5.2 Fringe Localization - optics.hanyang.ac.kroptics.hanyang.ac.kr/~shsong/ch5-Coherence and Fringe localizatio… · Quasimonochromatic extended sources can be treated as a collection

Coherence and Fringe Localization 41

©2006 Tom D. Milster

is the mutual intensity. In this case, is the normalized mutual intensity. Noticethat, if pinhole positions and are coincident, = = is theirradiance at that point in the light field. Also, the self coherence of the light fieldis given by .

5.2 Fringe LocalizationWaves traveling through space interfere and produce visible fringes if the

conditions are right. In particular, the waves must have some degree of spatial andtemporal coherence over a region of space. Fringe localization defines the regionof space where interference occurs and fringes with reasonably good contrast areobserved. There is a wide variety of methods for producing fringes – ranging froma simple parallel plate to complex interferometers, and each has an associatedregion in space where high-contrast fringes may be observed. The location of thisregion relative to the components of the interferometer depends on properties ofthe source and geometry of the interferometer. In this section, basic properties offringe localization are illustrated for various types of sources and interferometers.

There are several different categories of fringe localization in interferometers.Some of the more common categories are

1) Localized everywhere: The fringes have high visibility everywhere in the observation space. (Also called non-localized or unlocalized.)

2) Localized: The fringes have high visibility over some surface in the observation space. The fringe localization surface can be curved.

3) Localized at infinity: A lens is used to transform the angular distribution of fringes into a spatial distribution of fringes on an observation plane with high visibility. The observation plane is usually at the back focus of the lens. Otherwise, fringe visibility is too low for observation. (Also called Haidinger’s fringes or fringes of equal inclination.)

4) Fringes of equal thickness: Localized fringes, where fringes correspond to contours of constant thickness between two surfaces.

5.2.1 Discussion of sources for illuminationSeveral types of sources are used for illumination in interferometers and other

optical systems. This section discusses basic coherence properties of four differenttypes of sources: monochromatic point source, polychromatic point source, quasi-monochromatic extended source and polychromatic extended source.

5.2.1.1 Monochromatic point source

Any interferometer using an ideal, monochromatic point of light as a sourceproduces high-visibility fringes everywhere in observation space. In this case,

μ12

P1 P2 J12 Γ11 0( ) I1

Γ11 τ( )

42 Chapter 5


two-point-source interference produces a hyperboloidal fringe field, as shown inFig. 5.30. The fringes occur where OPD0 is an integer number of wavelengths.These fringes are nonlocalized or localized everywhere or unlocalized. Of course,there is no truly ideal monochromatic point source of light. However, a goodapproximation is found by using high-quality laser sources, where the coherencelength due to the temporal bandwidth of the laser is much greater than themaximum . That is,

, (5.75)

where n is the refractive index in observation space.

5.2.1.2 Polychromatic point source

With a polychromatic point emitter in the source space, two-point-sourceinterference produces hyperboloidal fringe fields in observation space for eachfrequency component of the source’s power spectrum. Portions of the fringe fieldin the observation space with high OPD0 between the two sources exhibit reducedcontrast due to wavelength-dependent fringe variations. Fringe visibility dependson where the observation is made. For example, if and n = 1, fringevisibility near and beyond the m = +/- 4 fringes in Fig. 5.30 would be very poor.

Fig. 5.30. Hyperboloidal fringe field generated from a pair of coherent point sources. Fringe surfaces are hyperboloids of revolution around the symmetry line that passes through both points. The fringe order m increases or decreases away from the plane of zero OPD0, which is the m = 0 fringe. Asymptotic lines are given by

.

OPD0

cnΔv---------- OPD0( )max»

a1

a2

m = 1

m = 2

m = 0

m = 3m = 4m = 5m = 6

m = -1

m = -2m = -3m = -4m = -5m = -6

Asymptote

SymmetryLine

d m 2=

θtan mmλ

d2 m2λ2–---------------------------=

Δv c 4λ⁄=



5.2.1.3 Quasimonochromatic extended source

Quasimonochromatic extended sources can be treated as a collection ofindependent emitters oscillating with wavelength .1 This ideal case might be agood approximation for a highly filtered tungsten lamp or a spectroscopic lamp.The total fringe field in observation space is simply the addition of fringe fieldsfrom individual source points. The net effect is usually a decrease in fringevisibility, but the characteristics of the visibility reduction depend greatly on thegeometry of the interferometer.

