Coherence and Fringe Localizationoptics.hanyang.ac.kr/~shsong/ch5-Coherence and... · the fringe pattern that changes as a result of the type of source is the fringe visibility.1

9

5

©2006 Tom D. Milster

Coherence and FringeLocalization

T. D. Milster and N. A. Beaudry

A simple plane wave is sometimes used to describe a laser beam’s electromag-netic disturbance, such as

, (5.1)

where the disturbance is known over all time and at every location in space. In Eq.(5.1), k = 2π/λ is the propagation constant, is the radian frequency and t is time.Many theoretical formulations in physical optics assume this ideal form of amonochromatic electromagnetic wave. In fact, much insight is gained and manyphenomena are adequately described using ideal waves. While an ideal wave canbe a close approximation to a very good laser beam, it is never a completely truerepresentation of nature. Real wave fields exhibit phases, frequencies andwavelengths that vary randomly with respect to time. The random nature may bea very small fraction of the average frequency, as in a high-quality laser beam, orit may be much more significant, like the field emitted from a tungsten light bulb.Due to this random nature, the statistical description of light plays an extremelyimportant role in determining the behavior of many optical systems.

While it is clear that some form of statistical description is necessary, in mostcases a complete statistical description is not required, and a second order modelis sufficient [5.1]. The second order average of the statistical properties of anoptical field is often referred to as a coherence function. The field of study knownas coherence theory is simply the study of the coherence function. A large amount

E z t,( ) E0ej kz ωt–( )=

10 Chapter 5


of information is derived from these second-order statistics, including the abilityof the optical field to create interference patterns.

As a simple example, consider the interference pattern created when a screenis placed in an optical field , as shown in Fig. 5.1. The screen has two smallpinholes that transmit light, but it is otherwise opaque. The pinholes are spaced bya distance equal to d. Cartesian vectors s1 and s2 describe the positions anddistances of the pinholes from a point on the source. Pinholes P1 and P2 are smallenough so that they act as point sources on transmission, and an interferencepattern is produced on the observation screen. The interference pattern consists ofbright and dark fringes, where the irradiance of the total field is the measurablequantity. The optical path length between P1 and (x0,y0) is vt1, where v is wavevelocity in the observation space and is the time required for the wave to travelfrom P1 to (x0,y0). Similarly, the optical path length between P2 and (x0,y0) is vt2.The optical path difference OPD is the difference in optical path lengths, or OPD= . A bright fringe is produced where OPD = and m is an integer.A dark fringe is produced where OPD = .

The characteristics of the fringe pattern on the observation screen depends onthe type of source used to create field . The most dramatic characteristic ofthe fringe pattern that changes as a result of the type of source is the fringevisibility.1 The fringe visibility is measured locally, and it is given by

Fig. 5.1. General geometry for coherence analysis. A fringe pattern is produced when a light source illuminates two pinholes in an opaque screen. Properties of the fringe pattern depend on the type and location of the source.

E r t,( )

t1

v t2 t1–( ) mλm 1 2⁄+( )λ

Observation PlanePinhole Plane

Observation SpaceSource Space

z

y0y

vt2 s′2=

vt1 s′1=

z0

P2

P1

s2

s1

E r t,( )

Fringe pattern observed inx0,y0 plane.

Source

d r1 r2–=

r1

r2

E r t,( )

Coherence and Fringe Localization 11


, (5.2)

where and are the maximum and minimum irradiances of the localfringe modulation, as shown in irradiance profiles along (0,y0) in Fig. 5.2. A high-visibility fringe pattern has , where is much smaller than . Fringesare clearly observed for , while it is difficult to discern the bright and darkportions of a fringe for . Note that V = 0 does not imply zero localirradiance, but it does imply that the irradiance in the region is void of modulationalong y0.

If the source of the optical field is a high quality laser beam, the twopinholes in Fig. 5.1 produce high-visibility fringes along yo. The analysis of thisproblem is identical to the analysis of the fields radiating from two coherent pointsources, where a hyperboloidal fringe field is generated at the observation plane.If the laser beam illuminates the pinholes symmetrically (lengths = in Fig.5.1), the fringes are hyperboloids that simplify to straight, equally spaced fringesalong the central region of yo when , as shown in Fig. 5.2(A).

If the source is extended but nearly monochromatic, like light from a highlyfiltered light bulb, the fringe pattern exhibits a significant reduction in visibility,as shown in Fig 5.2(B). The visibility is inversely proportional to the source size.That is, as the source size increases, visibility reduces. Notice that the fringevisibility is not a function of yo. The behavior of visibility reduction as a functionof source size is a result of the effect known as spatial coherence.

If the source used to create field is a centered polychromatic pointsource, like a distant arc lamp, fringe visibility varies as a function of yo as shownin Fig. 5.2(C). Notice that fringe visibility is higher near yo = 0 and decreases asyo increases. Behavior of fringe visibility as a function of the wavelength distri-bution is the result of the effect known as temporal coherence.

In this chapter, the characteristics of both temporal and spatial coherence areexplored. Also, derivations for basic coherence formulas are provided, whichusually result in simple Fourier transform relationships. The chapter is dividedinto three sections. Section 5.1 is an introduction to coherence theory with asimplified development. Section 5.2 discusses the topic of fringe localization,which concerns the spatial location of high-visibility fringes in different interfer-ometers. Section 5.3 extends the mathematical development of coherence theory.For further reading, suggested references are [5.2,5.3, and 5.4].

1. Fringe visibility is also called fringe contrast in some reference material.

VImax Imin–

Imax Imin+-------------------------=

Imax Imin

V 1≈ Imin ImaxV 0.5>V 0.2<

E r t,( )

s1 s2

d z0«

E r t,( )

12 Chapter 5


5.1 Basic coherence

5.1.1 The mutual coherence function

Mathematically, the instantaneous electric field arriving at some position(xo,yo) in the observation plane is given by

, (5.3)

where and are constants that describe the fraction of energy transmitted byeach pinhole to the observation screen, and and are transit times required forthe optical wave to travel from the pinholes to the observation point. Irradiance ofthe pattern on the observation screen is proportional to

Fig. 5.2. Various characteristics of fringe visibility. A fringe pattern is produced when a light source illuminates two pinholes in an opaque screen. Properties of the fringe pattern depend on the type and location of the source.

I y0( )

I y0( )

Imax

Imin

Imax

Imax

Imin

Imin

VImax

Imin

–

Imax

Imin–---------------------------=

(B) Highly filtered extended source (V = 0.2)

(C) Polychromatic point source [V = f(y0)]

I y0( )

(A) Laser source (V = 1)

y0

y0

y0

Etotal r1 r2 t t1 t2, , , ,( ) K1E r1 t t1+,( ) K2E r2 t t2+,( )+=

K1 K2

t1 t2



, (5.4)

where denotes a time average. The subscript t that represents a time averageis suppressed throughout the remainder of the chapter and plain angle brackets imply a time average. Notice that the first two terms in Eq. (5.4) are simply irradi-ances at the observation point from each pinhole separately. The third termmodifies the total irradiance and is called the cross term or interference term. Ifwe assume that the constants and do not vary appreciably over the obser-vation region, it is the cross term that, when at negative maximum, determines

. When maximum positive, the cross term determines . Therefore, thecross term is a primary factor in determining the fringe visibility.

