properties of gaussian beam propagating in ring resonator

Research ArticleProperties of Gaussian Beam Propagating inRing Resonator Sensor

Long Jin Zhiqiang Yang and Qiang Zhang

Department of Basic Science Hubei University of Automotive Technology 167 Checheng West RoadShiyan City 442002 Hubei province China

Correspondence should be addressed to Long Jin crazyjinlong163com

Received 18 November 2018 Accepted 3 February 2019 Published 3 March 2019

Academic Editor Giancarlo C Righini

Copyright copy 2019 Long Jin 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

In this paper we deduce the paraxial analytical expression for the Gaussian beam propagating in the sandwich slab system whichcontained double negativematerial based on light transfer matrix and generalizedHuygens-Fresnel integral equation the evolutionproperties of emerging Gaussian beam contour graph intensity distribution on the receiver plane the relation between beam spotsize and negative refractive index coefficient and beam side transmission view in slab system changed with three negative refractiveindex parameters are illustrated through numerical examples What is more we propose a ring resonator sensor to measure theconcentration ofNaCl solution on the basis of above theory of which the operating principle is deliberatively analyzed the influenceof the concavemirror curvature radius on the emerging beam evolution is acquired the functional relation between the normalizedcentral intensity of the emerging beam the beam spot size and NaCl solution concentration is further developed by fit linearmethod and the mathematical statistics results reach high precision and linearity It is expected that the proposed ring resonatorsensor and the corresponding conclusions can be useful for precise optical measurement especially for food safety inspection andmedical services of health care

1 Introduction

The ring resonator [1 2] is one of the significant appliancesin laser optics It is a kind of traveling wave resonatorowning the advantages of low noise the single frequencywave output and good frequency characteristic Meanwhilethis ring resonator can effectively suppress the standing waveeffect and the spatial hole burning which can be widely usedin developing high precision laser measurement precisionspectroscopy frequency measurement and coherent infor-mation processing Ring resonator structure is an effectivemethod to obtain high power traveling wave laser in a varietyof optical precision detectionfieldsmost sensors such as fourfrequency differential laser gyros [3] and fiber interferometerfilter [4] are composed partially by making use of it Thesesensors which possess the properties of high accuracy strongability to resist severe environment long-lasting life spansand broad dynamic range [5 6] arewidely applied tomeasurethe physical quantity [7] biochemical quantity [8] biomassparameter [9] and so on

In recent years attractive attention has been paid tometamaterials the so-called negative indexmaterials (NIMs)[10ndash13] Double negative material (DNM) [14] with simulta-neously permittivity 120576 and permeability 120583 is also known asleft handed material (LHM) whose unusual electromagnetic(EM) properties were firstly proposed by Velesago [15] Inaddition there are single negative materials (SNMs) amongwhich only one of the EM parameters is negative thesematerials include the epsilon negative media (ENGM 120576 lt 0120583 gt 0) and the mu negative media (MNGM 120576 gt 0 120583 lt0) [16 17] NIMs have many markedly novel phenomenafor instance large negative lateral shift [18] beam focusingand phase compensation [19] negative refraction EM energy[20] reverse Doppler Effect [21] inverse Cerenkov radiation[22] etc These novel characteristics are widely employed indesigning periodic-structure like the photonic crystal [23]and optical thin film coating [24] For instance Pendry onceput forward a sandwich slab system in 2000 and he foundthat the most remarkable application of DNM in the middleof right handed materials (RHMs) is the concept of perfect

HindawiInternational Journal of OpticsVolume 2019 Article ID 3573976 10 pageshttpsdoiorg10115520193573976

2 International Journal of Optics

lens which can achieve subwavelength image beyond thediffraction limit [25] More recently Kang et al researchedthe resonant modes in periodic sandwiched photonic multi-ple quantum well structures (PSPMQWSs) with SNM theyconcluded that the omnidirectional resonance modes can berealized by applying two different photonic crystals whichcontained SNM and the resonance modes inside the zero-effective phase gap of the PSPMQWSs are insensitive tothe incident angle and the scaling of the barrier photoniccrystals [26] We note that the most commonly used incidentbeam for studying the interaction between light and sandwichslab system is the plane wave or linearly polarized beamwhile the resonator is one of the key components of a laserand the fundamental of the optical resonator theory is theGaussian beam (GB) GB and its research products have beenwidely applied to science and technology communicationand medicine and other fields with the development of hightechnology and physical optics [27ndash29]

In this letter we mainly study the optical properties ofthe emerging GB propagating in ring resonator sensor beamevolution in the sandwich slab system is also deduced asparticular case in the coming section The rest of the paper isorganized as follows firstly the general propagation formulaof the GB propagating through the paraxial optical system iscalculated in the space domain by light transfer matrix andgeneralized Huygens-Fresnel integral equation in Section 2Then in Section 3 the GB intensity profiles on emergentintersecting surface of slab system and side view of this beamtransmitting in each RHM and DNM unit are explored theemerging beam spot size changed with negative refractiveindex coefficient is acquired as well Furthermore a ringresonator sensor which can detect the concentration of NaClsolution is designed for what is convinced of to be the firsttime Finally some important conclusions are summarized inSection 4

2 Physical Model and Formula

The schematic drawing of periodic sandwich slab system isdepicted in Figure 1 it is made of two different kinds oftextures the medium layer with red colour stands for RHMand the light greenmaterial stuck between twoRHMs isNIMhere we choose isotropic DNM with its relative permittivityand permeability given by the Drude model [30]

120576119903 (120596) = 1 minus 12059621199011198901205962 (1)