5.2.1.4 Polychromatic extended source

Analysis of systems with a polychromatic extended source must take intoaccount both the temporal bandwidth of the source and its finite extent. In thesimplest case, the polychromatic nature can be considered separately. That is, firstconsider the source as a collection of point emitters at , then estimate theadditional visibility reduction due to the temporal bandwidth. However, somecomplicated sources may not be well described in this way. A more completediscussion of this topic is provided in Section 5.3

5.2.2 General procedure for determining fringe localizationThere are three basic steps in assessing fringe localization for most interferom-

eters, which are

1) Determine the fringe period and position in the observation plane given an ideal, monochromatic point source in the source plane;

2) Integrate fringe patterns resulting from positioning the point source over the spatial extent of the distributed source, and/or from the separate frequencies in the power spectrum; and

3) Define regions in the observation space where V is large enough (V > 0.2) to detect fringes.

Usually, results from Step (2) produce a relationship that can be expanded intoa closed-form solution for fringe visibility in observation space as a function ofsource properties and interferometer geometry. If a closed-form solution can’t befound, a computer simulation can provide the necessary information.

1. In order for the quasimonochromatic assumption to hold over the observation space, Eq. (5.75) must be satisfied.

λ

λ

44 Chapter 5


5.2.3 Application of fringe localization techniques to specific interferometers

In this section, several specific interferometers are analyzed with respect tofringe visibility for different types of sources. The interferometers examinedinclude Young’s double pinhole interferometer (YDPI), the plane parallel plate(PPP), a wedged plate, thin films, Fizeau interferometers, the Michelson interfer-ometer, the Twyman-Green interferometer, Lloyd’s mirror and Fresnel’s biprism.

5.2.3.1 Young’s double pinhole interferometer (YDPI)

If a monochromatic point source is used with the YDPI displayed in Fig. 5.1,pinholes produce a coherent pair of source points. The fringes are unlocalized inthe entire observation space, which is the entire region to the right of the pinholeplane.

If the YDPI is used with an on-axis polychromatic point source, it produces aregion of fringe localization close to the optical axis, as shown in Fig. 5.31, whichis specified by Eq. (5.28). The region of fringe localization is bounded by thefringes corresponding to +/- OPD0 that produce the minimum acceptable visibilityV.

Spatial coherence of a YDPI with an extended quasimonochromatic source isexamined in detail in Section 5.1.5. Visibility of the fringes in observation space

Fig. 5.31. Localization of a YDPI with an on-axis polychromatic point source. A YDPI is used with a polychromatic point source exhibiting a coherence length of . Fringes are localized near the axis where V > 0.2, which occurs until approximately OPD0 = . The gray region indicates where V < 0.2.

ys y0

OPD0 2=

Pinhole Plane

d

z0zs

V > 0.2

V < 0.2On-Axis Polychromatic Point Source

l 2=

Δl 2λ=

2λ



is a function of the wavelength, the size of the source and its distance to thepinhole plane. Larger sources and smaller distances produce lower visibility.Unlike temporal coherence with the YDPI, visibility due to an extended source isnot a function of position in the observation space. Equation (5.53) provides anexplicit relationship between the source distribution, interferometer parameters,and visibility.

5.2.3.2 Plane parallel plate (PPP)

Perhaps the simplest form of interferometer is the plane parallel plate, asshown in Fig. 5.32. Light from point source a reflects from the two surfaces of theplate toward the observation plane. The reflection of the real ray from the topsurface of the plate (surface 1) appears to originate from virtual source point a1.Part of the real ray refracts into the plate and reflects off the bottom surface(surface 2). When this ray exits the plate from the top surface, it appears tooriginate from virtual source point a2. The coherent point-source pair a1 and a2that are produced by the reflections have a symmetry line through a1a2 perpen-dicular to the surface of the plate, and they produce a hyperboloidal fringe field.Direct light from the real source a to the observation plane is blocked by an opaquescreen, in order not to cause additional interference fringes. Notice that obser-vation of the region with highest visibility along the m = 0 fringe is impossible,due to the position of the plate relative to the virtual sources.

If a is a monochromatic point source, fringe visibility is unity everywhere inthe observation space, and the fringes are unlocalized. For practical experiments,fringes can be observed everywhere in the observation region with good visibilityif , where t is the thickness of the plate, n is its index of refractionand is the bandwidth of the source power spectrum. If a is a polychromaticpoint source with , visibility reduces as the fringe order increases,because of increasing OPD0. Fringes of adequate visibility can be observed if theobservation plane intersects fringes where .

