Strehl ratio for optical systems with ultrafast illumination

Weichuan Gao; Tom Milster

doi:10.1364/OE.26.018028

1. Introduction

Due to advances in laser science, ultrafast laser pulses are now routinely generated in the sub-picosecond time scale. Given the advantage of high peak intensity and high temporal resolution, ultrafast lasers have found their way into numerous emerging applications, including ultrafast spectroscopy, multiphoton microscopy, and laser material processing [1–6]. Generally, pulses are delivered to the sample through optical systems, where non-optimized designs result in temporal broadening and poor spatial energy confinement. Effects of material dispersion and aberrations in pulsed source optical systems have been studied extensively using geometrical optics and Fourier analysis for optical components experiencing linear material interactions [7–20]. However, none of the existing analytical approaches has demonstrated both fast and accurate results that can be integrated into analysis and optimization in optical design software.

In this paper, we propose a general method to design and analyze optical systems used for ultrafast illumination. Further, the definition of Strehl ratio is extended to include the effects of the chromatic nature and temporal behavior of ultrafast laser pulses. Several important assumptions are made in the derivation of this method to dramatically increase the efficiency of the computation while maintaining accuracy for most systems, which include:

1. Analyses are performed in the framework of scalar theory. Such a scalar theory is justified, as long as the numerical aperture (NA) of the systems under analysis is not too large (NA < 0.5). As the NA of the system increases beyond 0.5, scalar theory tends to over-estimate the Strehl ratio.
2. Nonlinear material interaction and linear effects from absorption, Fresnel reflection loss and scattering are not considered.
3. Propagation from the entrance pupil to the exit pupil ignores diffraction. Since both entrance pupil and exit pupil are images of the same aperture stop, such assumption is generally accurate unless the pupils are highly aberrated. Large pupil aberrations alter the spatial presence of energy in pupils, such as is characteristic of wide-angle lenses, so derivations in this work are most accurate for optical systems with low pupil aberrations.
4. Frequency-independent path lengths are assumed, and thus temporal dispersion based on angular dispersion, e.g. from prism/grating pairs, requires a separate analysis.
5. Frequency-independent efficiencies are assumed for diffractive surfaces. For systems with diffractive surfaces, frequency-dependent diffraction efficiencies can distort the spectral profiles of the output pulses and introduce pulse broadening.

In Section 2, the definition of Strehl ratio is extended to incorporate chromatic nature and temporal behavior of ultrafast laser pulses. Section 3 revisits the aberration function including 4th order Seidel aberrations and chromatic aberrations up to the second order. Section 4 provides a discussion on the impact of material dispersion and aberrations on Strehl ratio. Section 5 gives examples of real optical system designs and analysis, where the results are compared with physical optics simulations generated in VirtualLab Fusion. Section 6 lists conclusion from this work.

2. Theory

Within the framework of scalar theory, the complex electric field at the entrance pupil of an optical system is expressed as:

U_{E P} (x_{p}, y_{p}; ω) = P_{E P} (x_{p}, y_{p}) A_{E P} (x_{p}, y_{p}; ω) \exp [j ϕ_{E P} (x_{p}, y_{p}; ω)],

where x_p and y_p are normalized pupil coordinates defined at the entrance pupil. P (x_p, y_p) is the spatially varying amplitude transmittance that includes the obscuration of the optical system, if any, A_EP (x_p, y_p; ω) is the spectral amplitude that includes apodization at the entrance pupil, and ϕ_EP (x_p, y_p; ω) is the spectral phase of pulses at the entrance pupil.

As shown in Fig. 1, electric field $U_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; ω)$ at the exit pupil reference sphere is described by:

Fig. 1 Illustration of pupil coordinates and image plane of an optical system.

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U_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; ω) = P_{X P} ({x^{'}}_{p}, {y^{'}}_{p}) A_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; ω) \exp [j ϕ_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; ω)] .

In the case of negligible pupil aberrations, entrance and exit pupil coordinates are related by a constant factor indicating the lateral scaling of the beam as it propagates through the optical system. In terms of normalized pupil coordinates, the scaling factor is 1, and:

{x^{'}}_{p} = x_{p}, {y^{'}}_{p} = y_{p} .

Thus, P_XP = P_EP and A_XP = A_EP. For analytical convenience, transform-limited Gaussian input pulses are assumed, and the temporal amplitude profile is expressed as:

A_{E P} (x_{p}, y_{p}; t) = F_{t}^{- 1} {A_{E P} (x_{p}, y_{p}; ω)} = A_{E P} (x_{p}, y_{p}) \exp [- 2 {(\frac{t}{τ})}^{2}],

where A_EP (x_p, y_p) is the apodization parameter of the input pulses, the subscript t indicates a temporal inverse Fourier transform, and τ is the 1/e pulse duration of the irradiance.

