Ultrashort Laser Pulses I
Description of pulses
Intensity and phase
The instantaneous frequency and group delay
Zero th and first-order phase
Prof. Rick Trebino Georgia Tech The linearly chirped Gaussian pulse www.frog.gatech.edu An ultrashort laser )
t I( t ) (
pulse has an intensity X and phase vs. time.
Neglecting the spatial dependence for field Electric now, the pulse electric field is given by: Time [fs] X ∝1 ω −φ + (t )2 I( t ) exp{ it [0 (t) ]} cc. .
Intensity Carrier Phase frequency
A sharply peaked function for the intensity yields an ultrashort pulse. The phase tells us the color evolution of the pulse in time. The real and complex )
t I( t ) pulse amplitudes ( X X
Removing the 1/2, the c.c., and the exponential factor with the carrier frequency yields the complex
amplitude, E(t), of the pulse: field Electric
E(t )∝I( t ) exp{ − i φ(t ) } Time [fs]
This removes the rapidly varying part of the pulse electric field and yields a complex quantity, which is actually easier to calculate with.
I( t ) is often called the real amplitude, A(t), of the pulse. The Gaussian pulse
For almost all calculations, a good first approximation for any ultrashort pulse is the Gaussian pulse (with zero phase). = − τ 2 EtE( )0 exp ( t /HW 1/ e ) = − τ 2 E0 exp 2ln 2( t /FWHM ) = − τ 2 E0 exp 1.38( t /FWHM )
τ τ where HW1/e is the field half-width-half-maximum, and FWHM is the intensity full-width-half-maximum.
2 The intensity is: ∝ − τ 2 ItE()0 exp 4ln2(/ t FWHM ) ∝2 − τ 2 E0 exp 2.76( t /FWHM ) Intensity vs. amplitude
The phase of this pulse is constant, φ(t) = 0, and is not plotted here.
The intensity of a Gaussian pulse is √2 shorter than its real amplitude. This factor varies from pulse shape to pulse shape. Calculating the intensity and the phase
It’s easy to go back and forth between the electric field and the intensity and phase: Also, we’ll stop writing “proportional The intensity: to” in these expressions and take E, I(t) = |E(t)|2 X, I, and S to be the field, intensity, and spectrum dimensionless shapes vs. time. The phase:
Im[E ( t )] E(ti) φ(t ) = − arcta n Im Re[E ( t )] −φ (ti) Equivalently, Re φ(t) = − Im{ln[ E(t)]} The Fourier Transform
To think about ultrashort laser pulses, the Fourier Transform is essential.
∞ ω XX(ω )= (t ) exp( − i t ) dt % ∫−∞
1 ∞ XX(t )= % ( ω ωω ) exp( itd ) 2π ∫−∞
We always perform Fourier transforms on the real or complex pulse electric field, and not the intensity, unless otherwise specified. The frequency-domain electric field
The frequency-domain equivalents of the intensity and phase are the spectrum and spectral phase . Fourier-transforming the pulse electric field:
X =1 ω −φ + (t )2 I( t ) exp{ it [0 (t) ]} cc. .
yields: Note that φ and ϕ are different!
X ω =1 ω −ω −ϕ ω −ω + %( )2 S( 0 ) exp{i [(0 ) ]} 1 −−ω ϕω + −−ω ω 2 S(0 ) exp{ i [ ( 0 )] }
Note that these two terms are not complex The frequency-domain electric field has conjugates of each other positive- and negative-frequency components. because the FT integral is the same for each! The complex frequency-domain pulse field
Since the negative-frequency component contains the same infor- mation as the positive-frequency component, we usually neglect it.
We also center the pulse on its actual frequency, not zero.
So the most commonly used complex frequency-domain pulse field is:
X%(ω )≡S(ω ) exp{ − iϕ(ω ) }
Thus, the frequency-domain electric field also has an intensity and phase. S is the spectrum, and ϕ is the spectral phase. The spectrum with and without the carrier frequency
Fourier transforming X (t) and E(t) yields different functions.
E%(ω ) X%(ω )
We usually use just this component. The spectrum and spectral phase
The spectrum and spectral phase are obtained from the frequency-domain field the same way the intensity and phase are from the time-domain electric field.
2 S(ω) = X%(ω)
Im[X% (ω )] ϕ ω( ) = − arctan Re[ X%(ω )] or ϕ ω( ) = − Im{ ln[ X%(ω )] } Intensity and phase of a Gaussian
The Gaussian is real, so its phase is zero.