For a quasimonochromatic extended source, the PPP produces a reflection ofeach source point for each surface of the plate, as shown in Fig. 5.33. Each virtualsource pair produces a hyperboloidal fringe pattern in observation space. Theindividual fringe patterns are offset in the horizontal direction according to theposition of the point of origin in the source. If the source length is greater than thefringe spacing, the hyperboloidal pattern washes out, and zero visibility ismeasured in observation planes close to the plate. In Fig. 5.33, fringes from onlyfive equally-spaced source points are shown, but the reduction in fringe visibilityis obvious.

In the particular geometry of Fig. 5-33, a useful characteristic of the fringepatterns are that asymptotes for each fringe order m are parallel. This conditionexists because the virtual images of the source, a1b1 and a2b2, are parallel to each

2t n⁄ c Δv⁄«Δv

2t n⁄ c Δv⁄>

OPD0 mλ c Δv⁄<=

46 Chapter 5


other. If the observation plane is displaced far enough in the vertical direction sothat the distance between adjacent fringe orders is larger than the length of thesource, fringes with acceptable visibility can be measured. A practical way ofachieving a long effective distance between the source pairs and the observationplane is to use a lens positioned as shown in Fig. 5.34, where the observation planeis at the back focus of a lens. In this configuration, the image of the observationplane is at infinity with respect to the plate. Each fringe order at angle

focuses to a circle in the observation plane, due tothe symmetry of the hyperboloidal fringe field. The radius of the circle is givenby rm = f cot θm. The combination of all circles yields a concentric fringe pattern.Like with two-point-source interference, the maximum OPD0 of this patterncorresponds to the innermost fringe at the center of the circular pattern. Anysignificant width of the source power spectrum washes out the fringe in this

Fig. 5.32. PPP with polychromatic point-source illumination. Point source a is shown positioned a distance l above a plane parallel plate (PPP) . A coherent pair of virtual point sources a1a2 is created from the two Fresnel surface reflections from the plate. The coherent point-source pair produces a hyperboloidal fringe field in the observation space. If a is polychromatic, fringe visibility reduces as fringe order number m increases. For example, the observation-space region where V < Vmin is shown in gray. An opaque beam block is used to block direct source light from reaching the observation plane and creating additional fringes.

a

a1

a2

d 2t n=

t

l

l

n

Plane Parallel Plate (PPP)

Surface 1Surface 2

Polychromatic Point Source

BeamBlock

Observation Plane

Real-Ray Path

Intersectionof Real Rays FromEach Surface

m = 0Virtual Point- Source Images

θmmλd

-------asin mλ2t n⁄------------asin= =



central region, due to temporal coherence effects. The circular fringes shown inFig. 5.34 are called Haidinger’s fringes. Because the fringes are only observed ata sufficiently large distance from the plate, the fringes are said to be localized atinfinity. Since the each fringe results from many source points, but only from aspecific angular range, the fringes are also called fringes of equal inclination.

5.2.3.3 Wedged plate, thin films and Fizeau interferometers

Closely related to the plane parallel plate is the simple wedge, as shown in Fig.5.35 with polychromatic point-source illumination. The virtual point sources a1and a2 now exhibit a tilted symmetry line, and their separation d is a function ofsource a’s position along the x axis. With this geometry, the m = 0 fringe can

Fig. 5.33. PPP with quasimonochromatic extended-source illumination. Extended source ab is shown positioned a distance l above a plane parallel plate (PPP). A coherent pair of virtual point sources is created for each point in the extended source from the two Fresnel surface reflections of the plate. Each coherent point-source pair produces a hyperboloidal fringe field in the observation space. The hyperboloids are shifted with respect to each other according to the source-point position. A reduction in visibility is observed in the observation space, due to overlap of the fringes. (In the figure, the fields from only five of the source points are shown.) An opaque beam block is used to block direct source light from reaching the observation plane and creating additional fringes.

a1

a2

d 2t n=

t

l

l

n

Plane Parallel Plate (PPP)

Surface 1Surface 2

ExtendedQuasi-Monochromatic Source

BeamBlock

Observation Plane

Real-R

ayPath

Intersectionof Real Rays FromEach Surface

m = 0

b1

b2

ba

Extended SourceImages

48 Chapter 5


intersect the observation plane. Since OPD0 = 0 at this point, the fringe visibilityis maximum. As shown in Fig. 5-36, OPD0 at the observation plane between thereal-ray paths originating at a on the source and reflecting from each surface of thewedge is approximately

