如图12-4-1所示，用插针法找出射光线AO对应的出线O′B，定出射点O′，画出折射光线OO′，然后测量出入射角θ1和折射角θ2，根据 n=sinθ1 计算玻
Vector wave analysis for nonnormal incident rays in epimicroscopic refractive index proﬁle measurements
Seung Bum Cho, 1, *Cheng Liu, 1Mats Gustafsson, 2and Dug Young Kim 1 1Department of Information and Communications, Gwangju Institute of Science and Technology, 1Oryong-dong, Buk-gu, Gwangju 500-712, Republic of Korea
2Department of Bioengineering, University of California, San Francisco, California, USA
Received 15August 2007; revised 7October 2007; accepted 2November 2007;
posted 8November 2007(Doc.ID 86417); published 7January 2008
We have investigated the effects of nonnormal incident rays in calculating the refractive index proﬁle of
a dielectric sample using the reﬂectance measurement data obtained with a scanning confocal epimicro-
scope and also by solving three-dimensional vector wave equations for linearly polarized light. The
numerically calculated reﬂection data of tightly focused Gaussian beams with different numerical aper-
tures (NAs)on planar surfaces with various refractive indices conﬁrm that the reﬂectance increases with
an increase in the NA of a focusing objective lens. This is due to the nonnormal incident ray components
of a Gaussian beam. We have found that the refractive index obtained with the assumption of a normal
incident beam is far from the real value when the NA of a focusing lens becomes larger than 0.5, and thus
the variation in the reﬂectance for different angular components in a Gaussian beam must be taken into
consideration while using a larger NA lens. Errors in practical refractive index calculation for an optical
ﬁber based on a normal incident beam in reﬂectance measurements can be as large as 1%in comparison
to real values calculated by our three-dimensional vector wave equations. ©2008Optical Society of
OCIS codes:060.2300, 060.2270, 180.1790.
The refractive index proﬁle is one of the most signif-icant factors in optical ﬁbers and waveguides . Ever since the development of optical ﬁbers and waveguides, many kinds of methods have been intro-duced to measure its refractive index proﬁles; the refracted near-ﬁeld (RNF)method, the transmitted near-ﬁeld (TNF)method, and the transverse inter-ferometer (TI)method are the conventional methods used for this purpose [2–7].However, these methods are not suitable for measuring the index of specialty ﬁbers with a complex structure .The refractive index proﬁler, based on measuring the power change in the reﬂected light, can be a good alternative for the ﬁber index proﬁling. Such a method does not need a complicated sample preparation and can take the proﬁle image without considering its structures. At the same time, this technique is applicable to general optical ﬁbers as well as special ﬁbers with complex and axially nonsymmetric structures. The reﬂectance of a planar surface can be related to its refractive index by the Fresnel equation, and this is always used to convert the obtained reﬂectance proﬁle into the refractive index proﬁle by simply assuming the focused light to be at normal incidence on the surface [2,3].Under this assumption the change in reﬂec-tance is almost linearly dependent on the refractive index change ,and thus the index retrieval is re-markably simpliﬁed in calculation. However, the fo-cused Gaussian beam is indeed not a planar wave, and its higher order angular components become re-markable with the increasing numerical aperture (NA)of the objective. For index proﬁling, the incident light should be considered as an angular spectrum representation of plane waves with each angular component having a different reﬂection coefﬁcient .It is widely known that when the NA of the ob-jective is higher than 0.5, the higher order angular
©2008Optical Society of America
10January 2008͞Vol. 47, No. 2͞APPLIED OPTICS 157
components of the focused ﬁled cannot be
