An analysis of group-velocity dispersion in graded-index (GRIN) lenses is presented. ... and grating-pair methods for producing negative dispersion, the method ...
September 1, 1992 / Vol. 17, No. 17 / OPTICS LETTERS
1177
Adjustable negative group-velocity dispersion in graded-index lenses An-Chun Tien, Railing Chang, and Jyhpyng Wang Institute of Atomic and Molecular Sciences,Academia Sinica, and Department of Electrical Engineering, National Taiwan University,Taipei, Taiwan Received April 15, 1992
An analysis of group-velocity dispersion in graded-index (GRIN) lenses is presented. The analysis shows that continuously adjustable negative group-velocity dispersion up to hundreds of square femtoseconds can be produced by propagating the optical beam off the axis of a GRIN lens. Compared with the well-known prism-pair and grating-pair methods for producing negative dispersion, the method described here is advantageous because it is 100-foldsmaller, it has low insertion loss, and it is compatible with integrated optics.
In the generation and application of ultrashort
laser pulses, optical elements with negative groupvelocity dispersion (GVD) have played important roles. In linear applications, negative dispersion is used to compensate for ubiquitous positive dispersion in most optical materials or to restore optical
pulses after chirped
amplification,"2
whereas in
nonlinear optics negative dispersion has been utilized with self-phase modulation to compress optical pulses.3 In addition, dispersion compensation is particularly important in long-distance optical communication. At present, two methods for generating negative dispersion are commonly used. One is the grating-pair method, which provides large negative dispersion but suffers high loss.4 The other is the prism-pair method, which has negligible loss and a smaller range of negative dispersion.' The prismpair method has the additional advantage of being able to adjust continuously from negative to positive dispersion. This property, together with low loss, makes the prism pair an important device in femtosecond lasers.6 A common disadvantage of the above methods is that the physical space required ranges from several to several tens of centimeters. For example, the quartz prism pair requires a 30-cm prism spacing to compensate for the positive dispersion of a few millimeters of quartz. This requirement makes it difficult to apply the method in small optical devices. In this Letter we study the GVD in graded-index (GRIN) lenses and describe a novel
persion is determined by the beam offset a from the optical axis of the GRIN lens, and positive dispersion comes from material dispersion. By adjusting a, it is possible to tune the dispersion continuously from positive to negative values, just like adjusting 1
in prism pairs.
The index of refraction of a GRIN lens can be written as n2= no2(1 - Ar2). (1) The wave equation in a parabolic index medium represented by Eq. (1) is V2 E(x,y,z) +
2W2 2E(x,y~z)
(a)
(b)
method for producing negative dispersion. The
method has all the advantages of the prism pair yet occupies 100-fold less space. Such a device may be utilized conveniently in integrated optics. The reason we investigated GRIN lenses for negative dispersion is shown in Fig. 1. The lens array in Fig. 1(b) is similar to the prism pairs in Fig. 1(a), and a GRIN lens [Fig. 1(c)] can be considered as a continuum of the lens array. In the prism pairs, negative dispersion is determined by the separation of the prisms 1,and positive dispersion is determined by the total amount of material dispersion. Similarly, as we shall see, in GRIN lenses negative dis0146-9592/92/171177-03$5.00/0
(2)
= 0.
(c)
Fig. 1. (a) Prism pairs for producing negative GVD. (b) A device similar to that shown in (a), with prisms replaced by lenses. (c) A GRIN lens, which can be considered as a continuum
of (b).
C 1992 Optical Society of America
1178
OPTICS LETTERS
/ Vol. 17, No. 17 / September 1, 1992
Propagation of Gaussian beams in parabolic index media has been analyzed before.7 '12 However, all the analyses were done under the paraxial approximation, and the aspect of adjustable negative GVD has not been discussed. In this Letter we solve Eq. (2) without using a paraxial approximation, and from the solution we indentify a geometric source of negative GVD similar to that in grating and prism pairs. We also show that for solutions sufficiently off axis for producing negative GVD, a paraxial approximation does not render the correct phase of the wave and hence is not a correct estimation of GVD, even though it does give the correct amplitude of the wave. Thus for our purposes it is essential to solve
where v = -[(Aa2 k/2) + \/A] and u =V/§3a2k/2.