Fig. 5.34. Haidinger’s fringes. Haidinger’s fringes or fringes of equal inclination are formed when images of the extended source are parallel to each other in observation space. Although fringes have low visibility near the source images, a lens with focal length f can be used to form the characteristic concentric-ring fringe pattern with good visibility. The pattern forms because fringe asymptotes of corresponding fringe orders are parallel. Since the lens effectively sets the observation at an infinite conjugate with respect to the source images, the fringes are said to be localized at infinity.

a1a2

b1b2

mb = i + 2mb = i + 1mb = i

ma = ima = i + 1 ma = i + 2

y0

x0y0

Observation Plane

f

Lens

Focus form = i +2fringes

m = i +1m = i

m = i + 2fringe



, (5.76)

where n1 is the refractive index of the incident medium, n2 is the refractive indexof the plate, t(x) is the thickness of the plate at the point of reflection, and is theangle of refraction inside the wedge. For a bright fringe at the observation plane,

, where is the phase difference between Fresnel reflectionsfrom the surfaces. For example, if n1 = 1, n2 = 1.5 and n3 = 1, .

Now consider the combination of a wedged plate with an extended, quasimo-nochromatic source, as shown in Fig. 5.37. Surfaces of the plate reflect light fromthe source, so two source images are formed in the observation space. Eachcoherent pair in the images produces hyperbolic fringes. Unlike the cases studiedso far, the images are not parallel, and the symmetry lines between coherent pairsare tilted with respect to each other. In addition, the distances d between coherentpairs are not equal, unlike with the PPP. This angled geometry complicates thediscussion of fringe localization. The shifts, tilts and spacing differences producehyperboloidal fringe fields that are also tilted, shifted and scaled.

Typically, the wedge under test is very thin with respect to the distance fromthe source to the top surface. Effectively, the wedge is a thin film. In addition, thewedge angle is small. Figure 5.38 shows the fringe fields in observation spacebelow the wedge for the geometry of Fig. 5.37 with t = 1 μm, wedge angle = ,wavelength = 500 nm, n = 1.5, l = 200 mm, and separation between source pointsa and b of 10mm. The fringe fields from coherent pairs are tilted and displacedwith respect to each other. However, the fringe fields converge with like ordersoverlapping at the wedge. That is, the mth order of the fringe field from source paira1a2 overlaps the mth order of the fringe field from source pair b1b2 at the wedge.Therefore, the visibility is maximum at the wedge and the fringes are said to belocalized at the wedge. An external optical system or the eye of an observer canbe used to reimage the wedge in order to detect high visibility fringes. Notice thatthe m = 0 fringe of each coherent pair intersects the wedge. Therefore, highvisibility is obtained near the intersection of the m = 0 fringe and the wedge, even

OPD0 n2 AB CB+( ) n1AD–=

n22t x( )

θ2cos-------------- n1AC θ1sin–=

n22t x( )

θ2cos-------------- n12t x( ) θ2 θ1sintan–=

n22t x( )

θ2cos-------------- n12t x( )

θ2sin

θ2cos--------------

n2 θ2sin

n1

-------------------–=

2n2t x( )θ2cos

------------------ 1 sin2θ2–[ ]=

2n2t x( ) θ2cos=

θ2

OPD0 mλ= λφ2π------+ φ

φ π=

0.1°

50 Chapter 5


with a broad bandwidth light source. Over a wider expanse of fringe orders, thelocalization surface is a sphere with diameter l, as shown in Fig. 5.39.

The OPD0 between real-ray reflections from wedge surfaces determines fringepositions according to Eq. (5.76). When , a bright fringe isobserved, and when , a dark fringe is observed. Noticethat OPD0 in Eq. (5.76) is directly proportional to the thickness of the wedge t(x).Therefore, fringes map out contours of constant wedge thickness. These fringesare called fringes of equal thickness,

If the wedge is not a simple linear function, the fringes are not distributed instraight, equally spaced lines. For example, the geometry shown in Fig. 5.40shows a Fizeau interferometer, where a linear air wedge is formed between a testpiece and a reference flat. For most of the test surface, the wedge is linear andfringes are straight and equally spaced. In the region of the bump, fringes move

Fig. 5.35. Wedged plate with polychromatic point-source illumination. Point source a is shown positioned a distance l above a wedged plate. A coherent pair of virtual point sources a1a2 is created from the two Fresnel surface reflections of the plate. The coherent point-source pair produces a hyperboloidal fringe field in the observation space. If a is polychromatic, fringe visibility reduces as fringe order number m increases. For example, the observation-space region where V < Vmin is shown in gray. An opaque beam block is used to block direct source light from reaching the observation plane and creating additional fringes.

