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Research Article Analysis of the Interference Modulation Depth in the Fourier Transform Spectrometer Rilong Liu School of Engineering, Jiangxi Agricultural University, Nanchang 330045, China Correspondence should be addressed to Rilong Liu; [email protected] Received 21 August 2015; Revised 14 September 2015; Accepted 20 September 2015 Academic Editor: Michele Norgia Copyright © 2015 Rilong Liu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on the principle of the Michelson interferometer, the paper briefly describes the theoretical significance and calculates and deduces three expressions of the interference modulation depth. e influence of the surface shape error of plane mirror on modulation depth is analyzed, and the tolerance of error is also pointed out. Moreover, the dependence of modulation depth on the reflectance change of beam splitter interface is also analyzed, and the curve is given. It is concluded that this paper is of general significance for the Fourier transform spectrometer based on the principle of the Michelson two-beam interference. 1. Introduction e Fourier transform spectrometer is an instrument which is used to analyze the spectral distribution of the light source by means of the interference effect based on the Fourier transform technology. It has incomparably more advan- tages than conventional spectral instruments in acquiring higher spectral resolution. ere are many kinds of Fourier transform spectrometers, including Michelson interferome- ter, Fabry-Perot interferometer, Sagnac interferometer, and double refraction interferometer [1, 2]. e classical Michelson interferometer consists of a mov- ing mirror and a fixed mirror. e two beams of lights, reflected by these two mirrors, make up the phase difference. When the moving mirror moves, the intensity of interference signals is modulated to form the interferogram. e mod- ulation depth is also known as the interference modulation efficiency. It is one of the most important factors influencing the signal-to-noise ratio of the interferogram, and a change in it will significantly affect the sensitivity of the spectrometer [3, 4]. Based on the Michelson type Fourier transform spectrom- eter, this paper explores the optical principle of interference system through a quantitative analysis of influencing factors of surface shape of mirror and performance change of beam splitter. It is hoped that suggestions can be made for the obtaining of ideal interferogram. 2. The Modulation Depth of Fourier Transform Spectrometer e most classical Fourier transform spectrometer is the Michelson interferometer, which is composed of a fixed mirror, a moving mirror, a beam splitter, and a detector. Its basic structure is shown in Figure 1. In the Michelson interferometer, the fixed mirror and the moving mirror are perpendicular to each other, with an angle of 45 between the beam splitter and two mirrors. e collimated beam shines the interference system, and the beam splitter separates the incident light into two beams with roughly the same reflectance and transmittance. One reflected beam by the fixed mirror then goes through the beam splitter again, whereas the other transmitted beam is reflected by the moving mirror and divided by the beam splitter. Because these two beams of light are produced by the same beam, they have a constant phase difference. When these two beams of light travel through a convergent lens, the interference is formed on the detector. Upon arrival at the detector, the two coherent beams have an optical path difference for the shiſt of the moving mirror. e optical path difference then shows periodic changes with round-trip translation of the moving mirror. According to the principle of interference spectroscopy, the interference signals can be obtained on the detector. Hindawi Publishing Corporation Advances in OptoElectronics Volume 2015, Article ID 472852, 4 pages http://dx.doi.org/10.1155/2015/472852
5

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Page 1: Research Article Analysis of the Interference Modulation ...downloads.hindawi.com/archive/2015/472852.pdf · Hence, the interference modulation depth is 1 = exp ... 1 Amplitude F

Research ArticleAnalysis of the Interference Modulation Depth in the FourierTransform Spectrometer

Rilong Liu

School of Engineering Jiangxi Agricultural University Nanchang 330045 China

Correspondence should be addressed to Rilong Liu ckjx312aliyuncom

Received 21 August 2015 Revised 14 September 2015 Accepted 20 September 2015

Academic Editor Michele Norgia

Copyright copy 2015 Rilong Liu This 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

Based on the principle of the Michelson interferometer the paper briefly describes the theoretical significance and calculatesand deduces three expressions of the interference modulation depth The influence of the surface shape error of plane mirroron modulation depth is analyzed and the tolerance of error is also pointed out Moreover the dependence of modulation depth onthe reflectance change of beam splitter interface is also analyzed and the curve is given It is concluded that this paper is of generalsignificance for the Fourier transform spectrometer based on the principle of the Michelson two-beam interference

1 Introduction

The Fourier transform spectrometer is an instrument whichis used to analyze the spectral distribution of the light sourceby means of the interference effect based on the Fouriertransform technology It has incomparably more advan-tages than conventional spectral instruments in acquiringhigher spectral resolution There are many kinds of Fouriertransform spectrometers including Michelson interferome-ter Fabry-Perot interferometer Sagnac interferometer anddouble refraction interferometer [1 2]

The classical Michelson interferometer consists of a mov-ing mirror and a fixed mirror The two beams of lightsreflected by these two mirrors make up the phase differenceWhen the moving mirror moves the intensity of interferencesignals is modulated to form the interferogram The mod-ulation depth is also known as the interference modulationefficiency It is one of the most important factors influencingthe signal-to-noise ratio of the interferogram and a change init will significantly affect the sensitivity of the spectrometer [34] Based on theMichelson type Fourier transform spectrom-eter this paper explores the optical principle of interferencesystem through a quantitative analysis of influencing factorsof surface shape of mirror and performance change of beamsplitter It is hoped that suggestions can be made for theobtaining of ideal interferogram