For an optical system with N optical surfaces, the spectral phase at the exit pupil is expressed as:

ϕ_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; ω) = ϕ_{E P} (x_{p}, y_{p}; ω) + \sum_{i = 1}^{N} [k_{i} (ω) \cdot P L_{i} (x_{p}, y_{p})],

where k_i (ω) is the wavenumber and PL_i (x_p, y_p; ω) is the path length the carrier frequency travels between surfaces i and (i + 1). For convenience of analytical expression, input pulses are assumed to exhibit constant ϕ_EP for each frequency component. Within the limited spectral bandwidth of the Gaussian pulse, the wavenumber is expressed as a Taylor series expansion about the carrier frequency ω₀, as:

k (ω) ≃ \frac{ω_{0}}{c} n_{0} [1 + a_{1} Δ ω + a_{2} {(Δ ω)}^{2}],

where Δω = ω - ω₀, n = n(ω₀) and c is the speed of light in vacuum. The a₁ and a₂ coefficients in the Taylor series expansion are

a_{1} = \frac{1}{ω_{0}} + \frac{1}{n_{0}} {\frac{d n}{d ω} |}_{ω = ω_{0}} = \frac{1}{k_{0} v_{g 0}}

and

a_{2} = \frac{1}{ω_{0} n_{0}} {\frac{d n}{d ω} |}_{ω = ω_{0}} + \frac{1}{2 n_{0}} {\frac{d^{2} n}{d ω^{2}} |}_{ω = ω_{0}} = \frac{1}{2 k_{0}} G V D,

where v_g0 is group velocity of carrier frequency and GVD is group velocity dispersion of the medium.

From Eqs. (5)-(8):

ϕ_{X P} (x_{p}, y_{p}; ω) ≃ \frac{ω_{0}}{c} \sum_{i = 1}^{N} {n_{i, 0} [1 + a_{i, 1} Δ ω + a_{i, 2} {(Δ ω)}^{2}] \cdot P L_{i} (x_{p}, y_{p})} .

In order to separate physical effects more clearly, Eq. (9) is rewritten as:

ϕ_{X P} (x_{p}, y_{p}; ω) ≃ α + β (Δ ω) + γ {(Δ ω)}^{2} .

To further simplify the evaluation, without loss of generality, the origin of the time reference is moved to the exit pupil by defining t = 0 when the carrier frequency component of the chief ray passes through the exit pupil. The expression for α becomes:

α = \frac{ω_{0}}{c} \sum_{i = 1}^{N} {n_{i, 0} [P L_{i} (x_{p}, y_{p}) - P L_{i} (0, 0)]},

which has units of radians, indicating the wavefront deviation across the exit pupil. Also,

β = \frac{ω_{0}}{c} \sum_{i = 1}^{N} {n_{i, 0} a_{i, 1} [P L_{i} (x_{p}, y_{p}) - P L_{i} (0, 0)]} = \sum_{i = 1}^{N} [\frac{P L_{i} (x_{p}, y_{p}) - P L_{i} (0, 0)}{v_{g 0, i}}],

which has units of time and represents the pulsefront deviation. Lastly,

γ = \frac{ω_{0}}{c} \sum_{i = 1}^{N} [n_{i, 0} a_{i, 2} P L_{i} (x_{p}, y_{p})] = \frac{1}{2} \sum_{i = 1}^{N} [{GVD}_{i} \cdot P L_{i} (x_{p}, y_{p})] = \frac{1}{2} GDD,

which has units of time squared and represents the group delay dispersion (GDD) of the optical system.

The time domain complex field at the exit pupil reference sphere is calculated by the inverse temporal Fourier transform of Eq. (2). By applying the convolution theorem:

U_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; t) = P (x_{p}, y_{p}) F_{t}^{- 1} {A_{X P} (x_{p}, y_{p}; ω)} \otimes_{t} F_{t}^{- 1} {e x p [j ϕ_{X P} (x_{p}, y_{p}; ω)]},

where

\otimes_{t}

denotes a temporal convolution. From Eq. (10) to (13), the latter inverse Fourier transform in Eq. (14) is expressed as:

F_{t}^{- 1} {e x p [j ϕ_{X P} (x_{p}, y_{p}; ω)]} = \frac{1}{2 \sqrt{- j π γ}} \cdot \exp (j α) \cdot \exp [- j \frac{{(t - β)}^{2}}{4 γ}] .

