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Littrow-mounteddiffraction grating cavity

Haim Lotem

Grating resonators are not characterized by a single cavity length, and thus the cavity-mode spacing cannot simply be obtained from the standing-wave pattern. This problem is studied in a Littrow grating cavity with geometric ray tracing and a result of the scalar diffraction theory for the phase of a plane wave diffracted at a grating. The round-trip phase in the cavity is considered, and it is shown that a grating wavelength deviation from the resonant Littrow wavelength, and the mirror separation, LC, is equal to the grating-cavity length at the center of the aperture. The model shows that the cavity axial mode separation may be determined from the standard expression, c/2Lc. An effect of finesse decrease and mode broadening, which are linearly dependent on wavelength deviation from the central Littrow wavelength, are predicted. A passive grating cavity was experimentally studied with an interferometric method and a tunable laser to demonstrate the discussed hypotheses.

cavity may be modeled by a tilted-mirror Fabry-Perot cavity. The tilt magnitude depends linearly on the

Diffraction gratings have long been used in laser resonators to tune and narrow the lasing bandwidth.1 2 A common grating cavity consists of a front plane mirror and a high-reflectivity grating mounted at the Littrow configuration. In these classical resonators the spectral selectivity results from the high angular dispersion and high resolving power of high-linedensity gratings. At the Littrow configuration, gratings are usually considered as effective plane retroreflectors, and thus the performance of a grating cavity

is assumed to be similar to that of a plane-parallel

passive grating cavity was experimentally studied with an interferometric method to demonstrate the discussed hypotheses. Frequency tuning of a grating cavity is obtained by

rotating the grating and thus changing the retrore-

Fabry-Perot cavity with narrowband mirror coating.3 4 In this configuration, however, the effective cavity length, a parameter that controls the cavitymode spacing (CMS), is ambiguous. In this paper the effective length of the cavity is discussed using a model of a tilted plane mirror Fabry-Perot cavity, with a wavelength-dependent tilt and with mirror separation that is defined as the central axis length of the cavity. The suggested model is useful in the evaluation of the CMS and the finesse of the longitudinal modes. The present study of a steady-state system is based on the treatment of a single-mode grating cavity by McNicholl and Metcalf.5 The grating cavity is studied with ray-tracing and round-trip phase analysis of a plane wave, taking into account the phase change on diffraction at the grating.5 A

The author is with the Lasers Group, Nuclear Research CenterNegev, P.O. Box 9001, Beer-Sheva 84190, Israel. Received 17 February 1993; revision received 21 July 1993.

flected resonant Littrow wavelength, AL. Fine tuning is achieved by varying the resonator length or by tilting an intracavity compensating plate.' 7 In the simple case of a plane-parallel Fabry-Perot cavity, the phase contribution of a compensating plate to the optical path is independent of the orientation of the plate axis of tilt and of the sign of the tilt angle, because of the axial symmetry of the system. A plane-parallel Fabry-Perot cavity with a compensating plate is schematically shown in Fig. 1(a). In such a cavity the accrued phase for a single round trip as a

function of plate angle, a, in cycles, is given by an

expression that is symmetrical with respect to the

angle, a:

Phs(u.) = 2

A +





with P(ci) = 2T{1 - n + [n



tD1994 Optical Society of America. 930 APPLIED OPTICS / Vol. 33, No. 6 / 20 February 1994

where Lc is the distance between the mirrors, P(a) is the plate phase contribution, T is the plate thickness, n is the refractive index, and a = asin[sin(u)/n]. It is assumed that no intraplate reflections occur.8 The grating cavity [Fig. 1(b)], exhibits a lower symmetry than the plane-parallel Fabry-Perot cavity, and there-



Delayed beam

resonant Littrow wavelength, basic grating equation:

= XL, is obeying the

In I V\'I


+ d[sin(OINC)

in(OREF)] =





b)_ (o)




where OC and REF are the incident and reflected angles upon the grating, respectively, with L = OINC = OREF; d is the grating line spacing; and M is the diffraction order. A priori, the asymmetry in tilt angle of the resulting phase in Eq. (2) demonstrates an apparent performance difference between a plane grating and a mirror, contradicting the assumption of equivalence of a plane mirror and a grating. This contradiction is lifted if a phase contribution of the grating to the phase of the reflected wave is correctly included in the calculation. This phase shift may be obtained by using the scalar diffraction theory.5 It may be seen in the explicit expression of the grating-complex reflection coefficient for a plane wave at diffraction orderM: 5