2 The Modulation Depth of FourierTransform Spectrometer

The most classical Fourier transform spectrometer is theMichelson interferometer which is composed of a fixedmirror a moving mirror a beam splitter and a detector Itsbasic structure is shown in Figure 1

In the Michelson interferometer the fixed mirror andthe moving mirror are perpendicular to each other withan angle of 45∘ between the beam splitter and two mirrorsThe collimated beam shines the interference system and thebeam splitter separates the incident light into two beamswith roughly the same reflectance and transmittance Onereflected beam by the fixed mirror then goes through thebeam splitter again whereas the other transmitted beam isreflected by the moving mirror and divided by the beamsplitter Because these two beams of light are produced bythe same beam they have a constant phase difference Whenthese two beams of light travel through a convergent lens theinterference is formed on the detector

Upon arrival at the detector the two coherent beams havean optical path difference for the shift of the moving mirrorThe optical path difference then shows periodic changes withround-trip translation of the moving mirror According tothe principle of interference spectroscopy the interferencesignals can be obtained on the detector

Hindawi Publishing CorporationAdvances in OptoElectronicsVolume 2015 Article ID 472852 4 pageshttpdxdoiorg1011552015472852

2 Advances in OptoElectronics

Fixed mirror

Moving mirror

Converging lens

Detector

Beam splitter

Figure 1 Basic structure of Michelson interferometer

According to the interference theory the intensity ofinterference signals generated by the two-beam interferenceis [5]

119868 = 1198681+ 1198682+ 2radic119868

11198682cos 120575

= (1198681+ 1198682) (1 +

2radic11986811198682

1198681+ 1198682

cos 120575)

(1)

where 1198681and 1198682are respectively the intensity of two coherent

beams and 120575 is the phase difference of two coherent beamsThe definition of the modulation depth of interference

signals namely the visibility of interference signals119872 is

119872 =119868max minus 119868min119868max + 119868min

(2)

This equation indicates the degree of bright dark contrast offringes in the interference field where 119868max and 119868min standfor the maximum intensity and the minimum intensity ofinterference signals We then get the following equation

119872 =119868max minus 119868min119868max + 119868min

=2radic11986811198682

1198681+ 1198682

(3)

Then (1) can be rewritten as

119868 = (1198681+ 1198682) (1 + 119872 cos 120575) (4)

From (3) we can have the coefficient of the cosinenamely the modulation depth after we obtain the distribu-tion of the cosine light intensity and normalize the constantterm If and only if 119868

1= 1198682

= 1198680 that is to say when

the intensity of two coherent beams of light is equal themodulation depth reaches a maximum and also an ideal119872 = 1 However with practical applications taken intoconsideration the modulation depth 119872 will usually be lessthan 1

3 The Surface Shape Error of Plane Mirror

Thefixedmirror and themovingmirror of Fourier transformspectrometer are composed of plane mirrors with a rectan-gular aperture In the process of making plane mirrors thereare a number of factors modulating the surface error such aspolishing coating and mirror bending It follows that thesemirrors are not ideal surface shape and there is always acertain surface error Therefore in a spectrometer composedof these mirrors the modulation depth is often not ideal Inwhat follows we will analyze the influence of curve error andpolishing error of plane mirror on interference modulationdepth Suppose polishing error of mirror obeys normaldistribution with zero mean and its maximum fluctuation is120576max In addition curve error of mirror is subject to uniformdistribution and its maximum curvature is 120589max Assumealso that other optical components in Fourier transformspectrometer are ideal

When there is a surface shape error of plane mirrorthe reflection of two beams of coherent light will generatewavefront distortion which results in a change in optical pathdifference of interferometer The phase difference 120575 of thetwo beams of coherent light changes accordingly For a smallchange 120585 of the surface shape error the change of the phasedifference is

Δ120575 = 2120587]120576 (5)

where ] is thewave number of the incident light Suppose119891(120576)

is the probability density function of surface shape error If120576 isin (120576 Δ120576) then we obtain surface 119889119878 = 119891(120576)119889120576 Because ofthe effect of surface shape error the intensity of interferencesignals is [6]

119868 =

int119878

1198681015840

(120575) 119889119878

int119878

119889119878

(6)

where 1198681015840

(120575) represents the distribution of interference inten-sity in any one point The probability density function is

119891 (120576) =1

radic2120587120590

expminus1205762

21205902 (7)

where 120590 is the standard deviation of 120576When 119889119878 = 119891(120576)119889120576 is substituted to (6) we can get the

intensity of interference signals

119868

=

int+infin

minusinfin

21198680(1 + cos (120575 + 2120587]120576)) (1radic2120587120590) exp (minus120576

2

21205902

) 119889120576

int+infin

minusinfin

(1radic2120587120590) exp (minus120576221205902) 119889120576

= 21198680[1 + exp(minus

(2120587]120590)2

2) cos 120575]

