low-coherence interferometry with synthesis of coherence function

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Low-coherence interferometry with synthesis ofcoherence function

Yuichi Teramura, Keiichi Suzuki, Masayuki Suzuki, and Fumihiko Kannari

Synthesis of a coherence function by manipulation of the spectral phase of low-coherent light with asegmented liquid-crystal phase modulator and its application in a low-coherence interferometry aredescribed. Effects of space–time coupling caused at diffractive gratings and second-order dispersion atthe spatial light modulator on the coherence function synthesis are theoretically and experimentallyverified. Various coherence functions can be shaped with phase-only masks designed by simulatedannealing optimization algorithm. We utilized this technique for a novel optical low-coherence reflec-tometry without any mechanical movement for scanning optical delay. © 1999 Optical Society ofAmerica

OCIS codes: 070.2590, 100.4550, 100.5090, 110.4500, 120.2830.

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1. Introduction

Low-coherence interferometry ~LCI! is a noninvasivesensing method that provides depth information withhigh resolution and high sensitivity. It has beenapplied to optical coherence tomography1 ~OCT! orhree-dimensional shape measurements of microob-ects with large height discontinuity.2

The low coherent ~LC! light is often compared withan ultrashort optical pulse. The bandwidth of LClight is as broad as a subpicosecond pulse. Fouriertransform of a power spectrum of LC light representsthe autocorrelation waveform, whereas Fouriertransform of a complex amplitude spectrum of anultrashort pulse corresponds to the pulse waveform.

Recently, shaping of ultrashort pulses into desiredwaveforms has been extensively studied.3,4 Themost successful technique for ultrashort pulse shap-ing involves Fourier synthesis in which the opticalfield is modified by manipulation of its spatiallydispersed frequency components.3 Computer-controlled pulse shaping was achieved by a seg-mented liquid-crystal spatial light modulator ~SLM!3

and by acousto-optic modulators.5 Various appli-

The authors are with the Department of Electrical Engineering,Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi,Kohoku-ku, Yokohama, Kanagawa 223-8522 Japan. F. Kannari’se-mail address is [email protected].

Received 22 March 1999; revised manuscript received 25 June1999.

0003-6935y99y285974-07$15.00y0© 1999 Optical Society of America

5974 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999

cation with the ultrashort pulse shaping such ascode-division multiplexing access ~CDMA! communi-ation,6 dynamic space-to-time conversion,7 and

adaptive control of chemical reactions8 have beendemonstrated. The similar shaping techniques areapplicable to shaping a coherence function of LClight. Binjrajka and co-workers9 demonstrated thecoherence function shaping utilizing the same exper-imental setup used for subpicosecond optical pulses.With this coherence function shaping, a CDMA com-munication scheme was also proposed and demon-strated by use of LC light.10 The use of a LC lightsource instead of ultrashort pulses makes a systemmore compact and less expensive. Moreover, sinceless nonlinear optical effects affect during propaga-tion through a fiber, concepts with LC light are moresuitable in applications with fiber delivery.

Separate developments on the coherence functionsynthesis have been performed by He and co-workers.11 They modulated the drive current of alaser diode and synthesized the coherence functioninto a delta-function-like peak at an arbitrary posi-tion. And the position of the peak can be scanned ata spatial resolution of a few centimeters by adjust-ment of the current modulation parameters. An in-terferometric system based on the coherence functionsynthesis and in which two-dimensional images canbe extracted from a three-dimensional object was in-vented.

In this paper, experiments of synthesis of the co-herence function of LC light in the range of a few tensof micrometers by programming the phase of the spa-tially dispersed frequency components is described.

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A simulated annealing ~SA! algorithm was used todesign phase-only filters to shape the coherence func-tions. We produced various shapes of coherencefunction and applied them to interferometric mea-surements of a multilayered object without any me-chanical movement. We also analyzed accuracy ofthe coherence function shaping taking account of ef-fects of space–time coupling caused at diffractivegratings and of second-order dispersion at a SLM.

