Fractal frequency spectrum in laser resonators and three-dimensional geometric topology
of optical coherent waves
J. C. Tung,1P. H. Tuan,1H. C. Liang,2K. F. Huang,1and Y. F. Chen1,*
1Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan 2Institute of Optoelectronic Science, National Taiwan Ocean University, 2 Pei-Ning Road, Keelung 20224, Taiwan
(Received 10 May 2016; published 3 August 2016)
We theoretically verify that the symmetry breaking in spherical resonators can result in a fractal frequency spectrum that is full of numerous new accidental degeneracies to cluster around the unperturbed degenerate cavity. We further experimentally discover that the fractal frequency spectrum excellently reflects the intimate connection between the emission power and the degenerate mode numbers. It is observed that the wave distributions of lasing modes at the accidental degeneracies are strongly concentrated on three-dimensional (3D) geometric topology. Considering the overlapping effect, the wave representation of the coherent states is analytically derived to manifest the observed 3D geometric surfaces.
DOI:10.1103/PhysRevA.94.023811
I. INTRODUCTION
Starting from Mandelbrot’s seminal discovery [1], self-similar and fractal structures have been observed in a variety of phenomena in nature [2] and have also been found in many branches of physics [3–5]. One of the prominent examples in quantum systems is the fractal conductance fluctuations in gold nanowires and in mesoscopic electron billiards [6–10]. Another prominent example of the self-similar phenomena is the plateau formation in the transverse Hall-resistance curve of a two-dimensional (2D) electron system at low temperatures in the presence of a strong perpendicular magnetic field, known as the quantum Hall effect [11,12]. More intriguingly, the Hofstadter’s fractal energy spectrum [13,14] for Landau levels in a 2D periodic lattice has been realized [15,16].
The remarkable analogy between optical and mechanical phenomena was fully developed in Hamilton’s ingenious opticomechanical theory that played a fundamental role in the development of ideas in quantum physics [17–20]. Thanks to the development of modern laser cavities, not only eigenvalues but also eigenfunctions have been analogously explored by using solid-state lasers for a 2D quantum har-monic oscillator [21–25] and semiconductor lasers for 2D quantum billiards [26–30]. Optical resonators with the same isomorphism clearly confirm that the level degeneracies in 2D mesoscopic quantum systems generally lead to the wave functions with intensities concentrated on classical periodic orbits [10]. Nevertheless, optical resonators have never been used to explore the energy spectrum in higher-dimensional quantum systems. In particular, the emergence of the ray geometry from the coherent wave in optical resonators is still an open and fascinating issue of active research in recent years. The attractive interest comes partly from the fundamental questions of light-matter interaction [31] and ray-wave correspondence [32–34], and partly from numerous applications, such as cavity spectroscopy [35–37], optical
*Department of Electrophysics, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan; yfchen@cc.nctu.edu.tw.
pattern formation [38–41], single-photon emitters [42], and ultralow threshold lasers [43,44].
One of the earlier reports on laser fractals was to demon-strate that the eigenmodes of unstable-cavity lasers have fractal structure [45]. The spatial fractal formation for laser transverse modes was later confirmed in so-called kaleidoscope lasers that include nontrivial transverse boundary conditions [46,47]. In contrast with the eigenmode fractal formation in the unstable cavity, we here explore the eigenfrequency fractal formation in the stable spherical cavity subject to the parasitic astigmatism. In this work we theoretically verify that the symmetry breaking induced by astigmatism leads to a fractal frequency spectrum in laser resonators. The fractal frequency spectrum is found to display numerous new accidental degeneracies at the cavity lengths near the unperturbed degenerate cavity. We further exploit the selective pumping to excite the astigmatic resonator and experimentally discover that the emission power variation on the cavity length exhibits the local maxima at the accidental degeneracies to form a fractal fluctuation corresponding to the fractal frequency spectrum. It is also observed that the wave distributions of lasing modes at the accidental degeneracies are remarkably concentrated on the three-dimensional (3D) geometric surfaces. Finally, we theoretically derive the coher-ent states to manifest the 3D geometric topology of the lasing modes. Based on the opticomechanical analogy, the present results can be directly applied to the 3D integrable quantum systems with symmetry breaking to explore the wave functions related to 3D geometric surfaces.