120583119903 (120596) = 1 minus 12059621199011198981205962 (2)

where 120596119901119890 and 120596119901119898 are the electric plasma frequency andthe magnetic plasma frequency respectively the frequency]=120596(2120587) is measured in 102 THz with 120596 being the angularfrequency of the input wave and the length of eachDNMunitis 119871 In general the formula for the refractive index of DNMis described as follows

119899119897 = minusradic120576119903 (120596) sdot 120583119903 (120596) (3)

GBs

O

RHM

DNM

z

Figure 1 Light path schematic drawing of the sandwich slab systemwhich contained DNM

With regard to RHM the refractive index and the length ofeach unit are 119899119903 and 119877 respectively

Considering a cluster of GBs that transmit from point119874 along the direction of arrow the electric field at originallocation (point 119874) can be described [31]

1198641 (1199090 1199100 119911 = 0) = exp(minus11990902 + 1199100212059620 ) (4)

where 1205960 is beam waist radius 1199090 and 1199100 are horizontal sizeand vertical size in the initial location The normalized beamintensity distribution in any paraxial intersecting surface ofthe slab system can be described by the generalized Huygens-Fresnel integral equation [32 33]

1198642 (119909 119910 119911) = ( minus119894120582119861) 119890119894119896119911∬1199041 1198641 (1199090 1199100 0)sdot e((1198941198962119861)[119860(11990920+11991020 )+119863(1199092+1199102)minus2(1199091199090+1199101199100)])1198891199090 1198891199100 (5)

where 120582=6328nm is the wave length of the GB and 119896=2120587120582 isthe wave number119860 119861 119862 and119863 are the elements of the totaltransfer matrix 119879 [34 35]

119879 = (119860 119861119862 119863) (6)

While each linear optical element also has its transmissionmatrix eg

119871 (119909) = (1 1199090 1) (7)

expresses the transfer matrix of wave traveling x distance inthe free space

119876 (119899119903 119899119897) = (1 00 119899119903119899119897) (8)

International Journal of Optics 3

denotes transfer matrix of RHM-DNM interface when beampropagates fromRHMintoDNMslab SimilarlyDNM-RHMinterface transfer matrix is

119876 (119899119897 119899119903) = (1 00 119899119897119899119903) (9)

119877 (119891) = ( 1 0minus 1119891 1) (10)

denotes transfermatrix ofwave reflecting in a concavemirrorwhere 119891=1199032 indicates its focal length and 119903 is curvatureradius of the concave mirror

By cross-producing thesematrices in accordance with theorder of the wave transmission we obtain the electric fielddistribution equation of GB propagating through the paraxialslab system in space domain

1198642 (119909 119910 119911) = (1198941198962119861) exp [minus (1198941198961198632119861) (1199092 + 1199102)]112059620 + 1198941198961198602119861sdot exp[minus(119896119909119861)2 + (119896119910119861)24 (112059620 + 1198941198961198602119861)]

(11)

The beam spot radius can be deduced as the following form

120596119911 = 1205960radic1198602 + [ 11986112058212058712059620 ]2 (12)

and the beam intensity is easily received by using complexconjugate of 1198642(119909 119910 119911)

119868 = 119899021198881205830 10038161003816100381610038161198642 (119909 119910 119911)10038161003816100381610038162 prop 10038161003816100381610038161198642 (119909 119910 119911)10038161003816100381610038162 (13)

where 119888 and 1205830 are the speed of light and permeabilityin vacuum and 1198990 denotes the refractive index of wavepropagating in free space

3 Results and Discussion

Firstly we exhibit transverse normalized intensity distribu-tion of GB before and after wave passing through the wholeslab system (ABA)119898 with three negative refractive indexeswhere A and B stand for RHM and DNM respectively 119898=1is for the sake of simulation Here we choose lossless DNMand its related parameters are set as 120596119901119890=120596119901119898=2120587] times radic22120587] times radic255 and 2120587] times radic31 where ]=119888120582=474times1014Hzleading to corresponding refractive indexes 119899119897=-100 -155and -210 respectively 119899119903=155 fixed1205960=01mm the Rayleighlength of this beam is 119885119877=12058712059602120582=49646cm and 119877=25ZRand 119871=50119885119877 remain unchanged It is easily recognized fromFigures 2(a2) and 2(b1) that the incidentGB and emergingGBoutline look exactly alike while DNM 119899119897= -119899119903=-155 When119899119897=-21 the emerging beam maximum intensity slightlydecreases along with the increased spot size seen from colour

bar of Figure 2(b2) With regard to 119899119897=-10 in Figure 2(b3)not merely the emerging beam maximum intensity decaysto 024 of the incident beam but also the spot size increasesconsiderably on output intersecting surface thus the outputbeam quality is relatively poor The emerging beam spotradius changed with negative refractive index coefficient isalso displayed in Figure 2(c)

In order to showhowwave transmits fromRHMtoDNMwe further display the side intensity view of GB propagatingin (ABA)1 with three negative refractive indexes in Figure 3pale yellow arrow lines show the material boundaries It iseasily discovered from Figure 3(a1) that the spot size enlargeswhile wave propagates in RHM when it passes through theRHM-DNM interface the GB undergoes negative refractionput another way both incident beam and refracted beam areon the same side of the interface normal then beam radiusgradually shrinks to its original feature and increases againin the rest of the first DNM When 119899119897= -119899119903=-155 119877=25119885119877and 119877 119871 119877=121 enlarged spot size could be compensatedexactly while GB transmits in the adjacent material hencethe intensity profiles of incident beam and emerging beamare completely the same the corresponding beam evolutionprinciple diagram is depicted in Figure 3(a2) As to 119899119897=-21119877=25119885119877 and 119871 =50119885119877 the location of the incident beamprofile reappeared in DNM is longer than 25119885119877 thereforethe emerging beam diverges slightly at the end of secondRHM in Figure 3(b1) Now we need to illustrate the accuratelocation while beam goes back to its original profile in themiddle DNM when GB passes through the first RHM andDNM the transfer matrix 119879 can be indicated as

