Internal reflections of the Gaussian beams in Faraday isolators

Internal reflections of theGaussian beams in Faraday isolators

Jerome Poirson, Jean-Charles Cotteverte, Albert Le Floch, and Fabien Bretenaker

The reflection coefficient of the Faraday isolator–mirror system in the presence of internal reflections ofthe incident Gaussian beam is theoretically and experimentally explored for three different architecturesof a Faraday isolator. In every case, these internal reflections are shown to alter widely the behavior ofthe system. In particular, we propose and test a design using a quarter-wave plate that can, in someexperiments, give better isolation ratios than conventional isolators. © 1997 Optical Society of America

Key words: Polarization, Faraday isolator, Gaussian beams

1. Introduction

The function of isolation is essential in many opticalsystems. Almost every application that uses semi-conductor laser sources requires an isolator device.Moreover, the performances required from thesecomponents are more and more stringent.1 For ex-ample, any application in which a reflective surfacefollows the laser, such as injection locking,2 requiresan isolator. Lord Rayleigh3 was the first to suggestthe use of Faraday rotation for optical isolators, along time before a real application with the laser wasfound in the early 1960’s.4–6 Early studies aimed atimproving the magnetic-field homogeneity7,8 and theFaraday material quality.9 Different isolator archi-tectures have also been explored. Apart from thestudies performed to miniaturize the isolator, thanksto multipass devices,10 or for fiber-optic applica-tions,11 great care has been taken to reduce the in-fluence of the Faraday rotator anisotropies12–15 andthe transverse inhomogeneity of the applied mag-netic field.16,17 In addition, the influence of internalreflections on Faraday rotators or isolators has beentheoretically investigated for plane waves.18–20

However, it was recently shown that the influence ofsuch internal reflections on the behavior of other op-tical components such as quarter-wave plates21 could

The authors are with the Laboratoire d’ElectroniqueQuantique–Physique des Lasers, Unite de Recherche Associee auCentre National de la Recherche Scientifique 1202, Universite deRennes I, Campus de Beaulieu, F-35042 Rennes Cedex, France.

Received 12 February 1996; revised manuscript received 10 July1996.

0003-6935y97y184123-08$10.00y0© 1997 Optical Society of America

be quantitatively explained only when the Gaussiannature of the incident beam is taken into account.Moreover, it is well known by experimentalists thatany commercial isolator always needs to be carefullyaligned to reach its best performance. One can thenwonder how to introduce this Gaussian nature of thebeam into the problem of the Faraday isolator in thepresence of internal reflections. One can also won-der whether these internal reflections, which are usu-ally considered detrimental, could not be used insome cases of experiments to improve the isolationratio of Faraday isolators, permitting one to reach,with a single isolator, the high performances reportedfrom the use of several cascaded isolators.22

Consequently, our aim in this paper is to analyzetheoretically and experimentally the behavior of op-tical isolators, taking internal reflections and theGaussian nature of the incident beam into accountand trying in each case to optimize the isolation ratio.After theoretically and experimentally consideringthe simplest isolator built with a Faraday rotator anda polarizer ~Section 2! and the usual isolator thatincludes a second polarizer ~Section 3!, we show thatcareful examination of the internal reflections per-mits us to use them to build a new kind of isolator~Section 4! that, in some cases, can exhibit a betterisolation ratio.

2. Basic Faraday Isolator

As stated in Section 1, our aim in this section is toanalyze the influence of the internal reflections of theGaussian beam on the efficiency of the most simpleFaraday isolator, composed of a polarizer and a Fara-day rotator of rotation angle u @see Fig. 1~a!#.

20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4123

A. Theoretical Predictions

We consider here the simple case of an uncoatedFaraday rod of thickness l with plane-parallel faces.For any angle of incidence i on this rod, we supposethat the magnetic field remains perpendicular to thefaces of the rod ~see the arrow in Fig. 2!, so that theincrease with i of the optical length inside the rod iscompensated for by the misalignment between themagnetic field and the beam direction. Hence u isindependent of i. Because the reflection coefficientson the faces of the rod are small and u is nearly equalto 45°, we consider here only the first internally re-flected beams, which are labeled 1 and 2 in Fig. 2.