II. FRACTAL FREQUENCY SPECTRUM
In a laser resonator with the gain medium, the inevitable symmetry breaking comes from the astigmatism induced by the birefringence of the gain medium and the angle of the beam divergence. As a consequence, the eigenfrequencies of the spherical cavity with parasitic astigmatism can be generally given by
ωm,n,= ωz+ (m + 1/2)ωt,x+ (n + 1/2)ωt,y, (1)
where is the longitudinal mode index, m and n are the transverse mode indices, ωzis the longitudinal mode spacing,
FIG. 1. Frequency spectrum ωm,n,of the ideal spherical cavity in the neighborhood of the indices (mo,no,o) as a function of the normalized
cavity length Lt/Rfor the range of|m − mo| 12, |n − no| 12, and | − o| 12.
y directions. For the concave-plano resonator, the transverse mode spacings are given by ωt,x = (ωz/π)sin−1
Lt,x/R
and ωt,y = (ωz/π)sin−1
Lt,y/R, where ωz= c/2Lopt, Lopt=
Lcav+ (nr− 1)Lg, nr is the refractive index of the gain
medium, Lg is the physical length of the gain medium, and
R is the curvature radius of the concave mirror. Due to the symmetry breaking, the effective cavity lengths can be generally expressed as Lt,x = Lt− d/2 and Lt,y = Lt+ d/2,
where Lt is the mean value of Lt,x and Lt,y and d I s the
difference between Lt,x and Lt,y.
Neglecting the astigmatic effect, Ltis given by Lt = Lcav+
[nr− (1/nr)]Lg and d= 0. Under this circumstance, ωt,x =
ωt,y = ωt = (ωz/π)sin−1
Lt/Rand the frequency spectrum
ωm,n,in the neighborhood of the indices (mo,no,o) is given
by ω ωz = [(m + n) − (mo+ no)] ωt ωz + ( − o), (2)
where ω= ωm,n,− ωmo,no,o is the frequency difference.
The frequency ratio ωt/ωz is a monotonically increasing
function of the effective cavity length Lt for a given R and
its value is between 0 and 1/2 for 0 Lt R. Note that the
relationship between ω/ωzand ωt/ωzin Eq. (2) is a simple
Diophantine equation. The occurrence of degeneracy is given by ω= 0 in Eq. (2). Figure1shows the frequency spectrum
ωm,n, in the neighborhood of the indices (mo,no,o) as a
function of the effective cavity length Ltfor R= 10 mm. It can
be seen that the degeneracies and gaps appear at the effective cavity length Lt = LP /Qthat leads to the frequency ratio of
ωt/ωzcorresponding to the rational number P /Q, where P
and Q are coprime integers. The family of the degenerate states can be in terms of the triple integers (p,q,s) to express the mode indices (m,n,) as m= mo+ pu, n = no+ qu, and
= o+ su, where the integer u is a common index given by
u= · · · − 2, −1, 0, 1, 2 · · · . Using Eq. (2), the triple integers (p,q,s) for the degenerate condition of ωt/ωz= P /Q can be
shown to satisfy a simple relation (p+ q)(P /Q) + s = 0 that indicates that p+ q = MQ and s = −MP , where M is an integer.
Figure2(a)shows the influence of the symmetry breaking on the frequency spectrum ωm,n, for an example of R=
10 mm and d= 50 μm. It is clear that the parasitic astigmatism gives rise to the level rearrangement and breaks the original
degeneracies at Lt = LP /Qin the unperturbed spherical cavity.
Nevertheless, there are numerous new degeneracies to appear in the neighborhood of Lt = LP /Q, as shown in Fig.2(b). To
reveal the new degeneracies near the region of Lt = LP /Q,
a dimensionless parameter ξ is introduced to express the effective cavity length as Lt= LP /Q+ ξ d, where |ξ| 1.