The primary quantitative expression in the cross term of Eq. (5.4) is the mutualcoherence function. Expressed in its most general scalar form, the mutualcoherence function is given by

. (5.5)

The mutual coherence function is a statistical property that, expressed inwords, is the temporal correlation of the electric field at two positions in spacewith respect to time. In the example of Fig. 5.1, Cartesian position vectors and

correspond to the pinhole locations.Consider the case of stationary fields. Stationary implies that the statistics of

the optical field are not affected by a shift in the time origin.[5.5,5.6] Stationaryneglects any transient response that occurs when the source is turned on andassumes the average power of the source is not fluctuating on time scales compa-rable to the temporal period of the optical field or the measurement time. In thiscase, statistics of the optical field only depend on a time difference. That is, and in Eq. (5.5) are replaced by t and , respectively. The mutualcoherence function becomes

. (5.6)

The time difference Δt = t2 - t1 corresponds to the optical path difference in the observation space of the pinhole interferometer. In Fig. 5.1,

the is a function of (xo,yo). The interferometer in Fig. 5.1 is called a Young'sdouble pinhole interferometer (YDPI). It is used throughout this chapter to refineconcepts of coherence.

I r1 r2, t1 t2,;( ) K1 E r1 t t1+,( ) 2⟨ ⟩ t K2 E r2 t t2+,( ) 2⟨ ⟩ tK1K2Re E r1 t t1+,( )E∗ r2 t t2+,( )⟨ ⟩ t[ ]

+

+

∝

⟨ ⟩ t ⟨ ⟩

K1 K2

Imin Imax

Γ r1 r2 t1 t2, , ,( ) E r1 t t1+,( )E∗ r2 t t2+,( )⟨ ⟩=

r1

r2

t t1+t t2+ t Δt+

Γ r1 r2 Δt, ,( ) E r1 t,( )E∗ r2 t Δt+,( )⟨ ⟩=

OPD vΔt=OPD

14 Chapter 5


5.1.2 The two-wavelength point source

Now we consider a point source that contains two wavelengths. An implemen-tation of the Fig. 5.1 YDPI two-pinhole screen is shown in Fig. 5.3 with an on-axis( = ) point source. The source emits spherical waves with wavelengths and . Since the point source is exactly halfway between the two pinholes, lightemitted from the first pinhole is exactly in phase with light emitted from thesecond pinhole. Pinholes are separated by distance d.

Light from the source arriving at each pinhole exhibits a complicatedmodulation due to the combination of the two wavelengths. Media on both sidesof the pinhole screen are non-dispersive, so the modulation components exhibitthe same velocity v. Therefore, behaviors of waves emitted by P1 and P2 andobserved at have fixed temporal relationships. If and ampli-tudes of each component wave have the value A, time-varying wave componentsat the observation point are

Fig. 5.3. An on-axis dual wavelength source in a YDPI. The source produces two independent fringe patterns in the observation plane with different periods and . Irradiances of the independent fringe patterns are added to produce the total irradiance.

d

Pinhole Plane

y

P2

P1

s2

s1

z0

vt1 s′1=

vt2 s′2=

y0

E r t,( )

Observation PlaneSource Plane Pinhole Plane

Point Source Λb

Λa

λb λa>

Λa Λb

s1 s2 λaλb

y0 K1 K2 1= =

0 y0,( )



(5.7)

where the first two terms result from P1 and the second two terms result from P2.Subscripts a and b refer to the first and second wavelengths, respectively. Whenthe total field from the addition of components in Eq. (5.7) is squared, the result is

Square of addition of field components: (5.8)

(5.8a)

(5.8b)

(5.8c)

(5.8d)

(5.8e)

, (5.8f)

where terms (5.8a) and (5.8b) simply result from the two wavelengths indepen-dently, and terms (5.8c) through (5.8f) are inter-modulation beat terms. The msubscripts in Eq. (5.8) imply modulation frequency or modulation wavenumber , where and . The constant is the difference in the phase terms of the original optical fields, where

. Notice that arguments of the cosines in Eqs. (5.8a) and (5.8b)do not depend on absolute time t. They depend only on wavelength, insomuch aswavelength is related to optical frequency by , and Δt, which is afunction of yo. Conversely, inter-modulation phase terms in Eqs. (5.8c) through(5.8f) are functions of absolute time t.

In order to find irradiance in the observation plane, a time average is performedover Eq. (5.8). A general relationship for averaging the sum of variables is

E1a A kas1′ ωat– ϕa+( ) ,cos=

E1b A kbs1′ ωbt– ϕb+( ) ,cos=

E2a A kas1′ ωa t Δt+( )– ϕa+[ ] ,cos=

and

E2b A kbs1′ ωb t Δt+( )– ϕb+[ ] ,cos=

Etotal2 A2 A2 ωaΔt( )cos+=

A2 A2 ωbΔt( )cos+ +

A2 2kms1′ 2ωmt 2β+–( )cos+

A2 2kms1′ 2ωmt 2β ωbΔt+ +–( )cos+

A2 2kms1′ 2ωmt 2β ωaΔt–+–( )cos+

A2 2kms1′ 2ωm t Δt+( )– 2β+( )cos+

+ high-frequency terms.

ωmkm ωm ωa ωb–( ) 2⁄= km ka kb–( ) 2⁄= β

β ϕa ϕb–( ) 2⁄=

λ 2πc ω⁄=

16 Chapter 5


. (5.9)

Application of Eq. (5.9) to Eq. (5.8) results in the first two terms Eqs. (5.8a)and (5.8f) remaining unchanged, because they are not functions of absolute timet. For example, averaging Eq. (5.8a) yields

. (5.10)

A different result is obtained when time averaging is performed on theremaining terms in Eq. (5.8). Since the cosine arguments are functions of absolutetime, integration over the full temporal period ( ) ofthe modulation yields a zero net result. For example, averaging Eq. (5.8c) yields

. (5.11)

The result of time averaging Eq. (5.8) is

. (5.12)

Higher-frequency terms in Eq. (5.8) effectively average to zero over a muchshorter integration time. Equation (5.12) shows that the irradiance pattern in theobservation plane is effectively the sum of irradiance patterns from individualwavelengths.

Physically, the time average represents the integration time of the detector usedto measure the energy at observation point yo. If the integration time ,inter-modulation and high-frequency terms in Eq. (5.8) average to zero. Forexample, if = 550 nm and = 570nm, seconds in order tosatisfy the integration-time requirement. It is usually a safe assumption to approx-imate the total irradiance by adding individual irradiance patterns produced fromindividual wavelengths, unless a detector is used for the measurement that has ashort time response. For example, cones in the human eye exhibit a time responseof around 0.010seconds, and a fast silicon detector exhibits a time response of

seconds.In the limit where , = ω, Eq. (5.8) reduces (before averaging)

to

. (5.13)

The leading factor defines coherent addition of the wavelengthsbased on their phase difference. No phase shift with β = 0 produces a coherent

C1 C2 C3 …+ + +⟨ ⟩ C1⟨ ⟩ C2⟨ ⟩ C3⟨ ⟩ …+ + +=

A2⟨ ⟩ A2 ωaΔt( )cos⟨ ⟩+ A2 A2 ωaΔt( )cos+=

nλaλb c λa λb–( )⁄ nλeq c⁄=

A2 2kms′1 2ωmt 2β+–( )cos⟨ ⟩ 0=

Etotal2⟨ ⟩ A2 A2 ωaΔt( )cos A2 A2 ωbΔt( )cos+ + +=

A2 2 ωaΔt( )cos ωbΔt( )cos+ +[ ]=

T π ωm⁄»

λa λb T 5.2 14–×10»

10 10–

λa λb→ ωa ωb≈

Etotal2 2A2 1 2βcos+( ) 1 ωΔtcos+( )=

1 2βcos+( )



sum, which is equivalent to the result obtained by simply increasing the sourceamplitude by a factor of two.