LOWER PART Fig. 1. Schematic description of the compensating-plate experiment (a) in a plane-parallel Fabry-Perot and (b) in a Littrow

grating cavity. R is a partially transparent plane mirror. A Zygo interferometer (at 633 nm) combined with the examined cavity is used to observe a reflection inteferogram from the cavity (c). A

GM(U) GM(p)exp[i(DO(p)exp(-i2TMU/d), =


glass slide is used as a compensating plate to tune the round-trip phase in the upper part of the cavity aperture. As the slide is tilted the fringe pattern always moves in the direction corresponding to cavity elongation.

fore the phase contribution of a compensating plate may be asymmetrical with the tilt angle and may also depend on the plate axis of tilt. It is interesting to analyze two cases with orthogonal tilt axes. In the first case, the axis is normal to the grating lines and to the cavity axis. In this case the phase term is symmetrical with the tilt angle because the beam shift versus tilt is symmetrical along the grating lines. The resulting expression for the round-trip phase is identical with the expression of a plane-parallel FabryPerot cavity; i.e., it is identical with Eq. (1). A more interesting case is that in which the plate axis is parallel to the grating lines. Here, when plate tilting occurs, the beam translation is normal to the grating lines, and the beam path is appreciably changed. As shown in Fig. 1(b), with a > 0, the beam path is longer than in the case with a = 0 and vice versa for a < 0. The accrued cavity round-trip phase through the plate and the grating is given by the plane-parallel Fabry-Perot expression, Eq. (1), plus an asymmetrical term, as follows: Ph(ot) = Phs(a) + 2T [


where the U-axis is along the surface of the grating perpendicular to the lines, with origin at a point p. 2 [GM(p)] is a real coefficient that expresses the intensity reflection coefficient and is independent of U; 2 [GM(p)] shows a slowvariation with X. The phase of the reflected wave is independent of wavelength, or equivalently, the diffraction angle, and depends only on beam location on the grating rulings. This dependence has long been used for frequency shifting of light by scattering from a moving grating.6 At normal incidence the frequency shift at diffraction order M, from a grating moving at velocity, V, is given by Fs = M(V/d). 6 This expression, which may be directly derived from Eq. (4), is well known for light scattering from acoustic travelling waves. Very recently, the phase-shift effect that is due to a moving grating, was demonstrated in an external gratingcavity diode laser. 9



In Eq. (2), the angle L, which corresponds to the

When the grating-phase contribution from Eq. (4) is added to the round-trip phase in Eq. (2), it is easy to show that the compensating plate asymmetric term in Eq. (2)is canceled. Thus the two discussed perpendicular axes of a plate in a grating cavity are expected to show an identical symmetric phase dependence on tilt angle, as given by Eq. (1). Because a general tilt of a plate in front of the grating is a linear combination of the two discussed orthogonal tilts, the symmetric expression for the Fabry-Perot cavity, Eq. (1), is suitable for the description of the compensating-plate effect in the general case. The solid curve in Fig. 2 shows the dependence of the round-trip phase on plate angle, as calculated with Eq. (1). The dashed steep line in the figure depicts the geometric phase term given by Eq. (2). The plate and cavity parameters used in the calculation are given below. The contribution of a tilted compensating plate to the round-trip phase was experimentally studied with an interferometric technique. Two cavities were

20 February 1994 / Vol. 33, No. 6 / APPLIED OPTICS 931


Un 50

_j 40


30 20





C/) 10 0


trum of a passive cavity. This may be done with an external tunable probe beam. For a grating cavity, the situation is more complicated because a single cavity length cannot be defined, and the resonant standing-wave pattern cannot be characterized by a single number of standing waves.5 The proper cavity parameter to consider here is the round-trip phase. In a cavity without a compensating plate, where the grating plane intersects the mirror plane at a point p [see Fig. 3(a)], the round-trip phase for a ray corresponding to an axial path length, L, in cycles, is 16 -12 -8 -4


LL -1 0






Fig. 2. Round-trip phase shift versus plate tilt angle in a FabryPerot cavity and a grating cavity based on Eq. (1), solid curve, and based on geometric considerations [Eq. (2)], dashed line. The plate parameters are T = 1.05 mm and n = 1.5; the grating is 1200 lines/mm, operated at M = 1 and XL = 633 nm. The observed phase in the grating cavity of Fig. 1(b) versus plate angle, U.,is indicated by filled circles and crosses for plate tilt axis parallel and

Ph(X, L) = - - (Md sin(OL)



normal to the grating lines, respectively.

