optical frequency-modulated continuous-wave interferometers

Optical frequency-modulated continuous-waveinterferometers

Jesse Zheng

I discuss optical frequency-modulated continuous-wave (FMCW) interferometers, including their prin-ciples, characteristics, specific requirements, procedure for their construction, optical configurations,primary applications, optical sources, resolution, measurement range, and stability. © 2006 OpticalSociety of America

OCIS codes: 260.0260, 260.3160, 120.0120, 120.3180.

1. Introduction

Frequency-modulated continuous-wave (FMCW) in-terference, which was originally investigated in radarhalf a century ago,1,2 has recently been introduced inoptics.3–7 Optical FMCW interference naturally pro-duces a dynamic signal, and thus it is easy to calibratethe fractional phase, distinguish the phase-shift direc-tion, and count the number of the full periods. There-fore, optical FMCW interference can offer higheraccuracy and a longer measurement range than thetraditional optical homodyne interferometers. Com-paratively, the classic homodyne interference couldhardly perform a long-range measurement because ofambiguity in the phase-shift direction. Moreover, be-cause the frequency of the beat signal is related to theoptical path difference (OPD), optical FMCW interfer-ence can also perform absolute OPD and velocity mea-surements.

In the first three parts of this series8–10 I an-alyzed the principles of optical FMCW interference,including sawtooth-wave optical FMCW interfer-ence, triangular-wave optical FMCW interfer-ence, sinusoidal-wave optical FMCW interference,multiple-beamopticalFMCWinterference,multiple-wavelength optical FMCW interference, and coher-ence of optical FMCW interference. In this part, Idiscuss optical FMCW interferometers—the appli-cation of optical FMCW interference.

2. Principles and Characteristics

The principles of optical FMCW interference weresystematically discussed previously.8–10 Here I giveonly a brief summary so that new readers can alsoread this paper. Optical FMCWs, as the name indi-cates, are optical waves whose frequencies or angularfrequencies are continuously modulated. If two opti-cal waves, which are derived from the same coherentoptical source whose angular frequency is modulatedwith a sawtooth waveform but have traveled alongdifferent paths, are recombined at a point in space,intensity I��, t� of the resultant electric field (or beatsignal) in any modulation period can be written as

I��, t� � I1 � I2 � 2�I1I2 cos��t � �0��

� I0�1 � V cos��bt � �b0��, (1)

where I1 and I2 are the intensities of the referenceand the signal waves, respectively, � is the opticalangular frequency-modulation rate, � is the delaytime of the signal wave with respect to the referencewave, t is the time, �0 is the optical angular frequencyat the center of the modulation period (also called thecentral optical angular frequency), I0 is the averageintensity of the beat signal �I0 � I1 � I2�, V is thecontrast of the beat signal �V � 2�I1I2��I1 � I2��, and�b and �b0 are the angular frequency and the initialphase, respectively, of the beat signal. Optical angu-lar frequency-modulation rate � is given by

� � ��Tm, (2)

where �� is the optical angular frequency-modulation excursion and Tm is the period of the mod-ulation signal (also called the modulation period).

The author ([email protected]) is with PhotonTech,1980 E51 Avenue, Vancouver, British Columbia V5P 1V9, Canada.

Received 18 May 2005; revised 22 November 2005; accepted 2December 2005; posted 6 December 2005 (Doc. ID 62239).

0003-6935/06/122723-08$15.00/0© 2006 Optical Society of America

20 April 2006 � Vol. 45, No. 12 � APPLIED OPTICS 2723

The quantities �b and �b0 are given by

�b � ��, (3)

�b0 � �0�. (4)

In the terms of frequency, wavelength and OPD,Eq. (1) can be rewritten as

I�OPD, t� � I0�1 � V cos�2��vvmOPDc t

�2�

0OPD

�I0�1 � V cos�2�vbt � �b0��, (5)

where �v is the optical frequency-modulation excur-sion ��v � ��2��, vm is the frequency of the modu-lation signal [also called the modulation frequency�vm � 1�Tm�], OPD is the optical path difference be-tween the two interfering waves �OPD � c�, where cis the speed of light in free space), 0 is the centraloptical wavelength in free space �0 � 2�c��0�, and vb

and �b0 are the frequency and the initial phase of thebeat signal, respectively, given by

vb ��OPD

2�c

��vvmOPD

c , (6)

�b0 ��0OPD

c

�2�OPD

0. (7)

Rearranging these equations, we have

OPD �c

�vvmvb, (8)

�OPD� �0

2��b0. (9)

Obviously, the frequency of the beat signal is relativeto the absolute value of the OPD, while the change inthe initial phase (i.e., phase shift) of the beat signal isrelative to the change in the OPD. In particular, if theOPD changes its wavelength, the beat signal willshift a period.