If the wavelengths are derived from physically distinct phenomena, like thebeams from two laser cavities, the phase β is random in time. The time average ofEq. (5.13) with a zero-mean β is

, (5.14)

because the time average of . Again, the total irradiance pattern ofEq. (5.8) is simply the addition of two irradiance patterns produced separatelyfrom the individual wavelengths, which both have equal amplitudes in this case.This argument can be expanded to apply to any two wavelengths in the spectrumof a polychromatic source.1

In the remaining sections of this chapter, total observation plane irradiance isderived by summing irradiance patterns resulting from each wavelength of thesource. Distance between adjacent fringe maxima corresponds to a change of onewavelength of optical path difference = vΔt. Equation (5.12) is written inthese terms as

, (5.15)

where and . Note that the cosine terms onthe top right side of Eq. (5.15) have slightly different spatial periods and for the same range of OPD, due to the different wavelengths. The first cosine termon the bottom right side of Eq. (5.15) exhibits high-frequency fringes with period

. The second cosine term on the bottom right side of Eq. (5.15) exhibitsa slow modulation corresponding to .

The central region of the observation screen along yo, which is important in thedevelopment of coherence functions in following sections, is defined as ,with the constraint that . As shown in Fig. 5.6, =

, where lengths and define distances from the obser-

vation point to P1 and P2, respectively. Under these conditions, is a linearfunction of position along yo. The period between bright fringes is .The cosine terms in Eq. (5.15) produce straight and equally spaced fringes in the

1. In fact, it is possible to phase two laser beams such that the beat frequency is only 1Hz. However, this laboratory experiment is very complicated and is not representative of naturally occurring sources.

Etotal2⟨ ⟩ 2A2 1 2βcos⟨ ⟩+( ) 1 ωΔtcos+( )=

2A2 1 ωΔtcos+( )=

2βcos⟨ ⟩ 0=

OPD

Etotal2⟨ ⟩ A2 2 kaOPD( )cos kbOPD( )cos+ +[ ]=

A2 2 22πλ

------OPD⎝ ⎠⎛ ⎞cos

πλeq

-------OPD⎝ ⎠⎛ ⎞cos+≈

λeq λaλb λa λb–⁄= λ λa λb+( ) 2⁄=Λa Λb

OPD λ=λeq

y0 z0«d z0« OPD s′1 s′2– d θsin

dy0 z0⁄ s′1 vt1= s′2 vt2=OPD

Λ z0λ d⁄=

18 Chapter 5


central region for wavelengths and with periods and , respectively,as shown in Fig. 5.4.

Figure 5.5 shows a gray-scale representation of fringes in the observationplane for a two-wavelength source. The observation range is restricted near = 0. Notice that the fringe modulation washes out periodically along . Figure5.7 shows line profiles of the component fringes, the combined fringe pattern andthe fringe visibility curve as functions of . In the exact center ofthe pattern where , and . The cosine terms are in phase,and V = 1. As increases away from zero, fringes become out of phase, andvisibility reduces. Fringes are π out of phase and when = = , where m and n are integers. The relationship between mand n at this is

. (5.16)

The example in Figs. 5.5 and 5.6 is shown for m = 5 and n = 4. Notice that thevisibility is a smooth envelope that decays to zero at the point where y0 = .The OPD corresponding to the distance between the maximum fringe contrast at

= 0 and the first zero of fringe visibility is the coherence length. Beyond thepoint of minimum visibility, fringes become more in phase, until V = 1 again. Thecombined pattern in Fig.5.6(b) has a periodic variation in visibility, between V =0 and V = 1, but the average irradiance is constant. The high-frequency fringesoccur with period in OPD and change phase between modulation lobes. The

Fig. 5.4. Approximation of OPD. When observation is made with small, d << z0 and z0 >> λ, the optical path difference in observation space OPD can be approximated by . Fringes are straight and equally spaced.

λa λb Λa Λb

OPD s1 s2– dy0

z0

-----=

y0s1

s2d

Pinhole Plane P1

P2OPD

y

d

z0

y0

OPD dy0 z0⁄≈

x0y0

y0

y0

OPD dy0 zo⁄=y0 0= Δt 0= OPD 0=

y0

V 0= mλan 1 2⁄+( )λb OPD

OPD

Δλλb------ n m– 1 2⁄+

m-----------------------------=

z0λeq d⁄

y0

λ



visibility curve in Fig.5.6(c) is periodic and corresponds to the modulation term inEq. (5.15). The coherence length is , which is approximately inOPD units. As the wavelength distribution of the source widens, the range of goodvisibility along the observation plane decreases. For example, Fig. 5.7 shows theresult of adding a third wavelength, and Fig. 5.8 shows the result for fivewavelengths.

5.1.3 The power spectrum of the source

The discrete wavelength character of the optical sources considered in Section5.1.2 is not found in nature. Thermal sources, like tungsten filaments, emit acontinuous distribution of wavelengths that varies as a function of the filamenttemperature. Gas discharge lamps can provide a narrow spectrum of radiationlimited by the Lorentzian linewidth of atomic transitions, but separate moleculesin the gas emit energy independently at frequencies distributed over the linewidth.Even with laser sources, a continuous description of the wavelength spectrum isrequired. Stimulated emission, like that found in laser cavities, can exhibit anarrow spectrum of less than one part in , but mechanical instabilities in thelaser cavity and other sources of instability result in slight variations of phase b(t)of the output. The phase variations cause tiny modulations of the wavelength,

Fig. 5.5. Fringes in the YDPI observation plane for small y0 with a dual-wavelength on-axis source. Fringes are straight and equally spaced. Regions of low V are easily observed. Notice that low V regions contain significant irradiance, but they do not contain clear modulation. The coherence length is defined as the distance in OPD units from the central fringe with maximum V to the first region of minimum V.

λeq 2⁄ λ2 2Δλ⁄

CoherenceLength

y0

x0

1013

20 Chapter 5


which can influence interference measurements. Fortunately, all of thesewavelength characteristics can be described in a straightforward formalism calledthe power spectrum, which is the more common name for the wavelengthspectrum.

Consider an optical wave that contains a collection of frequency componentswith differential amplitudes given by and phases given by , where ν isthe temporal frequency of the vibration. The total electric field is

. (5.17)

Notice that Eq.(5.17) restricts the temporal frequency of the field to positivefrequencies. This restriction is a consequence of relating the real-valued electro-magnetic disturbance to a complex quantity, which is known as the analyticsignal. (More detail concerning the analytic signal is provided in Appendix B.)Given , is found by

Fig. 5.6. Fringe pattern analysis for a YDPI two-wavelength source. A) Line profiles of the component fringes; B) The total irradiance, which is the linear sum of the component fringe irradiance. Notice that the modulation envelope of this pattern is the visibility function, and the high frequency fringes occur with period in units of OPD. The two-wavelength fringe visibility as a function of OPD shows a periodic variation. The coherence length is .

OPD

OPD

OPD

AverageIrradiance

I OPD( )

I OPD( )

I OPD( )

Out of phase, Vmin

In phase, Vmax

Imax

Imin

Loca lvalue of V

A) Componentfringe patterns

B) Combinedfringe pattern

C) Visibi l i tycurve

λa

λb

λ

Coherence Length

1

λeqλ

2

Δλ------≈

Vis i b i l i t yEnvelope

λ

λeq 2⁄ λ22Δλ⁄=

a ν( ) φ ν( )

E t( ) a v( )ejϕ v( )ej2πvt vd0

∞

∫=

E t( ) a v( )ejϕ v( )



. (5.18)

Irradiance I of the optical field represented by Eq. (5.17) is

Fig. 5.7. Fringe patterns due to three equally-spaced wavelengths in the source.

Fig. 5.8. Fringe patterns due to five equally-spaced wavelengths in the source.