(8)

Hence the interference modulation depth is

1198721= expminus

(2120587]120590)2

2 (9)

Advances in OptoElectronics 3

It can be seen that the interference modulation depthis inversely proportional to the variance of polishing errorof mirror which means that a good polishing surface canimprove the modulation depth If the interference modula-tion depth is set at 119872

1ge 09 the error tolerance of standard

deviation 120590 is

120590 le120582

14 (10)

In the same way when the mirror surface is curved theexistence of the curve error causes the intensity distributionof the interference signals to be

119868

=

int+120589max

minus120589max21198680(1 + cos (120575 + 2120587]120589)) (1 (120589max minus (minus120589max))) 119889120589

int+120589max

minus120589max(1 (120589max minus (minus120589max))) 119889120589

= 21198680[1 +

sin (2120587]120589max)

2120587]120589maxcos 120575]

(11)

It follows that when the surface of mirror is deformed theinterference modulation depth is

1198722=sin (2120587]120589max)

2120587]120589max (12)

Equation (12) shows that 1198722is a function of sinc and

the modulation depth 1198722decreases when the curvature 120589

increases If the interference modulation depth is set at1198722ge

09 the error tolerance of the curvature 120589 is

120589 le120582

8 (13)

Through the above analysis we know that the surfaceerror of mirror has an effect on the modulation depth Inorder to achieve better interference effect the surface errorof mirror should meet error tolerance requirements

4 The Reflectance Change ofBeam Splitter Interface

In the Michelson interferometer shown in Figure 1 the sepa-rating surface of the beam splitter is coated with transflectivefilm Suppose 119903

119894and 119905

119894represent the amplitude reflection

coefficient and the amplitude transmission coefficient ofbeam from the surrounding medium into surface 119894 of thebeam splitter respectivelyMeanwhile 1199031015840

119894

and 1199051015840

119894

are the ampli-tude reflection coefficient and the amplitude transmissioncoefficient of beam from surface 119894 of the beam splitter intothe surrounding medium respectively At the same time 119903

119891

is the amplitude reflection coefficient of fixedmirror and 119903119898is

the amplitude reflection coefficient of moving mirror and 119860

stands for the amplitude of incident light [7 8] The complexamplitude of the two beams of light reflected by Michelsoninterferometer respectively is

1198641= 11990511199031015840

2

1199051015840

1

11990311989111990511199051015840

2

119860 exp (1198952120587]1205941)

1198642= 11990511199051015840

2

1199031198981199032119860 exp (1198952120587]120594

2)

(14)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

120588

M3

Figure 2 Influence of reflectance change of beam splitter onmodulation depth

where ] is the wave number of the incident light while 1205941and

1205942represent optical path of two beams of light respectively

Generally speaking if the amplitude reflection coefficient ofthe fixedmirror equals that of themovingmirror that is 119903

119891=

119903119898 then we get

1198641

1198642

=1199031015840

2

11990511199051015840

1

1199032

exp (1198942120587] (1205941minus 1205942)) (15)

From (15) the modulation depth can be expressed as

1198723=

2100381610038161003816100381611986411198642

1003816100381610038161003816

1 +100381610038161003816100381611986411198642

1003816100381610038161003816

2

=

210038161003816100381610038161003816(1199031015840

2

11990511199051015840

1

) (1199032)10038161003816100381610038161003816

1 +1003816100381610038161003816(1199031015840

2

11990511199051015840

1

) (1199032)1003816100381610038161003816

2

(16)

According to the Fresnel formula the amplitude reflec-tion coefficient and the amplitude transmission coefficientsatisfy the following conditions

1199032= minus1199031015840

2

11990511199051015840

1

= 1 minus 1199032

1

(17)

And the interface reflectance of the beam splitter is

120588 = 1199032

1

(18)

Then substituting (18) into (17) we can get

11990511199051015840

1

= 1 minus 120588 (19)

Thus the interference modulation depth is

1198723=

2 minus 2120588

2 minus 2120588 + 1205882 (20)

From (20) it can be seen that the interference modulationdepth 119872

3will change with the interface reflectance 120588 of the

beam splitter Their relationship is captured in Figure 2From Figure 2 the smaller the interface reflectance 120588 of

the beam splitter is the greater the interference modulation

4 Advances in OptoElectronics

Figure 3 The surface shape of plane mirror

0 5 10 15 20 25 30

Time (120583s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de

Figure 4 The electrical signal of the laser interference

depth1198723becomes To put it another way if 120588 value is 0 then

1198723

= 1 When the interference modulation depth is set at1198723ge 09 120588 is

120588 le 0372 (21)

5 Experimental Results

The analysis of interference modulation depth indicates thatin order to realize the interference modulation depth 119872 ge

09 the maximum value of the curve error of mirror 120589

should be 120589max le 1205828 Figure 3 presents measured resultsof the surface of the plane mirror with the aid of Zygointerferometer As can be seen fromFigure 3 the planemirrorhas a good surface with the PV value 1205828

The interferometer is placed on the testbed and theinterference is formed by using theHe-Ne laserThe electricalsignal of the laser interference can then be acquired Figure 4provides the interference signal generated by the laser lightsource