If the frequency of the optical source is modulatedwith a triangular waveform, in the frequency-risingperiods the beat signal is exactly the same as thesignal of sawtooth-wave optical FMCW interference,but in the frequency-falling periods the frequency of

the beat signal is negative and the phase-shift direc-tion is opposite that in the frequency-rising periods.The most important property of triangular-waveFMCW interference is that the Doppler frequencyshift is the same in both rising and falling periodswhen the OPD is changing. Therefore, by measuringthe average beat frequencies in the rising and fallingperiods, vbr

�� and vbf��, respectively, separately we can

find the average Doppler frequency shift vD� from theequation

vD��12 �vbr��vbf��. (10)

The average Doppler frequency shift is related to theOPD by

vD��10

dOPD�t�dt . (11)

If the signal wave is a wave reflected from a movingobject, the speed of the moving object, s, will be

s �0

2n vD�, (12)

where n is the refractive index of air �n � 1�.Sinusoidal-wave FMCW interference is similar to

triangular-wave FMCW interference. The differenceis that the frequency of the beat signal in any mod-ulation period is a variable. Although the averagevalue of the beat frequency can be used to determinethe OPD, the measurement accuracy is relativelylower.

3. Specific Requirements

The essential requirement for an optical interferom-eter is that there should be an optical arrangement inwhich two or more beams, derived from the samelight source but that have traveled along differentpaths, can be recombined to interfere. According tothe analysis above, to construct an optical FMCWinterferometer one should satisfy the following addi-tional requirements:

(1) Parallel-beam interference: The OPD betweenthe two waves that interfere is generally spatiallydependent, and thus the intensity of the resultantelectric field of optical FMCW interference usually isa function of both the spatial and the temporal coor-dinates. If a photodetector with a certain sensingsurface (normally a planar surface) is placed in theinterference field, the beat signal will be proportionalto the sum of the intensities of the light projectingonto the sensing surface at all points. Because thelight intensities are normally not in phase at allpoints on the sensing surface owing to the spatialdependency of the OPD, the contrast of the beat sig-nal will be reduced. The only way to ensure that thelight intensities at all points on the sensing surface

2724 APPLIED OPTICS � Vol. 45, No. 12 � 20 April 2006

are in phase is to make the two beams collimated andparallel to each other and to place the photodetectoron the plane perpendicular to the propagation direc-tion of the beams. In this way, the spatial dependenceof the intensity of the resultant electric field can beremoved and a beat signal with the best contrast canbe obtained. Obviously, from one point of view therequirement for parallel-beam interference is a limi-tation on optical FMCW interferometers.

Now let us discuss a little further the effect of thespatial dependency of the OPD on the contrast of thebeat signal. For simplicity, we assume that the twoFMCW waves that interfere are collimated but notparallel to each other (the angle between the propa-gation directions of the two waves is �, as shown inFig. 1) and that the photodetector has a rectangularplanar sensing surface (the length of the sensing sur-face is L). The variation of OPD on the sensing sur-face can be written as

��OPD� � nx sin

� x , (13)

where x is the coordinate at the side along which theOPD varies.