I OPD

I OPDOPD

OPD

A.) Component

Fringe Patterns

B. ) Combined

Fringe Pattern

0

Coherence Length

I OPD( )

I OPD( ) OPD

OPD

A.) ComponentFringe Patterns

B. ) CombinedFringe Pattern

0

Coherence Length

a v( )ejϕ v( ) E t( )e j– 2πvt td∞–

∞

∫=

22 Chapter 5


. (5.19)

The quantity in Eq. (5.19) is proportional to the contribution to theirradiance from the frequency range . is the spectral density ofthe light vibrations, and is referred to as the power spectrum of thelight field. A typical power spectrum is shown in Fig. 5.9. The frequency range islimited to width Δ around mean frequency . A typical power spectrum of opticalsource, with mean frequency and bandwidth .

Section 5.1.2 establishes that, for the purpose of calculating irradiancepatterns, the patterns from two separate wavelengths are independent. Extrapo-lation of this result to a continuous wavelength distribution yields that eachspectral density component of the power spectrum is also independent. Therefore,the calculation of irradiance patterns for interference phenomena is accomplishedby a linear sum of the patterns from each spectral density component.Real measurements of the power spectrum are taken over non-infinite time

intervals. For example, if is the analytic signal taken over time interval

and are the spectral components,

. (5.20)

With the assumption of stationary and ergodic statistics,1

Fig. 5.9. A typical source power spectrum with mean frequency and width .

1. Stationarity is defined in Section 5.1.1, and ergodic statics imply that an ensemble average is equal to the corresponding time average involving a typical member of the ensemble.

I E t( )E∗ t( )∝ a2 ν( ) νd0

∞

∫ G ν( ) νd0

∞

∫= =

a2 ν( ) νdν ν dν+,( ) G ν( )

G ν( ) a2 ν( )=

vv Δv

a2

v( )

vv

Δv

v Δv

ET t( )t T< aT ν( )e

jϕr v( )

ET t( ) aT ν( )ejϕT ν( )

ej2πν t νd0

∞

∫=



, (5.21)

where the bar denotes an ensemble average over many samples. Therefore, themeasured power spectrum is actually

. (5.22)

5.1.4 Basic temporal coherence

Temporal coherence is the dependence of fringe visibility on the powerspectrum of the source. In Figs. 5.3 and 5.5 through 5.8, the power spectrum is asimple combination of discrete wavelengths. In this section, the functional depen-dence of fringe visibility for a continuous power spectrum is derived from a simpleargument.

Consider the YDPI of Fig. 5.3 in air with a power spectrum given by .Spectral component of the source power spectrum produces an inter-ference pattern at the observation plane given by

, (5.23)

where and are diffraction constants that depend on pinhole sizeand , where c is the speed of light in air (approximately ms-1

and vary slowly with wavelength. For this development, assumethat and are simple constants that do not vary with wavelength and are nota function of observation-plane position. The pinholes are extremely small, so thatdiffraction effects from the finite pinhole size can be neglected.

Irradiance patterns from individual source wavelengths are now combined toobtain the total fringe pattern. If Eq. (5.23) is integrated over the bandwidth of thesource,1

1. Conversion of the integration range to include is justified, because

in this range by definition.

IaT

2 ν( )T

-------------T ∞→lim νd

0

∞

∫=

G ν( )aT

2 ν( )T

-------------T ∞→lim=

a2 ν( )a2 ν0( )

I ν0 Δt;( ) a2 ν0( ) K1 y0( ) K2 y0( ) 2 K1 y0( )K2 y0( ) 2πν0Δt( )cos+ +[ ]≈

K1 y0( ) K2 y0( )Δt OPD c⁄= 3 8×10

K1 y0( ) K2 y0( )"K1 K2

v ∞– 0,( )∈

a2 v( ) 0=

24 Chapter 5


, (5.24)

where

(5.25)

is the integral of the source power spectrum,

(5.26)

is the normalized Fourier transform of the source power spectrum, and

(5.27)

is the phase of the Fourier transform.1 Fringe visibility is

. (5.28)

Usually,

, (5.29)

where is the mean frequency of the power spectrum. In this case,

1. Subscripts “12” on m, and are a reference to the sampling points of the source field

from which the coherence is measured. For example, the pinholes P1 and P2 define the sampling points of a YDPI.

I Δt( ) K1 K2+( ) a2 ν( ) νd0

∞

∫ 2 K1K2 a2 ν( ) 2πν0Δt( )cos νd

0

∞

∫+≈

K1 K2+( )IW 2 K1K2Re a2 ν( )ej2πv0Δt

νd∞–

∞

∫+=

K1 K2+( )IW 2 K1K2IWReFΔt a

2 v( )[ ]IW

--------------------------⎩ ⎭⎨ ⎬⎧ ⎫

+=

K1 K2+( )IW 2 K1K2IWm12 2πν0Δt( ) ϕ12 Δt( )[ ]cos+=

IW a2 ν( ) νd

0

∞

∫=

m12 Δt( ) FΔt a2 v( )[ ]IW

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

ϕ12 Δt( ) arg FΔt a2 v( )[ ]{ }=

ϕ βE r t,( )

VImax Imin–

Imax Imin+-------------------------

2 K1K2

K1 K2+--------------------m12 Δt( )= =

a2 ν( ) f v v–( )=

v



, (5.30)

where

. (5.31)

Equations (5.24) through (5.31) imply that integration of cosine fringe patternsfrom individual wavelengths in the power spectrum yields an aggregate fringepattern with small-scale oscillations having period and a visibilitythat depends on the Fourier transform of the power spectrum of the source. Fringeshift is observed if the power spectrum is asymmetric about the meanfrequency .

Example 5.1: Temporal coherence with a simple rectangular power spectrum

Consider a point source with a power spectrum given by a simple rectan-gular function, with a mean frequency and a bandwidth Δv. The source isused on axis in a YDPI, like the geometry of Fig. 5.3.

Mathematically, the power spectrum is given by

. (5.32)

Calculation of the Fourier transform of Eq. (5.32) yields

. (5.33)

From Eq. (5.25), we calculate that . Using Eqs. (5.26) and (5.27) withthe result from Eq. (5.33), and are found to be

, (5.34)

. (5.35)

Substitution of Eqs. (5.34) and (5.35) into Eq. (5.24) with OPD= in a YDPI yields the following interference pattern in theobservation plane

, (5.36)

or

ϕ12 Δt( ) 2πvΔt β12 Δt( )+=

β12 Δt( ) arg FΔt f v( )[ ]{ }=

1 v⁄ λ c⁄=

β12 Δt( )v

v

a2 ν( ) 1Δν------rect ν v–

Δν-----------⎝ ⎠⎛ ⎞=

FΔt a2 v( )[ ] sinc ΔvΔt( )ej2πvΔt=

IW 1=m12 Δt( ) ϕ12 Δt( )

m12 Δt( ) sinc ΔvΔt( )=

ϕ12 Δt( ) 2πvΔt sinc ΔvΔt( )[ ]arg+=

cΔt dy0 z0⁄≈

I y0( ) 1 sincdΔνy0

cz0

---------------⎝ ⎠⎛ ⎞ 2π

dvy0

cz0

----------- β12

Δνy0

cz0

------------⎝ ⎠⎛ ⎞+⎝ ⎠

⎛ ⎞cos+=

26 Chapter 5


, (5.37)

where the cosine term creates fringes with period and the sincterm is the modulation in visibility. The phase of the sinc function is formallyincluded in , as shown in Eq. (5.36), but it is sometimesconvenient to write the form shown in Eq. (5.37). A gray-scale picture of thefringe pattern is shown in Fig. 5.10, and plots of the irradiance and visibilityalong the y-axis are shown in Fig. 5.11.