As can be seen from Figure 4 the laser interference signalis a sinusoidal signal and the amplitude of the interferencesignal is stable and the fringe is clearly visible which meetsthe requirement of the interference modulation depth

6 Conclusions

In the Fourier transform spectrometer the Michelson inter-ferometer is the core optical system By focusing on the classicMichelson interferometer we have conducted a theoreticalanalysis of the influence of surface shape of mirror andperformance change of the beam splitter on the modulationdepth Theoretical deductions are obtained regarding toler-ance requirements for the error of surface shape mirror andthe reflection ratio of the beam splitter The results obtainedare applicable to the Fourier transform spectrometer withinthe framework of the Michelson dual beam interference

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The research was supported by the Program for Science andTechnology funded by the Education Department of JiangxiProvince (GJJ14316)

References

[1] B Saggin D Scaccabarozzi and M Tarabini ldquoInstrumentalphase-based method for Fourier transform spectrometer mea-surements processingrdquo Applied Optics vol 50 no 12 pp 1717ndash1725 2011

[2] Y Ferrec J Taboury H Sauer and P Chavel ldquoOptimal geom-etry for Sagnac and Michelson interferometers used as spectralimagersrdquo Optical Engineering vol 45 no 11 Article ID 1156012006

[3] B J Meers and K A Strain ldquoModulation signal and quantumnoise in interferometersrdquo Physical Review A vol 44 no 7 pp4693ndash4703 1991

[4] H Liang and P-G Wang ldquoEffects of low temperature on mod-ulation efficiency of michelson interferometerrdquo Acta PhotonicaSinica vol 38 no 4 pp 967ndash970 2009

[5] Z-C Bu C-M Zhang B-C Zhao and H-C Zhu ldquoAnalysisand calculation of the modulation depth of the Michelsoninterferometer with wide field chromatic compensation andthermal compensationrdquo Acta Physica Sinica vol 58 no 4 pp2415ndash2422 2009

[6] P K Katti and K Singh ldquoA note on the surface accuracy andalignment of the end mirrors in a michelson interferometerrdquoApplied Optics vol 5 no 12 pp 1962ndash1964 1966

[7] L Shi K Li Z-F Gao L-B Zeng and Q-S Wu ldquoDesign ofa compact structure interferometerrdquo Spectroscopy and SpectralAnalysis vol 33 no 8 pp 2294ndash2298 2013

[8] D Y Yu andH Y Tan Engineering Optics Mechanical IndustryPress Beijing China 2006

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Page 2: Research Article Analysis of the Interference Modulation ...downloads.hindawi.com/archive/2015/472852.pdf · Hence, the interference modulation depth is 1 = exp ... 1 Amplitude F

2 Advances in OptoElectronics

Fixed mirror

Moving mirror

Converging lens

Detector

Beam splitter

Figure 1 Basic structure of Michelson interferometer

According to the interference theory the intensity ofinterference signals generated by the two-beam interferenceis [5]

119868 = 1198681+ 1198682+ 2radic119868

11198682cos 120575

= (1198681+ 1198682) (1 +

2radic11986811198682

1198681+ 1198682

cos 120575)

(1)

where 1198681and 1198682are respectively the intensity of two coherent

beams and 120575 is the phase difference of two coherent beamsThe definition of the modulation depth of interference

signals namely the visibility of interference signals119872 is

119872 =119868max minus 119868min119868max + 119868min

(2)

This equation indicates the degree of bright dark contrast offringes in the interference field where 119868max and 119868min standfor the maximum intensity and the minimum intensity ofinterference signals We then get the following equation

119872 =119868max minus 119868min119868max + 119868min

=2radic11986811198682

1198681+ 1198682

(3)

Then (1) can be rewritten as

119868 = (1198681+ 1198682) (1 + 119872 cos 120575) (4)

From (3) we can have the coefficient of the cosinenamely the modulation depth after we obtain the distribu-tion of the cosine light intensity and normalize the constantterm If and only if 119868

1= 1198682

= 1198680 that is to say when

the intensity of two coherent beams of light is equal themodulation depth reaches a maximum and also an ideal119872 = 1 However with practical applications taken intoconsideration the modulation depth 119872 will usually be lessthan 1

3 The Surface Shape Error of Plane Mirror

Thefixedmirror and themovingmirror of Fourier transformspectrometer are composed of plane mirrors with a rectan-gular aperture In the process of making plane mirrors thereare a number of factors modulating the surface error such aspolishing coating and mirror bending It follows that thesemirrors are not ideal surface shape and there is always acertain surface error Therefore in a spectrometer composedof these mirrors the modulation depth is often not ideal Inwhat follows we will analyze the influence of curve error andpolishing error of plane mirror on interference modulationdepth Suppose polishing error of mirror obeys normaldistribution with zero mean and its maximum fluctuation is120576max In addition curve error of mirror is subject to uniformdistribution and its maximum curvature is 120589max Assumealso that other optical components in Fourier transformspectrometer are ideal