neglected [9,10].For index proﬁling, this means that the max-imum spatial resolution reachable is only at the scale of the wavelength used when the normal incidence assumption is adopted for index retrieval. For single mode ﬁbers, whose core has a diameter of only several micrometers, this resolution is obviously too low. To resolve such problems, we suggest using a tightly focused Gaussian beam through an objective of high NA to illuminate the specimen for high spatial resolution. The vector Fresnel equations are to be used for the analysis of off-normal incidence on the ﬁber surface .To be speciﬁc, the electric ﬁeld and the magnetic ﬁeld of the focused light incident on the ﬁber surface are calculated with a three-dimensional (3D)vector integral [11,12].The reﬂected ﬁelds in-cluding both the electric and magnetic ﬁelds are com-puted by considering the difference in the reﬂection for each angular component with the vector Fresnel equations. The reﬂectance of the surface, which is the ratio of the reﬂected energy to the incident energy, is determined by computing its Poynting ﬂux. With this method, we found that the practical reﬂectance al-ways increases with the NA of the objective used; furthermore, the practical reﬂectance is always larger than that predicted with normal incidence as-sumption. It is only when the NA is less than 0.5that these two values come closer. For a practical experi-ment with a given NA, the reﬂectance change with the material index can be simulated in advance with the above mentioned method. Thus the reﬂectance measured at any point during the scanning of the ﬁber surface can be converted to the corresponding refractive index, realizing index proﬁling at high ac-curacy. To verify the feasibility of this method, an experiment is performed using a confocal microscope with a NA of 0.7and a wavelength of 675nm. It is found that the measured refractive index with our suggested method is much more accurate than that obtained with normal incidence assumption, and the accuracy can be improved at least 1%.
2. Calculation of Incident and Reﬂected Fields
Under the assumption of normal incidence, the re-ﬂectance of a sample surface is determined by its refractive index using the Fresnel’s equation ex-pressed in Eq. (1),and the refractive index n ͑x , y ͒can be retrieved from the detected reﬂectance proﬁle R ͑x , y ͒with Eq. (1a):
R ͑x , y ͒ϭͩn Ϫ1ͪ
n ͑x , y ͒ϭ
These two equations are obtained by assuming a plane wave incident normally on an inﬁnitely wide interface. It is obvious that this requirement cannot be ﬁlled by an index proﬁler, where the incident light
should be considered as an angular spectrum repre-sentation of plane waves, and each angular compo-nent will take a different reﬂection coefﬁcient .When the NA is small, the higher order angular com-ponents are negligible, and Eq. (1a)can be used for index retrieval. However when a tightly focused Gaussian beam is used to illuminate the specimen for higher spatial resolution, the higher order angular components become remarkable, and accordingly Eq. (1a)becomes invalid. To measure the index value accurately, a retrieval method with a higher accuracy should be used.
Figure 1shows a schematic of focusing of a plane wave with an objective, where x , y , and z are three axes of the Cartesian coordinates. f is the focal length of the focusing objective lens, is the polar angle from the negative z axis, is the azimuthal angle from positive x axis. The incident ﬁeld is assumed to be a fundamental Gaussian beam before the lens and po-larized in the x direction. The magnitude of the Gaussian beam just after the focusing objective lens can be expressed as
E ͑ ͒ϭE 0exp ͓Ϫ͑f sin ͞w 0͒2͔,
where w 0is the beam waist. The focused electric ﬁeld of this Gaussian beam, which is used as the incident light in index proﬁling, can be calculated as
E inc ͑x , y , z ͒ϭC
͵0 m ͵
͑ , , ͒exp ͓jk ͑s x x ϩs y y ϩs z
ϫsin d d , (3)
where m is the maximum incident angle, is an azimuthal angle, C is a constant, and
Fig. 1. (Coloronline) Schematic of a linearly polarized collimated Gaussian beam focusing in a coordinate system (x , y , z ). The di-rection of polarization of the electric ﬁeld is along the x axis.
APPLIED OPTICS ͞Vol. 47, No. 2͞10January 2008
s x ϭsin cos , s y ϭsin sin ,
s z ϭcos .
͑ , , ͒are the x , y , and z components of the electric ﬁeld given by [11,12]
ϭexp ͓Ϫ͑f sin ͞w 0͒2͔͑cos ͒1͞2͓cos ϩsin 2 ͑1Ϫcos ͔͒, (5)
ϭexp ͓Ϫ͑f sin ͞w 0͒2͔͑cos ͒1͞2͑cos Ϫ1͒sin cos , (6) ϭexp ͓Ϫ͑f sin ͞w 0͒2͔͑cos ͒1͞2sin cos .
