resolution noncontact andfast processing characteristics And also any measurementsystem has measuring error Future research direction is tosolve the measurement errors which exist

Advances in Materials Science and Engineering 7N

umel

Height (nm)

250

200

150

100

50

0minus60 minus40 minus20 0 20 40 60


Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11272368) and the Natural ScienceFoundation of CSTC (cstc2013yykfB0198)

References

[1] D Gabor ldquoA new microscopic principlerdquo Nature vol 161 no4098 pp 777ndash778 1948

[2] D Gabor ldquoMicroscopy by reconstructed wavefronts 2rdquo Pro-ceedings of the Royal Society vol 64 pp 449ndash469 1951

[3] J W Goodman and R W Lawrence ldquoDigital image formationfromelectronically detected hologramsrdquoApplied Physics Lettersvol 11 no 3 pp 77ndash79 1967

[4] Y Yang B-S Kang and Y-J Choo ldquoApplication of the corre-lation coefficient method for determination of the focal planeto digital particle holographyrdquo Applied Optics vol 47 no 6 pp817ndash824 2008

[5] Y Yang and B-S Kang ldquoExperimental validation for thedetermination of particle positions by the correlation coefficientmethod in digital particle holographyrdquo Applied Optics vol 47no 32 pp 5953ndash5960 2008

[6] L Xu X Peng J Miao and A K Asundi ldquoStudies of digitalmicroscopic holography with applications to microstructuretestingrdquo Applied Optics vol 40 no 28 pp 5046ndash5051 2001

[7] G Pedrini and H J Tiziani ldquoQuantitative evaluation of two-dimensional dynamic deformations using digital holographyrdquoOptics and Laser Technology vol 29 no 5 pp 249ndash256 1997

[8] Y Yang and B Kang ldquoMeasurements of the characteristics ofspray droplets using in-line digital particle holographyrdquo Journalof Mechanical Science and Technology vol 23 no 6 pp 1670ndash1679 2009

[9] G Popescu L P Deflores J C Vaughan et al ldquoFourierphase microscopy for investigation of biological structures anddynamicsrdquo Optics Letters vol 29 no 21 pp 2503ndash2505 2004

[10] P Marquet B Rappaz P J Magistretti et al ldquoDigital holo-graphic microscopy a noninvasive contrast imaging technique

allowing quantitative visualization of living cells with subwave-length axial accuracyrdquoOptics Letters vol 30 no 5 pp 468ndash4702005

[11] A Ettemeyer ldquoApplications of digital holography tomicrostruc-turesrdquo in Proceedings of the SAE World Congress 2009 DetroitMich USA April 2009

[12] S Seebacher W Osten andW P O Juptner ldquoMeasuring shapeand deformation of small objects using digital holographyrdquo inProceedings of SPIE pp 104ndash115 July 1998

[13] B Bowe and V Toal ldquoWhite light interferometric surfaceprofilerrdquo Optical Engineering vol 37 no 6 pp 1796ndash1799 1998

[14] T Kreis Handbook of Holographic Interferometry Wiley-VCH2005

[15] RM Goldstein H A Zebker and C LWerner ldquoSatellite radarinterferometry two-dimensional phase unwrappingrdquo RadioScience vol 23 no 4 pp 713ndash720 1988

[16] D C Ghiglia and M D Pritt Two-Dimensional Phase Unwrap-ping Theory Algorithms and Software John Wiley amp SonsHoboken NJ USA 1998


[18] J Schoner A Ettemeyer U Neupert H Rottenkolber CWinter and P Obermeier ldquoNew approaches in interpretingholographic imagesrdquo Optics and Lasers in Engineering vol 14no 4-5 pp 283ndash291 1991

[19] D Kerr G H Kaufmann and G E Galizzi ldquoUnwrappingof interferometric phase-fringe maps by the discrete cosinetransformrdquo Applied Optics vol 35 no 5 pp 810ndash816 1996

[20] AEP Technology ldquoNanoMap 500LS 3D profilometer UserrsquosManualrdquo Revision B 2009

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MetallurgyJournal of


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MaterialsJournal of


Nano

materials


Journal ofNanomaterials


Reflector

Laser


Beam splitter prismTesting field

ObjectCCD

Computer

Reflector


Beam splitter prism

Figure 1 Off-line recording of digital hologram

Compared with other image processing and measuringtechniques digital holography has a number of importantadvantages [4] Firstly it does not requiremuch optical equip-ment especially changing the hologram plates Secondly itcaptures the image with noncontact nondestructive andfast height accuracy sensitivity and resolutionThirdly thereis no complex chemical processing in digital holographyso reproducing hologram and reconstructing the hologramare quicker than traditional optical holography Fourthlycomputer image processing allows to process the errors andnoise so as to revise additional optical phase for improvingthe quality of the reconstructed image

Holographic interferometry (HI) is a procedure whichenables static and dynamic displacements of objects withoptically rough surfaces to be measured with optical interfer-ometer accuracy [4 14] Then in digital holography we canaccurately measure the slight deformation of the objects bycalculating the change of the information recorded by digitalholograms [4] In this paper through digital holographyand image processing we can measure the height of thematerial surface Otherwise by recording the hologram thephase diagram can be obtained and the surface of materialcan be measured quickly and accurately making the three-dimensional visualization possible