In Example 5.1, visibility reduces as (and, thus OPD) increases from zero,as evidenced by the form of the sinc function in Eq. (5.37). When visibility dropsto zero, there is no fringe contrast. The OPD between maximum fringe visibilityat OPD = 0 and the first zero in visibility is the coherence length .Coherence time, which corresponds to coherence length, is given by

. All two-beam interferometers exhibit similar behavior,with the general result that coherence time is approximately the inverse of thewidth of the power spectrum. In practice, coherence length may not be definedwhere visibility reaches an absolute zero, but, rather, where visibility reachessome minimum value, like V = 0.2.

Fig. 5.10. Fringes in the YDPI observation plane for the rectangular power spectrum of Example 5.1.

I y0( ) 1 sincdΔνy0

cz0

---------------⎝ ⎠⎛ ⎞ 2π

dvy0

cz0

-----------⎝ ⎠⎛ ⎞cos+=

Λ cz0 dv⁄=

β12 Δνy0( ) cz0( )⁄[ ]

Coherence

Length

y0

x0

Δt

Δl c Δν⁄=

Δtc cΔl 1 Δν⁄= =



Example 5.2: Temporal coherence properties of a Twyman-Green interferometer with a narrow Gaussian power spectrum.

Consider the power spectrum of a frequency-stabilized HeNe laser ( =632.8 nm) with a Gaussian power spectrum and v = 300 kHz. The applicationis for a Twyman-Green interferometer, where the flatness of a mirror is tobe tested, as shown in Fig. 5.12. What is the maximum distance that can be setfor the mirror separation if the minimum resolvable visibility is V = 0.2?

By the peculiar property of a Gaussian function, the Fourier transfor-mation of the Gaussian power spectrum is also in the form of a Gaussianfunction. If we express the power spectrum = when

= , the calculation of its Fourier transformation yields

. (5.38)

Fig. 5.11. Line profiles of the fringe pattern in Fig. 5.10, where the envelope of the fringe pattern is a sinc function.

I y0( )

I y0( )

Imax

Imin

Imax

Imax

Imin

Imin

VImax

Imin

–

Imax

Imin–---------------------------=

(B) Highly filtered extended source (V = 0.2)

(C) Polychromatic point source [V = f(y0)]

I y0( )

(A) Laser source (V = 1)

y0

y0

y0

λ

Δyo

a2 ν( ) gaus ν v–( ) Δν⁄[ ]gaus ν( ) πν2–( )exp

FΔt a2 v( )[ ] Δνgaus ΔνΔt( )e 2πvΔt–=

28 Chapter 5


From Eq. (5.26), the normalized Fourier transform of the source powerspectrum is

, (5.39)

after normalization with . The fringe visibility V is proportional to theabsolute of the normalized Fourier transform. That is,

. (5.40)

After replacing (because light in a Twyman-Green interferometer passes each arm twice) and manipulating the contrast tobe larger than 0.2, we obtain the maximum mirror separation:

. (5.41)

For V > 0.5, where the fringes can be clearly observed, the maximum mirrorseparation is 235m. Due to the long coherence length originating from the

Fig. 5.12. The Twyman-Green amplitude division interferometer uses a collimated laser beam that is divided by a beam splitter. Each split beam reflects off a mirror and recombines at the observation plane. The two beams form an interference pattern in the observation plane that is determined from OPD.

Reference Mirror

HeNe Laser

Beam

Splitter

Test M

irror

y0

Observation Plane

E r t

d1

d2

OPD 2 d1 d2– 2 y0–=

E1 r tE2 r t OPD c+

M1

M2y0

m12 Δt( ) gaus ΔvΔt( )=

IW

V gaus ΔνΔt( )=

Δt OPD c⁄ 2 c⁄( ) d1 d2–= =

d1 d2– maxc

2Δν---------- 5ln

π-------- 358m≅=



narrow spectrum width of the HeNe laser, the fringe visibility does not changesignificantly over the small OPD changes associated with the mirror imperfec-tions .

Example 5.3: Temporal coherence properties of a multiple-mode laser diode

Consider the power spectrum shown in Fig. 5.13 of a typical laser diodewith a mean wavelength = 649.2nm ( = ), whereparameters of the diode laser include = 3.5, L = 200 µm,

= and = 25 MHz. is the bandwidthof an individual laser mode. A mathematical description of the powerspectrum is

, (5.42)

where (*) denotes one-dimensional convolution. Hence,

. (5.43)

Figure 5.14 shows the visibility curve versus , and Figure 5.15shows the irradiance.

If a single-mode version of this laser is used in a Twyman-Green interfer-ometer, the visibility only depends on , which leads to a 12m coherencelength. However, the multiple-mode nature of the power spectrum in Fig. 5.13

Fig. 5.13. Power spectrum of a multiple-mode laser diode.

Δ y0( )

λ v c λ⁄= 4.621 14×10 Hzncavity

Δv 3= c 2ncavityL⁄ 6.4 11×10 Hz Δvm Δvm

a2 ν( ) A0rect ν v–Δv

-----------⎝ ⎠⎛ ⎞ comb ν

c 2ncavityL⁄---------------------------⎝ ⎠⎛ ⎞ *gaus ν

Δνm----------⎝ ⎠⎛ ⎞=

m12 Δt( ) FΔt a2 v( )[ ]IW

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

sinc ΔtΔv( )* comb cΔt2ncavityL--------------------⎝ ⎠⎛ ⎞gaus ΔνmΔt( )=

OPD cΔt=

v

Δv

Δvm

c2ncavityL-----------------------

Δνm

30 Chapter 5


dramatically shortens the coherence length to under 0.5mm. The visibilityreturns periodically at = , where m is an integer.

Example 5.4: Fourier Transform Spectroscopy

Often, it is necessary to measure the power spectrum of a source. Adetailed description of the power spectrum can yield information about thechemical composition of a star, energy levels of atomic transitions,atmospheric pollution and other applications. The device often used to

Fig. 5.14. Visibility function associated with Fig. 5.13 and the parameters of Example 5.3, where n = ncavity.

Fig. 5.15. Profile of the fringe pattern observed for Example 5.3.

V OPD( )

OPD

Full width of envelope (at 0.043amplitude) ~2c Δvm⁄ 24m=

1

1.4mm

0 2nL 4nL2– nL4– nL

2cΔv------ 0.93mm=

OPD0

I OPD( )

2nL 4nL-2nL-4nL

OPD 2mncavityL



measure the power spectrum is a Fourier Transform Spectrometer. A simpleFourier Transform Spectrometer is shown in Fig. 5.16, which is basically aTwyman-Green Interferometer in which the path length of one arm is adjustedwith a movable scan mirror. d1 and d2 are the lengths of each arm, respec-tively. Thus, the optical path difference ( ) is

. (5.44)

When observing a distant point source, the fringe visibility is recorded as afunction of . For example, Fig. 5.17 shows a hypothetical fringe patternand visibility measured by the detector that is characteristic of astronomicalobservations. The spectral power density is found from the inverse Fouriertransform of Eq. (5.26), or

, (5.45)

where V is the visibility. The spectral power density calculated from the datain Fig. 5.17 is shown in Fig. 5.18. For more information on Fourier TransformSpectroscopy, the reader is directed to References 5.8 and 5.9...

5.1.5 Basic Spatial Coherence

Section 5.1.4 assumes that the electric field reaching the observation pointfrom P2 is simply a delayed copy of the electric field reaching the observationpoint from P1. This condition occurs only if the two electric fields originate fromprecisely the same position in space, i.e. a perfect point source. However, if theelectric field originates from a source of finite size, there are additional spatialfluctuations in the electric field.

Fig. 5.16. Twyman-Green interferometer used as a Fourier Transform Spectrometer.