If the optical source is modulated with a sawtoothwaveform, the beat signal produced by the photode-tector should be

I�t� ��0

L

�i1 � i2��1 �2�i1i2

i1 � i2

� cos��bt � �b �2�

0xdx

� �i1 � i2��L �2�i1i2

i1 � i2

0

� sin��L

0

� cos��bt � �b ��L0

� �i1 � i2�L�1 �2�i1i2

i1 � i2sinc��L

0

� cos��bt � �b ��L0

� �I1 � I2��1 �

2�I1I2

I1 � I2sinc��L

0

� cos��bt � �b ��L0

� I0�1 � V cos��bt � �b �

�L0

, (14)

where i1 and i2 are the linear intensity densities ofthe two waves �i1 � I1�L, i2 � I2�L� and V is thecontrast of the beat signal, given by

V �2�I1I2

I1 � I2sinc��L

0 . (15)

Obviously, the contrast of the beat signal is closelyrelated to the angle of the two waves. In particular, if � 0 (i.e., if the two waves are parallel), the contrastof the beat signal will be

V �2�I1I2

I1 � I2. (16)

If � 0�L (i.e., if there are exactly one bright fringeand one dark fringe on the sensing surface in homo-dyne interference), the contrast of the beat signal willbe

V � 0. (17)

Note that the size of the sensing surface generally iskept as large as possible to produce a big signal.

Moreover, because of the effect of the coherencelength of the practical optical source,10 the real con-trast of the beat signal will be

V �2�I1I2

I1 � I2 sinc��

lcOPD sinc��L

0 , (18)

where lc is the coherence length of the optical source.The curves in Fig. 2 show the contrasts of the beat

signals under various conditions. Curve a representsthe contrast of the beat signal with an ideal opticalsource when I1 � I2 and � 0. Curve b representsthe contrast of the beat signal with an ideal lasersource when I1 � I2�4 and � 0. Curve c representsthe contrast of the beat signal with an ideal lasersource when I1 � I2�4 and � 0�2L. Curves d, e, andf stand for the contrasts of the beat signals of a, b, andc, respectively, but with a practical optical source.

(2) Unbalanced structure: In order that the beat

Fig. 1. Spatial dependency of the OPD on the sensing surface ofa photodetector.


signal can be self-calibrated, the beat signal shouldinclude the values that correspond to the phases 0and � [or 2k� and �2k � 1��, where k is an integer].Note that it is not always true that the maximum andminimum values of a beat signal correspond tophases 0 and �. Only under the condition that vb

� vm, the maximum and minimum values of the beatsignal are related to phases 0 and �, respectively.

Figure 3 shows the waveforms of the beat signal ofsawtooth-wave FMCW interference with differentbeat frequencies. Waveform (a) represents a beat sig-nal whose frequency is less than the frequency of themodulation signal �vb � vm�, and this signal cannot beaccurately calibrated owing to the lack of phase ref-erence. Waveform (b) represents a beat signal whosefrequency is equal to the frequency of the modulationsignal �vb � vm�, and obviously it is easy to process

this signal. Waveform (c) represents a beat signalwhose frequency is larger than the frequency of themodulation signal �vb � vm�, and the phase shift ofthis signal can also be accurately calibrated.

The frequency of the beat signal is proportional tothe product of the OPD and optical frequency-modulation excursion. To obtain a beat signal withthe proper frequency requires that the optical FMCWinterferometers be unbalanced and the light sourcehave the proper frequency-modulation excursion. Forinstance, in the case of sawtooth-wave FMCW inter-ference, if the frequency of the beat signal is desiredto be equal to or larger than the frequency of themodulation waveform �vb � vm; i.e., the beat signalhas at least one full wavelet in each modulation pe-riod), according to Eq. (8) the OPD between the twointerfering beams should be

OPD � c��v. (19)

In addition, an optical FMCW interferometer shouldinclude a photodetector to convert the optical signalinto an electric signal, so the time-domain interferencepattern can be viewed by use of an electronic oscillo-scope or processed by use of an electric circuit or com-puter.

4. Procedure for Construction

Optical FMCW interferometers have specific fea-tures, and therefore constructing an optical FMCWinterferometer needs a different skill. The procedurefor constructing an optical FMCW interferometer canbe as follows:

(1) Construct a parallel-beam unbalanced homo-dyne interferometer with a laser without frequencymodulation.

(2) Adjust the optical elements until only onebright fringe is left in the view field, and place aphotodetector at the center of this bright fringe.

(3) Modulate the frequency of the laser with aproper modulation waveform, and use an electronicoscilloscope to view the pattern of the electric beatsignal.

5. Configurations and Applications

The configurations of optical FMCW interferometersactually are quite simple. In this section some double-beam amplitude-division FMCW interferometersthat are derived from classical interferometers, suchas the Michelson interferometer, the Mach–Zehnderinterferometer, and the Fabry-Perot interferometer,are briefly described.