From Eqs. (4), (14) and (15), an expression for the complex field at the exit pupil reference sphere is written as:

U_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; t) = \frac{1}{2 \sqrt{- j π γ}} P \cdot A_{E P} \cdot \exp (j α) {\exp [- 2 {(\frac{t}{τ})}^{2}] \otimes_{t} \exp [- j \frac{{(t - β)}^{2}}{4 γ}]},

where the temporal convolution inside the braces is simplified to

\exp [- 2 {(\frac{t}{τ})}^{2}] \otimes_{t} \exp [- j \frac{{(t - β)}^{2}}{4 γ}] = \sqrt{\frac{4 π γ τ^{2}}{8 γ + j τ^{2}}} \exp [- 2 \frac{{(t - β)}^{2}}{τ^{2} - 8 j γ}] .

From Eqs. (16) and (17), the complex field at the exit pupil reference sphere is:

U_{X P} ({x^{'}}_{p}, {y^{'}}_{p}; t) = \frac{τ}{τ^{'}} P \cdot A_{E P} \cdot \exp (j α) \exp [- 2 {(\frac{t - β}{τ^{'}})}^{2}],

where

τ^{'} = \sqrt{- j 8 γ + τ^{2}} = \sqrt{- j 4 GDD + τ^{2}}

indicates the broadened pulse duration caused by the 2nd order dispersion of the optical system. The imaginary part of τ’ is related to the linear chirp of the pulse. P, A_EP, α, β and τ’ are all functions of the normalized pupil coordinates x_p and y_p.

In the framework of Fourier optics, the instantaneous point spread function (PSF) $h_{i n s t} (x, y; t)$ at the reference sphere center of curvature is the squared Fraunhofer diffraction pattern of U_XP and is written as:

h_{i n s t} (x, y; t) = {| F_{y}^{- 1} F_{x}^{- 1} {U_{X P} (ξ, η; t)} |}^{2},

with spatial frequencies

ξ = \frac{- x_{p}}{λ_{0} r}

and

η = \frac{- y_{p}}{λ_{0} r}

, where λ₀ is the wavelength of the carrier frequency and r is the radius of the reference sphere. The irradiance at the center of the PSF is obtained by the spatial integral

{| h (0, 0; t) |}^{2} = {| \iint_{ξ η} U_{X P} (ξ, η; t) d ξ d η |}^{2} .

From Eqs. (18) and (20) with

x_{r} = λ_{0} ξ r

and

y_{r} = λ_{0} η r

, irradiance at the center of reference sphere is:

I_{α β γ} (t) = {| \iint_{ξ η} \frac{τ}{τ^{'}} P \cdot A_{E P} \cdot \exp (j α) \exp [- 2 {(\frac{t - β}{τ^{'}})}^{2}] d ξ d η |}^{2} .

For an unaberrated, non-dispersive optical system where α and β are constant, τ = τ’ and γ = 0, the central irradiance becomes:

I_{0} (t) = {| \iint_{ξ η} P \cdot A_{E P} \cdot \exp [- 2 {(\frac{t}{τ})}^{2}] d ξ d η |}^{2},

and the maximum of Eq. (22) is:

I_{0, \max} = {| \iint_{ξ η} P \cdot A_{E P} d ξ d η |}^{2} .

When the optical systems of interest are illuminated with lasers, apodization at the entrance pupil A_EP is often present, and it is an important system consideration affecting focusing quality as shown in Eq. (23).

The classical Strehl ratio for continuous-wave (CW) illumination is the ratio between the central irradiance for an apodized and aberrated wavefront relative to its value for an unapodized and unaberrated wavefront [21,22]. For practical systems of interest in this study, the classical Strehl ratio is defined as:

S R = \frac{{| \iint_{ξ η} P \cdot A_{E P} \cdot \exp (j α) d ξ d η |}^{2}}{{| \iint_{ξ η} P \cdot A_{E P} d ξ d η |}^{2}} .

For well-corrected optical systems, SR ≥ 0.8, and SR is not time dependent. For optical systems illuminated with ultrafast laser pulses, the classical Strehl ratio is no longer sufficient to represent characteristics of the optical system in the spatiotemporal domain. The time-dependent Strehl ratio SR(t) is defined by the ratio between (21) and (23), where:

S R (t) = \frac{I_{α β γ} (t)}{I_{0, \max}} = \frac{{| \iint_{ξ η} \frac{τ}{τ^{'}} P \cdot A_{E P} \cdot \exp (j α) \exp [- 2 {(\frac{t - β}{τ^{'}})}^{2}] d ξ d η |}^{2}}{{| \iint_{ξ η} P \cdot A_{E P} d ξ d η |}^{2}} .