Using the parameter ξ and the property d R, the transverse frequencies near Lt = LP /Qcan be derived as
ωt,x ωz = P Q+ β ξ −1 2 , ωt,y ωz = P Q+ β ξ+1 2 , (3) where β= d/[πR sin(2πP /Q)]. Substituting Eq. (3) into Eq. (1) and using the triple integers (p,q,s) with the identity of (p+ q)(P /Q) + s = 0, the frequency spectrum near Lt =
LP /Qwith respect to ωmo,no,ocan be simplified as
ω ωz = βu (p+ q)ξ − 1 2(p− q) . (4)
Note that the relationship between ω/ωzand ξ in Eq. (4)
is also a Diophantine equation. The criterion ω= 0 in Eq. (4) gives the new accidental degeneracies to occur at the effective cavity lengths Lt = LP /Q+ ξ d with
ξ = 1
2 (p− q)
(p+ q). (5)
Numerous integer pairs (p,q) for the new degeneracies are shown in Fig.2(b). Substituting Eq. (5) into Eq. (3), it can be found that ωt,x ωz = P Q− β q p+ q, ωt,y ωz = P Q+ β p p+ q. (6)
As discussed later, Eq. (6) plays an important role in manifesting the topological geometry of the coherent states.
Now we discuss the fractal dimension of the frequency spectrum shown in Fig.2. The frequency spectrum is related to the degeneracy distribution of the cavity length. Without the astigmatic effect, d= 0, there are no additional degenerate points in the frequency spectrum; consequently, the fractal dimension is zero. When d= 0, the value of the fractal dimension is between 0 and 1, similar to that of the Cantor set. Using the definition that fractal dimension D equals the log of the number of pieces N divided by the log of the magnification factor r, the fractal dimension of the frequency spectrum can
FIG. 2. (a) Influence of the symmetry breaking on the frequency spectrum ωm,n,for an example of R= 10 mm and d = 50 μm for the
range of|m − mo| 12, |n − no| 12, and | − o| 12. (b) Partial magnification of the frequency spectrum in the neighborhood of
Lt = LP /Qwith P /Q= 1/4.
be expressed as D= log(N)/ log(r). As shown in Fig.2, the magnification factor is given by r = R/d. From Eq. (6), the number of pieces N can be estimated to be N= P /(MQβ), where the multiplication factor M is related to p+ q = MQ. The maximum multiplication factor is usually not greater than 5. For a typical example of R= 10 mm and d = 50 μm, the mean fractal dimension D is calculated to be 0.745 for
M= 1, 0.594 for M = 2, and 0.506 for M = 3. It can be
seen that the overall characteristics of the fractal dimension are rather close to that of the Cantor set [48].
III. EXPERIMENTAL OBSERVATIONS
To explore the effect of the fractal frequency spectrum, we exploited the off-center selective pumping to excite extremely high-order modes in a solid-state laser, as shown in Fig.3for the setup. The cavity was formed by a concave mirror and a gain medium. For the concave mirror, the radius of curvature is R= 10 mm and the reflectivity is 99.8% at the wavelength of 1064 nm. The concave mirror was precisely controlled to vary the cavity length in the range of 3.8–8.8 mm. The gain medium was an a-cut 2.0−at.% Nd3+: YVO
4 crystal with a length of 2 mm. One side of the gain medium was coated for antireflection at 808 nm and 1064 nm (reflection <0.1%)
and the other side was coated to be an output coupler with a transmission of 0.5% at 1064 nm. The pump source was an 808-nm fiber-coupled laser diode with a core diameter of 100 μm, a numerical aperture of 0.16, and a maximum output power of 3 W. A focusing lens was used to reimage the pump beam into the gain medium with the off-center displacement of
x = 0.7 mm and y = 0.56 mm. The emission power was
systematically recorded by changing the cavity length with a precise step of 10 μm.
FIG. 3. Experimental laser setup with the off-center selective pumping to excite extremely high-order transverse modes for ex-ploring the effect of the fractal frequency spectrum.
FIG. 4. (a) Experimental result for the emission power as a function of the cavity length under a pump power of 1.0 W. (b) Partial magnification of the power variation in the region of P /Q= 1/4. (c) Experimentally tomographic images of the lasing modes at the main peaks of the emission power.