119879 = (119860 119861119862 119863)= (1 00 119899119897) times (1 1198710 1) times (

1 00 119899119903119899119897) times (1 1198770 1)times(1 0

0 1119899119903)(14)

Substituting 119877=25119885119877 119899119903=155 and 119899119897=-210 into (14) theanalytical transmission matrix element 119861 can be expressed as

119861 = minus1021 times 119871 + 625 sdot 12058724521 (15)

and thus the accurate location to compensate beam diver-gence in RHM is 35119885119877 and the length of total DNM is119871=2times35119885119877=70119885119877 we depict the sandwich slab system with119877= 25119885119877 and 119871=70119885119877 arranged in Figure 3(b2) it is clearlyseen that the emerging beam intensity gets back to its originalfeature at the end of RHM Inversely the accurate locationfor 119899119897=-10 119877=25119885119877 is shorter than 25119885119877 while beam goesback to its original profile in DNM and the spot size morethan double increases at the end of DNM if 119871=50119885119877 fixedtherefore the emerging beam could not come back to itsoriginal feature as shown in Figure 3(c1) By substituting


minus1 0 1x (m)

01

02

03

04

05

06

07

08

09

1 Normalized beam intensity distribution

times10minus4

minus22 minus20 minus18 minus16 minus14 minus12 minus10

10

12

14

16

18

20

22

(b1) (b2)

(a1) (a2)

(c)Refractive index

(b3)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)

Normalized beam intensity distribution

y (m

)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

x (m)


y (m

)2

1

0

minus1

minus2210minus1minus2

055

05

045

04

035

03

025

02

015

01

005

times10minus4

times10minus4

x (m)


y (m

)

2

1

0

minus1


02

015

01

005

Spot

size

(0)

Figure 2 Optical parameters of GB versus different negative refractive indexes with119898=1 1205960=01times10minus3m 119899119903=155 119885119877 =49646cm 119871=50119885119877119877=25ZR and 120582=6328nm fixed (a1) incident beam distribution 1D (a2) incident beam distribution 2D (b1) emerging beam distributionDNM 120596119901119890=120596119901119898=2120587] times radic255 (b2) emerging beam distribution DNM 120596119901119890=120596119901119898 =2120587] times radic31 and (b3) emerging beam distribution DNM120596119901119890=120596119901119898 =2120587] times radic2 (c) beam spot size distribution curve


Object plane Image plane

(a2)(a1)

(b2)(b1)

(c2)(c1)

nr=155nl=-155nr=155

3

2

1

0

minus1

minus2

minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02

025

03 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015 0

002

004

006

008

01

012

z (m)

Figure 3 Side view of GB propagating in each RHM and DNM unit successively with three kinds of DNMs 1205960=01times10minus3m 119899119903=155119885119877=49646cm and 120582=6328nm fixed (a1) DNM 120596119901119890=120596119901119898=2120587]timesradic255 (a2) principle diagram of the emerging GB comes back to its originalfeature (b1) DNM 120596119901119890=120596119901119898=2120587]timesradic31 (b2) beam transmission in sandwich slab system 119877=25119885119877 119871=70119885119877 (c1) DNM 120596119901119890=120596119901119898=2120587]timesradic2and (c2) beam transmission in sandwich slab system 119877=25119885119877 119871=34119885119877


He-Ne laser

Mirror B

Mirror D

Mirror A

Mirror C

CCDPC

He-Ne laser

Collimating lensFocusing lens

()1

Figure 4 Light path schematic diagram of ring resonator sensor

L1=25l-2ZR L2=15l-2ZR

(a)

l2l L3=05l-2ZR L3=05l-2ZR

(b)

Figure 5 Equivalent lens sequence diagram of ring resonators for two kinds of curvature radius 1205960=01times10minus3m 119899119903=155 120596119901119890=120596119901119898=2120587] timesradic255 119885119877=49646cm 119871=2119885119877 119877=119885119877 119897=02m and 120582=6328nm fixed (a) 119903 997888rarr infin (b) 119903=02m

119877= 25119885119877 119899119903=155 and 119899119897=-100 into (14) analytical transfermatrix element 119861 can be also written as

119861 = minus119871 + 625 sdot 12058724521 (16)

and thus the length of themiddleDNM is119871=2times17119885119877=34119885119877The side intensity view of GB propagating in this structuralmodel with 119877= 25119885119877 and L=34119885119877 is demonstrated inFigure 3(c2) the simulation result agrees well with ourtheoretical analysis

In this subsection we propose a ring resonator sensorto measure the concentration of NaCl based on the abovetheories In general there are two independent-positiveand negative-traveling wave modes in a bidirectional ringresonator unidirectional ring resonators are also widelyavailable and have higher output power [36] Rotatory com-ponents which form and split the left and right circularpolarizationmodes in ring cavity [37] ensure the stable outputsignal of laser with lower loss and backscatter Besides theprism including planemirror and concavemirror and activemedium [38] are also indispensable elements for this ringresonator Concave mirror can realize beam focusing andcavity stabilization while the plane mirror is used to changethe direction of wave path We choose the He-Ne lasercollimating lens and focusing lens to guide the GB into thisring resonator the schematic diagram of this ring resonatoris shown in Figure 4 [39] Mirror A and Mirror B represent

plane mirrors Mirror C Mirror D are the concave mirrorswith radius of curvature 119903=02m the single cavity length is119897=02mHe-Ne laser with its output wavelength 120582=6328nm isinstalled outside of thewhole cavity the sandwich slab system(ABA) 1 is located in the center of the two concave mirrorshere A is the space resounded with NaCl solution B standsfor DNM with its 119899119897=-155 fixed 119871=2119885119877 and 119877=119885119877 so as tomatch the standard size of ring resonator sensor