Fig. 1. Experimental arrangements used to study the internalreflections inside the three types of Faraday isolator: ~a! simplestisolator built with a polarizer P whose axis is aligned with x and aFaraday rotator FR, ~b! conventional isolator with an output po-larizer whose axis is oriented by an angle a with respect to x, ~c!novel isolator including a quarter-wave plate ~QWP! whose fastaxis is oriented by an angle b with respect to x. BS, beam splitter;D, detector; L, collecting lens; M’s, mirrors that simulate any back-reflecting component.

Fig. 2. First-order internal reflections inside a tilted 45° Faradayrotator. ~a! Forward direction. Internal reflections give rise to asecond beam labeled 1 whose polarization is perpendicular to theone of the main beam labeled 0. ~b! Backward direction. Whencoming back, internal reflections create another secondary beamlabeled 2. The arrow at the bottom of the rod symbolizes themagnetic-field direction.

4124 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997

Beam 1 comes from the forward traversal of the rodand beam 2 from the backward trip through the rod.The amplitude transmission coefficients for oneround trip through the isolator for the three beamslabeled 0, 1, and 2 are then given by

t0 5 T 2 cos~2u!, (1a)

t1 5 R T 2 cos ~4u!, (1b)

t2 5 R T 2 cos ~4u!, (1c)

where we have omitted the common phase factor andR and T are the intensity reflection and transmissioncoefficients, respectively, of both faces of the Faradayrod, i.e.,

R 5 Sn 2 1n 1 1D

2

, (2a)

T 54n

~n 1 1!2 , (2b)

where n is the refractive index of the Faraday mate-rial. R and T are supposed to be independent of ibecause we restrain to nearly normal incidences.The calculation of the reflected intensity, in which theGaussian nature of the beam and spatial walk-off ofbeams 1 and 2 with respect to beam 0 and for anyvalues of t0, t1, and t2 are taken into account, is per-formed in Appendix A. Equation ~A5! then gives thebackward relative intensity:

II0

5 t02 1 t1

2 1 t22 1 2t0~t1 1 t2! cos w

3 expS2Dx2

2w02D 1 2t1t2 expS2

2Dx2

w02 D , (3)

where C and Dx are, respectively, the phase and thespatial shifts between main beam 0 and secondarybeams 1 and 2. These parameters depend on the tiltangle i of the Faraday rod:

C 54p

lnl cosFarcsinSsin i

n DG , (4)

Dx 5 2l cos i tanFarcsinSsin in DG . (5)

Equation ~3! contains three different terms. Thefirst one, t0

2 1 t12 1 t2

2, is the sum of the intensitiesof the three beams. The second term characterizesthe interferences between main beam 0 and both sec-ondary beams 1 and 2. Finally, the third term holdsfor the interferences between beams 1 and 2.

Figure 3 reproduces the results obtained from Eq.~3! together with Eqs. ~1! with l 5 632.8 nm, w0 5230 mm, n 5 1.68, and l 5 2 cm. Figure 3~a! corre-sponds to the case of an ideal Faraday rotation u 545°. In this case, main beam 0 is completely stoppedby the polarizer @see Eq. 1~a!#. Then, in Eq. ~3!, thesecond term vanishes, so that the signal becomes

independent of C. This can be seen in Fig. 3~a! bythe fact that no oscillations occur. Nevertheless, thethird term in Eq. ~3! that corresponds to constructiveinterferences between beams 1 and 2 is not zero. Itleads to the monotonous decrease in Fig. 3~a!, whichcorresponds physically to the decrease of the overlap-ping between beams 1 and 2 when i increases.Hence, when i increases, the signal decreases from4R2T4I0 ~corresponding to an isolation ratio of 19.5dB! for i 5 0 to 2R2T4I0 ~corresponding to 22.5 dB! for2Dx .. w0.

When one now takes into account the fact that u isnot strictly equal to 45°, one obtains curves similar tothe one of Fig. 3~b!, obtained with u 5 42°. Then themain-beam transmission coefficient t0 is differentfrom 0, leading to quickly varying interferences be-tween this beam and beams 1 and 2 @second term inEq. ~3!#. However, when one increases i, the spatialwalk-off Dx between main beam 0 and each secondarybeam 1 or 2 increases, leading to a decrease of thecontrast of these interferences, as can be seen in Fig.3~b!. In such curves, the average value of the back-ward intensity and the amplitude of the oscillationsdepend on R and u. When one slightly increasesupy4 2 uu, both the average value of the signal and itsoscillation amplitude increase. One can then opti-mize the isolation ratio by choosing the value of u forwhich the oscillations compensate for the averagevalue. This corresponds to u 5 py4 6 R, as chosenin Fig. 3~b!. In this figure, we have moreover chosenC [ 0 ~2p! for i 5 0, so that this compensation occursfor the first oscillation, leading to a very good isola-tion ratio. Our aim in the next subsection is to ver-ify these experimental predictions. In particular,we discuss the experimental feasibility of the pro-posed optimization.