a

a1

a2

t(x)

l

l

n

Wedged Plate

Surface 1

Surface 2

Polychromatic Point Source

BeamBlock

Observation Plane

Real-Ray Path

m = 0

Virtual Images of Point Source

Intersectionof Real Rays From Each Surface

x

OPD0 m φ2π------+⎝ ⎠

⎛ ⎞ λ=OPD0 m 1

2--- φ

2π------+ +⎝ ⎠

⎛ ⎞ λ=



to positions of equal OPD0, which are toward the open side of the wedge. Theotherwise straight fringe lines are distorted in a shape characteristic of the bump.

For the case of nearly normal incidence ( ~ 0) and n = 1, OPD0 = 2t(x), andeach fringe represents one-half wavelength multiples of surface departure. For thebump shown in Fig. 5.40, the height of the defect is approximately the fractionalfringe deviation multiplied by the surface departure between fringe lines. That is,

. In order to see the fringes, the wedge is imaged onto an obser-vation plane. In a practical experiment, the observation plane can be a CCD array,or the lens and observation plane can be the lens and retina of the observer, respec-tively. Notice that localization of the fringes at the wedge is an advantage to theobserver. Spurious fringes caused by reflections from other surfaces typically havevery low visibility and do not distract from the measurement of interest.

The simplified example of a thin wedge is a very useful demonstration offringe localization, but it is also useful to develop a general technique to determinefringe localization in other types of interferometers. A simple construction can beused to determine fringe localization for two-beam interferometers using quasi-monochromatic extended sources, as shown in Fig. 5.41. Steps in the constructionare:

1) Determine positions of the two source images in observation space;

2) Locate coherent pairs a1a2 and b1b2 on ends of the source images;

3) Draw symmetry lines between a1 and a2 and between b1 and b2;

Fig. 5.36. Magnified view of real-ray intersection at the wedged plate for calculation of OPD0.

A

B

D1

t(x)

x

n1

n2

Surface 1

Surface 2

2C

n3

θ1

height λΔ2S-------=

52 Chapter 5


4) Construct the perpendicular bisectors to each symmetry line. (These bisectors are the m = 0 fringes for the coherent pairs.); and

5) Find the intersection of the m = 0 fringes. This line in space is the center of the localized fringes. The localization surface is a sphere with diameter equal to the distance between the center of the source images and the intersection of the m = 0 fringes.

5.2.3.4 Michelson Interferometer

The basic Michelson interferometer is shown in Fig. 5.42. An extended sourceis used to illuminate a two-beam interferometer through a beam splitter BS. Atfirst, the Michelson interferometer is analyzed using a quasimonochromaticextended source, with endpoints a and b. After the basic concepts of fringe local-ization are explained, the analysis is extended to polychromatic extended sources.

The first beam of the interferometer is defined by the light reflected off BS andmirror M1, which then transmits through BS toward the observation plane. The

Fig. 5.37. Wedged plate with quasimonochromatic extended-source illumination. The extended source is positioned a distance l above the top surface of a wedged plate. A coherent pair of virtual point sources is created for each point in the source from the two Fresnel surface reflections of the plate. Each coherent point-source pair produces a hyperboloidal fringe field in the observation space.

a

a1

a2

t(x)

l

l

n

Wedged Plate

Surface 1

Surface 2

Quasimonochromatic Extended Source

BeamBlock

Observation Plane

Real-Ray Path

Virtual Images of Extended Source

Intersectionof Real Rays From Each Surface From Source Point a.

x

b

b1b2



total path for the first beam is d1 + d3 + d3 + d4. The second beam of the interfer-ometer is defined by the light that passes through BS, reflects off mirror M2 andthen reflects off BS toward the observation plane. The total path for the secondbeam is d1 + d2 + d2 + d4. An imaging system is used to look into the beam splitterand form images of the fringes on an observation plane.

An unfolded diagram of the interferometer is shown in Fig. 5.43. The sourceimages are formed by reflections off the mirrors. The images a1b1 and a2b2 are asfar behind the mirrors as the source is in front. As shown, the mirrors are parallelto the source, so the source images are also parallel to the source and each other.

Fig. 5.38. Fringe fields in observation space for points a and b on the extended quasimonochromatic source illuminating a thin wedge. The fringes are localized at the wedge.

