Get specific image size for small viewport width

Im trying to get this featured image only when wiewport is < 768:

add_image_size( 'img-movil', 660, false );

I have no much coding skills. I was reading about how to pass a variable from JS to PHP with jQuery $.ajax() but I don’t understand it completely. This is what I did:

First step. I try to set JS variable and send it to the server:

    $(document).ready(function(){
          var viewportWidth = $(window).width();
          if (viewportWidth < 768) {
            var modoView = 'movil';
            $.ajax({
               type: "POST",
               url: 'front-page.php',
               data: {
                   modoView : modoView
               },
               success: function(data) {
                   alert("success!");
               }
            });
    }

It doesn’t work (success allert doesn’t fire). I will appreciate any help with it.

Second step (when first step works), check if PHP variable is present (this code is in the body of front-page.php):

<?php
if (isset($_POST['modoView'])) {
  $my_image_size="img-movil";
}
?>

Is it right?
Thank you in advance.

After a few days reading about srcset attribute (Thanks Benoti for point to this!), definitely, I can see that this is the right way. This solution is simple, enough to solve this question, although srcset attribute with <picture> tag is much more powerfull. My solution:

Declare sizes in functions.php

add_image_size( 'c200x200', 200, 200, true );
add_image_size( 'c400x400', 400, 400, true );

Check a custom field with the size stored

I have two types of posts, big and small, with 200×200 and 400×400 px img size.

$tamano = get_field('tamano');
if ($tamano == '200x200') {
  $clase="c200x200";
} elseif ($tamano == '400x400') {
  $clase="c400x400";
} else {
  $clase="sin-clase";
}

Build the img tag

<?php
      $img_id = get_post_thumbnail_id();
      $img_src = wp_get_attachment_image_url( $img_id, 'c200x200' );
      $img_srcset = wp_get_attachment_image_srcset( $img_id, $clase );
      $alt_text = get_post_meta($img_id , '_wp_attachment_image_alt', true);
      ?>

      <img
          srcset="https://wordpress.stackexchange.com/<?php echo esc_attr( $img_srcset ); ?>"
          alt="<?php echo $alt_text; ?>"
          sizes="(min-width: 768px) 400px, 200px"
          src="https://wordpress.stackexchange.com/questions/247897/<?php echo esc_url( $img_src ); ?>"
      >

Background:

Responsive Images in WordPress 4.4

Responsive Images in Practice