Figure 2shows the components of the electric ﬁeld in the geometrical
focal plane in Fig. 1, where Figs. 2(a),2(b),and 2(c)are x , y , and z components, respec-tively. The focal length f and the NA of the objective are assumed to be 2mm and 0.7, respectively. Since a laser beam with a diameter of ϳ2.1mm is used in our practical experiment, the beam waist w 0in Eq. (3)
is assumed to take a ﬁxed value of 1.05mm. From Fig. 2we can ﬁnd that the focused ﬁeld is obviously depolarized, that is, except the x component; a strong y and z component can also be found in the focal plane. This result clearly shows that the focused light of this kind cannot be regarded as a normal incidence anymore. Though no z component is found in the light normally incident on the x – y plane, the amplitude of the z component shown in Fig. 2(c)is almost compa-rable to that of the x component, and accordingly, Eq. (1a)is invalid.
In calculating the incident light with Eq. (3),the focused light is regarded as the superposition of different angular components corresponding to var-ious . The reﬂected light from the surface can be determined with the same scheme. To be speciﬁc, the reﬂection of an angular component with a given incident angle can be exactly determined with relation to Fresnel’s formula. The overall reﬂected ﬁeld is treated as the superposition of the reﬂected ﬁeld of different angular components. To do so, each angular component is decomposed into its parallel and perpendicular components part with respect to the incident plane .With conventional Fresnel’s equations, the parallel component of the reﬂected ﬁeld e pr and the perpendicular component of the re-ﬂected ﬁeld e sr can be written as
e pr ϭϪ͑ cos ϩ sin ͒r p , (8)e sr ϭϪ͑ sin Ϫ cos ͒r s ,
where r p and r s are reﬂection coefﬁcients of TM and TE mode,
r p ϭ
n 2cos Ϫ͑n 2Ϫsin 2 ͒1͞2
n 2cos ϩ͑n 2Ϫsin 2 1͞2,
r s ϭ
cos Ϫ͑n 2Ϫsin 2 ͒1͞2
cos ϩ͑n 2Ϫsin 2 1͞2
The x , y , and z components of the reﬂected light can be written as
r ϭe pr cos ϩe sr sin , r ϭe pr sin ϩe sr cos ,
r ϭ r normal .
By introducing Eq. (11)into Eq. (3)and calculating the integral, the reﬂected electric ﬁeld is directly ob-tained. Figure 3shows the distributions of x , y , and z components of the reﬂected light on the interface of air and planar material with an index of 1.456. We can ﬁnd that, except for being weak in intensity, Fig. 3shows ﬁeld distributions similar to that of the in-cident light. For clarity, Fig. 4shows the intensity distribution of the calculated incident ﬁeld and the transmitted ﬁeld around the interface between the
Fig. 2. (Coloronline) (a),(c),and (e)are x , y , z components of the incident electric ﬁeld in the focal plane. The wavelength of the beam is 675nm, the focal length f is 2mm, the beam waist of fundamental Gaussian proﬁle, denoted by w 0, is 1.05mm, and the NA of the objective lens is 0.7.
10January 2008͞Vol. 47, No. 2͞APPLIED OPTICS
air and a planar material with an index of 1.456, where we can clearly ﬁnd that only a part of the light transmits through the interface. 3. Refractive Index Retrieval and Error Analysis
It is well known that the reﬂectance of a surface is the ratio of the energy of the reﬂected light to that of the incident light. That is
͵P reflected d x d y
P incident d x d y
͵͑ԽE x r H y r ԽϩԽE y r H x r Խ͒d x d y ͑ԽE x inc H y inc ԽϩԽE y inc H x inc Խ͒d x d y , (12)
where P reflect and P incident are the Poynting ﬂux of the reﬂected and the incident lights at the interface; the E x inc , E y inc , E x r , and E y r are the x , y components of the incident and reﬂected electric ﬁeld on the ﬁber surface, respectively, H x inc , H y inc , H x r , and H y r are the related x , y components of the incident and reﬂected magnetic-ﬁeld, which can be calculated with a similar scheme described in .