Fig. 5.39. Fringe fields in observation space over an extended range in x for many points on the extended quasimonochromatic source illuminating a wedge. The localization surface is a sphere with diameter l.

a1 a2

t(x)

l

Thin Wedged PlateSurface 1Surface 2

Coherent point-source pairs

x

b1 b2

ma = 0 mb = 0

mab = 0

l

Surface 1

Images of Extended Source

x

LocalizationSurface

54 Chapter 5


Fig. 5.40. A typical Fizeau interferometer. Fringes of equal thickness indicate contours of constant t(x) and are localized at wedge. The height of the defect, as indicated by the fringe deviation, is approximately .

Fig. 5.41. Localization procedure for extended sources. Intersection of m = 0 fringes from endpoints of the source images from Surface 1 and Surface 2 of the wedge determines the center of the localization region. Fringes are localized on a curved surface.

a

t(x)

n

BeamSplitter

Surface 1

Surface 2


Observation Plane (Image of Film Wedge)

Real-R

ay PathVirtual Images of Extended Source Below the Film Wedge

x

b

x0

y0

Image Location Along the Film

S

x0

OPD0 2nt x cos=

λΔ2S-------

a1

a2

Surface 1 Surface 2

b1

b2

ma = 0

mb = 0

Localization Surface



From the perspective of the observer, as shown in Fig. 5.44, the source images areexactly analogous to the geometry shown in Fig. 5.33, where an extended sourceis used with a plane parallel plate. In this case, the optical system in the obser-vation space can be used to observe Haidinger’s fringes at its back focus plane, asshown in Fig. 5.34 for the plane parallel plate. Separation of the coherent pairsincreases as the distance 2(d3 - d2) between the source images increases, and morefringes are observed in the concentric pattern. Like with the plane parallel plate,maximum OPD0 is found in the center of the pattern.

The Michelson interferometer can be configured so that the two mirrors aretilted, but not displaced, as shown in Fig. 5.45. The source images are also tilted,and they are displaced from each other. Application of the construction describedin Section 5.2.3.3 shows that the m = 0 fringes intersect at the mirrors, where thefringes are localized. If the mirrors are perfectly flat, straight and equally spacedfringes of equal thickness are observed when the mirrors are reimaged to theobservation plane. As the tilt angle decreases, separations of the coherent pairsdecrease, and fringe spacing increases. Fizeau-type fringes are observed if themirrors are not flat, where the fringe deviation from a straight line is proportionalto the surface feature.

If the two mirrors are tilted and displaced, as shown in Fig. 5.46, the center ofthe localization surface is now offset from the mirrors. If the offset is not large,

Fig. 5.42. A basic Michelson interferometer with a quasimonochromatic extended source.

BS

M1

M2


a

b

Observation Plane

d1 d2

d3

d4

Imaging System

θ

56 Chapter 5


high visibility fringes can be observed at the mirrors. Fringes intersecting themirrors are of high order m and some curvature due to the hyperbolic shape isobserved.

For the case of tilted mirrors with no displacement, as shown in Fig. 5.45, them = 0 fringe intersects the mirrors. Therefore, if the source is polychromatic, goodvisibility can be observed until , where is the bandwidth of thesource power spectrum. For example, if a white-light (400 nm to 700 nm)extended source is used, Hz, and beforefringe visibility fades. Since the average wavelength is 550 nm, about two fringes(m = +/- 2) can be observed with good visibility on each side of the m = 0 fringefor a total of five fringes. These fringes are often called white-light fringes. Fringe

Fig. 5.43. An unfolded Michelson interferometer.

Fig. 5.44. The observation-space description of a Michelson interferometer with parallel mirrors. Haidinger fringes are observed in the back focal plane of the imaging system.

d1

BS M1 M2


a

b

d1

d2

d3 d3

d2

d1

d1

a1

b1

a2

b2

Source Images

d1d1

d2

a1a2

b1b2

d3

d2d1

Source Images

d3

BSM1M2

ImagingSystem

ObservationPlane

OPD0 c Δv⁄∼ Δv

Δv 2.9 14×10∼ OPD0 1 6–×10 m 1μm=∼



shift due to the colored spectrum reduces fringe visibility for higher OPD0, asshown in Fig. 5.47.

A practical consequence of the beam splitter in the Michelson interferometeris that one of the beams must pass through some thickness of glass, as shown inFig. 5.48. The light from the source first transmits through the glass of BS and thenreflects off its back side toward M1. Then, it passes through the glass twice more

Fig. 5.45. The observation-space description of a Michelson interferometer with tilted mirrors. Fringes of equal thickness are observed when the mirrors are imaged onto the observation plane.