B. Experimental Verification and Discussion

The experimental verification is performed at 632.8nm with the experimental setup shown in Fig. 1~a!.

Fig. 3. Basic isolator @see Fig. 1~a!#. Theoretical evolution of theintensity of the residual backward beam versus the tilt angle i ofthe Faraday rotator. The Faraday rotation angle is ~a! u 5 45°, ~b!u 5 42°. The beam waist of the incident Gaussian beam is w0 5230 mm at wavelength l 5 632.8 nm. The refractive index of theFaraday glass is taken to be n 5 1.68 and its thickness is l 5 2 cmwith C [ 0 ~2p! for i 5 0.

The source is an intensity- and frequency-stabilizedHe–Ne laser. The beam waist of the Gaussian beamis 230 mm. The input polarizer of the isolator is aGlan–Thompson polarizer whose polarization rate isbetter than 50 dB. The Faraday rod ~diameter 4.8mm! is made of a Tb-doped glass of length l 5 2 cmand with parallel and uncoated faces. It can bemoved inside the magnet in order to vary the Faradayrotation from 0 to 45°. The magnet is actually com-posed of three sandwiched permanent magnets of re-verse magnetizations, an arrangement commonlyknown to be a valuable source of a homogenous mag-netic field.16 The whole rotator can be continuouslytilted from normal incidence to a few degrees with thehelp of a constant-speed electric motor. The back-ward Gaussian beams 0, 1, and 2 are reflected on theinput beam splitter and focused on a Si photodiode.In the first experiment, u is adjusted very near to 45°.The experimental evolution of the intensity with re-spect to the tilt angle of the Faraday rotator is repro-duced in Fig. 4~a!. We have overlooked the verysmall incidence angles i for which the beam reflectedby the input face of the rotator impinges on the de-tector. The corresponding theoretical curve ob-tained with u 5 44.6° is reproduced in Fig. 4~b! and isin good agreement with the experiment. In partic-ular, one can see the decreasing evolution of the av-erage intensity, as in Fig. 3~a!, which is superimposedwith relatively small oscillations because u is not

Fig. 4. Basic isolator @see Fig. 1~a!#. ~a! ~c! Experimental, ~b! ~d!theoretical evolutions of the intensity of the residual backwardbeam versus i. ~a! ~b! u 5 44.6° and C 5 20.6 rad for i 5 0°; ~c!~d! u 5 42° and C 5 0.4 rad for i 5 0. In the theoretical curves,the only adjusted parameter is the value of C for i 5 0.


strictly equal to 45°. As expected, the amplitude ofthese oscillations decreases with i, because of thespatial walk-off.

Besides, we have predicted in Subsection 2.A thatbetter isolation ratios could paradoxically be obtainedwith values of u still more different from 45°. Thiscan be seen from the experimental results of Fig. 4~c!obtained with u 5 42°, which is also in good agree-ment with its corresponding theoretical curve @Fig.4~d!#. The isolation ratio obtained here for the sec-ond minimum is 27 dB.

To summarize, in this section we have theoreticallyand experimentally analyzed the influence of internalreflections in the well-known simple isolator com-posed of a polarizer and a Faraday rotator. It hasbeen verified that they generally limit the isolationratio of the device, but can also be used to improvethis isolation ratio of typically 5 dB with our experi-mental setup. In Section 3, we wonder whetherelimination of some of the secondary beams can leadto better isolation ratios.

3. Conventional Isolator

As can be seen from Fig. 2~a!, beams 0 and 1 areperpendicularly polarized at the output of the Fara-day rod in the forward direction. This is why a sec-ond polarizer is usually introduced at the output ofthe rod to eliminate beam 1. To study the influenceof internal reflections in this case, we now considerthe arrangement of Fig. 1~b!.