When there is a surface shape error of plane mirrorthe reflection of two beams of coherent light will generatewavefront distortion which results in a change in optical pathdifference of interferometer The phase difference 120575 of thetwo beams of coherent light changes accordingly For a smallchange 120585 of the surface shape error the change of the phasedifference is

Δ120575 = 2120587]120576 (5)

where ] is thewave number of the incident light Suppose119891(120576)

is the probability density function of surface shape error If120576 isin (120576 Δ120576) then we obtain surface 119889119878 = 119891(120576)119889120576 Because ofthe effect of surface shape error the intensity of interferencesignals is [6]

119868 =

int119878

1198681015840

(120575) 119889119878

int119878

119889119878

(6)

where 1198681015840

(120575) represents the distribution of interference inten-sity in any one point The probability density function is

119891 (120576) =1

radic2120587120590

expminus1205762

21205902 (7)

where 120590 is the standard deviation of 120576When 119889119878 = 119891(120576)119889120576 is substituted to (6) we can get the

intensity of interference signals

119868

=

int+infin

minusinfin

21198680(1 + cos (120575 + 2120587]120576)) (1radic2120587120590) exp (minus120576

2

21205902

) 119889120576

int+infin

minusinfin

(1radic2120587120590) exp (minus120576221205902) 119889120576

= 21198680[1 + exp(minus

(2120587]120590)2

2) cos 120575]

(8)

Hence the interference modulation depth is

1198721= expminus

(2120587]120590)2

2 (9)

Advances in OptoElectronics 3

It can be seen that the interference modulation depthis inversely proportional to the variance of polishing errorof mirror which means that a good polishing surface canimprove the modulation depth If the interference modula-tion depth is set at 119872

1ge 09 the error tolerance of standard

deviation 120590 is

120590 le120582

14 (10)

In the same way when the mirror surface is curved theexistence of the curve error causes the intensity distributionof the interference signals to be

119868

=

int+120589max

minus120589max21198680(1 + cos (120575 + 2120587]120589)) (1 (120589max minus (minus120589max))) 119889120589

int+120589max

minus120589max(1 (120589max minus (minus120589max))) 119889120589

= 21198680[1 +

sin (2120587]120589max)

2120587]120589maxcos 120575]

(11)

It follows that when the surface of mirror is deformed theinterference modulation depth is

1198722=sin (2120587]120589max)

2120587]120589max (12)

Equation (12) shows that 1198722is a function of sinc and

the modulation depth 1198722decreases when the curvature 120589

increases If the interference modulation depth is set at1198722ge

09 the error tolerance of the curvature 120589 is

120589 le120582

8 (13)

Through the above analysis we know that the surfaceerror of mirror has an effect on the modulation depth Inorder to achieve better interference effect the surface errorof mirror should meet error tolerance requirements

4 The Reflectance Change ofBeam Splitter Interface

In the Michelson interferometer shown in Figure 1 the sepa-rating surface of the beam splitter is coated with transflectivefilm Suppose 119903

119894and 119905

119894represent the amplitude reflection

coefficient and the amplitude transmission coefficient ofbeam from the surrounding medium into surface 119894 of thebeam splitter respectivelyMeanwhile 1199031015840

119894

and 1199051015840

119894

are the ampli-tude reflection coefficient and the amplitude transmissioncoefficient of beam from surface 119894 of the beam splitter intothe surrounding medium respectively At the same time 119903

119891

is the amplitude reflection coefficient of fixedmirror and 119903119898is

the amplitude reflection coefficient of moving mirror and 119860

stands for the amplitude of incident light [7 8] The complexamplitude of the two beams of light reflected by Michelsoninterferometer respectively is

1198641= 11990511199031015840

2

1199051015840

1

11990311989111990511199051015840

2

119860 exp (1198952120587]1205941)

1198642= 11990511199051015840

2

1199031198981199032119860 exp (1198952120587]120594

2)

(14)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

120588

M3

Figure 2 Influence of reflectance change of beam splitter onmodulation depth

where ] is the wave number of the incident light while 1205941and

1205942represent optical path of two beams of light respectively

Generally speaking if the amplitude reflection coefficient ofthe fixedmirror equals that of themovingmirror that is 119903

119891=

119903119898 then we get

1198641

1198642

=1199031015840

2

11990511199051015840

1

1199032

exp (1198942120587] (1205941minus 1205942)) (15)

From (15) the modulation depth can be expressed as

1198723=

2100381610038161003816100381611986411198642

1003816100381610038161003816

1 +100381610038161003816100381611986411198642

1003816100381610038161003816

2

=

210038161003816100381610038161003816(1199031015840

2

11990511199051015840

1

) (1199032)10038161003816100381610038161003816

1 +1003816100381610038161003816(1199031015840

2

11990511199051015840

1

) (1199032)1003816100381610038161003816

2

(16)

According to the Fresnel formula the amplitude reflec-tion coefficient and the amplitude transmission coefficientsatisfy the following conditions

1199032= minus1199031015840

2

11990511199051015840

1

= 1 minus 1199032

1

(17)

And the interface reflectance of the beam splitter is

120588 = 1199032

1

(18)

