Waves

CalcpadCE treats classical wave interference and diffraction as plain superposition problems: each emitter contributes a sinusoidal field, and the worksheet sums them on a 2D grid via $Map, with the result shown as a heat-map straight inside the page.

Two complementary viewpoints recur throughout the category — a general view that maps the full \(\Psi(x,\, y)\) field over the whole region, and a screen view that collapses the same physics onto an intensity profile \(I(x)\) along a distant detector. The water-wave examples introduce the simplest two-source case and add an animated variant where time is swept frame-by-frame; the single-slit diffraction sheets approximate the slit by 7 line sources; and the multi-slit interference sheets scale the same formulation to 2, 4 and 6 coherent emitters.

Wavelength, slit spacing and observation distance are exposed at the top of every sheet, so each worksheet doubles as a sandbox for fringe-spacing and angular-resolution studies.

Waves 2D on Water Interference 🎬

Time-swept animation of two circular surface waves with different amplitudes and frequencies, summed on a 2D grid via $Map and rendered as a sequence of CSS-toggled SVG frames.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja fal na wodzie
#rad
λ = 0.8m
k = (2*π)/λ
r(x; y) = sqrt(x^2 + y^2)
a = 1.6m
v = 2m/s
T = λ/v','f = 1/T|Hz
ω = (2*π)/T','ω_1 = ω','ω_2 = 0.5*ω
Ψ_0 = 0.1*m','Ψ_01 = Ψ_0','Ψ_02 = 1.5*Ψ_0
Δt = 0*T/4
Ψ_2D_1(x; y; t) = Ψ_01*sin(k*r(x - a/2; y) - ω_1*t)
Ψ_2D_2(x; y; t) = Ψ_02*sin(k*r(x + a/2; y) - ω_2*(t + Δt))
Ψ_2D(x; y; t) = Ψ_2D_1(x; y; t) + Ψ_2D_2(x; y; t)
'<!--'PlotHeight = 300','PlotWidth = 300','PlotSVG = 1','PlotPalette = 6','n = 10'-->
#val
'Animation - slowed down by 'kT = 3' times
'<style>[id^="fr-"]{display:none;}</style>
#for i = 1 : n
    '<div id="fr-'i'">
    $Map{Ψ_2D(x; y; 2*i/n*T) @ x = -2m : 2m & y = -2m : 2m}
    '</div>
#loop
'<script>var i='n';var fr=$("#fr-"+i);setInterval(function(){fr.hide();if(++i>'n')i=1;fr=$("#fr-"+i);fr.show();}, '1000*kT*T/n');</script>

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja fal na wodzie

λ = 0.8 m

k = 2 · πλ = 2 · 3.140.8 m = 7.85 m^-1

r ( x; y )  =    √ x² + y²

a = 1.6 m

v = 2 ms

T = λv = 0.8 m2 m ∕ s = 0.4 s , f = 1T = 2.5 Hz

ω = 2 · πT = 2 · 3.140.4 s = 15.71 s^-1 , ω₁ = ω = 15.71 s^-1 , ω₂ = 0.5 · ω = 0.5 · 15.71 s^-1 = 7.85 s^-1

Ψ₀ = 0.1 m , Ψ₀₁ = Ψ₀ = 0.1 m , Ψ₀₂ = 1.5 · Ψ₀ = 1.5 · 0.1 m = 0.15 m

Δt = 0 · T4 = 0 s

Ψ_{2D_1} ( x; y; t )  = Ψ₀₁ · sin(k · r (x − a2; y) − ω₁ · t)

Ψ_{2D_2} ( x; y; t )  = Ψ₀₂ · sin(k · r (x + a2; y) − ω₂ ·  ( t + Δt ) )

Ψ_2D ( x; y; t )  = Ψ_{2D_1}  ( x; y; t )  + Ψ_{2D_2}  ( x; y; t ) 