A. Michelson FMCW Interferometer

The Michelson FMCW interferometer is shown sche-matically in Fig. 4. A FMCW laser beam is first col-limated by a collimating lens and then divided intotwo beams by a beam splitter (BS). The referencebeam propagates along path l1 and is reflected by afixed mirror �M1�, while the signal beam propagatesalong path l2 and is reflected by a moving mirror �M2�.

Fig. 2. Contrast of beat signals under different conditions: a, idealoptical source, I1 � I2, � � 0; b, ideal optical source, I1 � I2�4, � �0; c, ideal optical source, I1 � I2�4, � � �0�2L; d, practical opticalsource, I1 � I2, � � 0; e, practical optical source, I1 � I2�4, � � 0;f, practical optical source, I1 � I2�4, � � �0�2L.

Fig. 3. Waveforms of the beat signals of sawtooth-wave FMCWinterference with three beat frequencies: (a) vb vm, (b) vb � vm,(c) vb � vm.


These two reflected beams are recombined by thesame beam splitter, producing a beat signal. Finally,the beat signal is detected by a photodetector. Theoptical isolator is used to prevent feedback light fromdisturbing the radiation of the laser.

The OPD between the two interfering beams can bewritten as

OPD � 2n�l2 � l1�, (20)

where n is the optical refractive index of air �n � 1�and l1 and l2 are distances from the beam splitter tomirrors M1 and M2, respectively.

The Michelson FMCW interferometer can measurethe displacement, distance, or velocity of an object.For instance, if the frequency of the laser is modu-lated with a sawtooth waveform, according to Eq. (7)the initial phase of the beat signal �b0 can be writtenas

�b0 �4�n�l2 � l1�

0. (21)

Hence by measuring the phase shift of the beat sig-nal, ��b0, one can obtain the displacement of movingmirror �l2:

�l2 �0

4�n ��b0. (22)

Figure 5 shows another version of the MichelsonFMCW interferometer, which employs a big beamsplitter (BS) to split and recombine the light beamsand uses two large retroreflectors �R1 and R2) toreflect the reference beam and the signal beam, re-spectively, back to the system. The advantages of thisconfiguration include the facts that the reflectedbeams are free from the effects of the positions of theretroreflectors because they always reflect the beamsby 180° and that there is no feedback light to affectthe laser because the forward-propagating beamsand the reflected beams travel along different paths.

Therefore, for this configuration, the optical isolatorcan be omitted.

B. Mach–Zehnder FMCW Interferometer

The Mach–Zehnder FMCW interferometer, as shownin Fig. 6, employs two beam splitters �BS1 and BS2)and two mirrors �M1 and M2) to divide and recom-bine the beams. First a collimated FMCW laser beamis divided into two beams by BS1. One is reflected byM1 and passes through BS2, propagating toward thephotodetector; another is reflected by M2 and BS2consecutively. These two beams mix coherentlybehind BS2, and the beat signal produced is receivedby a photodetector.

The OPD between the two interfering beams can bewritten as

OPD � n�l2 � l1�, (23)

where l1 is the total geometrical path length from BS1to M1 and BS2 and l2 is the total geometrical pathlength from BS1 to M2 and BS2, assuming that thebeam splitters BS1 and BS2 are identical.

The advantage of the Mach–Zehnder FMCW inter-ferometer is that all the laser beams propagate for-ward, so there is no feedback light to affect theradiation of the laser. The disadvantage of the Mach–Zehnder FMCW interferometer is that it is noteasy to remove the mirrors without disturbing theparallel-beam interfering arrangement. Therefore

Fig. 4. Michelson FMCW interferometer.Fig. 5. Michelson FMCW interferometer with retroreflectors.

Fig. 6. Mach–Zehnder FMCW interferometer.


the Mach–Zehnder FMCW interferometer is not suit-able for displacement, distance, or velocity measure-ments, but it can be used to measure the opticalrefractive index of a gas or a liquid in a transparentchamber that is placed in its path.

