Fractal light from lasers

Fractals, complex shapes with structure at multiple scales, have long been observed in nature: as symmetric fractals in plants and sea shells, and as statistical fractals in clouds, mountains, and coastlines. With their highly polished spherical mirrors, laser resonators are almost the precise opposite of nature, and so it came as a surprise when, in 1998, transverse intensity cross sections of the eigenmodes of unstable canonical resonators were predicted to be fractals [G. P. Karman et al., Nature (London) 402, 138 (1999)]. Experimental verification has so far remained elusive. Here we observe a variety of fractal shapes in transverse intensity cross sections through the lowest-loss eigenmodes of unstable canonical laser resonators, thereby demonstrating the controlled generation of fractal light inside a laser cavity. We also advance the existing theory of fractal laser modes, first by predicting three-dimensional self-similar fractal structure around the center of the magnified self-conjugate plane and second by showing, quantitatively, that intensity cross sections are most self-similar in the magnified self-conjugate plane. Our work offers a significant advance in the understanding of a fundamental symmetry of nature as found in lasers.

Figure 1

Imaging inside an unstable canonical resonator. The two spherical mirrors, ${\mathrm{M}}_1$ (focal length $f$) and ${\mathrm{M}}_2$ (focal length $F$), perform geometric imaging. We define two longitudinal coordinates, $z$ and $Z$. Both the $z$ and the $Z$ axes coincide with the optical axis, but the $z=0$ plane coincides with the plane S, the magnified self-conjugate plane, and the $Z=0$ plane coincides with the plane of the mirror ${\mathrm{M}}_2$. We use $z$ in our theoretical analysis, $Z$ in the experimental part. The transverse coordinates are $x$ (not shown) and $y$. Three-dimensional imaging during one round trip is indicated by an object in the shape of the letter “P” (shown in black) and its image, which looks like a horizontally elongated letter “b” (shown in grey): the “P” has turned into a “b” because the transverse magnification, $M$, is negative, and so the image of the “P” is upside down; the “b” is horizontally elongated because the longitudinal magnification, $M_l$, is positive and its magnitude is greater than that of the transverse magnification. $A$ is an aperture immediately in front of ${\mathrm{M}}_2$. The figure is drawn for the particularly simple case of a confocal resonator (length $F+f$; the plane S then coincides with the common focal plane) with $M=-2$ and $M_l=+4$.

Figure 2

Self-similarity of the simulated lowest-loss eigenmode's intensity distribution in the magnified self-conjugate plane, S. The frames show the intensity after 20 round trips, starting with a uniform plane wave. The self-similarity of the pattern is demonstrated by showing its center at different magnifications ($2\times , 4\times , 8\times$), resulting in a similar pattern (rotated by ${180}^{\circ }$ after each magnification by a factor of 2 due to the resonator's transverse magnification, $M$, being negative); the dotted white square in the center of the frame marked $\times 1$ shows the outline of the area shown in the next frame. The horizontal dotted line is the orthographic projection of the lateral self-conjugate plane $S_L$, in which we demonstrate, below (see Fig. 3), the three-dimensional (3D) self-similarity of the light. The figure is calculated for light of wavelength $\lambda =632.8\mathrm{nm}$ in a resonator of the type shown in Fig. 1 with $F=16.5\mathrm{cm}, f=8.25\mathrm{cm}, M=-2$, and a seven-sided regular polygonal aperture of circumradius $r_0=2.4\mathrm{mm}$. The beam's transverse cross sections were represented by a $1024\times 1024$ array of complex numbers sampled over a physical area of size $1\mathrm{cm}\times 1\mathrm{cm}$. For further details of the way this simulation, and indeed all the other simulations in this paper, was performed, see Appendix pp1.

Figure 3

Self-similarity of the intensity distribution in a lateral self-conjugate plane, $S_L$, which contains the optical axis and intersects the plane S horizontally in Fig. 2. The vertical dotted line is the orthographic projection of the plane S. Vertically, the plots are centered on the optical axis, $z$. The beam is the same as that shown in Fig. 2. After each magnification horizontally by a factor 4 and vertically by $-2$, the pattern looks similar again, which is shown for different magnifications. The dotted box in the center of the frame marked $\times 1$ marks the outline of the area shown in the next frame ($\times (-2)$ vertically, $\times 4$ horizontally). The $\times 1$ frame represents a physical area of size 2 m (horizontally) by 10 mm (vertically).

Figure 4

Self-similarity of the intensity distribution in the lateral plane of a strip resonator of the type shown in Fig. 1. The different frames show the center of the intensity distribution, successively magnified by a factor $M$ in the vertical direction and by $M^2$ in the horizontal direction. The $\times 1$ frame represents a physical area of size 20 m (horizontally) by 2.82 cm (vertically), centered on the magnified self-conjugate plane and the optical axis in the horizontal and vertical direction, respectively. The dotted box shown in the top left frame outlines the area shown in the top right frame. The figure was calculated for light of wavelength $\lambda =632.8\mathrm{nm}$, resonator parameters $F=70.7\mathrm{cm}$ and $f=50\mathrm{cm}$ (corresponding to transverse magnification $M=-\sqrt{2}$), and the aperture $A$ was a slit of width 2.08 cm. Each beam cross section was represented on a 4096-element array of complex numbers, representing a physical width of 4 cm.