Since the emission power is generally proportional to the number of the excited cavity modes, the emission power has an intimate connection with the degeneracy. Figure4(a)shows the experimental result for the emission power as a function of the cavity length under a pump power of 1.0 W. It is clear that the emission power variation on the cavity length exhibits the local maxima at the accidental degeneracies to form a fractal fluctuation corresponding to the fractal frequency spectrum. Figure 4(b) shows the partial magnification of the power variation in the region of P /Q= 1/4. The experimentally tomographic images of the lasing modes at the main peaks of the emission power are shown in Fig.4(c). The tomographic images of the lasing modes were obtained by controlling the exposure time of the camera to selectively capture the transverse patterns inside the cavity along the longitudinal propagation. More specifically, a pair of relay lenses was used to reimage the laser modes onto a paper screen that was moved by tracking the camera to record the tomographic images. The transverse patterns can be clearly seen to be localized on the Lissajous curves with specific indices (p,q) related to the degeneracies shown in Fig. 2(b). In the following, it is theoretically shown that the formation of the lasing modes can be nicely represented by using the quantum coherent states.
IV. MANIFESTING 3D GEOMETRIC TOPOLOGY OF LASER MODES
Considering the paraxial approximation and the astigma-tism between the x and y directions, the eigenmodes for the laser cavity with a concave mirror at z= −Lopt and a plane mirror at z= 0 can be divided into two waves traveling in opposite directions: m,n,= [(+)m,n,−
(−)
m,n,]/
√
2, where
(±)m,n,(x,y,z,t)= Xm(x,z)e−i(m+1/2)θ
(±)
t,x(˜z,t)Y n(y,z)
× e−i(n+1/2)θt,y(±)(˜z,t)e−iθL(±)(˜z,t), (7)
with Xm(x,z)= 2/π /[wx(z)2mm!]e− ˜x 2/2 Hm( ˜x), (8)
Yn(y,z)=2/π /[wy(z)2nn!]e− ˜y
2/2
Hn( ˜y), (9)
θt,x(±)(˜z,t)= (ωt,x/ωz)θz(±)(˜z,t)∓ θG,x(z), (10)
θt,y(±)(˜z,t)= (ωt,y/ωz)θz(±)(˜z,t)∓ θG,y(z). (11)
θz(±)(˜z,t)= ωz(t± ˜z/c), Hm(·) are the Hermite polynomials
of order m, ˜x=√2x/wx(z), ˜y= √ 2y/wy(z), wx(z)= wo,x 1+ (z/zR,x)2, wy(z)= wo,y 1+ (z/zR,y)2, wo,x = 2zR,x/km,n,, wo,y = 2zR,y/km,n,, zR,x= Lt,x(R− Lt,x), zR,y= Lt,y(R− Lt,y), ˜z= z{1 + [x2/ 2(z2+ z2 R,x)]+ [y2/2(z2+ zR,y2 )]}, θG,x(z)= tan−1(z/zR,x),
and θG,y(z)= tan−1(z/zR,y).
Based on the completeness of the basis states, the general representation for the wave function can be expressed as
(±)(x,y,z,t)=
n
mcbnam(±)m,n,(x,y,z,t), where
am and bn are the amplitude coefficients for the transverse
orders m and n and c is the amplitude coefficient for the
longitudinal order . It has been shown that the longitudinal modes in an end-pumped standing-wave cavity are primarily related to the spatial hole burning (SHB) effect [49]. The strength of the SHB effect is mainly determined by the separation between the gain medium and the input mirror. The stronger the SHB effect, the more the longitudinal lasing modes. Here it is conveniently assumed that there are 2N+ 1 longitudinal modes to be excited and c= 1/(2N + 1) for | −
o| N and c= 0 for | − o| > N. It has been shown [34]
that the coefficients ambnare mainly controlled by the spatial
overlap between the transverse mode Xm(x,z)Yn(y,z) and the
distribution of the pump source F (x,y); i.e.,
ambn=
Xm(x,zc)Yn(y,zc)F (x,y)dxdy, (12)
where zcis the location of the gain medium. Considering a selective pumping with the transverse displacements x and
y in the x and y directions, the pump distribution F (x,y) can be modeled as [34] F(x,y)= 2 π 1 wx(zc)wy(zc) e−(x−x)2/w2x(zc)e−(y−y)2/w2y(zc). (13)
Substituting Eqs. (8), (9), and (13) into Eq. (12) and using the generating function of the Hermite polynomials [50], the coefficients can be derived as am= (mo)m/2e−mo/2/
√
m! and bn= (no)n/2e−no/2/
√
n!, where mo= [x/wx(zc)]2 and
no= [y/wy(zc)]2. Note that the values of the parameters mo
and nosignify the magnitudes of the off-center displacements
in the x and y directions, respectively. The expression for am
and bn is exactly the form of the square root of the Poisson
distribution. For convenience, we take the parameters moand
noto be the integers closest to the values of [x/wx(zc)]2and
[y/wy(zc)]2, respectively.