Animation - slowed down by 3 times

Waves 2D on Water Interference

Static snapshot of two coherent point sources on a water surface: the field \(\Psi(x,\, y)\) is plotted as a heat-map showing the canonical hyperbolic interference fringes.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja fal na wodzie
#rad
λ = 0.2m
k = (2*π)/λ
r(x; y) = sqrt(x^2 + y^2)
a = 0.6m
v = 1m/s
T = λ/v
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*m
t = 0.0s
Δt = 0*T/4
Ψ_2D_1(x; y; t) = Ψ_0*sin(k*r(x - a/2; y) - ω*t)
Ψ_2D_2(x; y; t) = Ψ_0*sin(k*r(x + a/2; y) - ω*(t + Δt))
Ψ_2D(x; y; t) = Ψ_2D_1(x; y; t) + Ψ_2D_2(x; y; t)
PlotHeight = 400','PlotWidth = 400
$Map{Ψ_2D(x; y; t) @ x = -1m : 1m & y = -1m : 1m}
$Map{Ψ_2D(x; y; t)^2 @ x = -1m : 1m & y = -1m : 1m}
$Map{abs(Ψ_2D(x; y; t)) @ x = -1m : 1m & y = -1m : 1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja fal na wodzie

λ = 0.2 m

k = 2 · πλ = 2 · 3.140.2 m = 31.42 m^-1

r ( x; y )  =    √ x² + y²

a = 0.6 m

v = 1 ms

T = λv = 0.2 m1 m ∕ s = 0.2 s

f = 1T = 5 Hz

ω = 2 · πT = 2 · 3.140.2 s = 31.42 s^-1

Ψ₀ = 0.5 m

t = 0 s

Δt = 0 · T4 = 0 s

Ψ_{2D_1} ( x; y; t )  = Ψ₀ · sin(k · r (x − a2; y) − ω · t)

Ψ_{2D_2} ( x; y; t )  = Ψ₀ · sin(k · r (x + a2; y) − ω ·  ( t + Δt ) )

Ψ_2D ( x; y; t )  = Ψ_{2D_1}  ( x; y; t )  + Ψ_{2D_2}  ( x; y; t ) 

PlotHeight = 400 , PlotWidth = 400

Diffraction of Light on Single Slit - General View

Spatial view of single-slit diffraction: the slit of width \(a = 1\, μm\) is approximated by 7 coherent line sources, summed on a 2D grid to reveal the central lobe and side fringes.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Dyfrakcja światła na pojedynczej szczelinie - widok przestrzenny
#rad
λ = 500nm
k = (2*π)/λ
r(x; y) = sqrt(x^2 + y^2)
a = 1μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
'The slit is divided into 7 zones:
Ψ_2D_1(x; y; t) = Ψ_0*sin(k*r(x - 3*a/7; y) - ω*t)
Ψ_2D_2(x; y; t) = Ψ_0*sin(k*r(x - 2*a/7; y) - ω*t)
Ψ_2D_3(x; y; t) = Ψ_0*sin(k*r(x - a/7; y) - ω*t)
Ψ_2D_4(x; y; t) = Ψ_0*sin(k*r(x; y) - ω*t)
Ψ_2D_5(x; y; t) = Ψ_0*sin(k*r(x + 1*a/7; y) - ω*t)
Ψ_2D_6(x; y; t) = Ψ_0*sin(k*r(x + 2*a/7; y) - ω*t)
Ψ_2D_7(x; y; t) = Ψ_0*sin(k*r(x + 3*a/7; y) - ω*t)
Ψ_2D(x; y; t) = Ψ_2D_1(x; y; t) + Ψ_2D_2(x; y; t) + Ψ_2D_3(x; y; t) + Ψ_2D_4(x; y; t) + Ψ_2D_5(x; y; t) + Ψ_2D_6(x; y; t) + Ψ_2D_7(x; y; t)
PlotHeight = 200','PlotWidth = 600','PlotStep = 1
$Map{Ψ_2D(x; y; t)^2 @ x = -1m : 1m & y = 0m : 1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Dyfrakcja światła na pojedynczej szczelinie - widok przestrzenny