We begin to discuss the influence of the concave mirrorcurvature radius on the evolution of the emerging beamwhile GB propagates a circle in the ring resonator along thearrow line marked in Figure 4 The equivalent lens sequencediagramof ring resonator for 119903 997888rarr infin and 119903=02m is depictedin Figure 5 As we know the concave mirror will convert intothe planemirror in the condition of 119903 997888rarr infin at this time thetransfer matrix in (10) degrades into

119877 (119891 997888rarr infin) = (1 00 1) (17)

Hence the GB transmission characteristics in the ring res-onator are installed with four plane mirrors more like that inthe sandwich slab except for the extra transmission distancein free space as shown in Figure 5(a) The correspondingbeam intensity distribution for 119903 997888rarr infin and 119903=02m isdemonstrated in Figure 6 as well We can come to theconclusion that the smaller the curvature radius is the


1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1minus1 minus05 0 05 1

x (m)

y (m

)

times10minus3

times10minus3 Normalized beam intensity distribution

1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)

Normalized beam intensity distribution8

7

6

5

4

3

2

1

(b)

Figure 6 Emerging beam intensity distribution when GB propagates a circle in the ring resonator for two kinds of curvature radius1205960=01times10minus3m 119899119903=155 120596119901119890=120596119901119898=2120587] times radic255 119885119877=49646cm 119871=2119885119877 119877=119885119877 119897=02m and 120582=6328nm fixed (a) 119903 997888rarr infin (b) 119903=02m

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)

Figure 7 Normalized beam intensity distribution of emerging beam on plane Mirror B cross section for different concentration of NaClsolution 1205960=01times10minus3m 120596119901119890=120596119901119898=2120587]timesradic255119885119877=49646cm 119871=2119885119877 119877=119885119877 119897=02m 119903=02m and 120582=6328nm fixed (a) purified water and(b) 25 NaCl solution

larger the maximum beam intensity is formed because thefocusing ability of a concave mirror is strengthened with thedecreasing of curvature radius

Now let us discuss how to measure the concentrationof NaCl by using the ring resonator sensor When GBpropagates a circle along the allow line marked in Figure 4the contour profiles for purified water and 25NaCl solutionon output intersecting surface plane ofMirror B are exhibitedin Figure 7 Visually the central bright intensity of 25 NaClsolution is larger than that of purified water discerned fromcolour bar With regard to spot size the quantity value ofpurified water is a bit bigger which can be detected by CCDor oscillograph

In order to have a better understanding of the operatingprinciple of this ring resonator sensor let us expand theanalysis as follows

The empirical formula between NaCl concentration andits refractive index is [40]119873 = 13331 + 000185 times 119862 (18)

where 119873 is refractive index and 119862 is NaCl concentrationWhen various kinds of NaCl solution are filled with space Athe relation between normalizedmaximum intensity the spotsize and solution refractive index is displayed in Figure 8It is acquired that the central maximum intensity graduallybecomes larger with the increasing of refractive index ofNaCl


133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)

Figure 8 Relation between optical parameters of GB and NaCl solution refractive index 1205960=01times10minus3m 119899119903=155 120596119901119890=120596119901119898=2120587] times radic255119885119877=49646cm 119871=2119885119877 119877=119885119877 119897=02m 119903=02m and 120582=6328nm fixed (a) beam intensity distribution curve (b) beam spot size curve

0 5 10 15 20 25 30075

076

077

078

079

080

081

082

NaCl solution concentration ()

Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)

Figure 9 Relation between optical parameters of GB and NaCl solution concentration 1205960=01times10minus3m 119899119903=155 120596119901119890=120596119901119898=2120587] times radic255119885119877=49646cm 119871=2119885119877 119877=119885119877 119897=02m 119903=02m and 120582=6328nm fixed (a) beam intensity distribution curve (b) beam spot size curve

solution by contrast the larger the refractive index of NaClsolution is the smaller the spot size is formed indicatingthat the NaCl concentration can affect the evolution of theGB as well as other RHM What is more both the curvesin Figure 8 possess good linear features which providesreliable guarantee for next precise measurements In thenext moment the normalized central GB intensity andbeam spot size on plane Mirror B cross section changedwith a series of NaCl concentration are illustrated as wellThe ultimateness is exhibited in Figure 9 and reaches highlinearity in coincidence with our assumption We make useof fit linear method to exhibit the function equation betweennormalized maximum intensity of emerging GB intensityits spot size value and NaCl solution concentration thequantitative function formulae are119868 = 000198 times 119862 + 075733 (19)

where 119868 signifies normalized central intensity The statisticsof Adjusted R-square and Residual Sum of Squares of thisformula reach 099997 and 767857times10minus8 And

120596119911 = 491625 minus 000606119862 (20)

and the statistics parameters of above formula are 099983and 356536times10minus6 respectively Just as we described abovethis ring resonator sensor demonstrates excellent linearcorrelation and statistics we are certain that the rational usingof it in concentration detection will become a necessary anduseful supplement to the chemical sensor

