Dispersal and species’ responses to climate change
Summary (3 min read)
1. Introduction
- Semiconductor lasers are sensitive to feedback effects, which occur when a portion of the emitted light re-enters the active laser cavity [1,2].
- Techniques to improve feedback immunity include surface relief gratings [3], increased cavity mirror reflectivity [4,5], and optical isolators.
- Numerical solution of this equation provides important insight into the way a self-mixing sensor functions.
- Lasers under feedback exhibit complicated phenomena, including hysteresis effects and the presence of multiple possible laser oscillation modes, some of which are unstable [28].
- Because of this, a series of considered steps must be followed to obtain physically meaningful solutions from the excess-phase equation.
2. Theory
- The self-mixing model developed here is based on the work by Petermann [30].
- The authors then provide a description of important considerations and relevant additional equations.
- The laser’s terminal voltage also varies in response to feedback [41] providing an alternative signal source that contains equivalent information to the optical power signal [13,42].
- This approximation often holds in practice, especially when light is scattered from rough surfaces (e.g., [45] reports an intensity ratio of 10−5).
A. Weak Feedback, C ≤ 1
- When the feedback parameter is less than or equal to one, the right-hand side (r.h.s.) of Eq. (6) is monotonic, and a unique solution can be found for ϕ.
- The solution can be found in a robust manner using a bounded root finding algorithm, such as bisection, between known bounds that the authors denote ϕmin and ϕmax.
- The bounds are obtained by considering the periodicity of the sine function that has a maximum value of 1 andminimum value of −1.
- Therefore ϕmin is found by substituting 1 for the sine function in Eq. (6), and ϕmax by substituting −1.
B. Moderate/Strong Feedback, C > 1
- When the feedback parameter is greater than one, there may be multiple solutions that satisfy Eq. (6).
- Figure 3 shows ϕmin and ϕmax corresponding to the trough and peak locations for the left-most solution.
- Solutions corresponding to possible lasing modes are indicated by circles and crosses in Fig. 3, but the modes indicated with crosses are not stable; thus their solutions are not contained here.
- The modal stability also can be formally verified though linear stability analysis [28].).
- The next step is to find the possible values of m where a valid solution exists.
1. Lower Bound
- The authors obtain mlower by finding the left-most peak position that is greater than zero.
- This is obtained by substituting the equation for the peak from Eq. (11) into Eq. (6) and rearranging form.
2. Upper Bound
- The authors obtain mupper in a similar fashion to the lower bound.
- It is obtained by finding the right-most trough position that is less than zero.
C. Path Dependence (Hysteresis)
- Initially, any valid solution interval could contain ϕ.
- Moreover, with a periodic stimulus (such as harmonic motion), ϕ will behave periodically from the second period onward.
- These solutions are plotted using the thin lines in Fig. 5(b); the thick lines in the figure will be used to illustrate the path dependence of the solutions.
- The path of solutions to the excess-phase equation are then plotted in Fig. 5(b).
- Relating these assumptions on the laser behavior to the equations derived above, if a solution previously existed for a given integer value of m, the solution will remain in the same region until m falls outside of mlower;mupper .
E. Pseudocode for Solving Self-Mixing Equations
- The values at the bounds must differ in sign.
- Algorithm 2 presents pseudocode to synthesize self-mixing signal waveforms for given round-trip phase samples in vector ϕ0, which demonstrates how to use the selmixpower function from Algorithm 1.
4. Applications
- This section demonstrates how to apply Algorithms 1 and 2 developed in the previous section to typical self-mixing sensor configurations.
- The examples presented aim to provide a starting point for self-mixing laser modeling that can be extended to other selfmixing sensor applications with little effort.
A. Target Displacement
- Measuring target displacement is an application often presented in the published self-mixing literature, as it is easy to understand andmakes use of simple stimuli.
- This section will describe how to generate synthetic self-mixing target displacement signals.
- This process permits the parameters associated with experimental signals to be estimated, such as the feedback parameter C [36].
- The authors can observe trends in the evolution of the self-mixing signal for increasing values of the feedback parameter C; the hysteresis effect becomes greater, waveform asymmetry becomes more pronounced, and the number of fringes is decreased.
B. Absolute Distance
- The distance to a fixed target can be obtained from the self-mixing sensor by frequency modulating the laser.
- The frequency modulation is typically achieved by modulating the laser bias current with an ostensibly triangular waveform [6].