A. Theoretical Calculation of the Backward Intensity

If a is the angle of the axis of the second polarizerwith respect to x, the calculation of the transmissioncoefficients of the three beams in this case leads to

t0 5 T 2 cos~a 2 u! cos~a 1 u!, (6a)

t1 5 RT 2 cos~a 2 3u! cos~a 1 u!, (6b)

t1 5 RT 2 cos~a 2 u!cos~a 1 3u!. (6c)

In Eqs. ~6! the first cosine term is due to the influenceof the output polarizer in the forward direction andthe second one is due to the input polarizer in thebackward direction.

Figure 5 then reproduces the results obtained fromEqs. ~3! and ~6! with the same parameters as in Sub-section 2.A. Figure 5~a! corresponds to the case ofan ideal Faraday rotation u 5 45°. In this case,main beam 0 is completely stopped by the input po-larizer @see Eq. 6~a!# and beam 1 by the output polar-izer if a 5 45°. Then, in Eq. ~3!, the second and thethird terms vanish, so that the signal contains onlythe intensity of beam 2. This can be seen in Fig.5~a!, which exhibits a constant value equal to R2T4,corresponding in our case to an isolation ratio of 25.5dB. Note that this result can be obtained for anyvalue of u by choosing a 5 py2 2 u. However, thisreduces the single-pass transmission coefficient of theisolator. This technique is consequently currentlyused only for uu 2 45°u ,6°,19 permitting us to com-pensate for uncertainties on u.


When now one supposes that u is not strictly equalto 45° but keeps a 5 45°, the interferences amongbeam 0 and beams 1 and 2 are restored, leading tocurves similar to the one of Fig. 5~b!, computed withu 5 42°. The optimum isolation ratio is again ob-tained for u 5 py4 6 R, as can be seen in Fig. 5~b!.Note here that, contrary to the case of Section 2, suchan optimization can be obtained for any value of u byadjusting the value of a ~for example u 5 45° and a 548°!. Of course, this leads to an increase of the in-sertion losses of the isolator. Our aim in the nextsubsection is to verify these experimental predictionsand to discuss the experimental feasibility of the pro-posed optimization.


Practically, the verification is made with the samematerial as in Section 2, except for a second Glan–Thompson polarizer located behind the Faraday ro-tator @see Fig. 1~b!#. In a first experiment, u isadjusted very near to 45° with a 5 45°. The exper-imental evolution of the intensity with respect to thetilt angle of the Faraday rotator is reproduced in Fig.6~a!. The corresponding theoretical curve obtainedwith u 5 44.6° and a 5 45° is reproduced in Fig. 6~b!and is in good agreement with the experiment. Inparticular, one can see the constant value of the av-erage intensity, as in Fig. 5~a!, which is superimposedwith relatively small oscillations because u is notstrictly equal to 45°. As expected, the amplitude ofthese oscillations decreases with i, because of thespatial walk-off.

Here again, we have predicted that better isolationratios could be obtained with values of u still moredifferent from 45°. This can be seen from the exper-imental results of Fig. 6~c! obtained with u 5 42° anda 5 45°, which is also in good agreement with itscorresponding theoretical curve @Fig. 6~d!#. The iso-lation ratio obtained here for the first minimum is 30dB. We have also verified that if one takes a 5 48°,one obtains a signal independently from i and corre-sponding to 25.5 dB, as theoretically expected.

Moreover, there is one way to get a better resultwith this setup. Indeed, when using a laser source

Fig. 5. Same as Fig. 3 for the conventional isolator with twopolarizers @see Fig. 1~b!#.

with a larger w0, we expect the interferences amongthe three beams to be more efficient for a given tiltangle of the Faraday rod, as one can note from theexponential terms in Eq. ~3!. This phenomenon isillustrated in Fig. 7, with the same values of u and aas in Fig. 6, except that w0 is now equal to 460 mm.It is obvious that the oscillations are more satisfac-tory in this case. In particular, the minimum inten-sity in Figs. 7~c! and 7~d! is three times smaller thanthat in Fig. 6~c! and 6~d! and leads to an isolationratio of ;35 dB. This isolation ratio is 10 dB largerthan the nominal one obtained when the Faradayrotation is exactly 45° and the orientation of the out-put polarizer is also 45°.

Finally, this subsection has permitted us to provetheoretically and experimentally that internal reflec-tions of the Gaussian beam govern the behavior of theusual isolator studied here.