2 General Principles of Digital Holography

21 Recording of Digital Hologram Digital holographicrecording optical path and the traditional optical holographyare consistent only replace holographic plate with a CCDcamera as the recording medium The concept of outlinedigital hologram recording is shown in Figure 1

The object wave 119874(119909 119910) and the reference wave 119877(119909 119910)

interference occur to form a hologram on the CCD surfaceIn the hologram plane the recorded intensity distribution119868119867(119909 119910) can be printed as follows [4]

119867(119909 119910) = |119874|2

+ |119877|2

+ 119877119874lowast

+ 119877lowast

119874 (1)

where119877119874lowast and119877

lowast

119874 represent the conjugation image and theoriginal image respectively

So the intensity distribution of the recorded hologram byCCD is written as

119867(119898 119899) = 119867 (119909 119910) rect( 119909

119871119909

119910

119871119910

) comb(119909

Δ119909

119910

Δ119910

) (2)

Object plane

Laser beam

Recording

Reconstruction plane

Reconstruction

y

x

d

120585998400120585120588

120578998400120578

CCD

d

120588

Figure 2 Coordinate system for image reconstruction

The light intensity is collected by the data acquisition cardand quantity The 3D information of object is saved by thecomputer and formed as a digital hologram

22 Reconstruction of Digital Hologram In traditional holog-raphy the reconstruction is carried out by means of illumi-nation of the hologram intensity with the reference wave Avirtual image and a real image of the object are reconstructedThe digital hologram simulates the traditional optical repro-duction processes and its numerical reconstruction can begained by computer

The laser beam goes through the object and the part ofthe beam diffracted by objects and reaching the recordingsurface is considered the object beam while the beam fromthe laser arriving without any distortion is considered asthe reference beam Two beams superposition create aninterference pattern on the CCD sensor This diffraction canbe described by the Fresnel-Kirchhoff integral as [4]

119877 (1205851015840

1205781015840

)

=119894

120582∬

infin

minusinfin

ℎ (119909 119910) 119864119877(119909 119910)

exp (minus119894 (2120587120582) 120588)

120588119889119909 119889119910

(3)

with

120588 = radic(1205851015840 minus 119909)2

+ (1205781015840 minus 119910)2

+ 1198892 (4)

The coordinates in (3) and (4) are illustrated in Figure 2119877(1205851015840

1205781015840

) is the wave fields of the reconstruction image andℎ(119909 119910) is the hologram function The 120588 is the distancebetween a point in the hologramplane and the correspondingpoint in the reconstruction plane 119889 is the distance betweentwo adjacent planes and 120582 is the wavelength

Different numerical reconstruction algorithms have beenproposed such as Fresnel approximation algorithm and



119877 (1205851015840

1205781015840

)

= 119865minus1

119865 [ℎ (119909 119910)]

sdot 119865 [119894

120582( exp[ minus 119894

2120587

120582


+ (119910 minus 1205781015840)2

])


+ (119910 minus 1205781015840)2

)

minus1

]

(5)





ℎ (119909 119910) =119894

120582∬

infin

minusinfin

119874 (120585 120578) 119864119877(120585 120578)

exp (minus119894 (2120587120582) 120588)

120588119889120585 119889120578

(6)


ℎ (119909 119910) = ∬

infin

minusinfin

119874 (120585 120578) 119892 (120585 120578 119909 119910) 119889120585 119889120578 (7)

with

119892 (120585 120578 119909 119910)

=119894

120582


+ (120578 minus 119910)2

]

radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

(8)


ℎ (119909 119910)

= 119865minus1

119865 [119874 (120585 120578)]

sdot 119865 [119894

120582(exp [minus119894

2120587

120582

radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

])


+ (120578 minus 119910)2

)

minus1

]

(9)




119868 (120585 120578) =1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816

2

= Re2 1003816100381610038161003816119877 (120585 120578)1003816100381610038161003816 + Im2 1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816 (10)

Φ(120585 120578) = arctanIm 1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816

Re 1003816100381610038161003816119877 (120585 120578)1003816100381610038161003816

(11)





The wrapped phase

Sample index

Wra

pped

pha

se in

radi

ans

4

3

2

1

0

minus1

minus2

minus3

minus40 100 200 300 400 500 600

(a)

The unwrapped phase

Sample index

Unw

rapp

ed p

hase

in ra

dian

s

6

4

2

0

minus2

minus4

minus60 100 200 300 400 500 600

(b)





ℎ (120585 120578) =120582

4120587Φ (120585 120578) (12)





CCDMirror


MirrorBeam splitter











05

0

1

08

06

04

02

0

Hei

ght (

mm

)

1

0

02

04

06

08

1


Poin

t num

el

Height (nm)

1000

900

800

700

600

500

400

300

200

100

072 73 74 75 76 77 78 79 80














Run scan


Unload sample


5 Conclusions


100

80

60

40

20

0

minus20

minus40

minus60

minus80


60

50

40

30

20

10

y(n

m)

x (120583m)