Then substituting (18) into (17) we can get

11990511199051015840

1

= 1 minus 120588 (19)

Thus the interference modulation depth is

1198723=

2 minus 2120588

2 minus 2120588 + 1205882 (20)

From (20) it can be seen that the interference modulationdepth 119872

3will change with the interface reflectance 120588 of the

beam splitter Their relationship is captured in Figure 2From Figure 2 the smaller the interface reflectance 120588 of

the beam splitter is the greater the interference modulation

4 Advances in OptoElectronics

Figure 3 The surface shape of plane mirror

0 5 10 15 20 25 30

Time (120583s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de

Figure 4 The electrical signal of the laser interference

depth1198723becomes To put it another way if 120588 value is 0 then

1198723

= 1 When the interference modulation depth is set at1198723ge 09 120588 is

120588 le 0372 (21)

5 Experimental Results

The analysis of interference modulation depth indicates thatin order to realize the interference modulation depth 119872 ge

09 the maximum value of the curve error of mirror 120589

should be 120589max le 1205828 Figure 3 presents measured resultsof the surface of the plane mirror with the aid of Zygointerferometer As can be seen fromFigure 3 the planemirrorhas a good surface with the PV value 1205828

The interferometer is placed on the testbed and theinterference is formed by using theHe-Ne laserThe electricalsignal of the laser interference can then be acquired Figure 4provides the interference signal generated by the laser lightsource

As can be seen from Figure 4 the laser interference signalis a sinusoidal signal and the amplitude of the interferencesignal is stable and the fringe is clearly visible which meetsthe requirement of the interference modulation depth

6 Conclusions

In the Fourier transform spectrometer the Michelson inter-ferometer is the core optical system By focusing on the classicMichelson interferometer we have conducted a theoreticalanalysis of the influence of surface shape of mirror andperformance change of the beam splitter on the modulationdepth Theoretical deductions are obtained regarding toler-ance requirements for the error of surface shape mirror andthe reflection ratio of the beam splitter The results obtainedare applicable to the Fourier transform spectrometer withinthe framework of the Michelson dual beam interference

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The research was supported by the Program for Science andTechnology funded by the Education Department of JiangxiProvince (GJJ14316)

References

[1] B Saggin D Scaccabarozzi and M Tarabini ldquoInstrumentalphase-based method for Fourier transform spectrometer mea-surements processingrdquo Applied Optics vol 50 no 12 pp 1717ndash1725 2011

[2] Y Ferrec J Taboury H Sauer and P Chavel ldquoOptimal geom-etry for Sagnac and Michelson interferometers used as spectralimagersrdquo Optical Engineering vol 45 no 11 Article ID 1156012006

[3] B J Meers and K A Strain ldquoModulation signal and quantumnoise in interferometersrdquo Physical Review A vol 44 no 7 pp4693ndash4703 1991

[4] H Liang and P-G Wang ldquoEffects of low temperature on mod-ulation efficiency of michelson interferometerrdquo Acta PhotonicaSinica vol 38 no 4 pp 967ndash970 2009

[5] Z-C Bu C-M Zhang B-C Zhao and H-C Zhu ldquoAnalysisand calculation of the modulation depth of the Michelsoninterferometer with wide field chromatic compensation andthermal compensationrdquo Acta Physica Sinica vol 58 no 4 pp2415ndash2422 2009

[6] P K Katti and K Singh ldquoA note on the surface accuracy andalignment of the end mirrors in a michelson interferometerrdquoApplied Optics vol 5 no 12 pp 1962ndash1964 1966

[7] L Shi K Li Z-F Gao L-B Zeng and Q-S Wu ldquoDesign ofa compact structure interferometerrdquo Spectroscopy and SpectralAnalysis vol 33 no 8 pp 2294ndash2298 2013

[8] D Y Yu andH Y Tan Engineering Optics Mechanical IndustryPress Beijing China 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 3: Research Article Analysis of the Interference Modulation ...downloads.hindawi.com/archive/2015/472852.pdf · Hence, the interference modulation depth is 1 = exp ... 1 Amplitude F

Advances in OptoElectronics 3

It can be seen that the interference modulation depthis inversely proportional to the variance of polishing errorof mirror which means that a good polishing surface canimprove the modulation depth If the interference modula-tion depth is set at 119872

1ge 09 the error tolerance of standard

deviation 120590 is

120590 le120582

14 (10)

In the same way when the mirror surface is curved theexistence of the curve error causes the intensity distributionof the interference signals to be

119868

=

int+120589max

minus120589max21198680(1 + cos (120575 + 2120587]120589)) (1 (120589max minus (minus120589max))) 119889120589

int+120589max

minus120589max(1 (120589max minus (minus120589max))) 119889120589

= 21198680[1 +

sin (2120587]120589max)

2120587]120589maxcos 120575]

(11)

It follows that when the surface of mirror is deformed theinterference modulation depth is

1198722=sin (2120587]120589max)

2120587]120589max (12)

Equation (12) shows that 1198722is a function of sinc and

the modulation depth 1198722decreases when the curvature 120589

increases If the interference modulation depth is set at1198722ge

09 the error tolerance of the curvature 120589 is

120589 le120582

8 (13)

