We experimentally demonstrate the phase relations of 3-port gratings by
investigating 3-port coupled Fabry-Perot cavities. Two different gratings which
have the same 1st order diffraction efficiency but differ substantially in
their 2nd order diffraction efficiency have been designed and manufactured.
Using the gratings as couplers to Fabry-Perot cavities we could validate the
results of an earlier theoretical description of the phases at a three port
grating

A. Bunkowski

A. Tünnermann

Bunkowksi

Bunkowski

E.-B. Kley

K. Danzmann

Kley

Moharam

O. Burmeister

R. Schnabel

Schnabel

T. Clausnitzer

English

arXiv

We experimentally demonstrate the phase relations of three-port gratings by investigating three-port coupled Fabry-Perot cavities. Two different gratings that have the same first-order diffraction efficiency but differ substantially in their second-order diffraction efficiency have been designed and manufactured. Using the gratings as couplers to Fabry-Perot cavities, we could validate the results of an earlier theoretical description of the phases at a three-port grating [Opt. Lett. 30, 1183 (2005)

Bunkowski, A.

Burmeister, O.

Danzmann, K.

Schnabel, R.

Clausnitzer, T.

Kley, E.

Tünnermann, A.

MPG.PuRe

Demonstration of three-port grating phase relations

Crossref

Bunkowski, Alexander

Burmeister, Oliver

Danzmann, Karsten

Schnabel, Roman

Clausnitzer, Tina

Kley, Ernst-Bernhard

Tünnermann, Andreas

Max Planck Society eDoc Server

2384 OPTICS LETTERS / Vol. 31, No. 16 / August 15, 2006Demonstration of three-port grating phase relationsA. Bunkowski, O. Burmeister, K. Danzmann, and R. SchnabelAlbert-Einstein-Institut Hannover, Max-Planck-Institut für Gravitationsphysik and Universität Hannover,Callinstrasse 38, 30167 Hannover, GermanyT. Clausnitzer, E.-B. Kley, and A. TünnermannInstitut für Angewandte Physik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, GermanyReceived April 19, 2006; accepted May 24, 2006;posted May 31, 2006 (Doc. ID 69970); published July 25, 2006We experimentally demonstrate the phase relations of three-port gratings by investigating three-portcoupled Fabry–Perot cavities. Two different gratings that have the same first-order diffraction efficiency butdiffer substantially in their second-order diffraction efficiency have been designed and manufactured. Usingthe gratings as couplers to Fabry–Perot cavities, we could validate the results of an earlier theoretical de-scription of the phases at a three-port grating [Opt. Lett. 30, 1183 (2005)] © 2006 Optical Society of AmericaOCIS codes: 050.5080, 120.2230, 230.1360.Conventional interferometers rely on splitting andrecombining optical fields with partly transmissivebeam splitters. When transmission through opticalsubstrates is disadvantageous, diffractive reflectiongratings can also serve as beam splitters, allowing forall-reflective interferometry.1 As long as the gratingsplits an incoming beam into two outgoing beams,the phase relation at the grating, and hence the prop-erties of the interferometer built thereof, are analo-gous to the well-known ones of a transmissive two-port beam splitter. However, if a diffractive beamsplitter has more than two orders, the mirror analog,and thus the simple phase relation, no longer hold.Still, a knowledge of these relations at the diffractivebeam splitter is essential for an understanding ofmultiple-port interferometry. In a recent experiment,a grating in a second-order Littrow mount was usedto couple light into a Fabry–Perot cavity.2 In thiscase, the incoming beam was split into three outgoingbeams. The phase relations at the so-called three-port grating were analyzed theoretically and theinput–output relations for a Fabry–Perot cavity witha three-port coupler were derived.3 The theoreticalinvestigation of the phases was based solely on en-ergy conservation and reciprocity of the device, butthere has not yet been an experimental validation ofthe results.In this Letter we report an experiment that wasperformed to demonstrate the phase relations of op-tical three-port devices. Two different gratings weredesigned and manufactured for this purpose, andused as couplers to Fabry–Perot interferometers.Phase relations