Effects of material dispersion and temporal properties of the pulses are incorporated in the coefficients β, γ, τ and τ’. With $| τ^{'} | \geq τ$ , SR(t) of an optical system is always smaller than or equal to 1. SR(t) = 1 indicates that the peak irradiance is maximized at the focus as the pulse propagates through the optical system. SR(t) = 0 indicates that the central irradiance is zero, where either the energy cancels completely, due to the destructive interference from the pupil or no energy is present at the pupil. As shown in the following sections, maximum SR(t) is typically small (<0.2) for systems that are not properly designed, but performance is greatly improved by optimizing α, β, and γ. It is worth mentioning that optical systems with high classical Strehl ratio (e.g., SR = 1) do not necessarily exhibit high maximum SR(t). Therefore, maximum SR(t) is a measure of an optical design in obtaining high energy density at the image plane when illuminated with ultrafast laser pulses.

3. Aberration function

The aberration function is the optical path difference from the reference sphere to the wavefront [23]. The aberration function for rotationally symmetric optical systems is expressed as:

\begin{matrix} W_{a b} (\overset{⇀}{H}, \overset{⇀}{ρ}) = \frac{α λ}{2 π} = W_{040} {(\overset{⇀}{ρ} \cdot \overset{⇀}{ρ})}^{2} + W_{131} (\overset{⇀}{H} \cdot \overset{⇀}{ρ}) (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) + W_{222} {(\overset{⇀}{H} \cdot \overset{⇀}{ρ})}^{2}, \\ + W_{220} (\overset{⇀}{H} \cdot \overset{⇀}{H}) (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) + W_{311} (\overset{⇀}{H} \cdot \overset{⇀}{H}) (\overset{⇀}{H} \cdot \overset{⇀}{ρ}) \end{matrix}

where the coefficients W₀₄₀, W₁₃₁, W₂₂₂, W₂₂₀ and W₃₁₁ indicate spherical aberration, coma, astigmatism, field curvature and distortion, respectively.

\overset{⇀}{H}

is the normalized field vector and

\overset{⇀}{ρ}

is the normalized aperture vector. W_ab has units of length and is directly proportional to α.

The change of aberration coefficients with wavelength is known as chromatic aberrations. Up to second order, optical systems exhibit chromatic change of focus $δ_{λ} W_{020}$ and chromatic change of magnification $δ_{λ} W_{111}$ . Upon ultrafast illumination, $δ_{λ} W_{020}$ generates delay between the wavefront and the pulsefront with a radially quadratic spatial dependence, and the delay is usually referred to as propagation time difference (PTD) [6, 7, 11]. $δ_{λ} W_{111}$ generates delay with a linear spatial dependence, which is usually referred to as pulsefront tilt (PFT) [4, 5, 7]. In terms of the Seidel sum:

δ_{λ} W_{a b} (\overset{⇀}{H}, \overset{⇀}{ρ}) = δ_{λ} W_{020} (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) + δ_{λ} W_{111} (\overset{⇀}{H} \cdot \overset{⇀}{ρ}) = (PTD + PFT) \cdot c,

where PTD and PFT have units of time and c is the speed of light in vacuum, which translates the temporal delay into spatial units. PTD and PFT describe pulsefront deviation β in the optical system, where a detailed analysis showing their effects on SR(t) is given in Section 4.2.

4. Effects of α, β and γ on the time-dependent Strehl ratio

This section assesses effects of the variables α, β and γ, with uniform illumination A_EP and a carrier wavelength of 800nm. MATLAB programs used in the analysis are given in Code 1 (Ref [24].).

4.1 Non-dispersive optical systems (β = α/ω₀and γ = 0)

Systems consisting of reflective mirrors are non-dispersive systems if dispersion from mirror coatings and the air are negligible. Thus, v_g = c/n₀, β = α/ω₀, γ = 0 and Eq. (25) is simplified to:

S R (t) = \frac{{| \iint_{ξ η} P \cdot \exp (j α) \exp [- 2 {(\frac{t - α / ω_{0}}{τ})}^{2}] d ξ d η |}^{2}}{{| \iint_{ξ η} P d ξ d η |}^{2}} .

Wavefront and pulsefront deviations at the exit pupil are shown in Fig. 2, where the wavefront deviation is characterized by α and the pulsefront deviation is characterized by β. The exit pupil wavefront and pulsefront share the same radius of curvature, since coefficients α and β are linearly related in the absence of dispersion. The wavefront deviates from the reference sphere according to aberrations and is not a function of time. The pulsefront propagates, and its energy distribution in the exit pupil is a function of time.

Fig. 2 Wavefront and pulsefront deviation of a non-dispersive system. Note for non-dispersive optical systems, wavefront and pulsefront share the same radius of curvature when the pulsefront vertex is at the pupil.