Through the above analysis we know that the surfaceerror of mirror has an effect on the modulation depth Inorder to achieve better interference effect the surface errorof mirror should meet error tolerance requirements

4 The Reflectance Change ofBeam Splitter Interface

In the Michelson interferometer shown in Figure 1 the sepa-rating surface of the beam splitter is coated with transflectivefilm Suppose 119903

119894and 119905

119894represent the amplitude reflection

coefficient and the amplitude transmission coefficient ofbeam from the surrounding medium into surface 119894 of thebeam splitter respectivelyMeanwhile 1199031015840

119894

and 1199051015840

119894

are the ampli-tude reflection coefficient and the amplitude transmissioncoefficient of beam from surface 119894 of the beam splitter intothe surrounding medium respectively At the same time 119903

119891

is the amplitude reflection coefficient of fixedmirror and 119903119898is

the amplitude reflection coefficient of moving mirror and 119860

stands for the amplitude of incident light [7 8] The complexamplitude of the two beams of light reflected by Michelsoninterferometer respectively is

1198641= 11990511199031015840

2

1199051015840

1

11990311989111990511199051015840

2

119860 exp (1198952120587]1205941)

1198642= 11990511199051015840

2

1199031198981199032119860 exp (1198952120587]120594

2)

(14)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

120588

M3

Figure 2 Influence of reflectance change of beam splitter onmodulation depth

where ] is the wave number of the incident light while 1205941and

1205942represent optical path of two beams of light respectively

Generally speaking if the amplitude reflection coefficient ofthe fixedmirror equals that of themovingmirror that is 119903

119891=

119903119898 then we get

1198641

1198642

=1199031015840

2

11990511199051015840

1

1199032

exp (1198942120587] (1205941minus 1205942)) (15)

From (15) the modulation depth can be expressed as

1198723=

2100381610038161003816100381611986411198642

1003816100381610038161003816

1 +100381610038161003816100381611986411198642

1003816100381610038161003816

2

=

210038161003816100381610038161003816(1199031015840

2

11990511199051015840

1

) (1199032)10038161003816100381610038161003816

1 +1003816100381610038161003816(1199031015840

2

11990511199051015840

1

) (1199032)1003816100381610038161003816

2

(16)

According to the Fresnel formula the amplitude reflec-tion coefficient and the amplitude transmission coefficientsatisfy the following conditions

1199032= minus1199031015840

2

11990511199051015840

1

= 1 minus 1199032

1

(17)

And the interface reflectance of the beam splitter is

120588 = 1199032

1

(18)

Then substituting (18) into (17) we can get

11990511199051015840

1

= 1 minus 120588 (19)

Thus the interference modulation depth is

1198723=

2 minus 2120588

2 minus 2120588 + 1205882 (20)

From (20) it can be seen that the interference modulationdepth 119872

3will change with the interface reflectance 120588 of the

beam splitter Their relationship is captured in Figure 2From Figure 2 the smaller the interface reflectance 120588 of

the beam splitter is the greater the interference modulation

4 Advances in OptoElectronics

Figure 3 The surface shape of plane mirror

0 5 10 15 20 25 30

Time (120583s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de

Figure 4 The electrical signal of the laser interference

depth1198723becomes To put it another way if 120588 value is 0 then

1198723

= 1 When the interference modulation depth is set at1198723ge 09 120588 is

120588 le 0372 (21)

5 Experimental Results

The analysis of interference modulation depth indicates thatin order to realize the interference modulation depth 119872 ge

09 the maximum value of the curve error of mirror 120589

should be 120589max le 1205828 Figure 3 presents measured resultsof the surface of the plane mirror with the aid of Zygointerferometer As can be seen fromFigure 3 the planemirrorhas a good surface with the PV value 1205828

The interferometer is placed on the testbed and theinterference is formed by using theHe-Ne laserThe electricalsignal of the laser interference can then be acquired Figure 4provides the interference signal generated by the laser lightsource

As can be seen from Figure 4 the laser interference signalis a sinusoidal signal and the amplitude of the interferencesignal is stable and the fringe is clearly visible which meetsthe requirement of the interference modulation depth

6 Conclusions

In the Fourier transform spectrometer the Michelson inter-ferometer is the core optical system By focusing on the classicMichelson interferometer we have conducted a theoreticalanalysis of the influence of surface shape of mirror andperformance change of the beam splitter on the modulationdepth Theoretical deductions are obtained regarding toler-ance requirements for the error of surface shape mirror andthe reflection ratio of the beam splitter The results obtainedare applicable to the Fourier transform spectrometer withinthe framework of the Michelson dual beam interference

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The research was supported by the Program for Science andTechnology funded by the Education Department of JiangxiProvince (GJJ14316)

References

[1] B Saggin D Scaccabarozzi and M Tarabini ldquoInstrumentalphase-based method for Fourier transform spectrometer mea-surements processingrdquo Applied Optics vol 50 no 12 pp 1717ndash1725 2011

[2] Y Ferrec J Taboury H Sauer and P Chavel ldquoOptimal geom-etry for Sagnac and Michelson interferometers used as spectralimagersrdquo Optical Engineering vol 45 no 11 Article ID 1156012006