Fig. 5.46. The observation-space description of a Michelson interferometer with tilted and displaced mirrors. Fringes with hyperbolic curves are observed when the mirrors are imaged onto the observation plane.

Fig. 5.47. White-light fringes.

d1

a1

a2

b1

b2

Source Images

M1M2

ImagingSystem

ObservationPlane

ma = 0

mb = 0

decreases

d1

a1

a2

b1

b2

Source Images

M1 M2

ImagingSystem

ObservationPlane

ma = 0

mb = 0

FringesLocalization Surface

58 Chapter 5


before reaching the observation plane. On the other hand, light from the sourcethat transmits through BS and reflects off M2 only passes through the glass once.If the optical glass used to construct BS exhibits significant dispersion, OPD0contains additional wavelength-dependent terms that drastically reduce visibilitywhen the interferometer is used with polychromatic sources. In order to balanceOPD0 and remove effects of the dispersion, a compensator plate C is placed in thesecond beam of the interferometer before M2. The extra glass is fabricated fromthe same glass as BS, and it is the same thickness. With the compensator, the twobeams pass through the same type and thickness of glass before reaching theobservation plane.

A summary of fringe localization effects with a Michelson interferometer isshown pictorially in Fig. 5.49, where path difference increases outwardly from thecenter. The top row displays the case where mirrors M1 and M2 are parallel to aquasimonochromatic extended source, which forms Haidinger’s fringes. As thepath difference increases, fringes expand outwardly from the center. Note thatHaidinger’s fringes are not observable with a white-light source. The bottom rowdisplays the case where the mirrors are tilted and displaced. With no displacement,fringes of equal thickness are observed. If the m = 0 fringe can be imaged onto theobservation plane, white-light fringes can be produced. As the path difference

Fig. 5.48. Compensator plate C used to cancel dispersion from the glass of BS. With C in the system, each beam of the interferometer experiences three passes through the same thickness of glass.

BS

Light From Source

Light Reflected From M1

C

Light ReflectedFrom M2

Light DirectedTo M2

Light Directed To M1

Reflective Surfaceof BS

To Imaging System



increases with increasing displacement, the center of the fringe localizationsurface offsets from the mirrors, and curved fringes are observed. Larger mirrordisplacements offset the center of the localization surface so much that fringes areno longer observable.

5.2.3.5 Twyman-Green Interferometer

In the Twyman-Green Interferometer displayed in Fig. 5.12, the coherentpoints are effectively located at infinity. The observation space begins as soon asthe beams reflected from each mirror combine. This combination also occursthroughout any optical system that may be used to image the mirrors onto adetector plane. If a point source is used where Eq. (5.75) is satisfied, hi visibilityfringes are observed throughout the observation region. If a polychromatic pointsource is used where Eq. (5.75) is not satisfied, the change in visibility versusOPD0 is given by Eq. (5.28). Since OPD0 is a function of the mirror separation,visibility can change if the mirrors are moved. A Twyman-Green interferometer isgenerally not used with extended sources. In that case, the geometry must beanalyzed like a Michelson interfereometer.

5.2.3.6 Lloyd’s mirror

A simple mirror can be used as an interferometer with a polychromatic pointsource. For example, the Lloyd’s mirror interferometer produces a virtual imageof the point source, which is located as far below the mirror surface as the realsource is above it, as shown in Fig. 5.50. If the total separation is d and thedistance between the source plane and the observation plane is z0, Eq. (5.28) canbe used to predict the region of fringe localization with hyperbolic boundaries.One important difference in the fringe pattern is that the central fringe is dark,instead of bright, if there is a phase change in reflection off the mirror. Lloyd’smirror with a single point source is similar to the YDPI, in that a hyperboloidalfield is generated from two-point-source interference, except for the dark fringe atthe center of the field. Straight-line cosine fringes are observed near the opticalaxis in the observation plane with period .

When the Lloyd’s mirror is used with a quasimonochromatic extended source,the distance between individual points and the mirror is not constant, as shown inFig. 5.51. Therefore, distances between source points and their coherent imagesare not constant, and the fringe spacing is a function of the source-point positionys. For simplicity, a perfect mirror is assumed without any loss on reflection.When the total fringe field is calculated, the integration over the source spatialdistribution is

Fig. 5.49. Summary of fringe localization in a Michelson interferometer.