4. Destructively Interfering Isolator

In Sections 2 and 3, we have used two differentschemes to try and optimize the isolation ratio. Onthe one hand, in Section 3 we have removed beams 0and 1 thanks to a second polarizer, making an anglea 5 py2 2 u. However, this method does not removebeam 2 and leads to relatively modest isolation ratios.On the other hand, we have used the interferencesamong beam 0 and beams 1 and 2 to reduce the back-ward signal. However, this optimization stronglydepends on C, i.e., on the optical length of the rod,which may vary with, e.g., the temperature. We

Fig. 6. Same as Fig. 4 for the conventional isolator with twopolarizers @see Fig. 1~b!# and with C 5 21.3 rad for i 5 0.

consequently wonder here how to make simulta-neously beams 1 and 2 interfere destructively andbeam 0 vanish.

A. Calculation of the Backward Intensity

Up to now, beams 1 and 2 came back from the isolatorwith the same phases and could consequently inter-fere only constructively. To make them interfere de-structively, we add a quarter-wave plate to the basicisolator, as shown in Fig. 1~c!. Then the transmis-sion coefficients of the three beams become

t0 5 T 2 cos~2b!, (7a)

t1 5 RT 2 cos@2~b 2 u!#, (7b)

t2 5 RT 2 cos@2~b 1 u!#, (7c)

where b is the orientation of the fast axis of thequarter-wave plate with respect to the x axis. Theseequations just express the fact that the round tripthrough the quarter-wave plate is equivalent to theaction of a half-wave plate, i.e., to a plane symmetry.From Eq. ~6a!, one can see that beam 0 is alwaysstopped by the input polarizer if b 5 45°. Thismakes the second term of Eq. ~3! vanish. In thiscase, Eqs. ~7b! and ~7c! become

t1 5 2t2 5 RT 2 sin ~2u!. (8)

Consequently, in Eq. ~3!, if exp~22Dx2yw02! ' 1, the

third term must compensate for the first one, leadingto a zero backward intensity. More generally, Eqs.

Fig. 7. Same as Fig. 6 ~conventional isolator! with w0 5 460 mminstead of 230 mm and with C 5 20.8 rad for i 5 0.


~8! and ~3! lead to the following expression of therelative backward intensity for this particular config-uration:

II0

5 2T4R2 sin2~2u!F1 2 expS22Dx2

w02 DG . (9)

Note moreover that the choice b 5 45° permits usto leave the polarization of the mean transmittedbeam linear, as in the basic and usual isolators.Figure 8~a! represents the computed evolution of therelative backward intensity versus the tilt angle ofthe Faraday rotator for the particular case in whichthe Faraday rotation u is exactly 45° and the beamwaist is w0 5 230 mm. As expected, beams 1 and 2interfere destructively and lead to a zero backwardintensity at normal incidence when exp~22Dx2yw0

2!is equal to 1, i.e., when Dx is zero and the beams areperfectly superimposed. When the tilt i angle in-creases, the exponential overlap term decreases.One can see from Eq. ~9! that this decrease can beslowed down by increasing w0, as shown in Fig. 8~c!.Moreover, one can also see from Eq. ~9! that thisspectacular decrease of the backward intensity can beobtained for any value of u.

Fig. 8. Destructively interfering isolator @see Fig. 1~c!#. ~a!, ~c!Theoretical, ~b!, ~d! experimental evolutions of the intensity of theresidual backward beam versus tilt angle i of the Faraday rotator.The Faraday rotation angle is u 5 45°. The beam waist of theGaussian beam is ~a!, ~b! w0 5 230 mm, ~c!, ~d! w0 5 460 mm atwavelength l 5 632.8 nm.



The experimental test of this new isolator is per-formed with the same setup as in Sections 2 and 3,with, in addition, a zero-order antireflection- ~AR!-coated quarter-wave plate at 632.8 nm located behindthe Faraday rotator @see Fig. 1~c!#. Figures 8~b! and8~d! reproduce the experimental evolutions of thebackward intensity versus the tilt angle of the Fara-day rotator for a Faraday rotation of 44.6° and twodifferent values of the beam waist, w0 5 230 mm @Fig.8~b!# and w0 5 460 mm @Fig. 8~d!#. One can see theexpected decrease of this intensity when i diminishes,i.e., when beams 1 and 2 gradually overlap and canconsequently efficiently interfere. The residual os-cillations in Figs. 8~b! and 8~d! are certainly due toimperfections of our quarter-wave plate, which arenot taken into account in our model. Nevertheless,a good agreement is observed between theory andexperiment. In particular, isolation ratios of 31.5dB in Fig. 8~b! and of 35 dB in Fig. 8~d! are obtained.As expected, the isolation ratio is better for w0 5 460mm than for w0 5 230 mm because of a better over-lapping of the beams. Besides, one can note that, asexpected, the fact that u is not strictly equal to 45°does not change the value of the backward intensity.Compared with the results obtained in sections 2 and3, we can see that the destructively interfering isola-tor leads to the best results. Moreover, this resultdoes not depend on C and is consequently more stablethan those of the preceding sections.