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At t = 0, the effect of apodization is weak for pulses with durations of a few tens of femtoseconds or longer, because well-corrected optical systems with low aberrations are only apodized at the beginning and end of the pulses. Examples are shown in Fig. 3, as a 25fs (FWHM = 40nm) input pulse passes through aberrated circular pupils with W₀₄₀ = 2 waves and W₁₃₁ = 2 waves, respectively. As SR(t) peaks near t = 0, its value falls off quickly following the temporal envelope of the input pulse. Apodizations at the exit pupil $P \cdot \exp [- 2 {(\frac{t - α / ω_{0}}{τ})}^{2}]$ are plotted at different times to help visualize impacts from the linearly related α and β. Note that the behavior of SR(t) shown in Fig. 3 is the same for any optical system up to the fourth order in Eq. (26). Classical Strehl ratios are 0.05 and 0.03 for W₀₄₀ = 2 waves and W₁₃₁ = 2 waves, respectively.

Fig. 3 SR(t) and apodization at the exit pupil due to aberrations for 25fs (FWHM = 40nm) pulses centered at 800nm, (a) SR(t) in presence of 2 waves of spherical aberration or 2 waves of coma aberration, (b) apodization at t = 0 and t = 5fs for 2 waves of spherical aberration, (c) apodization at t = 0 and t = 5fs for 2 waves of coma aberration.

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To further demonstrate this effect, the peak values of SR(t) upon ultrafast illumination and classical Strehl ratio with CW illumination are shown in Fig. 4, where the system exhibits defocus W₀₂₀. The two cases exhibit different oscillating trends as the defocus increases. For CW illumination, it is well known that, with an integer number of waves of defocus and an unapodized pupil, the axial irradiance is zero (SR = 0). The oscillation is understood through a Fresnel zone interpretation. For defocus, the radial boundary of each Fresnel zone is defined where the wavefront deviation reaches mπ, where m is a positive integer. In the case of W₀₂₀ = 1 wave, there are two complete Fresnel zones in the exit pupil, as shown in Fig. 4. With CW illumination, light waves from the two zones destructively interfere, and the net contribution is zero on axis. However, when illuminated with ultrafast pulses, the exit pupil is apodized due to the coupling between α and β. As a result, cancellation from two Fresnel zones is no longer complete, and axial irradiance is greater than zero. The effect of apodization increases as pulse duration decreases. With short (3fs) pulses, the oscillatory behavior is no longer present, and the maximum SR(t) decreases monotonically with increasing defocus.

Fig. 4 Comparison of the oscillatory behavior of Strehl ratio versus defocus between CW illumination and ultrafast illumination. With 3fs (FWHM = 400nm) pulse duration, maximum SR(t) decreases monotonically with increasing defocus. Wavefront deviation α, apodization at the exit pupil and the PSFs are plotted for W₀₂₀ = 1.

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4.2 Systems with chromatic aberrations

For rotationally symmetric systems consisting of thin lenses, the chromatic aberration coefficients in Eq. (27) are derived by defining the Abbe number with a modified material parameter representing dispersion at the carrier frequency, where:

V_{\partial λ} = \frac{n_{0} - 1}{ω_{0} {(d n / d ω) |}_{ω = ω_{0}}} = - \frac{n_{0} - 1}{λ_{0} {(d n / d λ) |}_{λ = λ_{0}}} .

The modified Abbe numbers for several materials at typical wavelengths are shown in Table 1. Equation (29) is used for diffractive lenses without further modification by following the work of Sweatt, where a linear relationship between the refractive index and the wavelength is assumed [25].

Table 1. Modified Abbe Number

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The coefficients for chromatic change of focus and chromatic change of magnification are:

δ_{λ} W_{020} = \frac{1}{2} \sum_{i = 1}^{j} (\frac{Φ_{i}}{V_{\partial λ, i}} y_{i}^{2})

and

δ_{λ} W_{111} = \sum_{i = 1}^{j} (\frac{Φ_{i}}{V_{\partial λ, i}} y_{i} {\bar{y}}_{i}),

respectively, where

Φ_{i}

is optical power of the lens at the carrier frequency, and

y_{i}

and

{\bar{y}}_{i}

are marginal ray height and chief ray height, respectively, at each lens.

As shown in Fig. 5, upon the presence of the first order dispersion dn/dω, the pulsefront deviation at the exit pupil decouples from the wavefront deviation. When the system contains no wavefront deviation (α = 0), the pulsefront deviation is expressed as:

Fig. 5 Wavefront deviation and pulsefront deviation at the exit pupil of a system with (a) chromatic change of focus, which is referred to as propagation time difference (PTD) and (b) chromatic change of magnification, which is referred to as pulsefront tilt (PFT).

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β = PTD + PFT= \frac{1}{c} [δ_{λ} W_{020} (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) + δ_{λ} W_{020} (\overset{⇀}{H} \cdot \overset{⇀}{ρ})] .