2. Design of Phase-only Filters

When two optical fields with the complex amplitudeof e1 and e2 are superposed with a time delay t, cor-responding to a certain optical path difference ~OPD!,he instantaneous complex amplitude eobj~t! at the

object plane is obtained by linear superposition of thecomplex amplitudes,

eobj~t! 5 e1~t! 1 e2~t 2 t!. (1)

At the observation plane, the optical intensity is de-scribed as

I 5 êobj~t!e*obj~t!&

5 ê1~t!e*1~t 2 t!& 1 ê2~t!e*2~t 2 t!&

1 ê1~t!e*2~t 2 t!& 1 ê*1~t!e2~t 2 t!&. (2)

where the angular brackets symbolize the operationof temporal average. The third angular bracket iscalled a ~mutual! coherence function G~t!:

G~t! 5 ê1~t!e*2~t 2 t!& 5 limT3`

12 *

2T

T

e1~t!e*2~t 2 t!dt. (3)

The fourth angular bracket of Eq. ~2! corresponds tothe complex conjugate of the coherence function, andthe first and the second ones are the optical intensityof each light, I1 and I2, respectively. Complex de-ree of coherence is defined by the normalized coher-nce function as follows:

g~t! 5G~t!

~I1 I2!1y2 ; ug~t!uexp@1if~t!#. (4)

Thus, the Eq. ~4! is described as Eq. ~5!:

I 5 I1 1 I2 1 2~I1 I2!1y2ug~t!ucos f~t!. (5)

Therefore the coherence function represents interfer-ence fringes, where ug~t!u is the envelope and f~t! ishe phase of fringes.

From Wiener–Khintchine theorem, Fourier trans-orm of the cross correlation is a mutual spectralensity function ~MSDF! G~v!,

G~v! 5 ^$g~t!% 5 E1~v!E2~v!, (6)

here E1 and E2 are the Fourier transform of the e1and e2, respectively.

Now, let us imagine a Mach–Zehnder interferom-eter with two optical arms. When light is equallysplit into two arms and spectral modulation M~v! ispplied in the reference arm, the light in the refer-

nce arm E2~v! is expressed as Eq. ~7! with the signallight E1~v!:

E2~v! 5 M~v!E1~v!. (7)

Then, MSDF is obtained as follows:

GM~t! 5 ^21$M~v!E1~v!E*1~v!% 5 ^21$M~v!I~v!%. (8)

Thus the coherence function is an inverse Fouriertransform of products of the spectral modulationM~v! and the power spectrum of the light source I~v!.

herefore one can produce arbitrary coherence func-ion when an appropriate mask in the frequency do-ain is designed.For example, when one desires to simply shift the

osition of the coherence function peak in the timeomain without changing its shape, the shift law inourier transform can be used:

G~t 2 t9! 5 *2`

`

@G~v!exp~2i2pvt9!#exp~2i2pvt!dv.

(9)

From this expression, a linearly increasing phasemask can shift the coherence function in the timedomain.

When one modulates both the spectral phase andthe amplitude of the light, the range and the accuracyof coherence functions that can be generated in theFourier synthesis significantly increase. However,when the reduction in optical power that necessarilyaccompanies amplitude filtering is undesirable for aparticular application, spectral phase-only filteringwould be the solution. In most applications of LClight, only an amplitude profile of the coherence func-tion is specified, and no constraint is placed on thephase profile. For the phase-only masks with a fi-nite number of segmented liquid-crystal phase mod-ulators to be realized, since we cannot analyticallydesign them, a certain optimization algorithm is nec-essary in the design procedure. We used the SAmethod that we used to design phase-only masks toshape both the amplitude and the phase of subpico-second optical pulses.12

3. Experimental Setup

The system configuration incorporates a Mach–Zehnder interferometer with a superluminescent di-ode ~SLD! as a light source, as shown in Fig. 1. Thepower spectral density of the SLD ~Anritsu:SD1D251C! and the measured autocorrelation func-tion are shown in Figs. 2 and 3, respectively. Thecenter wavelength is l0 5 790 nm with a spectralwidth of 14.82 nm ~FWHM! corresponding to coher-ence length of ;34 mm. The output beam from SLDis spatially coherent. The coherence function syn-thesizer placed in the reference arm consists of a pairof 2000-linesymm gratings placed at the outer focalplane of a unit magnification confocal lens pair. Thefocal lengths of these achromatic lenses are f 5 200

1 October 1999 y Vol. 38, No. 28 y APPLIED OPTICS 5975

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mm. At the inner focal plane, where the opticalspectrum is spatially dispersed, spectral filtering isaccomplished with a liquid-crystal SLM ~CambridgeTechnology: SLM-OPT! consisting of 128 individu-ally programmable pixels. Each 97-mm-width pixelis separated by a 3-mm gap where light goes throughwith no phase modulation.

