We derive generic phase and amplitude coupling relations for beam
splitters-combiners that couple a single port with three output ports or input
ports, respectively. We apply the coupling relations to a reflection grating
that serves as a coupler to a single-ended Fabry-Perot ring cavity. In the
impedance-matched case such an interferometer can act as an all-reflective ring
mode cleaner. It is further shown that in the highly undercoupled case almost
complete separation of carrier power and phase signal from a cavity strain can
be achieved

A. Bunkowski

Bunkowski

Clausnitzer

Heinzel

K. Danzmann

O. Burmeister

R. Schnabel

English

arXiv

We derive generic phase and amplitude coupling relations for beam splitters-combiners that couple a single port with three output ports or input ports, respectively. We apply the coupling relations to a reflection grating that serves as a coupler to a single-ended Fabry-Perot ring cavity. In the impedance-matched case such an interferometer can act as an all-reflective ring mode cleaner. It is further shown that in the highly undercoupled case almost complete separation of carrier power and phase signal from a cavity strain can be achieved

Schnabel, R.

Bunkowski, A.

Burmeister, O.

Danzmann, K.

MPG.PuRe

658 OPTICS LETTERS / Vol. 31, No. 5 / March 1, 2006Three-port beam splitters–combiners forinterferometer applicationsR. Schnabel, A. Bunkowski, O. Burmeister, and K. DanzmannMax-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Universität Hannover,Callinstrasse 38, 30167 Hannover, GermanyReceived September 19, 2005; accepted November 3, 2005; posted November 29, 2005 (Doc. ID 64890)We derive generic phase and amplitude coupling relations for beam splitters–combiners that couple a singleport with three output ports or input ports, respectively. We apply the coupling relations to a reflection grat-ing that serves as a coupler to a single-ended Fabry–Perot ring cavity. In the impedance-matched case suchan interferometer can act as an all-reflective ring mode cleaner. It is further shown that in the highly un-dercoupled case almost complete separation of carrier power and phase signal from a cavity strain can beachieved. © 2006 Optical Society of AmericaOCIS codes: 050.1950, 120.3180, 230.1360.Two-port beam splitters–combiners, for example, par-tially transmitting mirrors, are key devices in laserinterferometry. They serve as 50/50 beam splitters inMichelson interferometers and as low transmissioncouplers to cavities. Amplitude and phase relations oftwo-port beam splitters–combiners are well known.In the case of grating optics, diffraction orders of agreater number can couple to one input port. Re-cently a reflection grating with three diffraction or-ders was used for interferometer purposes; laser lightwas coupled into a linear high-finesse Fabry–Perotcavity by using the second-order Littrowconfiguration.1 The grating was built from a binarystructure. This property, together with the second-order Littrow configuration, provided a symmetryagainst the grating’s normal. The system was theo-retically analyzed in Ref. 2. It was shown that a newthree-port (3p) coupled Fabry–Perot interferometercan be designed such that resonating carrier light iscompletely backreflected toward the laser source.The additional interferometer port is then on a darkfringe and contains half of the interferometer strainsignal.In this Letter we first derive the generic couplingrelations of 3p beam splitters. This includes couplingamplitudes and coupling phases that are required forinterferometric applications. Our description in-cludes arbitrary gratings with three orders of diffrac-tion regardless of the groove shape and the diffrac-tion angles, as shown in Fig. 1. We then investigatethe 3p reflection grating coupled Fabry–Perot ring in-terferometer and show that for a resonating carrier adark port can be constructed that contains an arbi-trary high fraction of the interferometer’s strain sig-nal.Optical devices can be described by a scatteringmatrix formalism.3 In general the coupling of n inputand n output ports requires an nn scattering ma-trix S. The n complex amplitudes of incoming andoutgoing fields are combined into vectors a and b, re-spectively. For a lossless device S has to be unitary topreserve energy, and reciprocity demands SijSjifor all elements Sij of S. For a generic 3p device sixcoupling amplitudes and nine coupling phases are in-volved. Since three input and three output fields are0146-9592/06/050658-3/$15.00 ©considered, the number of phases can be reduced tosix without loss of physical generality; the remainingsix phases describe the phases of the six fields withrespect to a local oscillator field. Here we choose thephases such that the matrix S is symmetric, and b=Sa can therefore be written asb1b2b3 = 1ei1 4ei4 5ei54ei4 2ei2 6ei65ei5 6ei6 3ei3  a1a2a3 , 1where 0i1 for all i describes the amplitude andeii describes the phase of coupling. Figure 1 showsFig. 