π

Λ λz0 d⁄=

60 Chapter 5


, (5.77)

where

(5.78)

is the total power of the source and K is the fraction of the light emitted by the eachsource point that reaches the observation plane. If the mean distance from thesource distribution to the mirror is , an argument similar to that used for Eqs.(5.29) through (5.31) results in

Fig. 5.50. Lloyd’s mirror with a polychromatic point source.

d

Point Source (P1)

Mirror Surface

y0

Virtual Point Image (P2)

z0

I y0( ) 2K a2

ys( ) ysd∞∫ 2K a

2ys( ) 2π

λ------

2ysy0

z0

-------------⎝ ⎠⎛ ⎞ ysdcos

∞∫+=

2KIL 2KILRe

F2y0λz0--------

a2

ys( )[ ]

IL

------------------------------

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

+=

IL a2 ys( ) ysd∞∫=

ys



, (5.79)

where

, (5.80)

, (5.81)

and

(5.82)

Notice that visibility is equal to μ(yo), which is a function of the observation-space coordinates. This behavior is dramatically different than the result obtainedwhen an extended quasimonochromatic source is used with a YDPI, where the

Fig. 5.51. Lloyd’s mirror with an extended quasimonochromatic source.

I y0( ) 2KIL 2KIL

F2y0λz0--------

f ys( )[ ]

IL

-------------------------- 2πy0

2ys

λz0

-------- β y0( )+cos+=

2KIL 2KILμ y0( ) 2πy0

2ys

λz0

-------- β y0( )+cos+=

f ys( ) a2

ys ys–( )=

μ y0( )

F2y0λz0--------

f ys( )[ ]

IL

--------------------------=

β y0( ) F2y0λz0--------

f ys( )[ ]⎩ ⎭⎨ ⎬⎧ ⎫

arg=

da


Mirror Surface

y0

db

z0

a

b

a1

b1

ExtendedImage

62 Chapter 5


visibility is constant through the observation space. In fact, with a rectangularsource distribution, Lloyd’s mirror fringes may appear more like those shown inFig. 5.2(C).

5.2.3.7 Fresnel’s biprism

Yet another variation is Fresnel’s biprism, as shown in Fig. 5.52 with apolychromatic point source, where two virtual point images a1 and a2 areproduced by an offset pair of prisms with refractive index n. If the reflections fromthe prism faces are negligible, then the two virtual point images, one from eachprism, produce an interference pattern in much the same way as the YDPI. Thedistance z0 in Eq. (5.28) is measured from the virtual source plane, which is adistance nl behind the prisms.

Fresnel’s biprism with an extended quasimonochromatic source is similar toLloyd’s mirror, except that two virtual images of the source are formed, and theparities of the images are not reversed, as shown in Fig 5.53. That is, the virtualsources are in the same orientation with respect to the y axis, where a1b1 and a2b2are upright. (In the Lloyd’s mirror, ab interferes with the reversed b1a1.) InFresnel’s biprism, the distance d is a constant between coherent emitters, and thefringe spacing is not a function of the source-point position. However, the offsetin source-pair position produces an offset in the fringe position, which is similarto the YDPI with an extended, quasimonochromatic source. The fringe contrastfor Fresnel’s biprism is not a function of the observation-space position y0, but it

Fig. 5.52. Fresnel’s biprism with a polychromatic point source.

y0

z0

a

a1

a2

d

l

nl

n

OPD0 2=



is a function of z0, and d. The net result of applying the procedure listed inSection 5.2.2 is that the visibility of the fringes is found from the coherence factor

, (5.83)

where d is the spacing between coherent pairs and z0 is the distance between thesource images and the observation plane.

5.3 Advanced Topics

5.3.1 The Cross Spectral DensityIn Sections 5.1.5 and 5.1.6, a quasi-monochromatic distributed source

provides the basis for the derivation of spatial coherence. This section exploresthe case where the quasimonochromatic restriction is removed.

It is reasonable to expect that a source with a finite-width power spectrumproduces correlation that depends on temporal frequency. For example, resultswith a point source are listed in Section 5.1.4. More generally, the behavior of themutual coherence versus frequency is given by the cross-spectral density, whichis the Fourier transform of the mutual coherence. That is,

Fig. 5.53. Fresnel’s biprism with an extended quasimonochromatic source.

λ

μ dλz0

--------⎝ ⎠⎛ ⎞

F dλz0--------

a2

ys( )[ ]

IL

---------------------------------=

y0

z0

a

a1

a2

d

l

nl

n

b1

b2

b

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