[3] B J Meers and K A Strain ldquoModulation signal and quantumnoise in interferometersrdquo Physical Review A vol 44 no 7 pp4693ndash4703 1991

[4] H Liang and P-G Wang ldquoEffects of low temperature on mod-ulation efficiency of michelson interferometerrdquo Acta PhotonicaSinica vol 38 no 4 pp 967ndash970 2009

[5] Z-C Bu C-M Zhang B-C Zhao and H-C Zhu ldquoAnalysisand calculation of the modulation depth of the Michelsoninterferometer with wide field chromatic compensation andthermal compensationrdquo Acta Physica Sinica vol 58 no 4 pp2415ndash2422 2009

[6] P K Katti and K Singh ldquoA note on the surface accuracy andalignment of the end mirrors in a michelson interferometerrdquoApplied Optics vol 5 no 12 pp 1962ndash1964 1966

[7] L Shi K Li Z-F Gao L-B Zeng and Q-S Wu ldquoDesign ofa compact structure interferometerrdquo Spectroscopy and SpectralAnalysis vol 33 no 8 pp 2294ndash2298 2013

[8] D Y Yu andH Y Tan Engineering Optics Mechanical IndustryPress Beijing China 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 4: Research Article Analysis of the Interference Modulation ...downloads.hindawi.com/archive/2015/472852.pdf · Hence, the interference modulation depth is 1 = exp ... 1 Amplitude F

4 Advances in OptoElectronics

Figure 3 The surface shape of plane mirror

0 5 10 15 20 25 30

Time (120583s)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de

Figure 4 The electrical signal of the laser interference

depth1198723becomes To put it another way if 120588 value is 0 then

1198723

= 1 When the interference modulation depth is set at1198723ge 09 120588 is

120588 le 0372 (21)

5 Experimental Results

The analysis of interference modulation depth indicates thatin order to realize the interference modulation depth 119872 ge

09 the maximum value of the curve error of mirror 120589

should be 120589max le 1205828 Figure 3 presents measured resultsof the surface of the plane mirror with the aid of Zygointerferometer As can be seen fromFigure 3 the planemirrorhas a good surface with the PV value 1205828

The interferometer is placed on the testbed and theinterference is formed by using theHe-Ne laserThe electricalsignal of the laser interference can then be acquired Figure 4provides the interference signal generated by the laser lightsource

As can be seen from Figure 4 the laser interference signalis a sinusoidal signal and the amplitude of the interferencesignal is stable and the fringe is clearly visible which meetsthe requirement of the interference modulation depth

6 Conclusions

In the Fourier transform spectrometer the Michelson inter-ferometer is the core optical system By focusing on the classicMichelson interferometer we have conducted a theoreticalanalysis of the influence of surface shape of mirror andperformance change of the beam splitter on the modulationdepth Theoretical deductions are obtained regarding toler-ance requirements for the error of surface shape mirror andthe reflection ratio of the beam splitter The results obtainedare applicable to the Fourier transform spectrometer withinthe framework of the Michelson dual beam interference

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The research was supported by the Program for Science andTechnology funded by the Education Department of JiangxiProvince (GJJ14316)

References

[1] B Saggin D Scaccabarozzi and M Tarabini ldquoInstrumentalphase-based method for Fourier transform spectrometer mea-surements processingrdquo Applied Optics vol 50 no 12 pp 1717ndash1725 2011

[2] Y Ferrec J Taboury H Sauer and P Chavel ldquoOptimal geom-etry for Sagnac and Michelson interferometers used as spectralimagersrdquo Optical Engineering vol 45 no 11 Article ID 1156012006

[3] B J Meers and K A Strain ldquoModulation signal and quantumnoise in interferometersrdquo Physical Review A vol 44 no 7 pp4693ndash4703 1991

[4] H Liang and P-G Wang ldquoEffects of low temperature on mod-ulation efficiency of michelson interferometerrdquo Acta PhotonicaSinica vol 38 no 4 pp 967ndash970 2009

[5] Z-C Bu C-M Zhang B-C Zhao and H-C Zhu ldquoAnalysisand calculation of the modulation depth of the Michelsoninterferometer with wide field chromatic compensation andthermal compensationrdquo Acta Physica Sinica vol 58 no 4 pp2415ndash2422 2009

[6] P K Katti and K Singh ldquoA note on the surface accuracy andalignment of the end mirrors in a michelson interferometerrdquoApplied Optics vol 5 no 12 pp 1962ndash1964 1966

[7] L Shi K Li Z-F Gao L-B Zeng and Q-S Wu ldquoDesign ofa compact structure interferometerrdquo Spectroscopy and SpectralAnalysis vol 33 no 8 pp 2294ndash2298 2013

[8] D Y Yu andH Y Tan Engineering Optics Mechanical IndustryPress Beijing China 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 5: Research Article Analysis of the Interference Modulation ...downloads.hindawi.com/archive/2015/472852.pdf · Hence, the interference modulation depth is 1 = exp ... 1 Amplitude F

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of