Numerical simulations are shown in Fig. 6 with 25fs (FWHM = 40nm) input pulses centered at λ = 800nm. The exit pupil exhibits apodization due to pulsefront deviation β. When α is small, the strength of the apodization is related to PV(β)/τ, which is the ratio between the peak-to-valley (PV) value of β and the pulse duration τ [14]. Chromatic change of focus and chromatic change of magnification exhibit different apodization dynamics with respect to time. That is, at a specific time, only certain spatial frequencies in the pupil contribute to the focus, even when no wavefront deviation is present in the system. As a result, the instantaneous PSFs exhibit different spatial distributions. Figure 6(d) shows two instantaneous PSFs at different times when a system exhibits nonzero $δ_{λ} W_{020}$ at the exit pupil. As higher spatial frequency components of the light wave contribute to the focus, the PSF at the later time shows a smaller FWHM, despite having a smaller Strehl ratio. As shown in Figs. 6(f) and 6(g), asymmetric PSFs result from apodization effects of $δ_{λ} W_{111}$ . Note that the behavior of SR(t) and the PSFs shown in Fig. 6 are characteristic of any optical system affected by chromatic aberrations up to the second order in Eq. (27).

Fig. 6 Strehl ratio for system with chromatic aberrations (a) maximum SR(t) versus PV(β)/τ in presence of PTD and PFT, (b) SR(t) with PV(β)/τ = 4, (c)(d) apodization at the exit pupil and instantaneous PSFs at t = 40fs and t = 80fs, respectively. (e) SR(t) with PV(β)/τ = 4, (f)(g) apodization at the exit pupil and instantaneous PSFs at t = 0fs and t = 40fs, respectively.

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4.3 Achromatic systems

When a system is free from wavefront aberrations at the carrier frequency of the pulses (α = 0) as well as chromatic aberrations (β = 0), the system is regarded as well corrected [23,26]. However, with ultrafast illumination, maximum SR(t) decreases due to residual higher order material dispersion (γ ≠ 0) across the exit pupil.

Up to second-order dispersion, the accumulated GVD through the optical system is expressed as γ in Eq. (13) and broadens the pulses, which is referred to as linear chirping.

At the exit pupil, the broadened pulses experience a reduction in peak irradiance described by ${| \frac{τ}{τ^{'}} |}^{2}$ . The broadened pulse also interacts with residual wavefront deviation α and pulsefront deviation β to further reduce the maximum SR(t). Thus, to fully describe the influence of γ, a real optical system must be analyzed. An example analysis is shown in Section 5.

4.4 Rule of thumb for maximum SR(t) > 0.8

In this work, maximum SR(t) is defined by normalizing the central focus irradiance with the non-dispersive and unaberrated peak irradiance and is used here as a metric for evaluating optical designs. Although the computation of SR(t) depends on coupling among α, β and γ, a rule of thumb for maximum SR(t) is generated for each of the factors, separately.

With β = 0 and γ = 0, the amount of wavefront deviation allowed for maximum SR(t) = 0.8 follows the Marechal criterion, where the root-mean-square wavefront deviation (rms(αλ/2π)) across the pupil is 0.07λ. With α = 0 and γ = 0, pulsefront deviation (β) with its PV value equal to the pulse duration (PV(β)/τ = 1) yields maximum SR(t) of 0.8. With α = 0 and β = 0, maximum SR(t) is equal to ${| \frac{τ}{τ^{'}} |}^{2}$ when γ is uniform across the pupil. When γ < 0.1τ², maximum SR(t) = 0.8.

For optical systems in practice, maximum SR(t) > 0.8 is expected only when rms(α) < 0.07λ, PV(β)/τ < 1 and γ < 0.1τ² are satisfied simultaneously.

5. Optical design and analysis

In pursuit of high-peak irradiance at the focal plane, it becomes obvious that wavefront and dispersion properties of the optical system must be optimized at the same time. As a figure of merit to assess focus quality, the SR(t) calculation requires analysis of path length and material dispersion. The path length is obtained accurately through ray-tracing, and the material dispersion is evaluated following the work by O’Shea [27]. Commercial optical design software, such as OpticStudio and CODE V, are capable of computing α, β and γ for any optical system at its exit pupil reference sphere with Eqs. (11), (12), and (13), respectively. With additional knowledge about pulse duration τ, the SR(t) is calculated with Eq. (25). Lens designs and macros used in this section are provided in Code 1 [24].