The solid curve in Fig. 5shows the calculated re-ﬂectance changing with the NA of the objective used. In the calculation, the material index is assumed to be 1.456, and various NAs are obtained by ﬁxing the radius of the objective to be 1.4mm and varying its focus length from 28to 1.47mm. It is clearly shown that the reﬂectance always increases with the NA of the objective used. Since all angular components and its corresponding reﬂectance are taken into ac-count to derive the above formula, this calculated reﬂectance can be regarded as the real reﬂectance in practical experiments. In Fig. 5, the dotted line tangent to this curve is the reﬂectance predicted un-der normal incident assumption, and we can ﬁnd that, only when the NA of the objective used is less
Fig. 5. Reﬂectance change according to an increase in NA. The reﬂectance increases according to the NA of a focused Gaussian beam. The focal length f is 2mm. The beam waist, denoted by w 0, is 1.05mm.
Fig. 3. (Coloronline) (a),(c),and (e)are x , y , z components of the reﬂected electric ﬁeld in the focal plane. The wavelength of the beam is 675nm, the focal length f is 2mm, the beam waist of fundamental Gaussian proﬁle, denoted by w 0, is 1.05mm, and the NA of the objective lens is 0.7.
Fig. 4. (Coloronline) Refraction of focused light on the air–ﬁber interface.
160APPLIED OPTICS ͞Vol. 47, No. 2͞10January 2008
than 0.5, that the predicted index value with the normal incidence assumption can coincide with the practical reﬂectance well.
Figure 6(a)shows the reﬂectance changing with the material index when objectives of various NAs are used. The curve labeled with the star is the reﬂectance under paraxial approximation, and the other curves labeled with the circle, triangle, and square are the reﬂectance calculated by Eq. (12)with NAs of 0.7, 0.8, and 0.9, respectively. It is obvious that the difference between the calculated reﬂectance with Eq. (12)and that with normal incidence assump-tion becomes obvious with increasing material index. This means that the assumption of normal incidence can be adopted for index retrieval only when both the material index studied and the NA of the objective used are low enough. Unlike the assumption of nor-mal incidence, the relation between the reﬂectance and the refractive index in Eq. (12)is mathematically complex, and no simple analytic formula can be di-rectly used for index retrieval. For practical experi-ment with the given NA, the change of reﬂectance with material index can be simulated in advance, and
then the detected reﬂectance at any point can be converted into corresponding material index with this simulated curve during the scanning of the sam-ple surface, realizing accurate index proﬁling. For better understanding of the advantages of this sug-gested index retrieval method, Fig. 6(b)shows the relative errors of the index value obtained by normal incidence method with respect to that retrieved with our suggested method. In Fig. 6(b)n is the refractive index retrieved with our suggested method, which can be regarded as the practical index value due to its high accuracy, and n pa is the refractive index obtained with normal incidence assumption. In this ﬁgure, we can clearly ﬁnd that the relative error of the com-monly used method under the assumption of normal incidence can reach 1%when the objective with a NA of 0.7is used, and for a NA of 0.9, this relative error can reach 12%.