C. Fabry–Perot FMCW Interferometer

Figure 7 shows a Fabry–Perot (or Fizeau) FMCWinterferometer. A collimated FMCW laser beampasses through beam splitter BS and is reflected by apartially reflecting mirror (PM). The remainder of thelaser beam propagates through a length of air pathand then is reflected by mirror M. The two reflectedlaser beams propagate back to the interferometer tomix coherently. A portion of the produced beat signalis reflected by the beam splitter and detected by aphotodetector.

The OPD between the two reflected beams can bewritten as

OPD � 2nd, (24)

where d is the distance from mirror PM to mirror M.The Fabry–Perot FMCW interferometer is suited formeasuring the displacement, distance, or velocity ofan object. For instance, if the frequency of the laser ismodulated with a sawtooth waveform, according toEq. (6) and Eq. (7) the frequency of the beat signal, vb,can be written as

vb �2nd�vvm

c , (25)

and the initial phase of the beat signal, �b0, can bewritten as

�b0 �4�nd

0. (26)

Therefore the absolute distance from mirror PM tomirror M is

d �c

2n�vvmvb, (27)

and the relative displacement of mirror M is

�d �0

4�n ��b0. (28)

6. Optical Sources

Optical FMCW interferometers require coherent op-tical sources whose frequencies can be continuouslymodulated. In practice, only lasers can satisfy such aspecific requirement, and the most frequently usedlasers are semiconductor lasers.

Semiconductor lasers (also called laser diodes)have the advantages of compact size, light weight,and relatively low cost. The most attractive propertyof semiconductor lasers is that one can linearly mod-ulate their optical frequencies simply by modulatingthe drive current. The physical background is thatthe change in the drive current alters the carrierconcentration in the active medium in the laser cav-ity, which, in turn, changes the refractive index of thesemiconductor and the laser oscillation frequency.Note that the frequency modulation of a semiconduc-tor laser is always accompanied by an intensity mod-ulation. This is so because that change in the carrierconcentration in the laser cavity also changes thegain coefficient.

Figure 8 shows the waveforms of the real signalfrom the Mach–Zehnder FMCW interferometer un-der several conditions. The experiment is performedwith a single-mode semiconductor laser emitting a660 nm wavelength and a p–i–n photodiode with a1 ns rise time. If the driving current is below thethreshold of the semiconductor laser, because thelight is produced mainly by spontaneous emission thelight is basically incoherent, and thus the contrast ofthe beat signal is almost equal to zero, as shown inFig. 8(a). If the driving current is above the threshold,because the radiation process is dominated by stim-ulated emission the spectral bandwidth of the semi-

Fig. 7. Fabry–Perot FMCW interferometer.

Fig. 8. Waveforms of the real beat signal under several condi-tions.


conductor laser gets narrower, and thus the contrastof the beat signal becomes bigger, as shown in Fig.8(b). If the driving current exceeds the maximumvalue of the semiconductor laser, a phase interrup-tion or frequency hopping in the semiconductor laserwill occur, causing a phase interruption in the beatsignal, as shown in Fig. 8(c). If the driving currentapproaches the catastrophic limit of the semiconduc-tor laser, a serious nonlinear frequency-current re-sponse will occur, causing the frequency of the beatsignal to decrease rapidly, as shown in Fig. 8(d). Inthe last situation it is highly probable that the semi-conductor laser will be permanently damaged. Hence,in practice, the ideal drive current for the semicon-ductor laser should be in the middle region, 80–90%of the range between the threshold and the maxi-mum. The experimental results show that the opticalfrequency-modulation excursion can be as much as100 GHz without phase interruption or frequencyhopping and that the modulation frequency can be asmuch as 10 MHz.

7. Resolution, Measurement Range, and Stability

The resolution of the optical FMCW interferometer islimited mainly by the phase noise and the nonlinearfrequency modulation of the optical source. Thephase noise of the optical source makes the phase ofthe beat signal unstable, which in turn affects thephase measurement. Both phase noise and nonlinearfrequency modulation can affect the beat frequencymeasurement. In principle, the measurement rangeof the optical FMCW interferometer is limited by thecoherence length of the optical source. However, theactual measurement range of the optical FMCW in-terferometer is also related to the speed of response ofthe photodetector, the intensity of the beat signal,and the signal processing method.