The analysis is applied to two simple optical systems with 25fs (FWHM = 40fs) input pulses centered at 800nm. As shown in Fig. 7(a), an infinite conjugate parabolic mirror closely resembles a non-dispersive, unaberrated optical system. With no phase modulation at the exit pupil, the wavefront coincides with the reference sphere. As a non-dispersive system, β and γ are both 0 across the exit pupil. A maximum SR(t) = 1 is achieved, indicating that the full potential of ultrafast laser pulses is realized in this design configuration. In comparison, a singlet lens exhibits chromatic change of focus and non-zero second order dispersion, as shown in Fig. 7(b). The maximum SR(t) is greatly reduced from 1 to 0.15. To verify the results from this work, VirtualLab Fusion, a commercial physical optics design software, is used where the fields at the focus are evaluated twice, once with the actual system and the other through the ideal system with no dispersion or aberration. The SR(t) calculated from VirtualLab Fusion shows good consistency ( ± 5%) with the simulations in this work.

Fig. 7 Wavefront deviation α, pulse front deviation β and second order dispersion γ of simple optical systems with 25fs (FWHM = 40fs) input pulses centered at 800nm: (a) a parabolic mirror, (b) N-BK7 aspheric singlet. Maximum SR(t) observes a significant drop from 1 to 0.15 in the presence of pulse front deviation and residual second order dispersion.

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SR(t) also applies to optical system design as a figure of merit. However, from Eq. (25), the SR(t) calculation requires integration of the complex field at the exit pupil reference sphere and additional knowledge about the pulse duration, which can significantly increase the computation time during each optimization cycle. To maximize design efficiency, coefficients α, β and γ are individually minimized in the merit function as described in Code 1 [24].

The wavefront α is typically optimized by standard error functions to minimize wavefront aberrations. The pulsefront β is evaluated and minimized with the chromatic aberrations of the optical system in additional settings of standard merit functions. The second order dispersion γ is evaluated through a user-defined operand/constraint, where the γ experienced by a specific ray is calculated. Design examples are provided in the following paragraphs to show that the collective efforts of minimizing each of the three coefficients are effective in increasing the focusing quality of optical systems used ultrafast illumination.

Achromatic doublets correct chromatic aberrations by minimizing β across the pupil. A hybrid achromat with one diffractive-refractive surface is shown in Fig. 8 [19,28]. As indicated in Table 1, a positive diffractive surface contributes negative chromatic dispersion, which combines with the positive chromatic dispersion from the refractive lens to compensate chromatic change of focus. The maximum SR(t) is 0.64 with 25fs (FWHM = 40fs) input pulses centered at 800nm, which is greatly improved compared with the dispersive system shown in Fig. 7(b). Although a diffractive surface changes diffraction efficiency with wavelength, the analysis shown in Fig. 8 compares well with a more complete VirtualLab simulation that is provided in the Supplementary Material [24]. Systems using diffractive optics with pulse widths much smaller than 25fs (FWHM = 40fs) might show significant error in the SR(t) calculation proposed here, if the pulse broadens due to changes in diffraction efficiency over a wide spectral bandwidth.

Fig. 8 Wavefront and dispersion properties of a hybrid achromat with 25fs (FWHM = 40nm) input pulses centered at 800nm. Without chromatic aberrations, maximum SR(t) is recovered as pulse front deviation is eliminated.

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The SR(t) of an achromatic system is primarily limited by the residual γ across the pupil. Most optical materials exhibit positive GVD in the ultraviolet (UV) to near-infrared (NIR) wavelength region and change sign in the short-wave-infrared (SWIR). GVD for several materials at typical wavelengths are listed in Table 2. With only positive GVD material available, SR(t) for designs in the UV and NIR wavelengths are ultimately limited by γ of the optical system. Although minimizing the path length inside the material and material selection with lower GVD helps to reduce γ, the improvements are limited in practice. Pre-compensation units, such as grating or prism pairs, provide a constant negative γ across the pupil that help to cancel positive γ of the optical system.

Table 2. GVD of Optical Material ( $f s^{2} / m m$ )

View Table | View all tables in this article

For applications in the SWIR or longer wavelengths, materials with positive and negative GVD are available. Optimizing the optical material combination and adjusting path length inside each material significantly reduces residual γ of the overall system. In this range, standard merit functions for optical system optimization are expanded to include material dispersion. User-defined operands offer real-time access to overall dispersion of the system up to 2nd order by tracing rays from the entrance pupil to the exit pupil. Such analysis is also available off axis where systems with extended field of view are optimized. By optimizing α, β and γ in the optical system, time-dependent focusing properties of the system are engineered to meet the needs of a specific application.