Figure 5

(a) Axial and (b) transverse intensity cross section through the field around the self-conjugate point at the center of the plane S in the strip resonator from Fig. 4. Like in Figs. 2, 3, 4, the self-similarity is demonstrated by successive magnifications, each of which stretches the part of the curve between the vertical dotted lines to the full width. The width of the curves marked $\times 1$ represents a physical length of (a) 2 m and (b) 2.08 cm. The intensity range represented by the different curves has been adjusted so that corresponding features in the curves have roughly the same vertical size.

Figure 6

Evolution of the self-similarity of transverse intensity cross sections upon propagation for the eigenmodes shown in (a) Figs. 2 and 3 and in (b) Figs. 4 and 5. The self-similarity is quantified here by $-d^2$, the negative of the normalized squared Euclidean distance between the central rectangle of the intensity cross section in the transverse plane with the given $z$ coordinate, and the central rectangle of the same size of the intensity cross section after being stretched in the transverse directions by a factor $M$. The magnification was (a) $M=-3$ and (b) $M=-2$. The width and height of the central rectangle were arbitrarily chosen to be $1/4$ of the width and height of the calculated intensity cross section. The dotted vertical lines indicate the planes of the mirrors, ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$, and the magnified self-conjugate plane, S.

Figure 7

Experimental setup, (a) schematic and (b) photo with the laser running. (a) The cavity comprised two concave mirrors, ${\mathrm{M}}_1$ (radius of curvature $R_1$, corresponding to focal length $f=R_1/2$) and ${\mathrm{M}}_2$ (radius of curvature $R_2$, focal length $F=R_2/2$), and an output coupler (OC) angled at ${45}^{\circ }$ with a 99.8% reflectivity. The geometrical length of the cavity was $G$, and that of the Nd:YAG gain medium was $g$. S is the magnified self-conjugate plane. A polygonal aperture (A) was positioned in front of ${\mathrm{M}}_2$. The intensity cross section in the self-conjugate plane S was captured outside the cavity at a distance $R_2/2$ from ${\mathrm{M}}_2$ using a CCD camera. (b) Clearly visible is the gain-medium assembly (bright yellow, center) and the mounts for the end mirrors and the output coupler. The two tubes (light blue, bottom left) connecting to the gain-medium assembly are cooling tubes for the gain medium.

Figure 8

A few of the transverse intensity patterns recorded inside the resonator with parameters $F=250$ mm, $f=75$ mm, and $G=343$ mm, and a hexagonal aperture of circumradius $r_0=2$ mm. The effective length of the resonator, which takes into account the effect of the refractive index of the gain medium (see Appendix pp4), is $L=308.5$ mm. The self-conjugate plane S is positioned at $Z=Z_S=231.8$ mm, the magnification was $M=-3.32$.

Figure 9

Self-similarity of the intensity cross sections in different transverse planes near the predicted position of the self-conjugate plane, $Z=Z_S=231.8$ mm. The images in the top row show the central $\approx 4\mathrm{mm}\times 4\mathrm{mm}$ of the experimentally recorded intensity cross sections in the transverse planes $Z=220$ mm (left), $Z=230$ mm (center), and $Z=240$ mm (right). The images in the bottom row show the centers of the corresponding top-row images, stretched by a factor $M=-3.32$, that is, stretched by a factor of 3.32 and rotated by ${180}^{\circ }$. To show structure over a wider intensity range, the brightness of each pixel is proportional to the logarithm of the recorded intensity.

Figure 10

Intensity cross sections corresponding to horizontal cuts through the centers of the transverse intensity distributions shown in Fig. 9. In the curves marked $\times (-3.32)$, the part between the vertical dotted lines of the curve marked $\times 1$ is stretched to the full width. Like in Fig. 5, the intensity range represented by each curve has been adjusted so that corresponding features in the curves have roughly the same vertical size, and the curves have been shifted vertically to make them distinguishable.

Figure 11

Evolution of the self-similarity upon propagation of measured transverse intensity cross sections. Like in Fig. 6, the plot shows the negative normalized squared Euclidean distance, $-d^2$, as a function of axial coordinate $Z$. The predicted position of the self-conjugate plane S, $Z=Z_S=231.8$ mm, is indicated by a dotted black line surrounded by a grey area representing experimental uncertainties. The planes of the mirrors, ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$, and the self-conjugate plane, S, are indicated by vertical dotted lines.

Figure 12

Collage of experimentally obtained intensity cross sections measured close to the self-conjugate plane S inside a variety of laser cavities. The edge of the aperture in the central column is that of the third iteration of the Koch snowflake (the first iteration being an equilateral triangle), and the other apertures are regular hexagons. $r_0$ is the circumradius of the apertures, $M$ is the approximate transverse round-trip magnification, and $N_F$ is the Fresnel number. Like in Fig. 9, brightness is proportional to the logarithm of intensity.