1. (Color online) Two examples of the three-port (3p)beam splitters and/or combiners. Input fields ai and outputfields bi denote complex amplitudes of the electric field. (a)Asymmetric triangular grating in second-order Littrowconfiguration. (b) Binary grating in non-Littrowconfiguration.2006 Optical Society of AmericaMarch 1, 2006 / Vol. 31, No. 5 / OPTICS LETTERS 659two examples of 3p devices. In both cases the inputbeam splits into three beams, and, vice versa, threeinput beams can interfere to become a single one.However, one realizes that the rigorously definedscattering matrix for the device in Fig. 1(b) has a di-mension of 66, but this matrix contains null ele-ments because not six but only three ports couple,and the matrix can be reduced to the matrix as givenin Eq. (1).The unitarity condition S†S=1 entails the follow-ing set of equations:1 = 12 + 42 + 52, 21 = 22 + 42 + 62, 31 = 32 + 52 + 62, 4cos24 − 1 − 2 =5262 − 1242 − 224224212, 5cos25 − 1 − 3 =4262 − 1252 − 325225213, 6cos26 − 2 − 3 =4252 − 2262 − 326226223, 7cos6 + 4 − 5 − 2 =1242 − 2242 − 526222456, 8cos6 − 4 − 5 + 1 =3252 − 1252 − 426221456, 9cos6 − 4 + 5 − 3 =2262 − 4252 − 326223456. 10Equations (2)–(10) set boundaries for physicallypossible coupling amplitudes and phases of the ge-neric lossless 3p beam splitter–combiner. Equations(2)–(4) represent the energy conservation law andarise from the diagonal elements of the unitarity con-dition. Equations (5)–(10) arise from the off-diagonalelements. They are already simplified to contain justa single cosine term. However, it can be easily de-duced that up to three phases in the scattering ma-trix S can be chosen arbitrarily. In this analysis wechoose the phases 1 ,2 ,3 to be zero. This is a per-mitted choice without introducing any restriction onpossible coupling amplitudes. Then the phases of thescattering matrix can be written as1 = 2 = 3 = 0,4 = −12arccos1242 + 2242 − 52622212  − 2 ,45 =12arccos4262 − 1252 − 325225213  ,6 = −12arccos2262 + 3262 − 425226223  + 2 . 11It is interesting to note that the coupling relationsrestrict the possible values of i. Let us assume thata free choice of 42 and 62 is desired, which then im-mediately determines 22 according to Eq. (3). Substi-tuting 1 and 3 by using Eqs. (2) and (4), Eqs.(5)–(10) provide the following pair of inequalities thatrestricts the values of 5 and thereby also the valuesof 1 and 3:461 − 242 + 62  5 461 + 242 + 62 . 12We now apply a 3p beam splitter–combiner in in-terferometry. We focus on the device in Fig. 1(b) as acoupler to a Fabry–Perot ring cavity as shown in Fig.2. Laser light incident from the left is coupled accord-ing to 42 into the cavity, which is formed by the grat-ing and two additional highly reflecting cavity mir-rors. If both cavity mirrors are lossless, the cavityfinesse depends on the specular reflectivity 22 anddoes not rely on high values of first- or second-orderdiffraction efficiencies. Using high reflection dielec-tric coatings makes high-finesse values and high la-ser buildups possible, similar to the linear cavity in-vestigated in Ref. 1. However, here the cavity outputsdepend on 42 (into port c1) and 62 (into port c3) thatcan have different values.Assuming unity laser input and perfectly reflectingcavity mirrors, the system is described byFig. 2. (Color online) Three-port coupled grating in a ringFabry–Perot interferometer. The grating can be designedsuch that the laser input is completely sent into port c1 oncavity resonance. If the cavity is impedance matched thisdevice might serve as an all-reflective mode cleaner. An-other interesting case occurs in which the cavity is highlyundercoupled. Then almost the complete cavity strain sig-nals are sent to port c3. Such a device separates carrierlight from its modulation sidebands.660 OPTICS LETTERS / Vol. 31, No. 5 / March 1, 2006c1c2c3 = S  1c2 exp2i0 . 13Here =L /c denotes the detuning from cavity reso-nance, with L the cavity length,  the laser field an-gular frequency, and c the speed of light. Solving forthe reflected amplitudes yieldsc1 = 1 +42 exp	2i4 + 1 − 2 exp2i, 14c2 =4 expi41 − 2 exp2i, 15c3 = 5 expi5 +46 exp	i4 + 6 + 21 − 2 exp2i. 16From Eq. (14) it can be shown that, for a gratingwith 52 at its maximum value for given 42 and 62,and a cavity on resonance =0, no carrier light fromthe laser incidenting from the left is leaving the cav-ity to the left c3=0. This dark port is indicated inFig. 