Eq. (2) without using a paraxial approximation.
where 0 is the phase angle of the wave. For mul-
With the method of variation of parameters,' 3 we find that for an off-axis Gaussian beam propagating along the x-z plane, the solution of Eq. (2) can be written as E(x, y, z) = lI(x, y, z)f (z)exp(ikz), where T(x, y, z) is the main element of the solution and f (z) is a phase factor. We choose T(x, y, z) to be the known paraxial solution'4 and use f (z) to correct the phase error, Tx
We have omitted terms containing -qand 6 because they are negligibly small. Equation (4) has two independent solutions:
f(z) = exp{i[-(k + v) ± k2 + 2kv - u]z} =
2[\-1/
D=
co
(V
i
2+6
_
2
)x2
-
(6)
tiple half-integer pitches of medium length, z = L = N17-/\/A, N is an integer, and we have Dk A,2 2\)TcL[2
vA ak ia
a2f2 k2 -_ k
2
228A aQl n + / 2 aNvA aA _ _A__ + a V) aA 2 A aA
a2f
+-+-
A
aA2 A
aA
A
A
(aV\) 221 ax LA J
(7)
From practical data on GRIN lenses,'6 it is readily shown that only the first four terms of the above
equation are important; the other terms are at least
N/')
1000-fold smaller. In these four terms, the positive one is
+ q2 sin2(N/'Z)] X exp([COS2(VAZ)
x {14 sin(2\/A4z)[(1 -
aa20
2ATaanoav§AAIA-a2_ 2 rno
sin2(V z)2
z) +j
(5)
We choose the solution with the plus for forwardpropagating waves. GVD in units of square seconds is defined as
- [x - a cos(N/Az)]2 XW2[71 Cos2(\/Az) + -asin2(Az)]}
X exp
exp(iflz).
( A2 A227ra2no 2irTcI A
?I2a2]
aA
(8)
2
which represents the material dispersion, and the negative ones are a2N/A - a2 A2 \2 (7rno N/A_ 2ic/ \ A aA
+ rj2 ax sin(\4 z)})
x exp
(2
-ikVA(l _ 2)y2 sin(2N/Az) t4[cos2 (Vz) + 62 sin 2(N/z)]J
X exp[-
i(p+ )]
+
a2ft + 2 all _A-7
A aA
' (9)
which represent the geometric (waveguide) disper(3)
harmonic oscillator
where k = 2'wno/A, p = tan-'('q tan VA-z),q = tan-'(g tan \Az), a is the amplitude of undulation 2 A-"/ 4 . The soluof the beam path, and w = V2k-1/
tion in Eq. (3) corresponds to a Gaussian beam winding around the axis of the GRIN lens sinusoidally with its size varying periodically as it propagates. The size of the Gaussian beam is related to 7)and e, which are determined by the boundary condition. It is not surprising that the solution is identical to that of a squeezed-state wave packet in a parabolic potential well'5 because the two problems are mathematically equivalent (see Fig. 2). The equation for the phase factor f(z) can be extracted by substituting E(x, y, z) into Eq. (2). When the beam size is much smaller than a, the equation is reduced to A
,A2
2i(k + V)a f - (V2 + U)f + a 2f = 0, az
a~~~~~iZ2
(4)
Gaussian beam inGRINlenses
Fig. 2. Analogy of a Gaussian wave packet in quantum harmonic oscillators and an off-axis Gaussian beam in GRIN lenses. Both move back and forth between edges with their sizes varying periodically.