Intensity and Phase
Time domain:
A Gaussian transforms to a Gaussian Spectrum and Spectral Phase
Frequency domain:
So the spectral phase is zero, too. The spectral phase of a time-shifted pulse
Recall the Shift Theorem: Y {fta(− )} = exp( − iaF ω ) ( )
Intensity and Phase
Time-shifted Gaussian pulse (with a flat phase):
Spectrum and Spectral Phase So a time-shift simply adds some linear spectral phase to the pulse! What is the spectral phase?
The spectral phase is the phase of each frequency in the wave-form.
All of these ω frequencies have 1 zero phase. So ω this pulse has: 2 ϕ(ω) = 0 ω 3 Note that this ω wave-form sees 4 constructive interference, and ω 5 hence peaks, at t = 0 . ω 6 And it has cancellation 0 t everywhere else. Now try a linear spectral phase: ϕ(ω) = aω. By the Shift Theorem, a linear spectral phase is just a delay in time. And this is what occurs!
ϕ ω ( 1) = 0 ϕ ω π ( 2) = 0.2 ϕ ω π ( 3) = 0.4 ϕ ω π ( 4) = 0.6 ϕ ω π ( 5) = 0.8 ϕ ω π ( 6) =
t Transforming between wavelength and frequency The spectrum and spectral phase vs. frequency differ from the spectrum and spectral phase vs. wavelength. π ω = 2 c The spectral phase is ϕ λ = ϕ π λ λ easily transformed: λ ( ) ω (2 c / )
To transform the spectrum, note that the energy is the same, whether we integrate the spectrum over frequency or wavelength: ∞ ∞
ω Sdλ ω ω()λ λ = Sd () ∫−∞ ∫ −∞
−∞ ∞ Changing −2π c 2π c = λπ λ = ω (2π / λ) λ Sω (2 c / ) 2 d S c 2 d variables: ∫∞ λ ∫−∞ λ π d πω −2 c λ = π λ 2 c = ⇒ Sλ ( ) Sω (2 c / ) 2 d λλ 2 λ The spectrum and spectral phase vs. wavelength and frequency
Example: A Gaussian spectrum with a linear spectral phase vs. frequency
vs. Frequency vs. Wavelength
Note the different shapes of the spectrum and spectral phase when plotted vs. wavelength and frequency. Bandwidth in various units
In frequency , by the Uncertainty Principle, a 1-ps pulse has bandwidth:
δν = ~1/2 THz using δν δt ∼ ½
In wave numbers (cm-1), we can write: c ν = ⇒ δν λδ = c (1/ ) ⇒ δν λδ (1/ )= / c λ
So δ(1/ λ) = (0.5 × 10 12 /s) / (3 × 10 10 cm/s) or: δ(1/ λ) = 17 cm-1
−1 In wavelength : δλ λ δ (1/ ) = ⇒ δλ λ λδ = 2 (1/ ) λ 2
Assuming an δλ = (800nm)(.8× 10−4 cm)(17 cm − 1 ) 800-nm wavelength: or: δλ = 1 nm The Instantaneous frequency
The temporal phase, φ(t), contains frequency-vs.-time information. ω The pulse instantaneous angular frequency , inst (t), is defined as: dφ ωω (t ) ≡ − inst 0 dt
This is easy to see. At some time, t, consider the total phase of the wave. Call this quantity φ : 0 φ ω φ = − 0 0 t( t )
Exactly one period, T, later, the total phase will (by definition) increase to φ + 2 π: 0 φ πφ ω + = ⋅+− + 02 0 []()tT tT where φ(t+T ) is the slowly varying phase at the time, t+T . Subtracting these two equations: φπ φ ω = − +− 20T [( tT ) ()] t Instantaneous frequency (cont’d)
π ω Dividing by T and recognizing that 2 /T is a frequency, call it inst (t):
ω π ω φ φ inst (t) = 2 /T = 0 – [ (t+T ) – (t)]/ T
But T is small, so [φ(t+T ) – φ(t)]/T is the derivative, dφ /dt .
So we’re done!
Usually, however, we’ll think in terms of the instantaneous ν π frequency , inst (t), so we’ll need to divide by 2 :
ν ν φ π inst (t) = 0 –(d /dt ) / 2
While the instantaneous frequency isn’t always a rigorous quantity, it’s fine for ultrashort pulses, which have broad bandwidths. Group delay
While the temporal phase contains frequency-vs.-time information, the spectral phase contains time-vs.-frequency information.
(ω) So we can define the group delay vs. frequency , tgr , given by:
(ω) ϕ / ω tgr = d d
A similar derivation to that for the instantaneous frequency can show that this definition is reasonable.
Also, we’ll typically use this result, which is a real time (the rad’s cancel out), and never dϕ/dν, which isn’t.