OPD

OPD 2 d2 d1–( )=

OPD

a2 ν( ) IWFv1– V OPD( )[ ]=

Fixed Mirror

ScanningMirror

Beam Splitter

Source

Detector

d1

d2

32 Chapter 5


Spatial coherence is the dependence of fringe visibility on the spatial extent ofthe source. In this section, distributed sources are treated as a collection of pointsources that vibrate independently. The spread in wavelengths among the sourcesis small enough so that the coherence length is much greater than the range of OPDin the measurement. Therefore, fringe visibility is not affected by the bandwidthof the power spectrum. Although the sources have nearly the same wavelength, thetemporal beat frequency resulting from combination of light from the sources isbeyond the temporal bandwidth of the detector used to measure the aggregatefringe pattern. These conditions serve as an informal definition of a distributedquasimonochromatic source. A more formal definition of a quasimonochromaticsource is developed in Section 5.3.2.

Fig. 5.17. Fringe profile used in Example 5.4, which describes Fourier Transform Spectroscopy.

Fig. 5.18. Power spectral density calculated by the inverse transform of data displayed in Fig. 5-17.

Detector Current

OPD

Visibility Envelope

Fringes

Power SpectralDensity (A.U.)

Frequency

(1014HZ)5 5.5 64 4.53.5 6.5



For example, consider the effects of two quasimonochromatic point sourceslocated at and , which are used as separate sources in a Young's doublepinhole experiment, as shown in Fig. 5.19. Because the sources are independent,the interference pattern from one source adds in irradiance with the pattern fromthe other source. This behavior is similar to the result found in Section 5.1.2,where the irradiance patterns from different wavelengths are added independently,except that now the irradiance patterns resulting from displaced source points areadded.

The on-axis source at generates a centered cosine fringe pattern on theobservation plane. The off-axis source at generates a shifted fringe pattern dueto the additional optical path difference in source space

, (5.46)

where d is the spacing of the pinholes.1 The shift of the fringe pattern causedby OPDs is given explicitly by

, (5.47)

which is independent of source wavelength and pinhole separation. In fact, theshift is simply found by applying similar triangles to the geometry of Fig. 5.19.

Fig. 5.19. The effect of adding a second quasimonochromatic point source at is to create a second fringe pattern in the observation plane that is shifted by .

1. In this section, it is necessary to differentiate between source-space optical path difference OPDs and observation-space optical path difference OPD0.

yS1 yS2

OPD0

OPDs

yS1

Δy0

yS

yS2

P1

P2

y0Pinhole Plane

zs z0

Λ

yS2

Δy0

yS1

yS2

OPDsdyS2

zs----------=

Δy0

Δy0

z0

d----OPDs

z0ys2zs

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

34 Chapter 5


The component fringe patterns and the combined fringe pattern are shown inFig. 5.20 for two source points, where , and is the meanwavelength of the quasimonochromatic source. Notice that the combined patternhas reduced visibility, and the visibility is not a function of observation-spaceoptical path difference . Also, the center of the combined pattern is slightlyshifted off axis. If more point sources are added to the source distribution, thistrend continues. For example, in Fig. 5.21, the combined fringe pattern from threeequally-spaced point sources exhibits significantly degraded visibility. Like withtwo point sources, the visibility is not a function of . If the source distri-bution is wide enough such that the fringe centers fill one fringe period, the resultcan be V = 0, as shown in Fig. 5.22. Note that, although V = 0 in Fig. 5.22, thereis significant irradiance across the observation region.

Figures 5.20 through 5.22 illustrate basic properties of spatial coherence.Firstly, visibility is not a function of location in the observation space. Secondly,visibility is inversely proportional to the source size. These properties differentiatespatial coherence from temporal coherence in a YDPI.

Now, the idea of adding irradiance patterns from individual source points isextended to a continuous source distribution. First, the mathematical expressionfor fringes with a single point source is derived. Then, this expression is integrated

Fig. 5.20. The component fringe patterns from an on-axis source point and an off-axis source point is shown. The . The total irradiance is the sum of the two fringe patterns. Notice that visibility is reduced uniformly across the observation plane, where . The center of the combined pattern is slightly shifted, but it retains the same period as the component fringes.

OPDs λ 5⁄= λ

OPD0

OPD0

I OPD0( )

I OPD0( ) OPD0

A.) Component Fringe Patterns

B.) Combined Fringe Pattern

0

V = 0.84

OPD0

OPDs λ 5⁄=

OPD0 dy0 z0⁄=



to produce the relationship for a continuous distribution. Initially, a one-dimen-sional source distribution is evaluated.

Fig. 5.21. The component fringe patterns from three distributed source points is shown. The between the source positions. The total irradiance is the sum of the three fringe patterns. Notice that visibility is reduced uniformly across the observation plane, where . The center of the combined pattern is slightly shifted, but it retains the same period as the component fringes.

Fig. 5.22. The component fringe patterns from five distributed source points is shown. The between the source positions. The total irradiance is the sum of the five fringe patterns. Notice that visibility is reduced to zero uniformly across the observation plane, where . However, zero visibility does not mean zero irradiance.

OPD0



0

V = 0.55

OPD0

I OPD( )0

I OPD( )0

OPDs λ 5⁄=

OPD0 dy0 z0⁄=

I OPD0( )

I OPD0( ) OPD0



0

V = 0

OPD0

OPDs λ 5⁄=

OPD0 dy0 z0⁄=

36 Chapter 5


The fringe pattern for a point source with radiance inthe source plane is

(5.48)

where , and is the average wavelength of the quasimonochromaticsource. Integration over a one-dimensional continuous source spatial distribution

yields

(5.49)

where is the angular subtense of the pinholes as seen from the source.The source-dependent coefficient in front of the cosine that affects visibility is

, (5.50)

where

. (5.51)

The shift of the fringe pattern due to asymmetries in the source distribution isdetermined by the phase

. (5.52)

a2 ys( ) a02 δ ys ys0–( )=

I OPD0 ys0;( ) K1a02 K2a0

2 2 K1K2a02 k OPD0

dys0zs

----------–⎝ ⎠⎛ ⎞cos+ +=

k 2π λ⁄= λ

a2 ys( )

I OPD0( ) K1 K2+[ ]a2 ys( ) 2 K1K2a2 ys( ) k OPD0 OPDs–( )[ ]cos+{ } ysd

source∫=

K1 K2+[ ]a2 ys( ) 2 K1K2a2 ys( ) k OPD0

ysdzs

-------–⎝ ⎠⎛ ⎞cos+

⎩ ⎭⎨ ⎬⎧ ⎫

ysdsource∫=

K1 K2+( )IL 2 K1K2Re ejkOPD0 a2 ys( )e

jkθAys–ysd

source∫

⎩ ⎭⎨ ⎬⎧ ⎫

+=

K1 K2+( )IL 2 K1K2Re ejkOPD0

FθA λ⁄ a2 ys( )[ ]

IL---------------------------------

⎩ ⎭⎨ ⎬⎧ ⎫

+=

K1 K2+( )IL 2 K1K2ILμ12 θA λ⁄( ) kOPD0 β12 θA λ⁄( )+[ ]cos+=

θA d zs⁄=

μ12

θAλ

-----⎝ ⎠⎛ ⎞ FθA λ⁄ a2 ys( )[ ]

IL---------------------------------=

IL a2 ys( ) ysdsource∫=

β12

θAλ

-----⎝ ⎠⎛ ⎞ arg FθA λ⁄ a2 ys( )[ ]

⎩ ⎭⎨ ⎬⎧ ⎫

=



Note that the fringe visibility,

(5.53)

is not a function of position in the observation space. Fringe spacing in the observation plane is identical to the spacing of fringes in the

interference pattern resulting from an individual point in the source. This devel-opment results in a similar mathematical structure to that found in Section 5.1.4.However, although Eqs. (5.28) and (5.53) are similar; Eq. (5.53) does not dependon .