September 1, 1992 / Vol. 17, No. 17 / OPTICS LETTERS
sion. It can be readily shown that the last two terms of the above expression, which come from f(z), are as important as the first term in contributing to the negative GVD. Thus they should not be omitted as they are in the paraxial approximation. For the W-2.0 graded-index materials from NSG America, Inc.,' 6 zero dispersion occurs at a - 1.6 mm for A = 620 nm. Although a beam offset of 1.6 mm is larger than the radii of common GRIN lenses, recently large GRIN lenses with diameters •8 mm have become commercially available.' 7 At a = 2.0 mm, D = -1158 fS2 corresponds to the positive dispersion of 22 mm of quartz. Note that the dispersion is practically independent of the beam size, because in Eq. (7) terms containing q and 6 are negligibly small. Also note that the beam sizes in the x and y axes oscillate between q1'/2w and rf-"2w and '112 w and f- 1/2w, respectively.
In the numerical
example above, w - 25 /.tm. Therefore, if the minimum beam size is 5 ,um, the maximum beam size is -125 Ium,and the requirement of Eq. (4) that beam size be much smaller than a is always satisfied. GVD alone may not be sufficient to describe the propagation of an ultrashort pulse. The effect of third-order dispersion can be neglected only when 3 ) is much smaller than D 82(p/8C2 . 3 1/8c AjWG For the same parameters used above, at a = 2.0 mm, D/(83 4/8w)3) - 5.5 X 10"4s-' corresponds to the spectral width of a 4-fs transform-limited pulse. Hence the effect of third-order dispersion is not important when the pulse duration is longer than 80 fs. The solution shown in Eq. (3) is periodic in the z direction. One can increase the dispersion, positive or negative, with multiple-pitch GRIN lenses without changing the beam characteristics. This property is convenient in device applications because one can change the pitch of the lens without affecting other components. The existence of a guided wave for a large range of beam size is particularly important in interfacing with waveguides and optical fibers. It allows the GRIN lens to be inserted directly into the path of the guided wave, without interfacing optics to match the beam size. This cannot be effected with prism pairs because the wave in prism pairs is not guided.
Because the parameter A in Eq. (3) depends
weakly on wavelengths, it is not possible for all the frequency components to be at exactly the same pitches. Deviation from exact pitches not only introduces errors in Eq. (7) but also causes the exit angle of different frequency components to disperse. Fortunately, these errors and the angular dispersion
1179
are of no practical importance. For example, for a 100-fs pulse at 620 nm propagating through a half-pitch GRIN lens, the error in Eq. (7) for the frequencies at the spectrum wings is -3%o,and the dispersion of the exit angle is only -1.3 X 10-3 rad. In comparison, the new method described in this Letter retains all the advantages of the prism pair, namely, low loss, no transverse
displacement,
con-
tinuous adjustment through zero, and a collinear transmitted beam, as pointed out in Ref. 5. Additional advantages of our method are compactness (the device occupies only a few millimeters) and effortless interface with waveguides and optical fibers. Furthermore, because the y dimension is not important in our method, the device can be made into a thin slab, with the beam propagating in the z direction and winding around the z axis in the xz plane. Such a device is compatible not only in application but also in fabrication with integrated optics.
References 1. 0. E. Martinez, IEEE J. Quantum Electron. QE-23, 1385 (1987). 2. P. Maine, D. Strickland,
P. Bado, M. Pessot,
and
G. Mourou, IEEE J. Quantum Electron. 24, 398 (1988).
3. W J. Tomlinson, R. H. Stolen, and C. V Shank, J. Opt. Soc. Am. B 1, 139 (1984).
4. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
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9. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982). 10. S. G. Krivoshlykov, and L. N. Sissakian, Opt. Quantum Electron. 12, 463 (1980). 11. G. Eichmann, J. Opt. Soc. Am. 61, 161 (1971). 12. C. G6mez-Reino and J. Liflares, J. Opt. Soc. Am. A 4, 1337 (1987).
13. T. M. Apostol, Calculus, 2nd ed. (McGraw-Hill, New York, 1967), p. 331.
14. E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970). The solution can be obtained easily with the Green function given on p. 164. 15. H. P Yuen, Phys. Rev. A 13, 2226 (1976).
16. Selfoc product guide (NSG America, Inc., Bridgewater, N.J.). 17. Gradient Lens Corporation, 207 Tremont Street, Rochester, N.Y. 14608.