for three-port gratings with equaldiffraction efficiencies in the ± first orders can bewritten as3,40 = 0, 11 = − 1/2arccos12 − 202/200, 22 = arccos− 2/220, 310146-9592/06/162384-3/$15.00 ©where 0, 1, and 2 are the phase shifts for zeroth,first, and second-diffraction orders, respectively. In-terestingly, the coupling phases depend on the cou-pling amplitudes, which are given by 0, 1, and 2for the zeroth, first, and second-diffraction orders, re-spectively, and by 0 for the normal incidence reflec-tivity of the grating.Direct measurements of beam-splitter phase rela-tions are difficult. However, if the three-port beamsplitter is used to couple light into a cavity, the cavityproperties can be used to validate the phase rela-tions. Figure 1 shows the optical layout of a Fabry–Perot interferometer with a three-port grating cou-pler. The grating is used in a second-order Littrowmount, and light from a laser source is coupled to theinterferometer via the grating’s first order. The fieldamplitudes of the back-reflected light c1 andforward-reflected light c3 result from interference ofthe input field with the intracavity field and directlydepend on the phase relations between the gratingports. In Ref. 3, amplitude reflection coefficients forc1 and c3, as well as the amplitudes for the intracav-ity field c2 and the transmitted field t, were de-rived and are repeated here for convenience.c1 = 2 expi2 + 12 exp2i1 + d, 4c2 = 1 expi1d, 5c3 = 0 + 12 exp2i1 + d, 6Fig. 1. (Color online) Grating in second-order Littrowmount with naming convention given in text.2006 Optical Society of AmericaAugust 15, 2006 / Vol. 31, No. 16 / OPTICS LETTERS 2385t = i1c2 expi, 7where the amplitude reflectance and transmittanceof the cavity end mirror are given by 1 and 1, re-spectively. The resonance factor is given by d= 1−01exp2i−1, and the length L of the cavity is ex-pressed by the tuning parameter =L /c, where  isthe angular frequency and c the speed of light.One distinct feature of this type of grating cavity isthat the grating phase relations allow for reflectioncoefficients (as a function of ) that are not symmet-ric to the detuning of the cavity. Figure 2 shows thecalculated power back reflectance c12 of a cavitywith input coupling of 12=0.1 and an ideal end mir-ror 1=1 as a function of cavity tuning  for se-lected values of the second-order diffraction efficiency22. In all cases shown, the cavity finesse is the same.For an ideal (lossless) grating, the finesse depends onthe first-order diffraction efficiency 1= 1−0 /21/2only. For the minimal second-order diffractionefficiency3 2,min= 1−0 /2, all the light is reflectedback towards the laser source if the cavity is on reso-nance =0 mod . However, for maximal second-order diffraction efficiency 2,max= 1+0 /2, no lightis reflected back from a resonating cavity. Hence forthe extremal values of 2 the back-reflected port be-haves either exactly like the reflection port or thetransmission port of a conventional, impedance-matched, two-mirror Fabry–Perot cavity. For inter-mediate values of 2, the power reflectance is nolonger symmetric to the =0 axis, and the resonancepeaks are not of the usual Airy form, as can be seenfor the two exemplary curves, 22=0.15 and 22=0.8, inFig. 2.To verify the grating behavior, two gratings withessentially the same first-order diffraction efficiencybut substantially different second- and hence zeroth-order diffraction efficiency were designed and manu-factured. The gratings use a binary structure writteninto the top layer of a dielectric multilayer stack con-sisting of Ta2O5 and SiO2 placed on a fused silica sub-Fig. 2. (Color online) Calculated power back reflectancec12 for a cavity with coupling 12=0.1 and an end mirrorwith 1=1 as a function of cavity tuning  for selected val-ues of second-order diffraction efficiency 22.strate. We chose a grating period of p=1450 nm,which corresponds to a second-order Littrow angle of47.2° for the Nd:YAG laser wavelength of 1064 nmused. A rigorous coupled wave analysis5 was per-formed to design the grating. The ridge width is p /2,and the top layer consists of 880 nm of SiO2. Figure 3shows the calculated diffraction efficiencies for allthree diffraction