To the best of our knowledge this ring resonator sensorcan probe other solutions like sucrose Na2CO3 and so onin so far as the parameters of intercept and slope of (19)and (20) are adjusted It is expected that the proposed ringresonator sensor and the corresponding conclusions can be


useful for precise optical measurement especially for foodsafety inspection and medical services of health care

4 Summary

In conclusion we have researched the evolution of GBpropagating in the sandwich slab system based on the lighttransfer matrix and generalized Huygens-Fresnel integralequation It is revealed that the emerging GB can come backto its original feature by using the sandwich slab systemwhichcontained negative index material as long as the negativerefractive index 119899119897=- 119899119903 and each unit length 119877119885119877=121while nl = - 119899119903 the larger abs(119899119897) is the longer DNM isneeded to achieve emerging beam reconstruction and viceversa The emerging beam spot size changed with negativerefractive index coefficient is also acquired Meanwhile wepropose a ring resonator sensor tomeasure the concentrationof NaCl whose operating principle especially the influenceof the concave mirror curvature radius on the emergingbeam evolution is deliberatively analyzed The functionalrelation between normalized central intensity beam spotsize and NaCl solution concentration is given by (19) and(20) respectively and assuredly reaches high precision andlinearity It is expected that the proposed ring resonatorsensor and the corresponding conclusions can be usefulfor precise optical measurement especially for food safetyinspection and medical services of health care

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the financial support from theCollege Innovation Training Program of Hubei Universityof Automotive Technology in China (nos DC 2018090 andDC2018099)

References

[1] S I Bozhevolnyi V S Volkov E Devaux J Y Laluet andT W Ebbesen ldquoChannel plasmon subwavelength waveguidecomponents including interferometers and ring resonatorsrdquoNature vol 440 no 7083 pp 508ndash511 2006

[2] H Qiu T Kurihara H Harada et al ldquoEnhancing terahertzmagnetic near field induced by a micro-split-ring resonatorwith a tapered waveguiderdquo Optics Expresss vol 43 no 8 pp1658ndash1661 2018

[3] J Yuan M Chen Y Li Z Tan and Z Wang ldquoReanalysis ofgeneralized sensitivity factors for optical-axis perturbation innonplanar ring resonatorsrdquo Optics Express vol 21 no 2 pp2297ndash2306 2013

[4] J Xu L Huang S Yin B Gao and P Chen ldquoAll-fiber self-mixing interferometer for displacement measurement based onthe quadrature demodulation techniquerdquo Optical Review vol25 no 1 pp 40ndash45 2018

[5] G Sanders ldquoFiber optic gyros for space marine and aviationapplicationsrdquo Proceedings of SPIE -e International Society forOptical Engineering vol 283712 no 12 pp 61ndash71 1996

[6] Z Wang X Long and F Wang ldquoOverview of four-modedifferential laser gyrosrdquo Laser amp Optoelectronics Progress vol49 no 4 article 040005 2012

[7] Q Xu S Sandhu M L Povinelli J Shakya S Fan and MLipson ldquoExperimental realization of an on-chip all-optical ana-logue to electromagnetically induced transparencyrdquo PhysicalReview Letters vol 96 no 12 article 123901 2006

[8] C Ciminelli C M Campanella F DellrsquoOlio C E Campanellaand M N Armenise ldquoLabel-free optical resonant sensors forbiochemical applicationsrdquo Progress in Quantum Electronics vol37 no 2 pp 51ndash107 2013

[9] M Bahadoran A F A Noorden K Chaudhary et al ldquoModel-ing and analysis of a microresonating biosensor for detection ofsalmonella bacteria in human bloodrdquo Sensors vol 14 no 7 pp12885ndash12899 2014

[10] J Valentine S Zhang T Zentgraf et al ldquoThree-dimensionaloptical metamaterial with a negative refractive indexrdquo Naturevol 455 no 7211 pp 376ndash379 2008

[11] H J Lezec J ADionne andHAAtwater ldquoNegative refractionat visible frequenciesrdquo Science vol 316 no 5823 pp 430ndash4322007

[12] M Sadatgol S K Ozdemir L Yang et al ldquoPlasmon injection tocompensate and control losses in negative indexmetamaterialsrdquoPhysical Review Letters vol 115 no 3 article 035502 2015

[13] S Lan L Kang D T Schoen et al ldquoBackward phase-matchingfor nonlinear optical generation in negative-index materialsrdquoNature Materials vol 14 no 8 pp 807ndash811 2015

[14] M J Hossain M R I Faruque and M T Islam ldquoDesign andanalysis of a new composite double negative metamaterial formulti-band communicationrdquo Current Applied Physics vol 17no 7 pp 931ndash939 2017

[15] V G Veselago ldquoThe electrodynamics of substances with simul-taneously negative values of 120576 and 120583rdquo Soviet Physics Uspekhi vol10 no 4 pp 509ndash514 1968

[16] H N Hajesmaeili M Zamani andM H Zandi ldquoBi-gyrotropicsingle-negative magnetic materials in the presence of longitudi-nal magnetization a transfer matrix approachrdquo Photonics andNanostructures Fundamentals and Applications vol 24 no 5pp 69ndash75 2017

[17] W Liu Y Yuan T Tang et al ldquoBand gap characteristicsof photonic crystals consisted of left-handed material in themicrowave bandrdquo Acta Photonica Sinica vol 45 no 2 article216002 Article ID 216002 2016

[18] Y-Q Kang W Ren and Q Cao ldquoLarge tunable negative lateralshift from graphene-based hyperbolic metamaterials backed bya dielectricrdquo Superlattices andMicrostructures vol 120 no 8 pp1ndash6 2018