Rather than increasing w0, another method to im-prove the overlapping between beams 1 and 2 wouldbe to decrease Dx. From Eq. ~5!, one can see thatthis can be obtained by using a thinner Faraday rod.We have consequently tested a 550-mm-thick AR-coated BIG Faraday rod operating at 1.52 mm. Thisisolator is based on a commercial product ~Optics ForResearch Model IO-4-IR2!. The experimental setupis similar to the one of Fig. 1. The laser source isnow a He–Ne laser operating at 1.52 mm, with w0 5440 mm. In a first experiment, we realize the isola-tor of Fig. 1~b! by using two polarizers. With anangle a 5 45° for the second polarizer, we then obtainthe measurements represented by the filled circles inFig. 9. The corresponding theoretical curve labeled1 in Fig. 9 has been computed from Eqs. ~3! and ~6!with the following measured values of the parame-ters: u 5 43.5°, R 5 6 3 1024, and T 5 0.99. As thereflection coefficient is very small, the oscillations al-most vanish, leading to a constant value of 33 dB forthe isolation ratio. This isolation ratio is here es-sentially limited by the fact that u is not exactly equalto 45°. Hence, as expected from Section 2, we canimprove it by choosing as usual a 5 py2 2 u, leadinghere to an angle of 46.5°. With this value, we thencompute from Eqs. ~3! and ~6! the theoretical curvelabeled 2 in Fig. 9. This corresponds to a constantisolation ratio of 64 dB, which is much better than therejection rate of our polarizers. Consequently thecorresponding measurements, which are representedby filled squares in Fig. 9, exhibit only an isolation

ratio of 52 dB with the setup of Fig. 1~b! with a 5py2 2 u. Surprisingly, although this result is partlylimited by the quality of the polarizer and the Fara-day rod is AR coated, it can still be improved with theuse of the destructively interfering isolator of Fig.1~c!. Indeed, as can be seen by the filled triangles ofFig. 9, the use of a quarter-wave plate leads to anisolation ratio of 55 dB. Of course, this is far fromthe corresponding theoretical curve labeled 3 in Fig.9, which has been computed from Eqs. ~3! and ~7!.Nevertheless, it is satisfactory to note that even whenthe Faraday rod is AR coated, the remaining verysmall internal reflections still permit us to improvethe isolation ratio. Moreover, the theoretical curve 3shows that an extra improvement of the isolationratio could theoretically be expected near normal in-cidence with better polarizers.

5. Discussion and Conclusion

In conclusion, we have theoretically and experimen-tally analyzed the behavior of three types of opticalisolator, taking into account the internal reflectionsand the Gaussian nature of the beam. For thesethree types of isolator, namely the simplest one builtwith a polarizer and a Faraday rod, the conventionalone built with two polarizers and a Faraday rod, and,finally, the new one built with a polarizer, a Faradayrod, and a quarter-wave plate, the internal reflectionshave been shown to play a dramatic role in the iso-lation ratio. In particular, the result has beenproved to be independent from the position of theisolator along the Gaussian beam, but to depend ononly the value of the beam waist. In all cases, a goodagreement has been obtained between theory andexperiments. Besides, we have shown that internalreflections could be used to optimize the perfor-

Fig. 9. Evolution of the isolation ratio versus i for the 550-mm-thick AR-coated Faraday rod operating at l 5 1.52 mm. The threecurves correspond to theoretical calculations with R 5 6 3 1024

and T 5 0.99. The curves labeled 1 and 2 correspond to theisolator with two polarizers @see Fig. 1~b!# with u 5 43.5° and a 545° for curve 1 and a 5 46.5° for curve 2. Curve 3 corresponds tothe destructively interfering isolator of Fig. 1~c! with u 5 43.5° andb 5 45°. The corresponding experimental measurements areschematized by circles, dots, and triangles.