examined: one consisted of two parallel plane mirrors [Fig. 1(a)], and the second was composed of a

mirror and a reflecting plane grating (1200 lines/mm, oriented at Littrow at the first order, for XL = 633

nm); see Fig. 1(b). The cavities were coupled to a

phase-measuring Fizeau interferometer (Zygo Mark II), allowing an easy cavity alignment and observation of the reflection interferogram from the cavity, at 633 nm. After alignment of the cavity, a regular reflection interferogram, similar to that shown in the lower part of Fig. 1(c), was observed over the 50-mm cavity aperture. On insertion of a glass slide (thickness 1.5) in the L = 1.05 mm and refraction index n upper part of the cavity, the fringe pattern of that section was transformed into the pattern shown in Fig. 1(c), upper part. The two patterns were not identical because of the small wedge angle of the slide. When the slide was tilted, the induced phase change caused a corresponding shift of the fringe pattern shown in the upper part of Fig. 1(c). The observed fringe shift was found to be an even function of the angle, and it was identical in the two cavities. The direction of the shift was always along the direction corresponding to cavity elongation. The experimental results were matched with the theory by use of the fact that one fringe cyclecorresponds to one period, or 2rr rad, of the round-trip phase. The results of the phase measurement in the grating cavity for the two discussed tilt axes are presented in Fig. 2. They are in good agreement with the prediction of Eq. (1) for the magnitude and sign of the phase. In a general cavity that supports several longitudinal resonances, the frequency of the resonant modes and the cavity free-spectral range (FSR) are directly related to the cavity length, Lc, by the number of the 8 standing waves in the cavity. In a plane-parallel mirror cavity the FSR is given by FSR = c/(2Lc).' 0 The FSR may be obtained by observing the resonant frequencies of the transmission or reflection spec932 APPLIED OPTICS / Vol. 33, No. 6 / 20 February 1994

Here, the left-hand term is the optical path length in wavelength units, and the right-hand term is the explicit grating-phase contribution, from Eq. (4). At the Littrow wavelength, XL, it is easy to show from Eq. (6)that the round-trip phase is constant over the entire cavity aperture, and thus the grating acts as a retroreflector. At other wavelengths Eq. (6) results in a round-trip phase that is linear in L and in (X - XL). This spectral dependence is the origin for beam deflection from the Littrow direction by the diffraction. The deflection angle, F(X), is proportional to (X - XL) and may be deduced from Eq. (6). Because of the wave deflection, the cavity-resonant condition cannot be exactly satisfied simultaneously over the entire cavity aperture for wavelengths other



The resonant characteristics of a grating cavity were experimentally demonstrated with an interfero-






.. .

Lc__ 'TIy


A beam from a

Fig. 3. Schematic description of the resonance-frequency-dependence experiment in a Littrow grating cavity.

tunable cw dye laser probes the cavity resonance. As the probe frequency is scanned, the first-order-diffracted beam from the

grating is deflected by 1, and the fringe pattern on the screen (by

the zero-order reflection) is shifted. The amount of shift depends on the location within the cavity aperture. The grating cavity in

(a) may be modeled by a plane-plane tilted-mirror cavity with

mirror separation, Lc, as shown in (b). The described beams in the schematic model cavity correspond to a case where X <


he tilted-mirror model is useful to explain broadening effects

and the CMSin the cavity.

metric method by monitoring the phase shift at various values of axial path length, L, while changing X. Here again a grating cavity is compared with a plane-parallel Fabry-Perot cavity. The investigated grating cavity is shown in Fig. 3(a). It is similar to the cavity shown in Fig. 1(b), but a higher dispersion grating of 3000 lines/mm is used, and a narrowband tunable cw dye laser replaces the Zygo interferometer. At Littrow mounting for XL 607 nm, with a 16-mm-diameter probe beam, the grating-cavity axial length variation over the beam diameter is D[tg(OL)] 35 mm. The cavity response, as measured by the grating zero-order reflection, is obtained from an interferogram observed on a screen [see Fig. 3(a)]. During the experiment the probe-laser wavelength was scanned from XL to XL + 5.4 GHz, and the resulting fringe shift was recorded. The measurements were performed at three settings of gratingmirror separation. At each setting, the fringes were monitored at several points of the aperture corresponding to different axial lengths, L. In the reference Fabry-Perot cavity, an identical fringe shift was observed over the entire cavity aperture. Across the aperture of the grating cavity, however, the fringe shift was different. The observed shift was always the largest at the aperture edge where the cavity is longest. For example, in the shortest cavity setting, with central axis length LC = 20.5 mm, the observed fringe shift was 0.3 + 0.5 and 1.3 0.5 fringe at the shortest and longest cavity axes, respectively. The measured dependence of the phase shift on cavity length, corresponding to the 5.4-GHz probe-laser frequency scan, is shown by the circles in Fig. 4. Only the measurements at the extreme edges of the aperture at each grating-mirror setting are shown. At the aperture center, the observed fringe shift was approximately the mean value of the shift at the edges. The experimental results fit the theoretical prediction of Eq. (6) very well, as shown by the solid

line in Fig. 4.


