Birefringent tuning filters without secondary peaks

Volume

24, number

OPTICS COMMUNICATIONS

1

BIREFRINGENT

January

1978

TUNING FILTERS WITHOUT SECONDARY PEAKS

I.J. HODGKINSON * and J.I. VUKUSIC * Physics Dept., University of Otago,

Box 56, Dunedin, New Zealand

Received 15 September 1977 Revised manuscript received 24 October

1977

The properties of periodic birefringent filters for tuning dye lasers are discussed theoretically. This type of filter has no secondary peaks and the rejection between principal peaks is maximum for a given number of plates. Expressions are given for the finesse and transmittance, and changes which occur in these quantities when the filter is tuned are considered.

The resonant modes of a laser cavity containing a tilted birefringent plate are eigenfunctions of the retardation plate [l] . A stack of identical plates similarly aligned (fig. 1) has the same eigenfunctions as a single plate, and hence the effective single pass transmittance of a stack of K plates is TX, where T is the transmittance of a single plate [2]. As T for a single plate has no secondary peaks, it follows that a stack of identical plates has no secondary transmission peaks. This property makes the filter suitable for tuning a high gain dye laser over a significant part of its gain curve. In general the filter is characterized by two functions, the minimum transmittance To, and the finesse F which, for convenience, is defined as the free spectral range divided by the full width of a peak at height (1 t T0)/2. Consider first the case when the angle Q between the plane of incidence and the plane containing the optic axis and the refracted ray is 7~/4. For this value of oL the rejection between peaks is greatest and the transmittance TK of a stack of K plates can be expressed in terms of the phase retardation 26 of each plate and the loss parameter 4, the ratio of the Fresnel TE and TM transmission coefficients. For incidence at the Brewster angle, 4 = 2n/( 1 + n2), where n is the mean refractive index (4 = 0.911 for quartz), and [ 1,2]

Fig. 1. Periodic

TK = I(1 t$)

cos 6 + [(1+$)2

filter.

co&

-4q2]‘/212K

22K For a stack of ten quartz plates the transmittance spec. trum has the form shown in fig. 2. Expressions for the minimum transmittance and the finesse can be derived from the above equation. Thus T,, = q2K,

F = n/{2cos-’

[(c + q2/c)/(l

t s2)] },

where c = [(l t qZK)/2]‘/2K

* This work was done by the authors at Imperial College, Blackett Laboratory, Applied Optics Section, Prince Consort Rd, London, SW7. The permanent address of I.J. Hodgkinson is University of Otago, Department of Physics, Dunedin, New Zealand.

birefringent

.

The main advantage of the filter is the ease with which To can be controlled. A disadvantage is that independent control is not available over the finesse, which is a quasi-linear function of K, increasing from 20 to 30 as K increases from 1 to 20 for quartz plates. 133

Volume

24. number

1

I TC 0.5 PHASE

RETARDATION

n OF EACH

I ig. 2. Effective single pass transmittance identical quartz plates in a laser resonator

TT

PLATE

of a stack of ten (0 = n/4).

It is noted that in theory the finesse can be increased by increasing the factor c/. When the filter is tuned by rotating the unit as a whole about an axis perpendicular to the surfaces, the shape of the transmittance spectrum changes slightly. the finesse decreases, and the minimum transmittance increases. In a typical case the finesse decreases by only a few percent, but the increase in transmittance may be more significant. For a change Aol in the angle CY, ATo = X(1

-~ q ’ )q 2K-1 (Acw(.

As an example, consider a stack of IO quart/. plates each of thickness 1.73 mm and cut with the optic axis at 25” to the surface. This filter has a free spectral range of 50 nm centred about 600 nm, F = 34, and

134

January

OPTICS COMMUNICATIONS

I Y7X

To = 0.156. When it is tuned to 675 nm. IA&l = 2.5” and AT,, 1 0.025. It has been assumed that the birefringent plates ale the only polarizing components in the lasel- cavity. A dye cell with Brewster angle windows behaves in a similar manner to a single glass plate. In the above example the effect of the dye cell is to decrease the finesse by about 5% and to introduce small secondary peaks The latter effect is also insignificant as the secondary peaks do not exceed the previous transmittance 01‘ 0 .15 6. The periodic birefringent filter which we have discussed pl.ovides maximum rejection between peaks 101 a given number of plates. Other periodic filters cm he designed to have no secondary peaks. For example. the filter with period fonncd by i?z glass plates, a birefringent plate and I glass plates has no seconclary peaks in a ring resonator [‘_I However, this design is not considered further as the finesse is lowered by the PI-~sence of the glass plates. I.J. Hodgkinson wishes to acknowledge the award of a Royal Society and Nuffield Foundation C’ommoIlwealth Bursary.

References [ 11 A.L. Rloom, J. 121 I.J. llodgkinson Optics.

Opt. Sot. Am. 64 (1974) 447. and J.I. Vukusic.

wbmittetl

to Applied