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

r ( x; y )  =    √ x² + y²

a = 1 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

The slit is divided into 7 zones:

Ψ_{2D_1} ( x; y; t )  = Ψ₀ · sin(k · r (x − 3 · a7; y) − ω · t)

Ψ_{2D_2} ( x; y; t )  = Ψ₀ · sin(k · r (x − 2 · a7; y) − ω · t)

Ψ_{2D_3} ( x; y; t )  = Ψ₀ · sin(k · r (x − a7; y) − ω · t)

Ψ_{2D_4} ( x; y; t )  = Ψ₀ · sin ( k · r  ( x; y )  − ω · t ) 

Ψ_{2D_5} ( x; y; t )  = Ψ₀ · sin(k · r (x + 1 · a7; y) − ω · t)

Ψ_{2D_6} ( x; y; t )  = Ψ₀ · sin(k · r (x + 2 · a7; y) − ω · t)

Ψ_{2D_7} ( x; y; t )  = Ψ₀ · sin(k · r (x + 3 · a7; y) − ω · t)

Ψ_2D ( x; y; t )  = Ψ_{2D_1}  ( x; y; t )  + Ψ_{2D_2}  ( x; y; t )  + Ψ_{2D_3}  ( x; y; t )  + Ψ_{2D_4}  ( x; y; t )  + Ψ_{2D_5}  ( x; y; t )  + Ψ_{2D_6}  ( x; y; t )  + Ψ_{2D_7}  ( x; y; t ) 

PlotHeight = 200 , PlotWidth = 600 , PlotStep = 1

Diffraction of Light on Single Slit - Screen View

Far-field intensity profile \(I(x)\) for the same 7-zone single-slit model, evaluated on a screen at distance \(D\) via the path-difference \(\Delta r(α) = a\, \sinα\).

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Dyfrakcja światła na pojedynczej szczelinie - widok na ekranie
#rad
λ = 500nm
k = (2*π)/λ
D = 1m
a = 1.5μm
Δr(α) = a*sin(α)
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
α(x) = atan(x/D)
'The slit is divided into 7 zones:'
Ψ_2D_1(x; t) = Ψ_0*cos(k*(-3/7)*Δr(α(x)) - ω*t)
Ψ_2D_2(x; t) = Ψ_0*cos(k*(-2/7)*Δr(α(x)) - ω*t)
Ψ_2D_3(x; t) = Ψ_0*cos(k*(-1/7)*Δr(α(x)) - ω*t)
Ψ_2D_4(x; t) = Ψ_0*cos(k*0*Δr(α(x)/7) - ω*t)
Ψ_2D_5(x; t) = Ψ_0*cos(k*(1/7)*Δr(α(x)/7) - ω*t)
Ψ_2D_6(x; t) = Ψ_0*cos(k*(2/7)*Δr(α(x)/7) - ω*t)
Ψ_2D_7(x; t) = Ψ_0*cos(k*(3/7)*Δr(α(x)/7) - ω*t)
Ψ_2D(x; t) = Ψ_2D_1(x; t) + Ψ_2D_2(x; t) + Ψ_2D_3(x; t) + Ψ_2D_4(x; t) + Ψ_2D_5(x; t) + Ψ_2D_6(x; t) + Ψ_2D_7(x; t)
PlotHeight = 100','PlotWidth = 600','x_max = 75cm
$Plot{Ψ_2D(α; t)^2 @ α = -x_max : x_max}
$Map{Ψ_2D(x; t)^2 @ x = -x_max : x_max & z = 0m : 0.1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Dyfrakcja światła na pojedynczej szczelinie - widok na ekranie

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

D = 1 m

a = 1.5 μm

Δr ( α )  = a · sin ( α ) 