To investigate the cavity-resonant conditions versus frequency it is necessary to study the phase dependence on wavelength and cavity length. If Lc is the axial length at the cavity aperture center, the CMS may be obtained from the phase relationship:

Ph(Xr, Lc) - Ph(XL, Lc) = ± 1 cycles,



where Xris the wavelength of a resonance adjacent to

Using Eq. (6), one obtains 4 CMS(Lc) = Ic/Xr



= /(2Lc).


This expression, which is consistent with typical results reported for CMS in dye lasers,1" may be a proper description of the cavity performance over the entire aperture, when the variation of the CMS over

the aperture, 8 CMS(D), is small:

8 CMS(D)/CMS(Lc) << 1,


where D is the diameter of the aperture. This expression may be shown to be equivalent to the geometric relation


>> 1-


The validity of Relations (9) and (9a) is demonstrated by the CMS dependence on cavity length, L, as presented by the theoretical dashed curve, based on Eq. (8), in Fig. 4. The curve clearly shows that as the cavity length is increased the slope of the CMS(L)and the variation of the CMS across the aperture as a function of L decrease, approaching zero at large values of L. The squares in Fig. 4 represent the measured values of the CMS, which were obtained by dividing the 5.4-GHz scan frequency by the observed number of shifted fringe cycles on the constant 5.4-GHz scan. The experimental data are in a good agreement with the theoretical expression for the









0 ~~~~0

The expression for the round-trip phase in a grating cavity, Eq. (6), suggests that a grating cavity may be modeled by a tilted-mirror Fabry-Perot cavity (see Fig. 3). In this model the mirror spacing is equal to the central axis length of the grating cavity, and the mirror-tilt axis is parallel to the grating lines. The round-trip phase in the Fabry-Perot model cavity is identical with the grating-cavity case, i.e., Eq. (6), when the tilt angle is





XL)/[2d COS(OL)]




0.2 0.3







Fig. 4. Phase shift in cyclesversus cavity length, L, corresponding to 5.4 GHz frequency change of the probe laser in Fig. 3(a), as

calculated from Eq. (6), solid line, and as experimentally obtained

Even with broadband mirrors, because of the tiltangle dependence on wavelength, the tilted-mirror Fabry-Perot cavity forms a low-loss resonator only in the vicinity of the Littrow feedback wavelength, L. Sharp longitudinal resonances are restricted to a spectral range in which the round-trip phase variation, 8%(X), across the cavity aperture, D [see Fig.

3(a)], is small, obeying

8IDD(X) = DF(X)/X << 1,

(solid circles). The dashed curve corresponds to the theoretical value of the CMS, whereas the squares present the measured CMS data.



20 February 1994 / Vol. 33, No. 6 / APPLIED OPTICS

otherwise, because of walk-off effects, the standingwave pattern is faded out and the resonance is broadened.' 2 The inverse of the last phase term is the cavity finesse term that is due to tilt error, Tilt Finesse = /[email protected](D(X)'1, (la)