5.1.6 Interpretation of the van Cittert-Zernike Theorem

Equations (5.50) and (5.53) are elements of the van-Cittert Zernike theorem,which can be stated for a YDPI:

If a quasimonochromatic source is a considerable distance from the apertureplane and pinhole separation is small, fringe visibility from an extended source isproportional to the Fourier transform of the source's spatial distribution. Thetransform variable is the angular separation of the aperture-plane samplingpoints divided by the wavelength.

Example 5.5: Spatial coherence from a one-dimensional line source

Consider a quasimonochromatic one-dimensional source as shown in Fig. 5.23,which is given by

, (5.54)

where, if ,

. (5.55)

A plot of Eq. (5.55) is shown in Fig. 5.24 with . A reference line at is drawn to help illustrate variable dependencies. A

complete understanding of the van Cittert-Zernike theorem is obtained byrecognizing how visibility changes for each variable separately

Increasing zs or decreasing L has the effect of decreasing the maximumfringe shift and increasing visibility. The contrast is increased because therange of shifts in the component fringe patterns is reduced compared to the

V2 K1K2

K1 K2+--------------------μ12

θAλ

-----⎝ ⎠⎛ ⎞=

OPD0

Λ z0λ d⁄=

OPD0

a2 ys( ) rectysL----⎝ ⎠⎛ ⎞=

K1 K2 1= =

V μ12

θAλ

-----⎝ ⎠⎛ ⎞ sinc L

θAλ

-----⎝ ⎠⎛ ⎞= =

θA d zs⁄=Ld lλ⁄ LθA λ⁄ 0.6= =

38 Chapter 5


Fig. 5.23. One-dimensional source distribution with length L and angular substance for Example 5.5.

Fig. 5.24. Visibility for an extended source. Visibility is shown as a function of the normalized variables for the one-dimensional extended source in Example 5.5. In order to increase visibility from the starting point, or L must be decreased, or must be increased.

y

P2

P1Extended Source

ys y0

Λ

d

z0zs

L

θAdzs

----=

μ12

θA

λ----- Starting Point

1

0.8

0.6

0.4

0.2

00 0.5 1.5 2.0 2.5

LθA

λ-----

LθA λ⁄θA λ



fringe period. (Remember, the fringe period does not depend on zs or L.) Thus,decreasing the angular extent of the source as seen by the pinholes increasesvisibility. Increasing pinhole separation d or decreasing has the effect ofreducing the fringe spacing, which decreases visibility, because the range ofshifts in the observation area does not change.

Example 5.6: Spatial coherence with a lateral shearing interferometer

Consider a lateral shearing interferometer (LSI), as shown in Fig 5.25.This type of interferometer is typically used to test collimation of laser beamsand other optical systems. The LSI forms an interference pattern from tworeflections off a plane parallel glass plate. The first reflection interferes withthe second reflection, which is a shifted version of the same wavefront. Theshift is the shear distance S, which is a function of the tilt angle , thethickness T and refractive index n of the plate. The incident wavefront mustexhibit good spatial coherence over the shear distance in order to observe highvisibility fringes.

If the extended quasimonochromatic source in Fig. 5.25 is collimated bya lens of focal length f, the angular distribution of the light exiting the lensdetermines the spatial coherence properties. OPDs between points interferingin the shifted wavefront due to source points distributed along the ys axis is

, (5.56)

where

Fig. 5.25. Spatial coherence for a lateral shearing interferometer (LSI).

λ

ϕ

L

T

n

Shear Distance, S

OPDs

f

θs

θs

ys

S

OPDs S θssin=

40 Chapter 5


. (5.57)

The fringe shift term in Eq. (5.49) is now , so . Fora rectangular source distribution,

. (5.58)

Therefore, the extent L of the incoherent quasimonochromatic source must besmaller than to obtain good visibility. For example, if T = 12.5mm,n = 1.5 and θ = 45°, S = 5.2mm. If f = 100mm, the source size that firstproduces a minimum in visibility is

.

Example 5.7: Michelson Stellar Interferometer

The Michelson stellar interferometer is shown in Fig. 5.26. The fringe spacingon the observation screen depends on d. However, the visibility depends on h,the separation of the collection mirrors, because, like in Example 5.6,

. However, the source radiance is specified in terms of the angle asviewed from the center of the telescope, not as a physical distance in a plane.In this case, the development of Eq. (5.49) is modified to use with as the transform variable. Thus, the observer maps the fringe visibility bychanging h. The angular radiance distribution is calculated by takingthe inverse Fourier transform of measured visibility.

One application of the Michelson stellar interferometer is to measure theangular separation of binary stars. For this problem, the radiance distribution is

, (5.59)

where are the angles between the optical axis and the ray coming fromeach star, and is the radiance of each star. Thus, the visibility is

θssinysf----≈

OPDs Sys f⁄= θA S f⁄=

μ12

θAλ

-----⎝ ⎠⎛ ⎞ sinc L

θAλ

-----⎝ ⎠⎜ ⎟⎛ ⎞

=

sinc LSλf----

⎝ ⎠⎜ ⎟⎛ ⎞

=

fλ S⁄

L fλS---- 100 3–×10 m( ) 550 9–×10 m( )

5.2 3–×10 m------------------------------------------------------------------ 1.1 5–×10 m 11μm= = = =

OPDsh= θssin

a2 θs( ) h λ⁄

a2 θs( )

a2 θs( ) I0 δ θs Δθ+( ) δ θs Δθ–( )+[ ]=

Δθ±I0



. (5.60)

Therefore, if the separation of mirrors and wavelength are determined, and thevisibility is measured, the angular separation can be obtained. In theexample above, if h is the mirror separation that produces the first zero infringe visibility, Δ is given by

. (5.61)

For example, consider when the first zero of fringe visibility occurs at h = 27mwith = 500nm, Δθ = radians, and the angular separation of thebinary stars is about 2 milli-arc-seconds (mas), where 1 mas = 4.6 nanoradians( radians). For comparison, the angular resolution of the human eye isabout 12.5 arc-seconds or 58 microradians ( radians). The Hale obser-vatory has a resolution of about 25 mas, and the Naval Observatory MichelsonStellar Interferometer has a resolution of about 200 micro-arc-seconds.1 Insimple terms, this capability is like resolving the width of a human hair at a

Fig. 5.26. Michelson Stellar Interferometer for determining properties of distant stars.

Distant BinaryStar Fringe

Observation Plane

Telescope Lens

Mirrors

θs

d

Pinholes

h

V

Fhλ---a2 θs( )[ ]

2I0-----------------------------=

I0 ej2πhΔθ λ⁄ e j2πhΔθ λ⁄–+⎝ ⎠

⎛ ⎞

2I0----------------------------------------------------------------=

2πhΔθ λ⁄( )cos=

Δθ

Δθ λ4h------=

λ 4.6 9–×10

10 9–

10 6–

42 Chapter 5


distance of 25 miles. The optical schematic for the Mark III Stellar Interfer-ometer at the Naval Observatory is shown in Fig. 5.27.

5.1.7 The Concept of Coherence Area

The van Cittert-Zernike Theorem of Section 5.1.6 applied to a two-dimen-sional quasimonochromatic source distribution provides insight into the coherentnature of the light field. The concept of coherence area suggests that there is somearea over which the light field behaves coherently at various distances away fromthe source. For example, the sketch in Fig. 5.28 shows the outline of a sourcedistribution and the outline of it's associated Fourier transform in anaperture/pinhole plane. The aperture/pinhole plane could be the pinhole plane ofa YDPI, the input aperture of an interferometer or the object plane of a micro-scope. The distribution of the Fourier transform, appropriately scaled from Eq.(5.49), determines the visibility of interference fringes for separations d of pointsin the aperture plane. The coherence area is defined by the region where points inthe aperture plane can interference with high visibility. The maximum extent ofthis region is the boundary determined by the minimum visibility acceptable to theoptical system of interest. Usually, the zeros of the visibility function are theeasiest limits to consider, but a value of V = 0.2 is often used for experimental data.