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

α ( x )  = atan(xD)

The slit is divided into 7 zones:

Ψ_{2D_1} ( x; t )  = Ψ₀ · cos(k · -37 · Δr  ( α  ( x )  )  − ω · t)

Ψ_{2D_2} ( x; t )  = Ψ₀ · cos(k · -27 · Δr  ( α  ( x )  )  − ω · t)

Ψ_{2D_3} ( x; t )  = Ψ₀ · cos(k · -17 · Δr  ( α  ( x )  )  − ω · t)

Ψ_{2D_4} ( x; t )  = Ψ₀ · cos(k · 0 · Δr (α  ( x ) 7) − ω · t)

Ψ_{2D_5} ( x; t )  = Ψ₀ · cos(k · 17 · Δr (α  ( x ) 7) − ω · t)

Ψ_{2D_6} ( x; t )  = Ψ₀ · cos(k · 27 · Δr (α  ( x ) 7) − ω · t)

Ψ_{2D_7} ( x; t )  = Ψ₀ · cos(k · 37 · Δr (α  ( x ) 7) − ω · t)

Ψ_2D ( x; t )  = Ψ_{2D_1}  ( x; t )  + Ψ_{2D_2}  ( x; t )  + Ψ_{2D_3}  ( x; t )  + Ψ_{2D_4}  ( x; t )  + Ψ_{2D_5}  ( x; t )  + Ψ_{2D_6}  ( x; t )  + Ψ_{2D_7}  ( x; t ) 

PlotHeight = 100 , PlotWidth = 600 , x_max = 75 cm

Interference of Light - 2 Slits General View

Spatial view of Young two-slit interference: two coherent point sources separated by \(d = 3\, μm\) are summed on a 2D grid to plot the full \(\Psi(x,\, y)\) field.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja światła z dwóch szczelin - widok przestrzenny
#rad
λ = 500nm
k = (2*π)/λ
r(x; y) = sqrt(x^2 + y^2)
d = 3μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
Ψ_2D_1(x; y; t) = Ψ_0*cos(k*r(x - d/2; y) - ω*t)
Ψ_2D_2(x; y; t) = Ψ_0*cos(k*r(x + d/2; y) - ω*t)
Ψ_2D(x; y; t) = Ψ_2D_1(x; y; t) + Ψ_2D_2(x; y; t)
PlotHeight = 200','PlotWidth = 600','PlotStep = 1
$Map{Ψ_2D(x; y; t)^2 @ x = -1m : 1m & y = 0m : 1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja światła z dwóch szczelin - widok przestrzenny

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

r ( x; y )  =    √ x² + y²

d = 3 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

Ψ_{2D_1} ( x; y; t )  = Ψ₀ · cos(k · r (x − d2; y) − ω · t)

Ψ_{2D_2} ( x; y; t )  = Ψ₀ · cos(k · r (x + d2; y) − ω · t)

Ψ_2D ( x; y; t )  = Ψ_{2D_1}  ( x; y; t )  + Ψ_{2D_2}  ( x; y; t ) 

PlotHeight = 200 , PlotWidth = 600 , PlotStep = 1

Interference of Light - 2 Slits Screen View

Far-field intensity profile of the two-slit setup, projected onto a distant screen and showing the cosine-squared fringe pattern with spacing \(\lambda D / d\).

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja światła z dwóch szczelin - widok na ekranie
#rad
λ = 500nm
k = (2*π)/λ
D = 1m
Δr(α) = d*sin(α)
d = 3μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
α(x) = atan(x/D)
Ψ_2D_1(x; t) = Ψ_0*cos(k*0*Δr(α(x)) - ω*t)
Ψ_2D_2(x; t) = Ψ_0*cos(k*1*Δr(α(x)) - ω*t)
Ψ_2D(x; t) = Ψ_2D_1(x; t) + Ψ_2D_2(x; t)
PlotHeight = 100','PlotWidth = 600','x_max = 75cm
$Plot{Ψ_2D(α; t)^2 @ α = -x_max : x_max}
$Map{Ψ_2D(x; t)^2 @ x = -x_max : x_max & z = 0m : 0.1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja światła z dwóch szczelin - widok na ekranie