[19] Q Li L Huang H So H Xue and P Zhang ldquoBeam patternsynthesis for frequency diverse array via reweighted lminus1 iterativephase compensationrdquo IEEE Transactions on Aerospace amp Elec-tronics Systems vol 54 no 1 pp 467ndash475 2018

[20] D Dahal andG Gumbs ldquoEffect of energy band gap in grapheneon negative refraction through the veselago lens and electronconductancerdquo Journal of Physics and Chemistry of Solids vol100 no 1 pp 83ndash91 2017


[21] A V Chumak P Dhagat A Jander A A Serga and BHillebrands ldquoReverse doppler effect of magnons with negativegroup velocity scattered from a moving bragg gratingrdquo PhysicalReview B - Condensed Matter and Materials Physics vol 81 no14 article 140404 2010

[22] B Wu J Lu J Kong M Chen and Z Duan ldquoCherenkovradiation in anisotropic double-negative metamaterialsrdquoOpticsExpress vol 16 no 22 pp 18479ndash18484 2008

[23] S A El-Naggar ldquoPhotonic gaps in one dimensional cylindricalphotonic crystal that incorporates single negative materialsrdquoe European Physical Journal D vol 71 no 1 pp 11ndash16 2017

[24] Y-J Jen C-H Chen and C-W Yu ldquoDeposited metamaterialthin film with negative refractive index and permeability in thevisible regimerdquoOptics Expresss vol 36 no 6 pp 1014ndash1016 2011

[25] J B Pendry ldquoNegative refraction makes a perfect lensrdquo PhysicalReview Letters vol 85 no 18 pp 3966ndash3969 2000

[26] Y Kang and C Zhang ldquoResonant modes in photonic multiplequantum well structures with single-negative materialsrdquo Optik- International Journal for Light and Electron Optics vol 124 no22 pp 5430ndash5433 2013

[27] M C Gokce and Y Baykal ldquoAperture averaging and BERfor Gaussian beam in underwater oceanic turbulencerdquo OpticsCommunications vol 410 pp 830ndash835 2018

[28] K-S Cheng and R B Roemer ldquoBlood perfusion and thermalconduction effects in Gaussian beam minimum time single-pulse thermal therapiesrdquoMedical Physics vol 32 no 2 pp 311ndash317 2005

[29] Z Xiang Y Yuan C Li et al ldquoMultiple reflections and fresnelabsorption of gaussian laser beam in an actual 3d keyholeduring deep-penetration laser weldingrdquo International Journal ofOptics vol 2012 Article ID 361818 8 pages 2012

[30] S A Cummer ldquoDynamics of causal beam refraction in negativerefractive index materialsrdquo Applied Physics Letters vol 82 no13 pp 2008ndash2010 2003

[31] Y Gao Z An N Li W Zhao and J Wang ldquoOptical design ofGaussian beam shapingrdquoOptics amp Precision Engineering vol 19no 7 pp 1464ndash1471 2011

[32] B Lu Laser Optics-Beam Characterization Propagation andTransformation Resonator Technology and Physics Higher Edu-cation Press Beijing China 3rd edition 2003

[33] L Jin and X Zhang ldquoPropagation properties of airy beamthrough periodic slab system with negative index materialsrdquoInternational Journal of Optics vol 2018 Article ID 94784837 pages 2018

[34] S Wang and L Ronchi ldquoII principles and design of opticalarraysrdquo Progress in Optics vol 25 no 1 pp 279ndash348 1988

[35] J Zhou H Luo S Wen and Y Zeng ldquoABCDmatrix formalismfor propagation of Gaussian beam through left-handedmaterialslab systemrdquoOptics Communications vol 282 no 14 pp 2670ndash2675 2009

[36] C Du G Liao L Zhang et al ldquoDesign and experimentalresearch of laser with four-mirror ring resonatorrdquo Laser ampInfrared vol 42 no 1 pp 26ndash30 2012

[37] S Berg T Prellberg and D Johannsmann ldquoNonlinear contactmechanics based on ring-down experiments with quartz crystalresonatorsrdquo Review of Scientific Instruments vol 74 no 1 pp118ndash126 2003

[38] Y Honda H Shimizu M Fukuda et al ldquoStabilization of anon-planar optical cavity using its polarization propertyrdquoOpticsCommunications vol 282 no 15 pp 3108ndash3112 2009

[39] L Jin and X Zhang ldquoWave transmission characteristics in ringcavity containingmeta-materialsrdquo Semiconductor Optoelectron-ics vol 38 no 1 pp 66ndash69 2017

[40] Z Bai Z Liu and H Xu ldquoAn experienced formula aboutthe connection of refraction index and consistence of severalliquidrdquo Journal of Yanan University (Natural Science Edition)vol 23 no 1 pp 33ndash36 2004

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components


Shock and Vibration


High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

Antennas andPropagation

International Journal of




Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics


Volume 2018


Mathematical PhysicsAdvances in


ChemistryAdvances in


Journal of

Chemistry


Advances inPhysical Chemistry


RotatingMachinery



Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom


lens which can achieve subwavelength image beyond thediffraction limit [25] More recently Kang et al researchedthe resonant modes in periodic sandwiched photonic multi-ple quantum well structures (PSPMQWSs) with SNM theyconcluded that the omnidirectional resonance modes can berealized by applying two different photonic crystals whichcontained SNM and the resonance modes inside the zero-effective phase gap of the PSPMQWSs are insensitive tothe incident angle and the scaling of the barrier photoniccrystals [26] We note that the most commonly used incidentbeam for studying the interaction between light and sandwichslab system is the plane wave or linearly polarized beamwhile the resonator is one of the key components of a laserand the fundamental of the optical resonator theory is theGaussian beam (GB) GB and its research products have beenwidely applied to science and technology communicationand medicine and other fields with the development of hightechnology and physical optics [27ndash29]