For example, the source in Fig. 5.28 is longer in the dimension than in the dimension. The source distribution can be roughly approximated with a

rectangle with sides of length and . The Fourier transform of the rectangleis separable, and the zeros of the resulting sinc visibility function are used todefine the coherence area. For example, the maximum width of for good

1. See http://www.mtwilson.edu/Tour/NRL/mark3.html for more information about the Naval Observatory's instrument.

Fig. 5.27. The Mark III Stellar Interferometer at the Naval Observatory.

Mark III

Optical

InterferometerWavefronts

from

star

South

Delay

Line

North

Delay

Line

ysxs

Lx Ly



coherence (zeros of V) in the y-dimension is . Note that, because ofthe rectangular shape in the source plane, the coherence area is also wider in onedimension than in the other. The wide dimension of the coherence area corre-sponds to the narrow dimension of the source. That is, narrow source dimensionspresent a wide area coherence in the pinhole/aperture plane. In the example ofFig. 5.28, the coherence area is

Coherence area = . (5.62)

Example 5.8: Coherence from a small, round distributed source.

A 1mm pinhole is placed immediately in front of an extended, spatiallyincoherent source of average wavelength 500nm. The light passed by thepinhole is to be used in a diffraction experiment, for which it is desired toilluminate a distant 3 mm diameter aperture coherently. Calculate theminimum distance between the pinhole source and the diffracting aperture forgood coherence. State any assumptions being made.

Assuming that the source distribution is uniform over the pinhole, a one-dimensional approximation is . The coherence at theaperture is obtained by applying the van Cittert-Zernike Theorem. Thevisibility is proportional to the Fourier transform of the source distribution thatis,

Fig. 5.28. The concept of coherence area is an extension of the Van Cittert Zernike Theorem.

Propagation Distance zs

Fourier Transform of Source Distribution

Equivalent Rectangle of Vmin

Threshold

z

y

ΔxzsλLx

-------≈

ΔyzsλLy

-------≈

x

θAyΔyzs

------=

Lx

Ly

xs

ys

Actua lS o u r c eOutline

Eq u iva l en tRectangleof Source

Coherence Area ΔxΔyzs

2λ2

LxLy

-----------≈≈

Δy zsλ Ly⁄=

ΔxΔyzs

2 λ2

LxLy------------=

a2 ys( ) rect ys A⁄( )=

44 Chapter 5


, (5.63)

where A is the pinhole diameter, d is the diameter of the distant aperture and lis the distance from the pinhole to the aperture. Assume illumination iscoherent until the visibility goes to zero. The minimum distance for goodcoherence is obtained where the visibility is zero, which is

. (5.64)

5.1.8 Combination of Temporal and Spatial Coherence

The complete description of coherence properties resulting from a multiple-wavelength, extended source is provided in Section 5.3. However, it is possible tounderstand the basic character of coherence from a heuristic argument. Forexample, consider the off-axis polychromatic point source pictured in the YDPI ofFig. 5.29. With reference to Fig. 5.19, maximum fringe visibility is observed at thepoint on the observation screen where . The decrease in V awayfrom this point is governed by the bandwidth of the source. Narrow implies good V over a wide distance along . The fringe pattern associated withthe single source point is the point response of the YDPI system. Shifting thesource point simply shifts the point response along the observation screen. In thecentral region of the observation screen, the relationship is linear and shiftinvariant in . The relationship is always linear and shift invariant in .

Fig. 5.29. The point response of an off-axis polychromatic source.

V μ12

θAλ

-----⎝ ⎠⎛ ⎞ sinc L

θAλ

-----⎝ ⎠⎛ ⎞ sinc A d

lλ----⎝ ⎠

⎛ ⎞= = =

l Adλ--- 1 3–×10( ) 3 3–×10( )

0.5 6–×10-------------------------------------------- 6m= = =

OPDs OPD0=Δv Δν

y0

y0 OPD′

OPD0

OPDs

yS

P1

P2

y0Pinhole Plane

zs z0



If more than one source point illuminates the YDPI, the point response fringepatterns are summed to produce the total fringe pattern. In the limit of acontinuous source distribution, the total fringe pattern is found by convolving thepoint response with the appropriately scaled source distribution.

For example, consider a source distribution that is a simple centered about the optical axis. If L is small, the shifts of the point responses areless than the width of a fringe period. Therefore, the convolution results in auniform loss factor in visibility across observation screen, but the basic shape ofthe point response remains. At the point on the screen where = 0, the lossin visibility is due only to the spatial extent of the source, and not the temporalbandwidth. As a rough approximation, the total visibility is governed by therelationship

. (5.65)

5.1.9 Terminology Used in Coherence Theory

Equation (5.4) combined with Eq. (5.6) can be written in the following form

, (5.66)

where and are irradiances at the observation point from eachpinhole independently, is the mutual coherence function resulting frompositions and in the light field and delayed by time . is thenormalized mutual coherence function, which is given by

. (5.67)

It is understood that . The normalization limits values of so that. The term is called the complex degree of coherence.

Equation (5.66) can also be written as

, (5.68)

where

. (5.69)

When working with temporal coherence problems, the complex degree ofcoherence is related to the terms in Section 5.1.4 by

rect ys L⁄( )

OPD0

V μ12

θAλ

-----⎝ ⎠⎛ ⎞m12 Δt( )∝

I y0( ) I1 y0( ) I2 y0( ) 2 I1I2Re γ12 τ( )[ ]+ +=

I1 y0( ) I2 y0( ) y0

Γ12 τ( )P1 P2 τ Δt= γ12 τ( )

K1K2

I1I2-------------Γ12 τ( )

τ f y0( )= γ12 τ( )0 γ12 τ( ) 1≤ ≤ γ12 τ( )

I y0( ) I1 y0( ) I2 y0( ) 2 I1I2 γ12 τ( ) α12 τ( )[ ]cos+ +=

α12 τ( ) arg γ12 τ( )[ ]=

46 Chapter 5


, (5.70)

and

. (5.71)

When working with spatial coherence problems, the complex degree of coherenceis related to the terms in Section 5.1.5 by

, (5.72)

and

, (5.73)

where

(5.74)

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

5.2 Fringe Localization

Waves traveling through space interfere and produce visible fringes if theconditions are right. In particular, the waves must have some degree of spatial andtemporal coherence over some region of space. Fringe localization defines theregion of space where interference occurs and fringes with reasonably goodcontrast are observed. There is a wide variety of methods for producing fringes—ranging from a simple parallel plate to complex interferometers, and each has anassociated region in space where high-contrast fringes may be observed. Thelocation of this region relative to the components of the interferometer depends onproperties of the source and geometry of the interferometer. In this section, basicproperties of fringe localization are illustrated for various types of sources andinterferometers.

There are several different categories that describe fringe localization in inter-ferometers. Some of the more common categories are;

m12 Δt( ) m12 τ( ) γ12 τ( )K1K2

I1I2-------------Γ12 τ( )= = =

β12 Δt( ) β12 τ( ) arg γ12 τ( )[ ]= =

μ12 γ12 0( )K1K2

I1I2-------------Γ12 0( )

K1K2

I1I2-------------J12= = =

β12 arg γ12 0( )[ ]=

J12 Γ12 0( ) E1 t( )E2˜ ∗ t( )⟨ ⟩= =

μ12

P1 P2 J12 Γ11 0( ) I1

Γ11 τ( )

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Coherence and Fringe Localizationoptics.hanyang.ac.kr/~shsong/ch5-Coherence and... · the fringe pattern that changes as a result of the type of source is the fringe visibility.1