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

D = 1 m

Δr ( α )  = d · sin ( α ) 

d = 3 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

α ( x )  = atan(xD)

Ψ_{2D_1} ( x; t )  = Ψ₀ · cos ( k · 0 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_2} ( x; t )  = Ψ₀ · cos ( k · 1 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_2D ( x; t )  = Ψ_{2D_1}  ( x; t )  + Ψ_{2D_2}  ( x; t ) 

PlotHeight = 100 , PlotWidth = 600 , x_max = 75 cm

Interference of Light - 4 Slits General View

Spatial view of four coherent slits: the same superposition recipe scales to four point sources, sharpening the principal maxima and adding three secondary maxima between them.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja światła z czterech szczelin - widok przestrzenny
#rad
λ = 500nm
k = (2*π)/λ
r(x; y) = sqrt(x^2 + y^2)
d = 3μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
Ψ_2D_1(x; y; t) = Ψ_0*cos(k*r(x - 3*d/2; y) - ω*t)
Ψ_2D_2(x; y; t) = Ψ_0*cos(k*r(x - d/2; y) - ω*t)
Ψ_2D_3(x; y; t) = Ψ_0*cos(k*r(x + d/2; y) - ω*t)
Ψ_2D_4(x; y; t) = Ψ_0*cos(k*r(x + 3*d/2; y) - ω*t)
Ψ_2D(x; y; t) = Ψ_2D_1(x; y; t) + Ψ_2D_2(x; y; t) + Ψ_2D_3(x; y; t) + Ψ_2D_4(x; y; t)
PlotHeight = 200','PlotWidth = 600','PlotStep = 1
$Map{Ψ_2D(x; y; t)^2 @ x = -1m : 1m & y = 0m : 1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja światła z czterech szczelin - widok przestrzenny

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

r ( x; y )  =    √ x² + y²

d = 3 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

Ψ_{2D_1} ( x; y; t )  = Ψ₀ · cos(k · r (x − 3 · d2; y) − ω · t)

Ψ_{2D_2} ( x; y; t )  = Ψ₀ · cos(k · r (x − d2; y) − ω · t)

Ψ_{2D_3} ( x; y; t )  = Ψ₀ · cos(k · r (x + d2; y) − ω · t)

Ψ_{2D_4} ( x; y; t )  = Ψ₀ · cos(k · r (x + 3 · d2; y) − ω · t)

Ψ_2D ( x; y; t )  = Ψ_{2D_1}  ( x; y; t )  + Ψ_{2D_2}  ( x; y; t )  + Ψ_{2D_3}  ( x; y; t )  + Ψ_{2D_4}  ( x; y; t ) 

PlotHeight = 200 , PlotWidth = 600 , PlotStep = 1

Interference of Light - 4 Slits Screen View

Screen-view intensity for four coherent slits, evaluated as a 1D profile \(I(x)\) on the detector. The principal-maxima width narrows as roughly \(1/N\) relative to the two-slit case.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja światła z dwóch szczelin - widok na ekranie
#rad
λ = 500nm
k = (2*π)/λ
D = 1m
Δr(α) = d*sin(α)
d = 3μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
α(x) = atan(x/D)
Ψ_2D_1(x; t) = Ψ_0*cos(k*0*Δr(α(x)) - ω*t)
Ψ_2D_2(x; t) = Ψ_0*cos(k*1*Δr(α(x)) - ω*t)
Ψ_2D_3(x; t) = Ψ_0*cos(k*2*Δr(α(x)) - ω*t)
Ψ_2D_4(x; t) = Ψ_0*cos(k*3*Δr(α(x)) - ω*t)
Ψ_2D(x; t) = Ψ_2D_1(x; t) + Ψ_2D_2(x; t) + Ψ_2D_3(x; t) + Ψ_2D_4(x; t)
PlotHeight = 100','PlotWidth = 600','x_max = 75cm
$Plot{Ψ_2D(α; t)^2 @ α = -x_max : x_max}
$Map{Ψ_2D(x; t)^2 @ x = -x_max : x_max & z = 0m : 0.1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja światła z dwóch szczelin - widok na ekranie