which affects the resolution of a plane-parallel FabryPerot interferometer.1 2 The CMS in the model Fabry-Perot cavity may be simply calculated with a standard FSR expression, i.e., Eq. (8). In the model cavity the finesse of longitudinal modes far from the Littrow resonant mode is expected to be low, and the linewidth is expected to be large, because the tilt finesse is inversely dependent on the frequency deviation from Littrow [Eq. (10) and Relation (11)]. Therefore, in a short cavity with a large mode separation, the uncertainty in the value of the CMS is expected to be large. In addition to the discussed tilt finesse, which is linear with D-', a Fabry-Perot cavity is characterized by a finesse term that is related to the wave diffraction in the apertured cavity. The latter term may be expressed by D2 /XLc. The combined finesse resulting from tilt error and diffracI am indebted to M. Dagenais for his hospitality tion can be shown to be minimized for D3 = 2XLC/F(X). during my sabbatical stay at the University of MaryFrom this expression it is clear that the linewidth of land. The experimental support of D. Belker and G. the central cavity mode, where the tilt error is nulled, Ben Amar and the useful discussions with Uri Laor, is improved by expanding the aperture. If, however, G. Bialolenker, P. Blau, M. Lando, and R. Shuker are it is required to enhanced more than one longitudinal determined appreciated. mode, the optimized value of D may be from the above expression. In general, no modes are expected to resonate outside the central spectral References range in which the finesse is larger than = 1. Con1. F. P. Schafer, ed., Dye Lasers, 3rd ed., Vol. 1 of Topics in the corresponding range sidering only the tilt finesse, Applied Physics (Springer-Verlag, Berlin, 1990). as is defined by IX - XL I < Ld cos(OL)/2MD, obtained 2. F. J. Duarte and L. W. Hillman, Dye Laser Principles, 1st ed. from Eq. (10) and Relation (11). This range may be (Academic, San Diego, Calif., 1990), Chap. 4, pp. 134-144. shown to be equal to the resolution band of the 3. J. E. Bjorkholm, T. C. Damen, and J. Shah, "Improved use of grating, which is calculated by considering the angugrating in tunable lasers," Opt. Commun. 4, 283-284 (1971). lar dispersion of the grating as well as the diffraction 4. G. J. Ernst and W. J. Witteman, "Transition selection with of an apertured beam of diameter D. In the central adjustable outcoupling for a laser device applied to C0 2 ," range above, the number of longitudinal modes may IEEE J. Quantum Electron. QE-7, 484-488 (1971). 5. P. McNicholl and H. J. Metcalf, "Synchronous cavity mode and be calculated with the CMS from Eq. (8), to be

N = [LC + Dtg(0L)]/Dtg(0L)(12)

feedback wavelength scanning in dye laser oscillators with gratings," Appl. Opt. 24, 2757-2761 (1985).

In conclusion, using round-trip phase considerations, one may model a grating cavity by a tiltedmirror Fabry-Perot cavity, with a tilt angle that is linear with the wavelength deviation from the Littrow wavelength. The Fabry-Perot mirror spacing is equal to the grating-cavity length at the center of the aperture. With this model, cavity finesse and the linewidth of the modes in a passive cavity may be evaluated. The theory shows that apertured grating cavity may support several longitudinal modes when the cavity length is much longer than the axial length variation across the cavity aperture. The cavity mode spacing (CMS) is determined from the central cavity length with a standard FSR expression. The theoretical analysis as well as the experimental demonstration show that the phase contribution to a wave diffracted from a grating is essential in the understanding of a grating cavity. Additionally, the study shows that fine tuning of a grating cavity by a compensating plate may be performed at any arbitrary plate tilt-axis orientation in spite of the asymmetry of the cavity. The present conclusions may be 5 extended to other configurations of grating cavities.

The above expression for the number of potential longitudinal modes in a grating cavity shows that with a short cavity, where Lc = 1/2Dtg(OL), the cavity supports N = 1.5 modes. Thus, in principle, singlemode lasing may be achieved in a short-grating-cavity laser with no intracavity 6talon. In practice, only diode lasers with a short gain section may be operated in such a short external cavity. In general, in active cavities, the effective finesse is determined by the finesse of the passive system combined with the effects of the active gain medium. The latter factor compensates for cavity-loss effects on the finesse, but its phase noise contribution and wave-front distortion usually reduce the overall cavity finesse. Therefore the predicted mode broadening that is due to the effective tilt error may be too small to be observed in common laser systems.

934 APPLIED OPTICS / Vol. 33, No. 6 / 20 February 1994

6. W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 8, pp. 108-130. 7. R. Wyatt, K. H. Cameron, and M. R. Matthews, "Tunable narrow line external cavity lasers for coherent optical systems," Br. Telecommun. Technol. 3, 5-12 (1985). 8. D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, and Zs.

Bor, "Single-frequency dye laser with 50 ns pulse duration," in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-

Opt. Instrum. Eng. 912, 87-94 (1988).

9. M. de Labachelerie and G. Passedat, "Mode-hop suppression of

Littrow grating-tuned lasers," Appl. Opt. 32, 269-274 (1993).

10. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 11, pp. 432-435.

11. M. G. Littman and H. J. Metcalf, "Spectrally narrow pulsed dye laser without beam expander," Appl. Opt. 17, 2224-2227


12. G.J. Sloggett, "Fringe broadening in Fabry-Perot interferometer," Appl. Opt. 23, 2427-2432 (1984).


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