J. Eur. Opt. Society-Rapid Publ. 22, 24( 2026) 249
Fig. 3. Visual representation of symmetric pulse spacing( F1 = F2) on an Archimedean spiral, shown for different values of the product of scaling factors F1 and F2. Left: F1 F2 = 1, adjacent bubbles do not overlap in the two principal directions and diagonal directions leaving tissue bridges. Middle: F1 F2 = 2, adjacent bubbles largely overlap in the two principal directions and just overlap in the two diagonal directions leaving no tissue bridges. Right: F1 F2 = 1.41, A good compromise that enables tissue bridge-free bubble overlap while minimizing the dose.
Table 1. Corner solutions derived from the inequalities( Eqs.( 16) and( 18)). Each solution represents an extreme case of dose and asymmetry. The optimum lies at minimum dose with maximum asymmetry, whereas the combination of moderate dose and minimum asymmetry is the least advantageous of the optimum solutions.
Solution
Dose( F1 F2)
Asymmetry( F2 / F1)
Interpretation
1 |
2( Maximum) |
2 / p 3( low) |
Moderate dose, low asymmetry |
2 |
4 / p( Minimum) |
2 / p 3( low) |
F1 F2 = 2 ensures tissue bridge-free bubble overlap
F2 / F1 = 2 / p 3 is the smallest asymmetry of a hexagonal lattice
Minimum dose, low asymmetry
|
3 |
2( Maximum) |
2 p 3( high) |
F1 F2 = 4 / enables tissue bridge-free bubble overlap with minimum dose
F2 / F1 = 2 / p 3 is the smallest asymmetry of a hexagonal lattice
Moderate dose, high asymmetry( lower roughness)
|
4 |
4 / p( Minimum) |
2 p 3( high) |
F1 F2 = 2 ensures tissue bridge-free bubble overlap
F2 / F1 = 2 p 3 is the largest asymmetry of a hexagonal lattice
Minimum dose, high asymmetry( lower roughness)
|
|
|
|
F1 F2 = 4 / p enables tissue bridge-free bubble overlap with minimum dose
F2 / F1 = 2 p 3 is the largest asymmetry of a hexagonal lattice
|
F2 / F1( asymmetry). This rectangular domain naturally gives rise to four corner solutions, each representing an extreme combination of dose and asymmetry( Fig. 4). Among these, the most favorable setting is located at the corner of minimum dose and maximum asymmetry, which combines energy efficiency with smoother cuts. Conversely, the least desirable region corresponds to the corner of maximum dose and minimum asymmetry, which should be avoided. The optimal operating point therefore lies toward the lower-right corner of the solution space. The two intersecting lines highlight intermediate trade-offs: the vertical line represents moderate asymmetry across the full dose range, while the horizontal line represents moderate dose across the full asymmetry range. From this cross-section, a Proper Optimum was derived as the geometric mean of the vertical and horizontal solutions, lying near Solution 4. This calculated optimum( as indicated in Fig. 4) corresponds to an asymmetry of F2 / F1 = 2.63 and a dose of F1 F2 = 1.41.
In addition to the four corner solutions derived from the inequalities( Eqs.( 16) and( 18)) representing the extreme case of dose and asymmetry, the case of F1 = 1wasalso evaluated. The corresponding values of F1 and F2 for these combinations are summarized in Table 2, while the resulting ranges of F1 andF2 are provided in Table 3.
Following the Photodisruption model [ 9 ], and using the different exemplary combinations of dose( F1 F2) and Asymmetry( F2 / F1), the Spot Distance, Track Distance, and corresponding dose for each combination are calculated for a particular LIOB threshold( Eth = 50 nJ) and Single Pulse Energy( E p = 100 nJ), and presented in Table 4.
Using LIOB Threshold energies loosely representing commercial systems currently used in ultrashort-pulse corneal tissue cutting, and the calculated optimum Single Pulse Energies compatible with previous works [ 9 ], for the different exemplary combinations of a proper optimum for dose( F1 F2) and Asymmetry( F2 / F1), the Spot Distance, Track Distance and corresponding Dose is presented in Table 5.