orders in the second-order Littrowmount as a function of groove depth. The gratingswere produced by ultrafast high-accuracy electronbeam direct writing6 (electron beam writer ZBA23hfrom Leica Microsystems Jena GmbH) and etched bymeans of reactive ion beam etching. The etching pro-cess was stopped after reaching a groove depth of500 nm (G1) and 850 nm (G2), respectively.A sketch of the experimental setup used to verifythe grating phase relations is shown in Fig. 4. A beamof a diode-pumped Nd:YAG nonplanar ring oscillator(Model Mephisto from Innolight GmbH) was spa-tially filtered with a triangular ring cavity. The grat-ing (either G1 or G2) was illuminated at a second-order Littrow angle, and a cavity end mirror with 12=300 parts per million was placed parallel to thegrating’s surface. The cavity length could be con-trolled by a piezoelectric transducer, and the threeports of interest were monitored by photodetectors.Figures 5 and 6 show the measured signals fromthe three photodetectors for linear cavity scans overone free spectral range using G1 and G2, respec-tively. Also shown are the theoretical curves c12,Fig. 3. (Color online) Calculated diffraction efficiencies asa function of groove depth obtained with RCW calculationsfor the gratings used. The circles show the design values ofour gratings G1 and G2, respectively.Fig. 4. (Color online) Experimental setup: PZT, piezoelec-tric transducer; PD, photodetector.2386 OPTICS LETTERS / Vol. 31, No. 16 / August 15, 2006c32, and t2, which were obtained from Eqs.(4), (6), and (7) using measured efficiencies of the twogratings. Coupling to the cavity was measured to beidentical for both gratings within the measurementFig. 5. (Color online) Normalized powers at the three pho-todetectors for three-port coupler G1 as the cavity lengthwas linearly scanned (solid, blue curve) and the calculatedvalues (dashed-dotted, green curve).Fig. 6. (Color online) Normalized powers at the three pho-todetectors for three-port coupler G2 as the cavity lengthwas linearly scanned (solid, blue curve) and the calculatedvalues (dashed, green curve).accuracy of about 5% of the power meter used,12G1=12G2=0.10. For the first grating, a value of22G1=0.15 was measured, and for the second one, avalue of 02G2=0.10 was measured. The remainingvalues were calculated using the identities 02+12+22=1 and 02+212=1. We found the calculated val-ues within the error bars of direct measurements.Figures 5 and 6 show that the theoretical and mea-sured curves agree very well. The interference at thethree-port gratings could therefore be well describedby the phase relations according to Eqs. (1)–(3). Thesmall deviations are possibly due to imperfect modematching, and losses at the grating that may becaused by transmission, scattering, and diffractionfrom periodic grating errors. As predicted, the mea-sured intensities in the reflecting ports showed theasymmetric behavior around cavity resonances.In conclusion, we have designed and manufacturedtwo diffraction gratings that allowed the constructionof grating-coupled Fabry–Perot cavities with thesame finesse but with totally different properties ofthe two reflected ports, thereby confirming the phaserelations that were earlier derived theoretically. Ourexperimental results could be fully described byphase relations based on energy conservation andreciprocity and the knowledge of the grating’s diffrac-tion efficiencies. No further information about thegratings was required.This research was supported by the DeutscheForschungsgemeinschaft within the Sonderfor-schungsbereich TR7. A. Bunkowski’s e-mail addressis alexander.bunkowski@aei.mpg.de.References1. K.-X. Sun and R. L. Byer, Opt. Lett. 23, 567 (1997).2. A. Bunkowksi, O. Burmeister, P. Beyersdorf, K.Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley,and A. Tünnermann, Opt. Lett. 29, 2342 (2004).3. A. Bunkowski, O. Burmeister, K. Danzmann, and R.Schnabel, Opt. Lett. 30, 1183 (2005).4. R. Schnabel, A. Bunkowski, O. Burmeister, and K.Danzmann, Opt. Lett. 31, 658 (2006).5. M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. 72,1385 (1982).6. E.-B. Kley, T. Clausnitzer, M. Cumme, K. Zöllner, andB. Schnabel, in Proc. SPIE 4231, 116 (2000).