In this letter we mainly study the optical properties ofthe emerging GB propagating in ring resonator sensor beamevolution in the sandwich slab system is also deduced asparticular case in the coming section The rest of the paper isorganized as follows firstly the general propagation formulaof the GB propagating through the paraxial optical system iscalculated in the space domain by light transfer matrix andgeneralized Huygens-Fresnel integral equation in Section 2Then in Section 3 the GB intensity profiles on emergentintersecting surface of slab system and side view of this beamtransmitting in each RHM and DNM unit are explored theemerging beam spot size changed with negative refractiveindex coefficient is acquired as well Furthermore a ringresonator sensor which can detect the concentration of NaClsolution is designed for what is convinced of to be the firsttime Finally some important conclusions are summarized inSection 4

2 Physical Model and Formula

The schematic drawing of periodic sandwich slab system isdepicted in Figure 1 it is made of two different kinds oftextures the medium layer with red colour stands for RHMand the light greenmaterial stuck between twoRHMs isNIMhere we choose isotropic DNM with its relative permittivityand permeability given by the Drude model [30]

120576119903 (120596) = 1 minus 12059621199011198901205962 (1)

120583119903 (120596) = 1 minus 12059621199011198981205962 (2)

where 120596119901119890 and 120596119901119898 are the electric plasma frequency andthe magnetic plasma frequency respectively the frequency]=120596(2120587) is measured in 102 THz with 120596 being the angularfrequency of the input wave and the length of eachDNMunitis 119871 In general the formula for the refractive index of DNMis described as follows

119899119897 = minusradic120576119903 (120596) sdot 120583119903 (120596) (3)

GBs

O

RHM

DNM

z

Figure 1 Light path schematic drawing of the sandwich slab systemwhich contained DNM

With regard to RHM the refractive index and the length ofeach unit are 119899119903 and 119877 respectively

Considering a cluster of GBs that transmit from point119874 along the direction of arrow the electric field at originallocation (point 119874) can be described [31]

1198641 (1199090 1199100 119911 = 0) = exp(minus11990902 + 1199100212059620 ) (4)

where 1205960 is beam waist radius 1199090 and 1199100 are horizontal sizeand vertical size in the initial location The normalized beamintensity distribution in any paraxial intersecting surface ofthe slab system can be described by the generalized Huygens-Fresnel integral equation [32 33]

1198642 (119909 119910 119911) = ( minus119894120582119861) 119890119894119896119911∬1199041 1198641 (1199090 1199100 0)sdot e((1198941198962119861)[119860(11990920+11991020 )+119863(1199092+1199102)minus2(1199091199090+1199101199100)])1198891199090 1198891199100 (5)

where 120582=6328nm is the wave length of the GB and 119896=2120587120582 isthe wave number119860 119861 119862 and119863 are the elements of the totaltransfer matrix 119879 [34 35]

119879 = (119860 119861119862 119863) (6)

While each linear optical element also has its transmissionmatrix eg

119871 (119909) = (1 1199090 1) (7)

expresses the transfer matrix of wave traveling x distance inthe free space

119876 (119899119903 119899119897) = (1 00 119899119903119899119897) (8)



119876 (119899119897 119899119903) = (1 00 119899119897119899119903) (9)

119877 (119891) = ( 1 0minus 1119891 1) (10)




(11)


120596119911 = 1205960radic1198602 + [ 11986112058212058712059620 ]2 (12)


119868 = 119899021198881205830 10038161003816100381610038161198642 (119909 119910 119911)10038161003816100381610038162 prop 10038161003816100381610038161198642 (119909 119910 119911)10038161003816100381610038162 (13)






119879 = (119860 119861119862 119863)= (1 00 119899119897) times (1 1198710 1) times (

1 00 119899119903119899119897) times (1 1198770 1)times(1 0

0 1119899119903)(14)


119861 = minus1021 times 119871 + 625 sdot 12058724521 (15)



minus1 0 1x (m)

01

02

03

04

05

06

07

08

09


times10minus4


10

12

14

16

18

20

22

(b1) (b2)

(a1) (a2)

(c)Refractive index

(b3)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

x (m)


y (m

)2

1

0

minus1


055

05

045

04

035

03

025

02

015

01

005

times10minus4

times10minus4

x (m)


y (m

)

2

1

0

minus1


02

015

01

005

Spot

size

(0)




(a2)(a1)

(b2)(b1)

(c2)(c1)

nr=155nl=-155nr=155

3

2

1

0

minus1

minus2

minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02

025

03 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015 0

002

004

006

008

01

012

z (m)



He-Ne laser

Mirror B

Mirror D

Mirror A

Mirror C

CCDPC

He-Ne laser


()1


L1=25l-2ZR L2=15l-2ZR

(a)


(b)



119861 = minus119871 + 625 sdot 12058724521 (16)





119877 (119891 997888rarr infin) = (1 00 1) (17)



1

08

06

04

02

0

minus02

minus04

minus06

minus08


x (m)

y (m

)

times10minus3


1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


7

6

5

4

3

2

1

(b)


1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)








133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







119876 (119899119897 119899119903) = (1 00 119899119897119899119903) (9)

119877 (119891) = ( 1 0minus 1119891 1) (10)




(11)