Fig. 1. Schematic setup of the coherence function synthesis. BS,beam splitter.

Fig. 2. Power spectrum of the SLD used in our experiments.Broken lines indicate the low and upper limits of the SLM band-width.

Fig. 3. Measured autocorrelation function of the SLD.


The total transmission efficiency is ;60%, which isattributed mainly to the diffraction efficiency at twogratings. Part of the wings of SLD spectrum is fil-tered by the limited size of the SLM aperture, asshown in Fig. 2. During coherence function mea-surements, a high-reflectance mirror was placed nor-mal to the incident light at the object plane.Scanning the length of the signal arm with a linearmechanical stage allowed the coherence functionwaveforms to be detected by Si photodiode.

The resolution and accuracy of the coherence func-tion shaping are determined by the SLM structureand its minimum phase resolution. Let us considerthe case of linear coherence function shift in the time~or the OPD! domain. From Eq. ~9!, the shift of theoherence function peak in the OPD dx is obtainedith the phase dispersion offered with the SLM:

dx 5l0

2

2p

df

dl. (10)

The minimum dispersion ~dfydl!min is limited bythe spatial dispersion of the spectrum, dlydx, at theFourier plane and by the minimum phase shift con-trolled by the electric circuitry. In our experiments,the resolution of the phase shift in the SLM was0.011p rad. Therefore the minimum dx is 0.15 mm,which is small enough compared with the separationof interference fringe. On the other hand, the max-imum shift is obtained when neighboring pixels arealternatively shifted by f 5 6p ~an alternativep-shift mask!. The maximum linear shift is deter-mined simply by the spectral resolution for the pix-elized phase mask. SLM’s designed with smallerand more pixels can, therefore, increase the linearshift range as long as the beam-spot size at the Fou-rier plane is small enough. Thus in our SLM thelinear shift range is 21.80 mm # dx # 11.80 mm.

4. Results and Discussion

A. Linear Shifting

First, linear shifting of coherence function was dem-onstrated. Figure 4 shows the experimental resultsof the coherence functions shifted by 10.6 mm, 11.2mm and the maximum shift of 1.8 mm by the alter-native p-shift mask. Although the peak position ofthe coherence function is shifted as expected, thereare significant distortions in the shifted coherencefunctions. The distortions are caused by the follow-ing reasons:

1. The gap effect in which no phase modulation isgiven for light propagating through 3-mm gaps in theSLM. It caused a small peak around the OPD of 0mm.

2. Each pixel of the SLM acts as a rectangularwindow function in the spectral domain. Thus the

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maximum peak height of a coherence function is lim-ited by a sinc function,

U 5 sincScDl

2l02 tD , (11)

where c is the speed of light in vacuum and Dl isspectrum window width of 2.28 nm.

3. A discreet, sampled phase shift causes a replicawaveform, which is obvious at the maximum phaseshift ~Fig. 4, bottom waveform!.

4. Higher-order spatial dispersion of the spec-trum at the SLM.13

5. The effect of space–time coupling. Since thewave front of light is always tilted after the diffrac-tive grating at the input side, any newly added spa-tial frequency components at the Fourier plane by thephase mask also causes shaping in the spatial beamprofile at the output. The spatial distortion is nolonger compensated by the grating adjustment.

Here, we discuss terms 4 and 5 in detail. More-over, we construct a numerical simulation model,taking into account these terms.

Diffraction of a grating is given by

l 5sin ui 1 sin ud

G, (12)

here ui and ud are the angle of incidence and dif-fraction, respectively. G is the number of grooves

er unit length. Ideally, the lateral displacement xf a given wavelength l from the center wavelength0 at the Fourier plane is given by

x 5 f tan@ud 2 ud0#, (13)

where ud0 is the diffraction angle of the center wave-length l0. Then, the spatial dispersion of the spec-rum is obtained as follows:

dl

dx5

cos ud0

fG. (14)

Fig. 4. Experimental results of the coherence function shifted inOPD by 0.6 mm ~upper!, 1.2 mm ~middle!, and 1.8 mm by use of analternative p-shift mask ~bottom!.

In our experimental setup, this spatial dispersion is1.73 nmymm. Because of this spatial dispersion,both spectrum wings are cut at the SLM as shown inFig. 2.