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

D = 1 m

Δr ( α )  = d · sin ( α ) 

d = 3 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

α ( x )  = atan(xD)

Ψ_{2D_1} ( x; t )  = Ψ₀ · cos ( k · 0 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_2} ( x; t )  = Ψ₀ · cos ( k · 1 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_3} ( x; t )  = Ψ₀ · cos ( k · 2 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_4} ( x; t )  = Ψ₀ · cos ( k · 3 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_2D ( x; t )  = Ψ_{2D_1}  ( x; t )  + Ψ_{2D_2}  ( x; t )  + Ψ_{2D_3}  ( x; t )  + Ψ_{2D_4}  ( x; t ) 

PlotHeight = 100 , PlotWidth = 600 , x_max = 75 cm

Interference of Light - 6 Slits General View

Spatial view of six coherent slits, approaching the diffraction-grating limit where principal maxima dominate the field.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja światła z sześciu szczelin - widok przestrzenny
#rad
λ = 500nm
k = (2*π)/λ
r(x; y) = sqrt(x^2 + y^2)
d = 3μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
Ψ_2D_1(x; y; t) = Ψ_0*cos(k*r(x - 5*d/2; y) - ω*t)
Ψ_2D_2(x; y; t) = Ψ_0*cos(k*r(x - 3*d/2; y) - ω*t)
Ψ_2D_3(x; y; t) = Ψ_0*cos(k*r(x - d/2; y) - ω*t)
Ψ_2D_4(x; y; t) = Ψ_0*cos(k*r(x + d/2; y) - ω*t)
Ψ_2D_5(x; y; t) = Ψ_0*cos(k*r(x + 3*d/2; y) - ω*t)
Ψ_2D_6(x; y; t) = Ψ_0*cos(k*r(x + 5*d/2; y) - ω*t)
Ψ_2D(x; y; t) = Ψ_2D_1(x; y; t) + Ψ_2D_2(x; y; t) + Ψ_2D_3(x; y; t) + Ψ_2D_4(x; y; t) + Ψ_2D_5(x; y; t) + Ψ_2D_6(x; y; t)
PlotHeight = 200','PlotWidth = 600','PlotStep = 1
$Map{Ψ_2D(x; y; t)^2 @ x = -1m : 1m & y = 0m : 1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja światła z sześciu szczelin - widok przestrzenny

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

r ( x; y )  =    √ x² + y²

d = 3 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

Ψ_{2D_1} ( x; y; t )  = Ψ₀ · cos(k · r (x − 5 · d2; y) − ω · t)

Ψ_{2D_2} ( x; y; t )  = Ψ₀ · cos(k · r (x − 3 · d2; y) − ω · t)

Ψ_{2D_3} ( x; y; t )  = Ψ₀ · cos(k · r (x − d2; y) − ω · t)

Ψ_{2D_4} ( x; y; t )  = Ψ₀ · cos(k · r (x + d2; y) − ω · t)

Ψ_{2D_5} ( x; y; t )  = Ψ₀ · cos(k · r (x + 3 · d2; y) − ω · t)

Ψ_{2D_6} ( x; y; t )  = Ψ₀ · cos(k · r (x + 5 · d2; y) − ω · t)

Ψ_2D ( x; y; t )  = Ψ_{2D_1}  ( x; y; t )  + Ψ_{2D_2}  ( x; y; t )  + Ψ_{2D_3}  ( x; y; t )  + Ψ_{2D_4}  ( x; y; t )  + Ψ_{2D_5}  ( x; y; t )  + Ψ_{2D_6}  ( x; y; t ) 

PlotHeight = 200 , PlotWidth = 600 , PlotStep = 1

Interference of Light - 6 Slits Screen View

Screen-view intensity for the six-slit grating-like configuration: principal maxima become sharp peaks separated by five tiny secondary lobes, the canonical \(N\)-slit fingerprint.