Using the property of the Schr¨odinger coherent state and the expressions for the coefficients am, bn, and c, the intensity of
the wave-packet state I(±)=|(±)(x,y,z,t)|2can be derived as
I(±)(x,y,z,t)= 2
π wx(z)wy(z)
e−{ ˜x−√2√mocos[θt,x(±)(˜z,t)]}
2
e−{ ˜y−√2√nocos[θt,y(±)(˜z,t)]}
2sin[(2N+ 1)θ(±)
z (˜z,t)/2]
sin[θz(±)(˜z,t)/2]
2
. (14)
The analytical form in Eq. (14) can straightforwardly be exploited to establish the ray-wave connection. First of all, since the total number of longitudinal modes is fairly greater than 1, i.e., N 1, the term involving the sine function in Eq. (14) leads the intensity of the wave packet to concentrate at θz(±)(˜z,t)= 2πu for any integers u. Substituting θz(±)(˜z,t)= 2πu into Eqs. (10)
and (11), the time-averaged intensity of the wave-packet state I(±)= |(±)(x,y,z,t)|2in Eq. (14) can be deduced to be localized on the distribution Ic(±): Ic(±)(x,y,z)= 2 π wx(z)wy(z) u=0,1,2··· e−{ ˜x− √ 2√mocos[(ωt,xωz )2π u∓θG,x(z)]} 2 e−{ ˜y− √
2√nocos[(ωt,yωz )2π u∓θG,y(z)]}
2
. (15)
Equation (15) reveals that the spatial distribution Ic(x,y,z)
of the wave-packet state is formed by the assemblage of numerous backward and forward Gaussian beams. Under the degeneracy condition given by Eq. (6), the spatial distribution
Ic(x,y,z) can be generally found to correspond to the 3D geometric surface with the transverse topology of Lissajous curves. To be explicit, substituting ˜x=√2x/wx(z) and ˜y =
(-2,6) (-1,5) (0,4) (1,3) (2,2) (3,1) (4,0) (5,-1) (6,-2) (3,5) (1,7)
FIG. 5. Numerical results for 3D geometric surfaces inside the cavity for various (p,q) shown in Fig.2(b)with no= 50, mo= 50,
R= 10 mm, d = 0.087 mm, and β = 2.757×10−3.
√
2y/wy(z) into Eq. (15) and using Eq. (6) for the degenerate
cavity, the mathematical parametric form for the central maxima of the spatial distribution Ic(x,y,z) in Eq. (15) can
be expressed as x(z; u)=√mowx(z) cos P Q− β q p+ q (2π u)∓ θG,x(z) y(z; u)=√nowy(z) cos P Q+ β p p+ q (2π u)∓ θG,y(z) , (16) with u= 0,1,2 · · · . Equation (16) can be directly used to manifest the geometric surfaces inside the cavity. Figure 5
depicts the 3D geometric surfaces inside the cavity for various (p,q) shown in Fig.2(b)with no = 50, mo= 50, R = 10 mm,
d = 0.087 mm, and β = 2.757×10−3.
V. CONCLUSIONS
In summary, it has been theoretically verified that the symmetry breaking in spherical resonators can result in a fractal structure in the frequency spectrum as a function of the cavity length. In the fractal frequency spectrum, numerous new accidental degeneracies are found to cluster at the cavity lengths around the unperturbed degenerate cavity. Further-more, it has been experimentally discovered that the fractal frequency spectrum can lead to the emission power varying with the cavity length in astigmatic laser resonators under the selective pumping to display a striking fractal fluctuation. We have also derived the quantum coherent states by considering the overlapping effect to manifest the noticeable finding that the wave distributions of lasing modes at the accidental degeneracies are strongly concentrated on the 3D geometric
surfaces. The present exploration can be directly applied to the 3D quantum integrable systems with symmetry breaking to derive various topological geometries of 3D coherent states.
ACKNOWLEDGMENT
This work is supported by the Ministry of Science and Technology of Taiwan (Contract No. MOST-103-2112-M-009-016-MY3).
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