248
J. Eur. Opt. Society-Rapid Publ. 22, 24( 2026)
2.3 Calculation of optimum window for Spot and Track Distance
The bubble overlap denoted by the scaling factors F1 and F2 plays a pivotal role in determining cutting smoothness and efficiency. We extend our previous works [ 24, 26 ] to define an optimum window for the overlap factors that minimizes overall dose per treatment and corneal surface roughness.
Numerical simulations and graphical visualizations of overlapping disk geometries presented in the Results were generated using Python( CPython implementation) in a Jupyter-based execution environment provided by OpenAI, employing standard scientific libraries including NumPy, SciPy, and Matplotlib.
3 Results
3.1 Minimum Overall Dose per treatment
Following equation( 5), the Minimum Overall Dose per treatment( D Min) is related to energy of a single pulse( E p), Area of a single cavitation bubble( Bubble Area) and Bubble Diameter as follows:
Bubble Area ¼ p Bubble Diameter2 4
; ð9Þ
�
4
E p p
D min ¼
Bubble Diamater: ð10Þ 2
This is the Minimum Overall Dose per treatment( D Min) since using a lower number of pulses( i. e., smaller F1 F2 product)( all leading to that bubble area) would always result in tissue bridges( since the Bubble Area number of pulses would be smaller than the treatment area).
Comparing equations( 10) and( 8), the term for scaling factors for Spot and Track Distance for this Dose( D Min) should be as follows:
F1 F2
4: ð11Þ p
3.2 Maximum required Overall Dose per treatment
As shown in our previous work [ 26 ], the spot and track distance for which adjacent bubbles would largely overlap in the two principal directions and just overlap in the two diagonal directions leaving no tissue bridges, can be calculated as follows:
Spot Distance ¼
Bubble Diameter 2 0: 5; ð12Þ
Bubble Diameter Track Distance ¼: ð13Þ
2 0: 5
From equation( 8) combined with equations( 12) and( 13), the maximum required Dose( D Max) for this Spot and Track Distance is:
E p 2 D Max ¼
Bubble Diameter: ð14Þ 2
Comparing equations( 14) and( 8), the term for scaling factors for Spot and Track Distance from this Dose( D Max) should be as follows:
F1 F2 2: ð15Þ
This is the Maximum required Overall Dose per treatment( D Max) since using a greater number of pulses( i. e., larger F1 F2 product)( all leading to that Bubble Area) would result in tighter bubble overlap without reducing the amount of tissue bridges( since for F1 F2 = 2, adjacent bubbles would largely overlap in the two principal directions and just overlap in the two diagonal directions leaving no tissue bridges).
Figure 3 visually illustrates the impact of the product of scaling factors F1 andF2.
A window for scaling factors can be estimated comparing the Minimum and Maximum Dose terms( Eqs.( 11) and( 15)):
4
F1 F2 2: ð16Þ p
In the clinical context, the typical values of F1 andF2 are confined within upper and lower bounds. From both geometric necessity and clinical experience, meaningful contiguous cutting requires F 1; 2 > p 2 ffiffi p
1:13, which corresponds to the minimum spacing for planar area coverage by circularp
projections. Practical clinical limits are typically F 1; 2 < ffiffi
2 1: 41; which allows not very dense placement which can cause excessive mechanical disruption, although some commercial systems exceed this in specific protocols. In symmetric configurations, typical clinical values often fall in the range F 1; 2 2 1 = 4 to 2 1 = 2, corresponding to efficient overlap with limited energy redundancy [ 24 ]. Modern clinical systems increasingly employ F1 6¼ F2, for example, to optimize cut smoothness as simulations suggest that lower pulse energies( well above the LIOB threshold) combined with asymmetric spacings( spot-to-track distance ratio >> 1) may be effective to lower the roughness of laser cuts [ 24 ], such that the ratio of F2 / F1 shouldbe:
F1 > F2; ð17Þ
2 p ffiffi F2
3 F1 2 p ffiffiffi 3: ð18Þ
Representative numerical ranges are included in the examples below to make these quantities interpretable in practice.
3.3 Optimum window for Spot and Track Distance
From the two inequalities presented in equations( 16) and( 18), involving the scaling factors for Spot( F1) and Track Distance( F2), a bounded solution space can be defined( Table 1).
The feasible region is delimited by the minimum and maximum values of the product F1 F2( dose) and the ratio