One optical system commonly used in nonlinear microscopy and ultrafast laser material processing applications is a microscope objective. As shown in Fig. 9, two custom f = 10mm NA = 0.5 objectives are designed for large field of view of ± 0.75mm. One is designed with a standard merit function and the other is designed with the modified method as described above. Both systems exhibit high performance upon monochromatic illumination at 1550nm with SR = 0.99 on axis and SR = 0.92 at the edge of the field. When the system is illuminated with 50fs (FWHM = 85nm) pulses centered at 1550nm, the system that is optimized with the merit function from this work improves the maximum SR(t) from 0.86 to 0.99 on axis by minimizing residual γ through optimizing materials and lens thicknesses. At the edge of the field, the maximum SR(t) is improved from 0.59 to 0.74 and is primarily limited by the residual chromatic change of magnification (PFT). In practice, the residual chromatic change of magnification in the objective is often pre-compensated through the tube lens or relay lenses located before the objective [29]. This example demonstrates application of SR(t) optimization across the full field of the image plane.

Fig. 9 Wavefront and dispersion properties of a 10mm NA = 0.5 microscope objective design with 50fs (FWHM = 85fs) input pulses centered at 1500nm.

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Maximum SR(t) can be used for sensitivity analysis or tolerance of optical systems with ultrafast illumination. Like the conventional Strehl ratio, SR(t) changes due to defects in the fabrication and misalignment of the optical system. SR(t) also changes due to properties of input laser pulses, including bandwidth variations and the spatiotemporal distortion. Thus, SR(t) provides insight beyond standard tolerance criteria and leads toward more robust optical designs.

6. Discussion and conclusion

A new method for analyzing the focusing properties of optical systems with ultrafast illumination is proposed. The definition of Strehl ratio is extended in the time domain to include the chromatic nature and temporal behavior of ultrafast illumination. Analytical expressions to calculate the complex field at the exit pupil reference sphere are derived, and their relations with instantaneous PSF and SR(t) are given. With the help of commercial optical design software, α, β and γ are calculated for complex optical systems. For the first time, the analysis is extended to the full field of an optical system with a well-defined exit pupil. Compared with algorithms presented in previous literature, this method is faster, because ray-tracing is calculated only once and only is required for the center frequency. Accuracy of the method is quantified by comparing with results from a physical optics propagation algorithm.

Low-order wavefront aberrations and chromatic aberrations in terms of Seidel sum and their impact to SR(t) are studied. A modified Abbe number is defined to characterize chromatic aberrations of systems with ultrafast illumination. PTD and PFT, the two most commonly encountered pulsefront aberrations, are related to chromatic change of focus and chromatic change of magnification, respectively.

The accumulated GVD through the optical system is represented by coefficient γ and reduces SR(t), due to pulse broadening. By understanding of the dispersion properties of an optical system, an optimal dispersion correction strategy can be selected to correct the residual pulsefront aberrations. Particularly in the SWIR wavelength region, pulse broadening is minimized by optimizing the material selection with positive and negative GVD.

Acknowledgments

The authors would like to thank LightTrans for providing the evaluation license of VirtualLab Fusion for this study. The authors would like to thank Zemax, LLC, and Synopsis, Inc. for student licenses of OpticStudio and CODE V, respectively.

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	550nm	800nm	1550nm
N-BK7	18.7	32.2	25.8
N-SF11	7.5	16.3	26.7
Diffractive	−1	−1	−1

	550nm	800nm	1550nm
N-BK7	77.0	44.7	−24.6
N-SF11	360.3	187.5	35.9
CaF₂	44.9	27.9	−0.1

	550nm	800nm	1550nm
N-BK7	18.7	32.2	25.8
N-SF11	7.5	16.3	26.7
Diffractive	−1	−1	−1

	550nm	800nm	1550nm
N-BK7	77.0	44.7	−24.6
N-SF11	360.3	187.5	35.9
CaF₂	44.9	27.9	−0.1

Strehl ratio for optical systems with ultrafast illumination

Abstract

1. Introduction

2. Theory

3. Aberration function

4. Effects of α, β and γ on the time-dependent Strehl ratio

4.1 Non-dispersive optical systems (β = α/ω₀and γ = 0)

4.2 Systems with chromatic aberrations

4.3 Achromatic systems

4.4 Rule of thumb for maximum SR(t) > 0.8

5. Optical design and analysis

6. Discussion and conclusion

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (2)

Equations (32)

Optics Express

Abstract

1. Introduction

2. Theory

3. Aberration function

4. Effects of α, β and γ on the time-dependent Strehl ratio

4.1 Non-dispersive optical systems (β = α/ω0and γ = 0)

4.2 Systems with chromatic aberrations

4.3 Achromatic systems

4.4 Rule of thumb for maximum SR(t) > 0.8

5. Optical design and analysis

6. Discussion and conclusion

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (9)

Tables (2)

Equations (32)

Optics Express

4.1 Non-dispersive optical systems (β = α/ω₀and γ = 0)