Now, Eq. ~13! is extended as a power series in theangular frequency v,

x 5 fF]ud

]vUv5v0

~v 2 v0! 112

]2ud

]v2Uv5v0

~v 2 v0!2 1 . . .G

5 a1v# 1 a2v2 1 . . . , (15)

where, v# 5 ~v 2 v0!. The coefficient of the first-rder term is given by

a1 5 21v0

Gl0 fcos ud0

, (16)

and the second-order term is given by

a2 5 21

2v02

Gl0 fcos ud0

S1 1pcG

v

tan ud0

cos ud0D . (17)

In our case, the second-order term a2v# 2 of Eq. ~15!s ;3% of the first-order term a1v# at the edge of the

ask. Therefore, in most cases, the second term cane ignored. However, this second-order dispersionerm makes a slight shift of the coherence functioneak and also widens the coherence function whenne intends to give large temporal shifts to the coher-nce function along the shift law given by Eq. ~9!, ashown in the bottom curve in Fig. 4. We took intoccount this second-term dispersion in the numericalodel calculation. The initial beam profile at the

nput grating is a 2-mm-diameter Gaussian. Curveb! of Fig. 5 is the coherence function obtained with-ut both the higher-order spatial dispersion at theLM and the space–time coupling. Curve ~c! of Fig.is one calculated with the second-order dispersion

ffect. The coherence function spreads over the

Fig. 5. ~a! Measured coherence function shaped by an alternativep-shift mask. Calculated coherence functions: ~b! ignoring bothsecond-order dispersion and space–time coupling, ~c! with second-order dispersion, ~d! with space–time coupling, ~e! with both terms.


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OPD, and its peak slightly decreases and shifts to-ward the deeper depth.

Next, let us discuss the space–time coupling in thecoherence function synthesizer.14 The space–timeoupling in the case of continuous broadband lighturing free-space propagation has actually been theubject of much debate in the past ten years.15 The

propagation of spatially incoherent light is essen-tially equivalent to an aberration that causes theoutput beam to spread or to change shape. In thispaper, we concentrate the discussion of the space–time coupling only to that caused by the grating inthe coherence function synthesizer, since this effect ismuch more significant. Since the spatial coherenceof the beam from the SLD is enough to be approxi-mated with a plane wave, we ignored the space–timecoupling during the propagation outside the synthe-sizer.

Since the incoherent light of a SLD is spatiallycoherent, we assume the initial complex amplitude ofthe beam at the input diffraction grating to be

ein~x, t! 5 es~x!et~t!. (18)

Then, the beam profile at the phase mask is repre-sented by the spatial and the temporal Fourier trans-form of this initial complex amplitude:

Ein~j, v! 5 Es~j!Et~v!. (19)

On the Fourier plane, the spectral component ~v, j! isocalized at

x 5 a1v 1 bj, (20)

here b is spatial frequency dispersion given by

b 5l0 f2p

cos ui

cos ud0. (21)

In this description, we ignored the higher-orderterms of the dispersion. The complex amplitude ofthe beam after the phase mask is described as

Esyn~j, v! 5 M~a1v 1 bj!Es~j!Et~v!. (22)

hen, the shaped coherence function is obtained with

G~x, t! 5 ^esyn~x, t!e*in~x, t 2 t!&

5 ^t21@Esyn~x, v!E*in~x, v!#

5 ^t21@^x

21$M~a1v 1 bj!Es~j!Et~v!%e*s~x!E*t~v!#

5 ^t21@^x

21$M~a1v 1 bj!Es~j!%e*s~x!It~v!#.(23)

It is obvious from Eq. ~23! that the coherence functionvaries at different spatial position x.

We measured the spatially resolved beam profiles.As shown in Fig. 6, the phase mask that ideally shiftsthe coherence function causes a distortion along the xaxis. Figure 7 shows the spatial shifts of the coher-ence function obtained with the numerical model cal-culation. As the phase difference between theneighboring SLM pixel increases, the coherence func-


tion distribution shifts also in space. Therefore,when the coherence functions are measured at a fixedposition corresponding to the center of the unshapedcoherence function distribution, the amplitude of thecoherence function decreases at larger temporal~OPD! shifts.