(ω) not ω Always remember that tgr is the inverse of inst (t). Phase wrapping and unwrapping
Technically, the phase ranges from –π to π. But it often helps to “unwrap” it. This involves adding or subtracting 2π whenever there’s a 2π phase jump.
Example: a pulse with quadratic phase Note the scale!
Wrapped phase Unwrapped phase
The main reason for unwrapping the phase is aesthetics. Phase-blanking ω E( i) When the intensity is zero, the phase is Im meaningless. When the intensity is nearly zero, the −φ ω ( i) phase is nearly meaningless. Re Phase-blanking involves simply not plotting the phase when the intensity is close to zero.
Without phase blanking With phase blanking
Time or Frequency Time or Frequency
The only problem with phase-blanking is that you have to decide the intensity level below which the phase is meaningless. Phase Taylor Series expansions
We can write a Taylor series for the phase, φ(t), about the time t = 0:
t t 2 φ φ φ φ (t )=+ + + ... 0 11! 2 2! where φ φ = d 1 is related to the instantaneous frequency. dt t=0
where only the first few terms are typically required to describe well- behaved pulses. Of course, we’ll consider badly behaved pulses, which have higher-order terms in φ(t). Expanding the phase in time is not common because it’s hard to measure the intensity vs. time, so we’d have to expand it, too. Frequency-domain phase expansion
It’s more common to write a Taylor series for ϕ(ω):
ω ω− ()ω ω− 2 ϕϕ ϕ ω ϕ ( )=+0 +0 + ... 0 11! 2 2!
where ϕ ϕ = d 1 is the group delay! dω ω ω= 0 d 2ϕ ϕ = is called the group-delay dispersion . 2 ω 2 d ω ω= 0
As in the time domain, only the first few terms are typically required to describe well-behaved pulses. Of course, we’ll consider badly behaved pulses, which have higher-order terms in ϕ(ω). Zeroth -order phase: the absolute phase
The absolute phase is the same in both the time and frequency domains. φ ⊃ ω φ f (t)exp( i 0 ) F( )exp( i 0 ) An absolute phase of π/2 will turn a cosine carrier wave into a sine. It’s usually irrelevant, unless the pulse is only a cycle or so long.
Different absolute phases Different absolute phases for a four-cycle pulse for a single-cycle pulse
Notice that the two four-cycle pulses look alike, but the three single- cycle pulses are all quite different. First-order phase in frequency: a shift in time − ϕ ⊃ ω ωϕ By the Fourier-transform Shift Theorem, f (t 1) F ( )exp( i 1)
Time domain Frequency domain
ϕ = 1 0
ϕ = − 1 20 fs
ϕ Note that 1 does not affect the instantaneous frequency, but the ϕ group delay = 1. First-order phase in time: a frequency shift
By the Inverse-Fourier-transform Shift Theorem, φ ω φ − ⊂ − F(1 ) ft ()exp( it 1 )
Time domain Frequency domain
φ = 1 0 / fs
φ = − 1 .07 / fs
φ Note that 1 does not affect the group delay, but it does affect the φ instantaneous frequency = – 1. Second-order phase: the linearly chirped pulse
A pulse can have a frequency that varies in time.
This pulse increases its frequency linearly in time (from red to blue).
In analogy to bird sounds, this pulse is called a chirped pulse. The Linearly Chirped Gaussian Pulse
We can write a linearly chirped Gaussian pulse mathematically as: = − ω β τ 2 ( + 2 ) EtE( )0 exp ( t /G ) exp itt0
Gaussian Carrier Chirp amplitude wave
Note that for β > 0, when t < 0 , the two terms partially cancel, so the phase changes slowly with time (so the frequency is low). And when t > 0 , the terms add, and the phase changes more rapidly (so the frequency is larger). The instantaneous frequency vs. time for a chirped pulse
A chirped pulse has: ∝(ω φ − ) Et( ) exp it0 ( t )
where: βφ (t ) = − t 2 ω φω ≡ − The instantaneous frequency is: inst (t )0 d / dt
which is: ω βω = + inst (t )0 2 t
So the frequency increases linearly with time. The Negatively Chirped Pulse
We have been considering a pulse whose frequency increases linearly with time: a positively chirped pulse.
One can also have a negatively chirped (Gaussian) pulse, whose instantaneous frequency decreases with time.
We simply allow β to be negative in the expression for the pulse:
( )= exp −() ω β /τ 2 exp ( + 2 ) EtE0 tG itt0
And the instantaneous frequency will decrease with time:
ω β ω ω = + = − inst ()t0 2 t 0 2 t