- Δν tn ΔFTri tn ; (18) where Tri represents a triangle function, and ΔF is the frequency modulation coefficient.
- MATLAB code implementing the algorithm appears in absolute_distance.m with the resulting signal plotted in Fig. 8 for a target distance of 24 mm and a laser frequency sweep over a range of 46 GHz with a base frequency of ν00 c∕ 845 nm .
- The authors can perform a rough check on the synthetic signal using the result derived by Beheim and Fritsch [6]: L Nf c∕ 2ΔF with an uncertainty of c∕ 2ΔF where.
C. Absolute Distance and Velocity
- It is also possible to consider the previous absolute distance measurement with a target in motion.
- Substituting Eq. (23) into Eq. (21), the authors obtain the expression for the external phase, ϕ0 tn 4π L0 vtn 1 λ0 ΔFTri tn c : (24) The inclusion of the velocity term accounts for the Doppler effect caused by the target motion.
- The self-mixing sensor can also provide useful signals for nondeterministic stimuli, which is of interest, for example, for sensing the velocity of a rough moving target [12,45].
- The stimulus signal time samples, ψ tn , are in the complex domain and represent the amplitude and phase fluctuations of the stimulus.
- Nevertheless, this procedure provides means for extracting parameter values, such as the feedback parameter, C, by fitting experimentally acquired signals to the synthetic velocimetry signal.
E. Other Applications
- The procedure for solving the excess-phase equation presented in this article also can be applied to a range of other applications where various parameters change over time.
- One example is the selfmixing imaging sensor where an optical chopper is used to modulate the self-mixing signal [13].
- The chopper can be modeled by considering two states: the chopper obstructing the beam, and the chopper allowing the beam to pass through.
- These states would correspond to changing the feedback parameter (when the chopper obstructs the beam, the feedback would be reduced) and the optical path length (which would be equal to the distance between the laser and the chopper when the beam is obstructed by the chopper).
- Again, Algorithm 2 can be used to generate the synthetic signal for this case if the time series of the round-trip phase and feedback parameters are provided.
5. Conclusions
- This article presents a simple, systematic method for solving the excess-phase equation numerically to generate synthetic self-mixing signals for a range of feedback levels.
- The ability to synthesize self-mixing sensor signals can provide insight into the operation and performance of self-mixing sensors under different experimental conditions.
- Moreover, such synthetic signals can be fitted to experimentally observed signals, enabling the extraction of independent experimental system parameters.
Did you find this useful? Give us your feedback
Citations
493 citations
475 citations
291 citations
199 citations
Cites background from "Dispersal and species’ responses to..."
...Habitat fragmentation, climate change and their interactions create new evolutionary pressures on dispersal behaviour by altering its cost–benefit balance (Kokko & Lopez-Sepulcre 2006; Berg et al. 2010; Le Galliard et al. 2012a; Baguette et al. 2013; Travis et al. 2013)....
[...]
...…on the shape and strength of selection on dispersal is now particularly relevant because organisms are facing new selective pressures on dispersal due to habitat fragmentation (Baguette et al. 2012) and climate change (Le Galliard et al. © 2014 John Wiley & Sons Ltd/CNRS 2012a; Travis et al. 2013)....
[...]
185 citations
Cites background from "Dispersal and species’ responses to..."
...…are however a handful of examples looking at the evolution of settlement behaviours (typically probability of settling) based on habitat selection (Stamps, Krishnan & Reid, 2005), mate finding (Shaw & Kokko, 2014, 2015), prey (Travis et al., 2013a) and conspecific density (Poethke et al., 2011b)....
[...]
...…rules that may depend, for example, on environmental conditions, local density of conspecifics (McPeek & Holt, 1992; Travis, Murrell & Dytham, 1999; Poethke & Hovestadt, 2002; Kun & Scheuring, 2006), and local density of prey/parasites/predators (Travis et al., 2013a; Iritani & Iwasa, 2014)....
[...]
...Dispersal has a central role in life history and its evolution is fundamental in determining the consequences of land use change, habitat degradation, and climate change for species persistence, or a species’ invasive potential (Clobert et al., 2012; Travis et al., 2013a)....
[...]
References
7,089 citations
5,076 citations
"Dispersal and species’ responses to..." refers background in this paper
...Sy nt he si s Elith and Leathwick 2009, Dawson et al. 2011, McMahon et al. 2011, Bellard et al. 2012, Bocedi et al. 2012, Schurr et al. 2012)....