4. Experiments and Discussions
Figure 7shows the schematic of our experimental setup for refractive index proﬁling, which is a modi-ﬁed confocal scanning optical microscope .The wavelength used is 675nm, the focal length f of the objective is 2mm, the beam waist of the Gaussian beam incident on the objective is 1.05mm, and the NA of the objective used is 0.7, corresponding to a maximum incident angle m in Eq. (2)of 44.4270°(theobjective model is M Plan Apo 100ϫ, Mitutoyo). Since all of these parameters are the same as that used for calculating the reﬂectance curve labeled with the cir-cle in Fig. 6, this curve can be used for the index retrieval in our experiment. The experimental result is given in Fig. 8. For clarity in display, only the difference between the refractive index of the ﬁber core and that of the ﬁber cladding is proﬁled in Fig. 8, where the curve [a ]is the experimental result of our suggested methods, and the curve [b ]is the experi-mental result obtained with Fresnel’s equation under normal incidence approximation. We can ﬁnd that
Fig. 6. (Coloronline) (a)Reﬂectance according to the refractive index with a ﬁxed NA of the lens. The star shaped spot curve is the change in reﬂectance with a refractive index obtained by consid-ering the normal incidence. The other spot curves are the changes in reﬂectance with refractive indices while considering the angular spectrum representation. (b)Refractive index error; n is the re-fractive index in the angular spectrum representation, and n pa is the refractive index in the normal incidence method.
Fig. 7. (Coloronline) Schematic of modiﬁed ﬁber-type confocal scanning optical microscope. The wavelength of the beam is 675nm, and the focal length f is 2mm. The beam waist of funda-mental Gaussian proﬁle, denoted w 0, is 1.05mm, and the NA of the objective lens is 0.7.
10January 2008͞Vol. 47, No. 2͞APPLIED OPTICS
the maximum refractive index differences are 0.0140and 0.01368in curves [a ]and [b ],respectively, that is, the refractive index difference of the optical ﬁber that is calculated by considering the difference in the
reﬂectance for different angular spectrum com-ponents is ϳ2.5%larger than that obtained with normal incidence. Estimation on the experimental ac-curacy can be easily performed according to the curve labeled with the circle in Fig. 6(a),which corresponds to a NA of 0.7. Take the ﬁber cladding for example, which has a index of ϳ1.456, we can ﬁnd that the accuracy of this experimental result is proven to be ϳ3.98%compared to that under normal incidence. The relative refractive index calculated from the normal incidence is expected to be higher in Fig. 8based on the lower slope of the reﬂectance curve of normal incidence R Paraxial ͑n ͒compared with that of vector wave R Vector ͑n ͒shown in Fig. 6(a).However, there is another factor, which needs to be considered; there are offsets in reﬂectance curves in Fig. 6(a)for a given refractive index. Due to the presence of the pinhole, and the reﬂection of the light on the lens surface, it is very difﬁcult or impossible to obtain the absolute value of the reﬂected optical power from a surface in our measurement. We measure relative intensity differences of reﬂected optical power in the experiment assuming that the refractive index for a cladding is exactly known. Therefore, we need a cal-ibration constant, which relates measured reﬂected optical power and each reﬂectance curve in Fig. 6(a).As the measured reﬂected optical power I ͑n ͒is a func-tion of refractive index n in our measurement system, we can relate I ͑n ͒with reﬂectance curves R Vector ͑n ͒and R Paraxial ͑n ͒, which are calculated with vectorial and normal incidence wave analyses, respectively, by using two constants. These two constants can be ob-tained with
I ͑n ϭn clad ͒ϭR Vector ͑n ϭn clad ͒1ϭR Paraxial ͑n ϭn clad ͒
where n clad is the refractive index of the cladding of a ﬁber. As can be seen in Fig. 6(a),R Vector ͑n ϭn clad ͒ϾR Paraxial ͑n ϭn clad ͒, and then we have a 1Ͼa 2. Even though R Vector ͑n ͒has a larger slope than R Paraxial ͑n ͒in Fig. 6(a),a new calibrated reﬂectance function of R Vector ͑n ͒͞a 1has a lower slope than another calibrated reﬂectance function of R Paraxial ͑n ͒͞a 2. This is the rea-son why the index difference measured with normal incidence method is smaller than that obtained with the vectorial method in Fig. 8.