60

40

20

0

minus20

minus40

minus60


y(n

m)

x(120583m)

minus80




umel

Height (nm)

250

200

150

100

50



Acknowledgments


References





























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119877 (1205851015840

1205781015840

)

= 119865minus1

119865 [ℎ (119909 119910)]

sdot 119865 [119894

120582( exp[ minus 119894

2120587

120582


+ (119910 minus 1205781015840)2

])


+ (119910 minus 1205781015840)2

)

minus1

]

(5)





ℎ (119909 119910) =119894

120582∬

infin

minusinfin

119874 (120585 120578) 119864119877(120585 120578)

exp (minus119894 (2120587120582) 120588)

120588119889120585 119889120578

(6)


ℎ (119909 119910) = ∬

infin

minusinfin

119874 (120585 120578) 119892 (120585 120578 119909 119910) 119889120585 119889120578 (7)

with

119892 (120585 120578 119909 119910)

=119894

120582


+ (120578 minus 119910)2

]

radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

(8)


ℎ (119909 119910)

= 119865minus1

119865 [119874 (120585 120578)]

sdot 119865 [119894

120582(exp [minus119894

2120587

120582

radic1198892 + (120585 minus 119909)2

+ (120578 minus 119910)2

])


+ (120578 minus 119910)2

)

minus1

]

(9)




119868 (120585 120578) =1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816

2

= Re2 1003816100381610038161003816119877 (120585 120578)1003816100381610038161003816 + Im2 1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816 (10)

Φ(120585 120578) = arctanIm 1003816100381610038161003816119877 (120585 120578)

1003816100381610038161003816

Re 1003816100381610038161003816119877 (120585 120578)1003816100381610038161003816

(11)





The wrapped phase

Sample index

Wra

pped

pha

se in

radi

ans

4

3

2

1

0

minus1

minus2

minus3

minus40 100 200 300 400 500 600

(a)

The unwrapped phase

Sample index

Unw

rapp

ed p

hase

in ra

dian

s

6

4

2

0

minus2

minus4

minus60 100 200 300 400 500 600

(b)





ℎ (120585 120578) =120582

4120587Φ (120585 120578) (12)





CCDMirror


MirrorBeam splitter











05

0

1

08

06

04

02

0

Hei

ght (

mm

)

1

0

02

04

06

08

1


Poin

t num

el

Height (nm)

1000

900

800

700

600

500

400

300

200

100

072 73 74 75 76 77 78 79 80














Run scan


Unload sample


5 Conclusions


100

80

60

40

20

0

minus20

minus40

minus60

minus80


60

50

40

30

20

10

y(n

m)

x (120583m)


60

40

20

0

minus20

minus40

minus60


y(n

m)

x(120583m)

minus80




umel

Height (nm)

250

200

150

100

50



Acknowledgments


References





























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




The wrapped phase

Sample index

Wra

pped

pha

se in

radi

ans

4

3

2

1

0

minus1

minus2

minus3

minus40 100 200 300 400 500 600

(a)

The unwrapped phase

Sample index

Unw

rapp

ed p

hase

in ra

dian

s

6

4

2

0

minus2

minus4

minus60 100 200 300 400 500 600

(b)





ℎ (120585 120578) =120582

4120587Φ (120585 120578) (12)





CCDMirror


MirrorBeam splitter











05

0

1

08

06

04

02

0

Hei

ght (

mm

)

1

0

02

04

06

08

1


Poin

t num

el

Height (nm)

1000

900

800

700

600

500

400

300

200

100

072 73 74 75 76 77 78 79 80














Run scan


Unload sample


5 Conclusions


100

80

60

40

20

0

minus20

minus40

minus60

minus80


60

50

40

30

20

10

y(n

m)

x (120583m)


60

40

20

0

minus20

minus40

minus60


y(n

m)

x(120583m)

minus80




umel

Height (nm)

250

200

150

100

50



Acknowledgments


References





























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials








05

0

1

08

06

04

02

0

Hei

ght (

mm

)

1

0

02

04

06

08

1


Poin

t num

el

Height (nm)

1000

900

800

700

600

500

400

300

200

100

072 73 74 75 76 77 78 79 80














Run scan


Unload sample


5 Conclusions


100

80

60

40

20

0

minus20

minus40

minus60

minus80


60

50

40

30

20

10

y(n

m)

x (120583m)


60

40

20

0

minus20

minus40

minus60


y(n

m)

x(120583m)

minus80




umel

Height (nm)

250

200

150

100

50



Acknowledgments


References





























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










Run scan


Unload sample


5 Conclusions


100

80

60

40

20

0

minus20

minus40

minus60

minus80


60

50

40

30

20

10

y(n

m)

x (120583m)


60

40

20

0

minus20

minus40

minus60


y(n

m)

x(120583m)

minus80




umel

Height (nm)

250

200

150

100

50



Acknowledgments


References





























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




umel

Height (nm)

250

200

150

100

50



Acknowledgments


References





























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



research article measurements of the characteristics of...

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