120596119911 = 1205960radic1198602 + [ 11986112058212058712059620 ]2 (12)


119868 = 119899021198881205830 10038161003816100381610038161198642 (119909 119910 119911)10038161003816100381610038162 prop 10038161003816100381610038161198642 (119909 119910 119911)10038161003816100381610038162 (13)






119879 = (119860 119861119862 119863)= (1 00 119899119897) times (1 1198710 1) times (

1 00 119899119903119899119897) times (1 1198770 1)times(1 0

0 1119899119903)(14)


119861 = minus1021 times 119871 + 625 sdot 12058724521 (15)



minus1 0 1x (m)

01

02

03

04

05

06

07

08

09


times10minus4


10

12

14

16

18

20

22

(b1) (b2)

(a1) (a2)

(c)Refractive index

(b3)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

x (m)


y (m

)2

1

0

minus1


055

05

045

04

035

03

025

02

015

01

005

times10minus4

times10minus4

x (m)


y (m

)

2

1

0

minus1


02

015

01

005

Spot

size

(0)




(a2)(a1)

(b2)(b1)

(c2)(c1)

nr=155nl=-155nr=155

3

2

1

0

minus1

minus2

minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02

025

03 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015 0

002

004

006

008

01

012

z (m)



He-Ne laser

Mirror B

Mirror D

Mirror A

Mirror C

CCDPC

He-Ne laser


()1


L1=25l-2ZR L2=15l-2ZR

(a)


(b)



119861 = minus119871 + 625 sdot 12058724521 (16)





119877 (119891 997888rarr infin) = (1 00 1) (17)



1

08

06

04

02

0

minus02

minus04

minus06

minus08


x (m)

y (m

)

times10minus3


1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


7

6

5

4

3

2

1

(b)


1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)








133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






minus1 0 1x (m)

01

02

03

04

05

06

07

08

09


times10minus4


10

12

14

16

18

20

22

(b1) (b2)

(a1) (a2)

(c)Refractive index

(b3)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

1

09

08

07

06

05

04

03

02

01

minus1 0 1minus1

0

1

x (m)


y (m

)

times10minus4

times10minus4

x (m)


y (m

)2

1

0

minus1


055

05

045

04

035

03

025

02

015

01

005

times10minus4

times10minus4

x (m)


y (m

)

2

1

0

minus1


02

015

01

005

Spot

size

(0)




(a2)(a1)

(b2)(b1)

(c2)(c1)

nr=155nl=-155nr=155

3

2

1

0

minus1

minus2

minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02

025

03 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015 0

002

004

006

008

01

012

z (m)



He-Ne laser

Mirror B

Mirror D

Mirror A

Mirror C

CCDPC

He-Ne laser


()1


L1=25l-2ZR L2=15l-2ZR

(a)


(b)



119861 = minus119871 + 625 sdot 12058724521 (16)





119877 (119891 997888rarr infin) = (1 00 1) (17)



1

08

06

04

02

0

minus02

minus04

minus06

minus08


x (m)

y (m

)

times10minus3


1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


7

6

5

4

3

2

1

(b)


1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)








133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







(a2)(a1)

(b2)(b1)

(c2)(c1)

nr=155nl=-155nr=155

3

2

1

0

minus1

minus2

minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02

025

03 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

minus3 minus3

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015

02 0

002

004

006

008

01

012

z (m)

3

2

1

0

minus1

minus2

times10minus4

x (m

)

0

002

004

006

008

01

012

005

01

015 0

002

004

006

008

01

012

z (m)



He-Ne laser

Mirror B

Mirror D

Mirror A

Mirror C

CCDPC

He-Ne laser


()1


L1=25l-2ZR L2=15l-2ZR

(a)


(b)



119861 = minus119871 + 625 sdot 12058724521 (16)





119877 (119891 997888rarr infin) = (1 00 1) (17)



1

08

06

04

02

0

minus02

minus04

minus06

minus08


x (m)

y (m

)

times10minus3


1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


7

6

5

4

3

2

1

(b)


1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)








133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






He-Ne laser

Mirror B

Mirror D

Mirror A

Mirror C

CCDPC

He-Ne laser


()1


L1=25l-2ZR L2=15l-2ZR

(a)


(b)



119861 = minus119871 + 625 sdot 12058724521 (16)





119877 (119891 997888rarr infin) = (1 00 1) (17)



1

08

06

04

02

0

minus02

minus04

minus06

minus08


x (m)

y (m

)

times10minus3


1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


7

6

5

4

3

2

1

(b)


1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)








133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






1

08

06

04

02

0

minus02

minus04

minus06

minus08


x (m)

y (m

)

times10minus3


1

09

08

07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


7

6

5

4

3

2

1

(b)


1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3

x (m)


07

06

05

04

03

02

01

(a)

1

08

06

04

02

0

minus02

minus04

minus06

minus08


y (m

)

times10minus3

times10minus3x (m)


07

06

05

04

03

02

01

(b)








133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






133 134 135 136 137 138 139075

076

077

078

079

080

081

082

Refractive index

Nor

mal

ized

max

imum

inte

nsity

(a)

133 134 135 136 137 138 139470

475

480

485

490

495

Refractive index

Spot

size

(0)

(b)


0 5 10 15 20 25 30075

076

077

078

079

080

081

082


Nor

mal

ized

max

imum

inte

nsity

(a)

0 5 10 15 20 25 30470

475

480

485

490

495


Spot

size

(0)

(b)




120596119911 = 491625 minus 000606119862 (20)





4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







4 Summary


Data Availability




Acknowledgments


References













































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





properties of gaussian beam propagating in ring resonator

Documents