The reﬂection-type confocal microscopy of high NA is
suggested for the measurement of refractive index of the ﬁber with high accuracy and high spatial resolu-tion. The properties of this kind of system are ana-lyzed with a 3D vectorial formula. It is found that under the illumination of a tightly focused Gaussian beam, the Fresnel formula based on normal incidence cannot be used for the index retrieval anymore, be-cause the end surface of the ﬁber is essentially illu-minated by a cone of light in this case, and the variations in the reﬂectance for different angular components should be taken into consideration. The reﬂectance of the light on the planar surface always increases remarkably with the NA of the objective used, and only when the NA is less than 0.5can it coincide well with the predications of the commonly used Fresnel formula under normal incidence situa-tions. For a NA of 0.9, the accuracy of the index value retrieved with our suggested method based on 3D vectorial analysis is improved ϳ12%compared to that of the common method. Therefore, the reﬂection-type confocal scanning optical microscope is much more suitable for proﬁling the index of the ﬁber at high accuracy and high spatial resolutions.
This work was supported by Creative Research In-itiatives (3DNano Optical Imaging System Research Group) of MOST ͞KOSEF.
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Fig. 8. (Coloronline) Relative refractive index difference of a commercial dispersion compensated ﬁber, where [a]is the refrac-tive index difference proﬁle calculated by Fresnel’s equation using the angular spectrum representation and [b]is the refractive index difference proﬁle calculated by Fresnel’s equation at normal inci-dence using the paraxial approximation.
APPLIED OPTICS ͞Vol. 47, No. 2͞10January 2008
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用插针法定玻璃折射率 用插针法测定玻璃折率 用插针法测定玻璃折射
长方玻璃砖一块 白纸张，大头针四枚，铅笔一，直尺一把，量角器一个，图钉四颗 【实验步骤】1、在板上铺白纸并用钉固定，在白纸上画一直线aa、作为面，过aa、上的一点O画出界面的法线NN、。并画出一条线段AO入射光。 2、把方形玻璃砖放在白纸上，使的一aa、重合，再画出玻璃砖的另一bb、。 3、在线段AO上竖直地插上两枚大头针P1、P2，使它距离远一些。P2距离玻璃
6、用量角器量出入射角折射角的度数，查出它们的正弦值，并把这些数据记在表格里。 7、用上述方法别求出入角为15o 30o45o 60o 75o 时的折射角，查射角折射
实验次数 入射角θ 折射角θ sinθ sinθ 折射率
【误差分析】 【误差分析】 【误差分析】
(2)如图(乙)所示，乙同学利用插针法定入射光线、折射光线后，测得的入射角不受影响，但测得的折射角比真实的折射角偏大，因此测的折射率小. 2.在测定玻璃的折射率实验中，所用玻的两表面aa′和bb′不平行，如图示，得的折射率将( ) A(偏大 B(偏小 C(不变 D(无法确 【当堂训
A(做实验时，线是由空气射入璃的 B(玻璃的折射率为0.67 C(玻璃的折射率为1.5 D(玻璃临界角的正为0.67 2(在如下图所示“测定璃的折射率”的实验中，P1、P2、P3和P4是四枚大头针的位置，同学的验步骤如