Semiconductor lasers are easily affected by feed-back light. A small amount of light reflected from thesurface of an uncoated lens in the interferometer cansignificantly affect the quality of the laser beam,cause a much higher level of noise, or even create anextremely undesirable result. Therefore, in practice,most optical FMCW interferometers (especially ref-lectometric optical FMCW interferometers) employan optical isolator.

Another drawback of the semiconductor laser isthat the intensity and the optical frequency arestrongly dependent on the surrounding temperature.A rise in temperature reduces the output power andshifts the laser spectrum to the long-wavelength di-rection. The frequency drift of the semiconductorlaser greatly affects the long-term stability of theinterferometer. Therefore, in practice, a semiconduc-tor laser usually is mounted on a copper heat sink,and a Peltier heating element is used to stabilize thelaser temperature. For high-precision measurement,a phase-drift compensation system is necessary. Thesimplest compensation system uses an additional op-tical interferometer with a constant path imbalanceto measure the phase drift and then to rectify themeasurement data.

Figure 9(a) shows the relationship between thephase shift of the beat signal and the displacement ofmirror M in the Fabry–Perot FMCW interferometer,and Fig. 9(b) shows the relationship between the fre-quency of the beat signal and the distance from mir-ror M to partially reflecting mirror PM. In thisexperiment, the semiconductor laser is mounted on acopper heat sink and a Peltier heating element isused to stabilize the laser temperature within 20 °C� 0.01 °C. The experimental results demonstratethat a 0.01 �m resolution for relative displacementmeasurement and a 10 �m resolution for absolutedistance measurement can be obtained.

8. Conclusions

An optical frequency-modulated continuous-wave in-terferometer generates a beat signal whose initialphase and frequency are both directly proportional tothe optical path difference between the two interfer-ing waves. Hence optical FMCW interferometers canmeasure the relative displacement, absolute dis-tance, and velocity of a moving object. In particular,as the signal from an optical FMCW interferometer isa dynamic signal, it is quite easy to calibrate thefractional phase, identify the phase-shift direction,and count the number of full periods. Therefore op-tical FMCW interferometers usually can offer higheraccuracy and a much longer measurement rangethan traditional optical homodyne interferometers.

The intensity of the resultant electric field of opti-

Fig. 9. Experimental results of the Fabry–Perot FMCW inter-ferometer.


cal FMCW interference normally is a function of boththe spatial and the temporal coordinates. Because, inpractice, the photodetector normally has a planarsensing surface, the spatial dependency of the OPDwill reduce the contrast of the beat signal. To preventoccurrence of this problem, a practical way is to makethe two beams collimated and parallel to each otherand place the photodetector on the plane perpendic-ular to the propagation direction of the beams.

To calibrate the phase and frequency of the beatsignal accurately requires the frequency of the beatsignal to be equal to or larger than the frequency ofthe modulation waveform �vb � vm�. As the beat fre-quency is related to both the OPD and opticalfrequency-modulation excursion, optical FMCW in-terferometers should be properly unbalanced �OPD� c��v�.

In general, the Michelson FMCW interferometerand the Fabry–Perot FMCW interferometer are suit-able for measuring the displacement, distance, andvelocity of an object, while the Mach–Zehnder FMCWinterferometer is suitable for measuring the opticalrefractive index of a gas or a liquid in a transparentchamber that is placed in the path of the interferom-eter. From this point of view we can believe thatoptical FMCW interferometers are suitable for longi-tudinal measurement (such as the displacement of anobject), whereas traditional homodyne interferom-eters are suitable for transversal measurement (suchas the surface error of an optical lens).

Thus far, semiconductor lasers are believed to be thebest optical sources for optical FMCW interferometers.Semiconductor lasers have the advantages of directcurrent modulation, acceptable linear response, largemodulation excursion (as much as 100 GHz), highmodulation rate (up to 10 MHz), compact size, light

weight, and relatively low cost. However, semicon-ductor lasers are easily affected by the feedback lightfor the interferometers and by changes in the sur-rounding temperature. Therefore optical FMCWinterferometers usually use an optical isolator to pre-vent feedback light and use a temperature-controlsystem or a phase-drift compensation system to min-imize the effects of laser frequency drift.

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optical frequency-modulated continuous-wave interferometers

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