1. Historical Review
Fig. 1. UV-light-initiated corneal collagen crosslinking (CXL) [10-13]
Fig. 2. A corneal model system under UV light crosslinking for the epi-on (A) and epi-off (B) case (left figure); and the time-dependent RF concentration profiles versus depth (z) (right figure).at various time t=(0, 135, 270, 540, 810, 1080, 1350) seconds for curves (1,2,3,45,6,7), with an initial UV light intensity I0=10 mW/cm2. The red curves are for C/C0=0,135
The first human trial of PRK was conducted by Dr. Trokel in 1987, based upon the IBM patent (UV laser for organic tissue ablation) and the animal study in 1983. FDA approval PRK in 1995. The flying-spot scanning technology was invented by Dr. J.T. Lin in 19992 (US patent) who also introduced the customized procedure in 1996. The waveguide device was commercialized in 1999. FDA approved LASIK in 2002. During 1995-1999, various laser systems/procedures were developed including: LTK (using Ho:YAG), DTK (using diode laser), RF and CK designed for hyperopia corrections; slide state lasers (YAG-213 nm for PRK), YAG pico-second-PRK, Mini-Excimer for PRK etc. Technologies developed in the 2000’s include: eye-tracking device (Lai, Nvatek), microkeratome, Elevation map, topography-guided LASIK, wavefront for customized LASIK (Tracey), presbyopia treatment using SEB (Schachar) and laser scleral-ablation for presbyopia (Lin); accommodative IOL. More recently, femto-second lasers are developed for flat cutting, stroma ablation and cataract. UV-light and riboflavin activated corneal cross linking (CXL) has been developed for clinical use for various corneal deceases such as corneal keratoconus, corneal keratitis, corneal ectasia, corneal ulcers, and thin corneas prior to LASIK vision corrections. Combined technology of CXL=PRK, CXL-intra stroma-femtolaser pocket, CXL-phakic-IOL, CXL-IC-ring.
2. Eye lasers and applications [1-4]
As shown in Table 1, various eye lasers have been developed for both vision and non-vision treatments. Fig. 1 shows vision technology using various lasers action on the corneal surface, scleral, crystal lens or retinal for various applications. Also shown is the LASIK system using a flying spot scanning method and the solid state UV-213 nm system (patented by Lin, 1992).
3. Recent developments and new trends
New laser technology trends shall include: high-power and pulsed cross linking and its new applications; endo-assisted glaucoma and non-invasive-presbyopia lasers; femto-laser new applications (for cataracts, lens surface ablation), new solid-state UV-laser for LASIK, and multi-wavelength systems for multi-applications (such as 3-wavelength, green, blue and yellow lasers for retina, AMD treatments).
CXL systems have been commercialized for years for various corneal deceases such as corneal keratoconus, corneal keratitis, corneal ectasia, corneal ulcers, and thin corneas prior to LASIK vision corrections and other potential applications such as the scleral treatment in maligan myopia, scleromalacia and low tension glaucoma. The safety dose (E*) and the efficacy of CXL are governed by the absorption coefficients, concentration, diffusion depth of the riboflavin (RF) solution, the UV light dose, irradiation duration, the cytotoxic energy threshold of endothelial cells and the corneal thickness. Fig. 2 shows a corneal model and the RF concentration profiles inside the stroma collagen. The CXL procedures could be conduced (as shown by Fig. 2) either with epithelium removed (epi-off) with a 0,1% riboflavin-dextran solution or with epithelium intact (epi-on) with a 0,25% riboflavin aqueous solution. The concentration profiles show that the CXL process starts from the depletion f surface RF and takes longer time to deplete the RF inside the stroma. A safety (or maximum) dose is required to protect the endothelial cells from damage, while a minimum dose is also needed to achieve the efficacy. I have developed an analytic formula for the safety (or minimum) concentration (C0
) relating to the corneal thickness (z) and the RF diffusion depth (D) as follows C0
=(1,61-32z)/(517Gz) which is inverse proportional to the corneal thickness and D, where G(z)=1-0,25z/D. Greater detail and the theory of CXL may be found in a separate paper of this Conference Proceeding and other published papers by Lin et al. [10-13].
5. The mathematics of vision corrections [4-9]
The combined technologies of scanning laser, eye tracking, topography and wavefront sensor advance the corneal reshaping (the refractive surgery) one step further from the conventional ablation of spherical surface to the customized ablation of aspherical surface. Therefore, the theory (or mathematics) behind LASIK is also expanded from the simple paraxial formula to the high-order nonlinear formulas involving the change of the corneal asphericity and the LASIK-induced surface aberrations. Most of the existing LASIK monograms are based on spherical corneal surface. The customized nomograms require aspherical surface in order to minimize the optical aberrations.
The refractive error and AIOL [8, 9]
As shown by Fig. 3 for an eye model, the refractive error may be calculated by Lin’s Effective Eye Model (EYM, Lin, 2004)
D= (1336/X–Dc/Z – P’)Z2 , (2)
with Z= (1-SDc/1336). Dc and P’ are the corneal and lens power. For emmetropia (D=0), X=FZ, that is the focal point matching the retina position. The axial length L=S+X+aT, with a=0,045 and T=4,0 mm (lens thickness) and S=ACD=gT, with ADC being the anterior chamber depth (Lin, 2004).