11sinr,,1.5，所以B错，C对;sinC,,,0.67，D对(答案 ACD 0.67nsini
10.002,8.002,1.5. 答案 1.5 4.00
cosθ1sin,90?,θ1,答案 (1)(或) (2)大 cosθ2sin,90?,θ2,
1) 一只大于1m 见方的塑料箱,
7) 一把电式万能驼毛刷（核工业产品，CostaMesa CA）。用丙酮清洗刷
9Obreimov,I.V.and Trekhov,E.S. “Optical Contacting if Polished Surfaces, ”Sovier Physics JEPT,5,235-242(1957)
Paul,H.E. “Amateur Telescope Making ”Book 3,Seientific,Aerican,New
摘要：折射率是质的一个重要理量，不同的物质对不同波长的光折射率是不同的。文章提出一种利用劈干涉装置测定液体折射率的原理和方法，验表明，该简易可行，很有实用值。 关键词：劈尖干涉; 折射率; 光程差 中图分类号：献标识：文章
Interference to measure liquid refractive index
Jiang ling,Wang xindan,Fan yingying
(China University of Petroleum(Huadong) Department of applied physics,Shandong Tingdao 266555) Abstract: Refractive index is an important physical quantity, different material with different wavelengths of light refractive index is different. Using split peak interference device is presented in this paper the principle and method of measuring liquid refractive index, the experiments show that this method is simple and feasible, and is of great practical value. Keyword: Cleft tip interference, refractive index, optical path difference
根据的干涉理论，当频率相的两束光波在相遇点具有相同的振动方向和固定的位相 差时，们会发生光的干涉现象，产生明暗相的干涉条。光的干涉有分波阵干涉和分振幅干涉两种形式，劈尖干涉就是一种分振幅。 2.1等
如图1(a)折射率为n 的透明介质薄膜两界面Σ1和Σ2成夹角α的劈形膜, 用波长为λ的单色平行光照射, 在Σ1上就出现了平行于劈棱的明暗相间的直条纹, 如图1(b)所示, 图中线表示极小, 虚线表示极大。根据图1(c),C 点交的两反射光线1′和2′之间光程差 δ=n(AB+BC)-n1CD-λ2 (1.1) 劈形棱角α很小而膜很薄, 实用中往往线似于垂直射,A 点跟C 点分靠近, 它们所在的膜厚以视为相等, 设为h, 则可推导出δ=2h n2-n21sin2i-λ2(1.2) (式中i 为入射光线的入射角), 上式可以看, 对于确定的实验装置, δ由h 唯一确定, 即劈膜上凡有同厚度的, 其光程差相同, 位于同条条纹上, 叫等厚条纹。这种在膜表面上形成明相间的等条纹的现象称为等厚干涉, 形棱角α很小, 所又可称之
用两块表面十分平整的璃片(测量时玻璃片长约8cm) 的一端之间夹一发丝, 另一端压紧形成劈尖形薄膜, 图2所示在垂单色光源(即i=0)照射时, 由(1.2) 式得程差
δ=2nh-λ2 (2.1) 由上式不难看出, 相邻明条纹(或暗条纹) 位置的厚度
Δh=λ2n (2.2) 由于劈的棱角α十分小, 故相邻明条纹(或条纹) 的间
Δx=Δhtg α≈Δh α
将(2.2)式代入上式, 可得: Δx=λ2α (2.4) 对空气薄膜(n=1)则相邻明条纹(或纹) 的间距
Δx 空=λ2α (2.5) 两块玻璃片中充满折射详细为n 的液体薄膜时, 应的纹间距
Δx 液=λ2n α (2.6) 将(2.5)式以(2.6),可得液体折射率n=Δx 空/Δx 液 (2.7) 这样通过实验分别测出劈形膜为空气和待测液体时对应的条纹间距由,(2.7)就可计算 待测液体的折
(1) 将器按图3所示装置好, 直接用单色扩展源钠照明, 由光源S 发出的光照射到
片G 上, 使一部分由G 反射进入劈形薄膜上, 先用眼睛在坚直方向察, 调节玻璃片G 的高低倾角度(约与水平方向成45°角), 使移测显微镜从视场中能观察到色明的视
(2) 调节测显微镜M 的目镜, 使目镜中看到的丝为清晰, 将移测显微镜对准劈形膜中
上移动镜筒, 对干涉条进行调焦, 使看到的条纹尽可能清晰, 并与显微镜的量叉丝之间无视差。测量时, 显镜的叉丝调成其中一根叉丝显微镜的移动方向相垂直, 移测时始终保持这根与干条纹相
(3) 选干涉条纹的测量范围, 旋转移测显微的螺从左到右次测出10条间隔干涉亮
(4) 在两玻璃片间滴上1~2酒精或 蔗糖溶液, 成劈水膜, 并使水膜均匀, 无气泡,
1,2,3骤, 将测量结果记录于表格一中; (5)依同样方法, 重复以上步
文章出的是利用劈尖现成的验装置, 用干涉法来测量液体的折射率。此实验方法和计算公式教材上介绍的几种测量方法要简单得多。并且测量也很小。这既增加了尖装置的实验范围, 又为液体折射率的测量提供了一种方法。 参考
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