The effects of the second-order spatial dispersion atthe SLM and the space–time coupling are summa-rized in Fig. 5. Curve ~d! shows a coherence functionmeasured at the center of the unshaped coherencefunction distribution. In this calculation, the effectsof the second-order spatial dispersion at the SLM isexcluded. Curve ~e! shows a coherence function cal-culated when we took account of the second-orderspatial dispersion at the SLM and the space–timecoupling. Compared with our experimental result

Fig. 6. ~a! Measured spatial beam profile of a SLD light shaped byan alternative p-shift mask. ~b! Measured spatial beam profile of

original SLD light.

Fig. 7. Calculated coherence functions indicating the space–timecoupling: ~a! unshaped original light, ~b! shifted by 0.6 mm, ~c!shifted by 1.2 mm, ~d! shifted by 1.8 mm with an alternative p-shiftmask.

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s

shown by curve ~a!, there is a good agreement in thehape and in the amplitude. The effect of theecond-order spatial dispersion can be compensatedhen the phase mask is designed under the consid-

ration of the dispersion. However, the space–timeoupling cannot be avoided.

B. Various Shaping of Coherence Function and MultilayerDetection

To synthesize the coherence functions by a phase-only mask, we used SA algorithm to design themasks. The main architecture of the SA algorithmused in this research is almost the same as that in ourprevious paper on ultrashort pulse shaping.12 Sinceonly the amplitude of the coherence function is usedin most of interferometric measurements, we speci-fied only the amplitude. We demonstrate a rectan-gular envelope, three equal peaks, and three peakswith different heights. The theoretical calculationsand experimental results are compared in Fig. 8. Inthese calculations, the effects of the second-order spa-tial dispersion at the SLM and the space–time cou-pling are not included. The ripples appearing on therectangular function are caused by the limited band-width of the phase mask. Fairly good agreementsare obtained between the theory and the measure-ments.

Fig. 8. Comparison of calculated and measured coherence func-tions shaped ~a! to a rectangular envelope, ~b! three equal peaks, ~c!three peaks with different heights.

Using this coherence function synthesis technique,we demonstrate image detection of selective planes ofa multilayer object without any mechanical scanning.The object is schematically shown in Fig. 9, whichconsists of three 160-mm-thick glass plates piled upike stairs. Each of the three faces is numbered withlack ink. The object beam is expanded and colli-ated to illuminate the entire object. The reflected

Fig. 10. Experimental results of synthesized coherence functions~left! and corresponding CCD images of interferometry ~right! withthe object shown in Fig. 9: ~a! only the second glass plate surfaceis imaged with an original SLD light by adjustment of the opticaldelay; ~b! both the second and the third glass plate surfaces areimultaneously imaged with a dual-peak coherence function; ~c!

similar image for the first and the third planes; ~d! all the threeplanes are imaged with a triple-peak coherence function.

Fig. 9. Schematic of an object consisting of three glass plates piledup like stairs, which is used to measure the individual plane by thecoherence function shaping in interferometry. The numbers arepainted on each glass plate to absorb light.


2. B. S. Lee and T. C. Strand, “Profilometry with a coherence

5

light from this object is imaged and superposed withthe shaped beam at a CCD camera.

We shaped the coherence function so that theshaped peaks correspond to the surface of the object.The shaped coherence functions and the correspond-ing images detected by the CCD camera are shown inFig. 10. It is small enough to ignore the effect of thespace–time coupling in this region. The imageswere obtained by our removing the background noiseand enhancing the digit image in a personal com-puter. By changing the peak position of the coher-ence function by replacing the phase mask pattern onthe SLM, we can monitor more than two selectivepositions simultaneously. When selecting two posi-tions overlapping in a transparent object, for exam-ple, monitoring a biological near-surface structurewith OCT, one can obtain the sum of the two planeimage information. It is also applicable for obtain-ing the sum of the optical memories written onstacked multiplates.

5. Summary

The synthesis of LC light was accomplished by ma-nipulation of the spectral phase with a segmentedliquid-crystal SLM. The effects of the higher-orderspatial dispersion at the SLM and the space–timecoupling were theoretically studied. Using the tech-nique developed in this study, one can obtain some ofthe applications proposed for ultrashort opticalpulses with pulse shaping also by LC light sources,such as CDMA communication, space–time–spacetransfer, and spectrum holography, although the in-formation always has to be extracted by means ofcorrelation measurements with a reference beamsince the information is brought by means of a coher-ence function.

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low-coherence interferometry with synthesis of coherence function

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