2 by a dashed arrow. If the cavity moves awayfrom resonance, for example, caused by a cavitystrain, amplitude c3 is no longer zero. This field isgenerally termed a phase signal and might appear atsome sideband frequency 	 if the cavity is locked tothe time-averaged carrier frequency 0 with a lock-ing bandwidth smaller than 	. The phase signal gen-erated inside the cavity obviously leaves the cavityaccording to the magnitudes of 42 and 62 in two di-rections. From Eqs. (14) and (16) it is easy to provethat the power of the signal indeed splits according tothe ratio 42 /62. We now discuss two distinct ex-amples; in both of them we consider 52 to be designedclose to its maximum value. For 42=62 the cavity out-put coupling is twice the input coupling and the sig-nal is split into two equal halves. We term this case asymmetric or an impedance-matched 3p coupled cav-ity; this is in analogy to the lossless impedance-matched linear cavity whose output coupling is alsotwice the input coupling. However, due to the choiceof 52 all the carrier power is sent into port c1 if thecavity is on resonance as discussed above. Such a de-vice can serve as an all-reflective mode cleaner. For46 the 3p coupled lossless cavity can be termedovercoupled and for 46 undercoupled. As the sec-ond example we consider the highly undercoupledgrating cavity 426222 and explicitly choose thefollowing coupling coefficients:42 = 0.0001, 62 = 0.0099, 22 = 0.99,52 = 0.0394, 12 = 0.9605, 32 = 0.9507,1 = 0, 2 = 0, 3 = 0,4  − 3.1349, 5  1.5708, 6  1.5707.17For this set of measures again 52 is almost at itsmaximum value, and consequently 12 and 32 areclose to their minimum values. As in the impedance-matched case described above, again all the carrierpower is sent into port c1. Owing to the high asym-metry of the ratio between 42 and 62 the signal issent mainly into port c3. The special property of thehighly undercoupled grating Fabry–Perot interfer-ometer is therefore the possibility of separating car-rier light and phase signal. This is a remarkable re-sult. Separation of carrier light and phase signal iswell known for a Michelson interferometer operatingon a dark fringe. Such an interferometer sends allthe laser power back to the laser source. The anti-symmetric mode of phase shifts in the Michelsonarms is sent into the dark port. The symmetric modeis combined with the reflected laser power and senttoward the bright port. In the case of the highly un-dercoupled 3p grating Fabry–Perot interferometerthe almost complete phase signal is separated fromcarrier light and is accessible to detection, and the re-flected field in the bright port contains only a mar-ginal fraction of the signal 42 /62.We point out that all results obtained for theFabry–Perot ring interferometer using the 3p couplerin Fig. 1(b) also hold for a linear cavity using the 3pcoupler in Fig. 1(a). However, some distinctive prop-erties should be mentioned. Regardless of their dif-ferent topologies, the ring Fabry–Perot interferom-eter is content with only low efficiencies for greaterthan zero diffraction orders. All coupling amplitudesin Eqs. (17) with values close to unity describe specu-lar reflections. The production of such a grating withlow overall loss should be possible with standardtechnologies building on the concept used in Refs. 1and 4. In the case of the (highly undercoupled) linearFabry–Perot interferometer 12 and 32 do not describespecular reflections, and high diffraction efficienciesin the second-order diffraction are required. How-ever, especially in the second-order Littrow configu-ration, carrier and signal separation offers straight-forward extension by interferometer recyclingtechniques.5This work was supported by the DeutscheForschungsgemeinschaft within the Sonderfors-chungsbereich TR7. R. Schnabel’s e-mail address isroman.schnabel@aei.mpg.de.References1. A. Bunkowski, O. Burmeister, P. Beyersdorf, K.Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley,and A. Tünnermann, Opt. Lett. 29, 2342 (2004).2. A. Bunkowski, O. Burmeister, K. Danzmann, and R.Schnabel, Opt. Lett. 30, 1183 (2005).3. A. E. Siegmann, Lasers (University Science Books,1986), p. 401.4. T. Clausnitzer, E.-B. Kley, A. Tünnermann, A.Bunkowski, O. Burmeister, K. Danzmann, R.Schnabel, A. Duparré, and S. Gliech, Opt. Express 13,4370 (2005).5. G. Heinzel, K. A. Strain, J. Mizuno, K. D. Skeldon, B.Willke, W. Winkler, R. Schilling, A. Rüdiger, and K.Danzmann, Phys. Rev. Lett. 81, 5493 (1998).

Three-port beam splitters-combiners for interferometer applications

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