[...]
3,986 citations
2,834 citations
Related Papers (5)
Frequently Asked Questions (18)
Q2. What are the common models of dispersal?
Most connectivity models consider spatial dispersal processes as a simple function of distance instead of considering the dynamics of emigration, movement between patches and settlement decisions that together result in colonisation.
Q3. What is the impact of climate change on dispersal at a population level?
Climate change will also affect other aspects of life history such as fecundity and mortality, which will determine the effectiveness of dispersal at a population level.
Q4. What is the importance of dispersal-informed modelling?
An increased understanding of dispersal under climate change is critical to inform the deployment of effective climate change resilient conservation strategies.
Q5. What is the role of dispersal in the evolution of species’ ranges?
While the same selective forces that act on dispersal in stationary ranges, including kin competition and inbreeding depression, may still play a role at expanding margins, selection will now favour dispersal strategies which maximise the likelihood that some descendants follow the expanding margin (Travis et al.
Q6. What are the potential interventions for dispersal?
Recognition that contemporary conservation needs to facilitate the shifting of species’ biogeographic ranges and promote local adaptation has resulted in a number of potential interventions being suggested, including landscape management, assisted colonisation and genetic reinforcement (assisted adaptation) (Loss et al. 2011).
Q7. What is the method used to establish bounds of uncertainty in species’ range changes?
An initial method used to establish bounds of uncertainty in species’ range changes has been to run models assuming that species exhibit either unlimited or no dispersal.
Q8. What is the impact of climate change on the dispersal process?
Predicted impacts of climate change on means and variabilities of temperatures, rainfall, storm events, wind speed, snow and ice cover, CO2 concentrations, etc. (IPCC 2007) could affect the dispersal process directly, and also indirectly by changing the biophysical environment (e.g. habitat quality, availability of food resources, etc.) and the state of individuals (body size and morphology, body condition and rate of development).
Q9. What is the effect of climate change on dispersal?
Species which rely on other biota for dispersal, such as seeds carried by ants, will suffer if the phenology of the dispersal agent becomes asynchronous under climate change (Warren et al. 2011).
Q10. What is the common approach for including dispersal in species’ distribution models?
A more sophisticated and increasingly used approach for including dispersal in predictive species’ distribution models is to fit a statistical function (i.e. dispersal kernels) to observed dispersal data (Pagel and Schurr 2012, Schurr et al. 2012).
Q11. Why is dispersal distances a variable property of a species?
Because different internal (e.g. individual condition, sex) and external (e.g. the local environment) factors can alter individual dispersal processes (Clobert et al. 2009), the distribution of dispersal distances is unlikely to be a fixed property of a species.
Q12. What is the importance of a mechanistic approach?
a mechanistic approach also reduces the requirement for direct measurements of the rare long distance dispersal events that have a disproportionate impact on rates of spread (Neubert and Caswell 2000, Clark et al. 2001).
Q13. How can the authors predict species’ future distributions?
By using the outputs of global climate models, these so-called habitat suitability models can project species distributions onto future climatic conditions.
Q14. What is the way to incorporate dispersal in a species’ distribution model?
A simple way to incorporate dispersal has been to couple habitat suitability models with colonisation models that are based on nearest-neighbour dispersal whereby landscape grid cells that become climatically suitable can be colonised if a neighbouring cell is already occupied (Midgley et al.
Q15. What are the main characteristics of dispersal kernels?
Dispersal kernels(Bartoń et al. 2012); and foster investment in dispersal traits at the expense of other life-history attributes (Burton et al. 2010).
Q16. What is the impact of temperature on dispersal distance?
In one recent case that highlights a further potential complexity, the impact of temperature on dispersal distance was shown to interact with the degree of habitat fragmentation (Delattre et al. 2013): dispersal distance was greater at lower temperatures in fragmented landscape while, in more continuous landscapes, dispersal distance was greater under warmer conditions.
Q17. What is the scope of this perspectives article?
The conceptual scope of this perspectives article is therefore purposely broad, covering a number of topics such as observed ecological and evolutionary patterns, theory, models and conservation.
Q18. What is the role of the eutrophication resistance in the prediction of climate impacts?
evolution of egg desiccation resistance has been incorporated in a biophysical model to predict climate impacts on the range of the dengue fever vector Aedes aegypti (Kearney et al. 2009).