mances of the isolator. In the first two setups, thisoptimization is obtained by a choice of the angle ofincidence of the beam on the Faraday rod and of theFaraday rotation angle. However, in these twocases, the positions of these optimums strongly de-pend on the exact optical length of the Faraday rodand hence vary quickly with the temperature of therod and the angle of incidence. In the third setup wehave introduced here, we have shown that a stillbetter optimum could be obtained. Moreover, thisoptimum does not depend on the exact optical lengthand rotation angle of the Faraday rod and is conse-quently particularly stable with, e. g., temperaturevariations. Obviously the optimization of the isola-tion ratio with the present method is a little bit morecomplicated than simply using an usual isolatormade of a Faraday rod and two polarizers. More-over, the method does not apply to fibered opticalisolators. Besides, the experimental setup we haveused here makes use of a simple mirror to symbolizethe reflecting surface. Of course, in a real experi-ment, if the reflecting surface is slightly birefringent,the preceding results are modified. However, in thethird setup, one can show that the angle of thequarter-wave plate can be adjusted to compensate forany small phase anisotropy of the reflecting surfaceand then restore the present results. Finally, onecan argue that the use of an uncoated rod leads tosome drawbacks, such as transmission losses and thefact that the transmitted-mode profile is altered bythe presence of internally reflected beams. How-ever, we have seen experimentally that even in thecase in which the Faraday rod is AR coated and inwhich internal reflections could be expected to play aless significant role, the new type of isolator still ben-efits from the residual internal reflections and canlead to better results than usual isolators. To givean example of application, this type of isolator is wellsuited to experiments involving the use of high-finesse Fabry–Perot cavities.23

Appendix A

Here we derive the general expression of the residualbackward intensity for the three types of isolator.We suppose that the incident beam, before the firstpolarizer, is linearly polarized along the x axis, i.e.,along the axis of the polarizer. Before the Faradayrotator, the electric field associated with the incidentGaussian beam can then be described by the follow-ing expression:

Ei 5 F10G E0 expS2 x2 1 y2

w2 D expF jp

lr~x2 1 y2!G , (A1)

where E0 is a normalization factor, w is the radius ofthe Gaussian beam, r is the radius of curvature of itswave front, and l is its wavelength. We neglect theinfluence of the Gouy phase shift because we supposethat the considered lengths are small compared withthe Rayleigh range of the beam. Then we can writethe electric fields associated with the three backward


beams ~see Fig. 2! as

E0 5 F10G E0t0 expS2

x2 1 y2

w2 D expF jp

lr~x2 1 y2!G ,

(A2a)

E1 5 F10G E0t1 expF 2

~x 2 Dx!2 1 y2

w2 G3expH j

p

lr@~x2Dx!21y2#J , (A2b)

E2 5 F10G E0t2 exp~ jC! expF 2

~x 1 Dx!2 1 y2

w2 G3expH j

p

lr@~x 1 Dx!2 1 y2#J . (A2c)

In these expressions, we have assumed that the lengthl of the Faraday rod is small enough so that the threebeams have the same w and r. t0, t1, and t2 hold forthe transmission coefficients of the beams throughoutthe whole device and depend on which type of isolatoris considered. Because the optical isolators we con-sider here lead to only purely real Jones matrices forone round trip, these transmission coefficients are alsoreal. C is the extra phase shift undergone for oneround trip through the plate, given by

C 54p

lnl cosFarcsinSsin i

n DG , (A3)

where n is the refractive index of the material. Thewhole backward intensity, taking the three beamsinto account, is given by

I 5 * *2`

1`

u~E0 1 E1 1 E2! z xu2 dx dy, (A4)

where x is a unit vector along the x axis. Equation~A4! finally gives, together with Eqs. ~A2!;

II0

5 t02 1 t1

2 1 t22 1 2t0~t1 1 t2! cos C

3 expS 2Dx2

2w02D 1 2t1t2 expS 2

2Dx2

w02 D , (A5)

where w0 is the beam waist of the incident Gaussianbeam and I0 is its intensity. Note that higher-orderreflections are negligible because of both the lowvalue of R and their high rejection by the input po-larizer.

This research was partially supported by the Cen-tre National de la Recherche Scientifique, the Direc-tion de la Recherche et de la Technologie, and theConseil Regional de Bretagne.

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Documents

Internal reflections of the Gaussian beams in Faraday isolators