Code:

'Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

'Interferencja światła z sześciu szczelin - widok na ekranie
#rad
λ = 500nm
k = (2*π)/λ
D = 1m
Δr(α) = d*sin(α)
d = 3μm
c = 299792458m/s
T = λ/c
f = 1/T|Hz
ω = (2*π)/T
Ψ_0 = 0.5*sqrt(W/m^2)
t = 0s
α(x) = atan(x/D)
Ψ_2D_1(x; t) = Ψ_0*cos(k*0*Δr(α(x)) - ω*t)
Ψ_2D_2(x; t) = Ψ_0*cos(k*1*Δr(α(x)) - ω*t)
Ψ_2D_3(x; t) = Ψ_0*cos(k*2*Δr(α(x)) - ω*t)
Ψ_2D_4(x; t) = Ψ_0*cos(k*3*Δr(α(x)) - ω*t)
Ψ_2D_5(x; t) = Ψ_0*cos(k*4*Δr(α(x)) - ω*t)
Ψ_2D_6(x; t) = Ψ_0*cos(k*5*Δr(α(x)) - ω*t)
Ψ_2D(x; t) = Ψ_2D_1(x; t) + Ψ_2D_2(x; t) + Ψ_2D_3(x; t) + Ψ_2D_4(x; t) + Ψ_2D_5(x; t) + Ψ_2D_6(x; t)
PlotHeight = 100','PlotWidth = 600','x_max = 75cm
$Plot{Ψ_2D(α; t)^2 @ α = -x_max : x_max}
$Map{Ψ_2D(x; t)^2 @ x = -x_max : x_max & z = 0m : 0.1m}

Rendered Output:

Assoc. prof. Tadeusz Szumiata, PhD, Radom University, Poland

 

Interferencja światła z sześciu szczelin - widok na ekranie

λ = 500 nm

k = 2 · πλ = 2 · 3.14500 nm = 0.0126 nm^-1

D = 1 m

Δr ( α )  = d · sin ( α ) 

d = 3 μm

c = 299792458 ms

T = λc = 500 nm299792458 m ∕ s = 1.67×10^-15 s

f = 1T = 11.67×10^-15 s = 599584916000000 Hz

ω = 2 · πT = 2 · 3.141.67×10^-15 s = 3767303134617706.5 s^-1

Ψ₀ = 0.5 ·   Wm² = 0.5 kg^0.5 ∕ s^1.5

t = 0 s

α ( x )  = atan(xD)

Ψ_{2D_1} ( x; t )  = Ψ₀ · cos ( k · 0 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_2} ( x; t )  = Ψ₀ · cos ( k · 1 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_3} ( x; t )  = Ψ₀ · cos ( k · 2 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_4} ( x; t )  = Ψ₀ · cos ( k · 3 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_5} ( x; t )  = Ψ₀ · cos ( k · 4 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_{2D_6} ( x; t )  = Ψ₀ · cos ( k · 5 · Δr  ( α  ( x )  )  − ω · t ) 

Ψ_2D ( x; t )  = Ψ_{2D_1}  ( x; t )  + Ψ_{2D_2}  ( x; t )  + Ψ_{2D_3}  ( x; t )  + Ψ_{2D_4}  ( x; t )  + Ψ_{2D_5}  ( x; t )  + Ψ_{2D_6}  ( x; t ) 

PlotHeight = 100 , PlotWidth = 600 , x_max = 75 cm

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