Fig. 4 shows the dual-optics accommodative intraocular lens (AIOL) for the correction of presbyopia, where the red dot indicates the position of the image which could be accommodated for both near and far vision via the translation of the dual AIOL. The total accommodation amplitude (A) may be expressed by9
A = M1(dS1) + M2(dS2) (3)
where dS1 and dS2 represent, respectively, the amount of axial movement of the front and back optics.
LASIK refractive surgery 
In LASIK procedure, the refractive power change is defined by the difference of the preoperative (R) and postoperative (R') front surface radius of the cornea, given by D=377(1/R–1/R'), where D in diopter (or 1/m) and R and R' in mm. Therefore, myopia (D<0), R'>R and hyperipia (D>0), R'<R. For examples, for a preoperative R=7,7 mm, D=(-1; -5; -10; +1; +5; +10), for R’=(8,0; 8,6; 9,7; 7,4; 7,0; 6,4) mm. It should be noted that in LASIK procedure the change on the corneal (front) surface represents the refractive errors of the treated subject which is normally measured by a spectacle power, Ds, (at a typical vertex distance of V=12 mm) related to D (or the contact-lens power) by D= Ds/ [1-V Ds]. For examples, Ds=(-10; -5; +5; +10), for D=(-8,9; -4,7;+5,3;+11,4).
In a spherical corneal surface, the central ablation depth (by LASIK for myopia correction) of single-zone method is given by³ H0=-(DW²/3)(1+C), with C=0,19 (W/R)2 being the high-order term, W is the ablation zone (or diameter). Most LASIK systems use a multi-zone method, for example, in a 3-zone nomogram, H0 is revised as H(3-zone) = f H0(single-zone), where the reduction factor f=(0,70 to 0,85) depending on the algorithms defining the power and radius of each zone. For example, comparing to a single zone with W=6,5 mm, a multizone depth will reduces to 71,6 % (or f=0,716) when a smaller inner zone of 5,5 mm is used.
LASIK procedure speed
By defining T*=T/D, or the procedure time (in seconds) per diopter correction (D), one may obtain the following scaling law (Lin, 2007): T * ~ W²/(AHPR²), where W is ablation effective zone diameter, A is the ablation rate (um/pulse), H is laser repetition rate, P is laser power (in W), and R is the laser spot size (radius). The laser fluence is defined by the laser energy/pulse per unit area F=E/(πR²). The following examples may be obtained from above equation. For a typical system parameters of W=6,0 mm, H=100 Hz, P=100 mW, E=1,0 mJ/pulse and spot size of R=1,0 mm (diameter) and ablation rate of A=0,5 microns/pulse, we define a typical T*=5,0 seconds in myopia correction. T*=(2,5; 10) seconds for H=(200, 50) Hz. Therefore for H<100 Hz, a larger spot size of R>1.2 mm would be needed for reasonable T*. For fixed (A,P,H,W), T* ~ R², therefore T*=(20; 13,9; 3,47) seconds, for R=(0,5; 0,6; 1,2) mm. This is the major reason that a small spot system such as a diode-pumped laser system made by CustomVis having a small energy/pulse about 1,0 mJ and spot size of 0,6 mm, requires a very high repetition rate of H>500Hz. On the other hand, for lower H<100 Hz, larger spot of >1,2 mm is needed.
Optical aberration 
The human eye typically has a negative Q for corneal surface and positive Q for the lens surface, in which whole eye optical aberration may be partially balanced by these two opposite components, particularly in young eyes. The shape factor (p) is related to the asphericity (Q) by p=Q+1, where Q=0 (or p=1) representing a spherical surface. It was well known that the shape factor or Q increases after myopic-LASIK and decreases after hyperopic-LASIK. The amount of these changes also an increasing function of the power of corrections. For examples, Defining: Q1=pre-LASIK, Q2=post-LASIK, for a myopia -5,0 D correction, Q2=1,23(Q1+1)-1. Therefore, an initial Q1=-0,1 (prolate) will result in Q2=+0,11 (oblate); and initial Q1=-0,4 result in Q2=-0,26 (less prolate). Optical aberration may be defined by the Zernike polynomials.
It was known that the prime spherical aberration (PSA) contributed from the lens is normally negative, whereas cornea has positive contribution and has larger value than that of lens. Therefore, the PSA of the whole eye in general is positive and may be expressed as follows
W (whole eye) = W'(cornea) + W"(lens),
For typical mean value of W"=-0,026 (range -0,015 to -0,04) and W' mean of +0,032 (range of 0,02 to 0,04), one expects W(whole eye) is a positive mean value of 0,005 (range of 0,002 to 0,02) depending on the shape of the cornea and lens which are also age dependent. As reported by Smith et al that lens has negative SA, however, it is not clear what the causes are. They proposed three sources of SA: the front, back surface asphericity and the bulk refractive index distribution in addition to the age-related factors. Therefore, customized change (or control) of the corneal asphericity for minimal SA depends on the individual lenticular SA (W"). For higher negative W", smaller Q* (of the cornea) would be needed as shown by Eq.(25). This optimal value (Q*) for minimal whole eye SA also depends on the corneal front surface radius (R) and the contribution from its posterior surface which has a typical value about -0,6.
This review paper for vision corrections and the related mathematics may serve as the guidance for future improvements, clinically and theoretically, particularly for the CXL procedure which is still in its early stage requiring more human data and basic studies. Greater detail for each of the topic discussed in this paper may be found in the References listed which I have limited ot my own published work due to the page limitation of this Proceeding.